diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzmfky" "b/data_all_eng_slimpj/shuffled/split2/finalzzmfky" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzmfky" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nThe inflationary paradigm~\\cite{Guth:1980zm,Starobinsky:1980te} is a part of the standard cosmology. Especially, in slow-roll inflation models~\\cite{Linde:1983gd}, the primordial curvature perturbations are naturally produced by the quantum fluctuation of (at least) one scalar field called inflaton during inflation. The spectrum of scalar perturbation predicted by a slow-roll model is almost flat, which is compatible with the cosmic microwave background (CMB) observation results.\n\nInflation models in supergravity (SUGRA) have been studied through many works. It is interesting that SUGRA is an effective theory of superstring, which is a possible candidate for a unified theory including gravity. \n\nIn this work, we propose a new class of inflation models in SUGRA including higher derivative interactions. Higher derivative terms always arise from e.g. the Dirac-Born-Infeld action describing D-brane dynamics in string theory. Some models with higher derivative terms have been constructed in SUGRA~\\cite{Koehn:2012ar,Farakos:2012je}. Higher derivative terms in SUGRA can be described by using the supersymmetry~(SUSY) covariant derivative operator of SUSY multiplets. We call a SUGRA term including SUSY derivative operators a SUSY derivative term in this letter. The components of SUSY derivative terms contain the higher derivative interactions. \n\nIn the standard SUGRA action without SUSY derivative terms, dynamical chiral multiplets should be contained in a K\\\"ahler potential. However, as we will show, the kinetic terms of chiral multiplets also arise from SUSY derivative terms.\n\n We will discuss inflation models in which the inflaton multiplet is not included in the K\\\"ahler potential. Surprisingly, the so-called $\\eta$ problem, which spoils the flatness of the inflation potential, does not exist in our models. It is also remarkable that the effects of the higher derivative terms of the inflaton become negligibly small on the slow-roll trajectory. Consequently, the inflation is effectively driven by a single scalar field with a standard kinetic term and a scalar potential. We will show two inflation models, the chaotic inflation\\cite{Linde:1983gd} and the Starobinsky inflation~\\cite{Starobinsky:1980te}. Especially, the chaotic inflation in our model predicts a characteristic spectrum of the perturbation, which can be tested by forthcoming observations.\n\nThe remaining parts of this letter are consisted as follows. First, in Sec.~\\ref{setup}, we will show the action we will discuss. After that, choosing the two types of superpotential, we will construct the chaotic and the Starobinsky like inflation models, and show the cosmological parameters in those models in Sec.~\\ref{chaotic} and \\ref{Starobinsky}. Finally we will conclude in Sec.~\\ref{con}. In Appendix.~\\ref{A}, the cosmological parameters in our models are discussed.\n\\section{Inflation with higher derivative terms}\\label{model}\n\\subsection{Setup}\\label{setup}\nTo clarify the SUGRA system discussed below, we first show the corresponding action in superspace. It is described by \n\\begin{align}\nS=&\\int d^4\\theta K(\\hat{S},\\bar{\\hat{S}})+\\left(\\int d^2\\theta W(\\hat{\\Phi},\\hat{S})+{\\rm h.c.}\\right)\\nonumber\\\\\n&+\\int d^4\\theta C D_\\alpha\\hat{\\Phi} D^{\\alpha}\\hat{S}\\bar{D}_{\\dot{\\alpha}}\\bar{\\hat{\\Phi}}\\bar{D}^{\\dot{\\alpha}}\\bar{\\hat{S}},\\label{gSUSY}\n\\end{align}\nwhere $\\hat{S}$ and $\\hat{\\Phi}$ are chiral multiplets, $K$ and $W$ are K\\\"ahler and super-potential respectively, $C$ is a constant, $D_\\alpha$ and $\\bar{D}_{\\dot{\\alpha}}$ denote SUSY covariant derivatives, and $\\alpha,\\dot{\\alpha}$ are spinor indices. Here and the following, we use the unit~$M_{P}=1$ where $M_P=2.4\\times10^{18}$GeV is the reduced Planck mass. We will denote the scalar component of each multiplet by the character without a hat below. \n\nThe last term in Eq.~(\\ref{gSUSY}) is a SUSY derivative term. It leads ghost-free interactions with respect to the bosonic sector discussed in Ref.~\\cite{Koehn:2012ar}. In the well-known SUGRA models without SUSY derivative terms, a dynamical chiral multiplet should be contained in a K\\\"ahler potential $K$ because superpotential terms does not give kinetic terms for chiral multiplets. However, in our case, the SUSY derivative term in Eq.~(\\ref{gSUSY}) gives $S$ and $\\Phi$ not only higher derivative terms, but also standard kinetic terms in non-canonical forms as we will show. \n\nHere, we choose the following $K$ and $W$~\\cite{Kawasaki:2000yn,Kallosh:2010xz}, \n\\begin{eqnarray}\nK&=&|\\hat{S}|^2,\\label{K}\\\\\nW&=&f(\\hat{\\Phi}) \\hat{S},\n\\end{eqnarray}\nwhere $f(\\hat{\\Phi})$ is a holomorphic function of $\\hat{\\Phi}$. We emphasize that the K\\\"ahler potential~(\\ref{K}) does not contain $\\hat{\\Phi}$, which we identify as the inflaton multiplet. $\\hat{S}$ becomes the Goldstino multiplet during inflation. With a minimal K\\\"ahler potential $K=|\\hat{S}|^2$, $S$ often has its light or tachyonic mass during inflation, and therefore the quantum fluctuation of $S$ may become a source of the curvature perturbation as discussed in Ref.~\\cite{Kawasaki:2000yn,Demozzi:2010aj}. Recently, the authors of Ref.~\\cite{Antoniadis:2014oya} showed the absence of such a situation in the case that $\\hat{S}$ is the Volkov and Akulov supermultiplet~\\cite{Volkov:1973ix}, which satisfies $\\hat{S}^2=0$.\\footnote{We can construct a similar model with an unconstrained multiplet instead of $\\hat{S}$ if its scalar component has a sufficiently heavy mass. However, in that case, the equations of motions of auxiliary fields are complicated and the solutions are not determined uniquely~\\cite{Koehn:2012ar}.} Then, $S$ is identically $0$ because it is the bilinear of the fermionic component of $\\hat{S}$~\\cite{Antoniadis:2014oya}. \n\nIn conformal SUGRA~\\cite{Kugo:1982cu,Kugo:1983mv}, the action corresponding to one in Eq.~(\\ref{gSUSY}) can be expressed as follows, \n\\begin{align}\nS_{\\rm SG}=&\\frac{1}{2}\\left[S_0\\bar{S}_0 (-3e^{-\\frac{K}{3}})\\right]_D\\nonumber\\\\\n&+\\left[S_0^3W\\right]_F+[C\\Da \\Phi\\uDa S\\Dad \\bar{\\Phi}\\uDad \\bar{S}]_D,\\label{Sconf}\n\\end{align} \nwhere $[\\cdots]_{D,F}$ denote the superconformal D- and F-density formulae~\\cite{Kugo:1982cu}, $S_0$ is a chiral compensator, and $\\Da$ denotes the superconformal spinor derivative corresponding to $D_\\alpha$ in Eq.~(\\ref{gSUSY})~\\cite{Kugo:1983mv}.\\footnote{We note that the last term in Eq.~(\\ref{Sconf}) is manifestly K\\\"ahler invariant. The K\\\"ahler transformation, $K\\rightarrow K+\\Lambda+\\bar{\\Lambda}$, $W\\rightarrow We^{\\Lambda}$, is the redefinition of the compensator $S_0$ in conformal SUGRA. Obviously, the last term in Eq.~(\\ref{Sconf}) is not coupled to the compensator, and therefore it is inert under the K\\\"ahler transformation. However, the sigma model diffeomorphism, implies that $C$ in Eq.~(\\ref{Sconf}) should be the tensor of the target space.} In the following, we only discuss the bosonic action, and then we can put the usual superconformal gauge fixing conditions~\\cite{Kugo:1982cu,Kugo:1983mv}, e.g. $S_0=\\bar{S}_0=e^{K\/6}$ and obtain the SUGRA action in Einstein frame. \n\nAfter eliminating the compensator $S_0$ and the auxiliary field $A_\\mu$, which is one of the gauge fields of the superconformal symmetry, we obtain the following action, \n\\begin{align}\n\\mathcal{L}=&K_{S\\bar{S}}F^S\\bar{F}^{\\bar{S}}-3F^0\\bar{F}^0\\nonumber\\\\\n&+\\left(W_SF^S+{\\rm h.c.}\\right)\\nonumber\\\\\n&+32CF^\\Phi F^S\\bar{F}^{\\bar{\\Phi}}\\bar{F}^{\\bar{S}}-16C\\partial_\\mu\\Phi \\partial ^\\mu\\bar{\\Phi}F^S\\bar{F}^{\\bar{S}},\n\\end{align}\nwhere $F^{0,S,\\Phi}$ are the auxiliary components of $\\hat{S}_0,\\hat{S},$ and $\\hat{\\Phi}$ respectively and we used the condition $K=K_S=W_\\Phi=W=0$ because $S\\equiv 0$. The solutions for the equations of motion of the auxiliary fields are simply given by \n\\begin{align}\n&F^0=0,\\\\\n&F^{\\Phi}=0,\\\\\n&F^S=-\\frac{\\bar{W}_{\\bar{S}}}{1-16C\\partial_\\mu\\Phi\\partial^\\mu\\bar{\\Phi}},\n\\end{align}\nwhere we have used $K_{S\\bar{S}}=1$.\n\nBy substituting the on-shell expressions of F-terms into the action, we obtain the effective Lagrangian as follows,\n\\begin{align}\n\\mathcal{L}=\\frac{V}{A^2}\\tilde{X}-\\frac{V}{A}\\left(2-\\frac{1}{A}\\right),\\label{Leff}\n\\end{align}\nwhere $V\\equiv|W_S|^2=|f(\\Phi)|^2$, $\\tilde{X}\\equiv-16C \\partial_\\mu\\Phi\\partial^\\mu\\bar{\\Phi}$ and $A\\equiv (1+\\tilde{X})$. We can redefine the complex scalar $\\Phi$ as follows,\n\\begin{align}\n\\varphi\\equiv4\\sqrt{C}\\int f(\\Phi)d\\Phi.\n\\end{align}\nThen the Lagrangian~(\\ref{Leff}) can be rewritten as\n\\begin{align}\n\\mathcal{L}=\\frac{X}{A^2}-\\frac{V}{A}\\left(2-\\frac{1}{A}\\right),\n\\end{align}\nwhere $X\\equiv \\partial_\\mu \\varphi\\partial^\\mu\\bar{\\varphi}=V\\tilde{X}$. \n\nAlthough the action contains the higher order terms of $X$, the cosmological parameters are same with ones in the models without higher derivative action as shown in Appendix.~\\ref{A}. Therefore, on the slow-roll trajectory, we can approximate the Lagrangian as follows, \n\\begin{align}\n\\mathcal{L}\\sim X-V,\n\\end{align}\nwhere we used the approximation $A\\sim 1$. The approximated action is one with the standard kinetic and the potential terms of $\\varphi$, and we will use this action in the following discussion.\n\nIt is worth to remark the absence of the $\\eta$ problem in this model. In the standard SUGRA models, the F-term scalar potential is given by,\n\\begin{eqnarray}\nV_F=e^K\\left(K^{I\\bar{J}}D_IWD_{\\bar{J}}\\bar{W}-3|W|^2\\right),\n\\end{eqnarray} \nwhere $D_IW=W_I +K_IW$, and $K^{I\\bar{J}}$ is the inverse of $K_{I\\bar{J}}$. In the case that K\\\"ahler potential is given by $K=|\\Phi|^2+\\cdots$, the scalar potential becomes steep due to the factor $e^K$. In our case, however, the inflaton $\\Phi$ is not contained in the K\\\"ahler potential, and therefore the $\\eta$ problem does not exist.\\footnote{From the conformal SUGRA viewpoint, we can also understand the reason why the $\\eta$ problem does not exist in our model. In our case, the last term in Eq.(\\ref{Sconf}) does not contains the mixing between Ricci scalar and the inflaton. In this case, the $\\eta$ problem does not occur~\\cite{Abe:2014opa}. } \n\\subsection{Chaotic inflation}\\label{chaotic}\nWe consider the following function $f(\\Phi)$,\n\\begin{align}\nf(\\Phi)=\\lambda_n\\Phi^n,\n\\end{align} \nwhere $n\\geq1$, $\\lambda_n$ is a coupling constant. Then the scalar $\\varphi$ can be written as\n\\begin{align}\n\\varphi=\\frac{4}{n+1}\\sqrt{C}\\lambda_n\\Phi^{n+1}.\n\\end{align}\nWe identify $\\sqrt{2}|\\varphi|$ as the inflaton denoted by $\\phi$, and we can write down the scalar potential $V$ as follows,\n\\begin{align}\nV=\\tilde{\\lambda}_n^2\\phi^{2n\/(n+1)},\\label{Vc}\n\\end{align}\nwhere $\\tilde{\\lambda}_n^2\\equiv \\left(2^{-n}(16C)^{-n}(n+1)^{2n}\\lambda_n^2\\right)^{1\/(n+1)}$. Note that this model resembles to the running kinetic inflation model~\\cite{Nakayama:2010kt}, however, the scalar potential is highly restricted in our case.\n\nSurprisingly enough, the effective potential is restricted from the linear potential to the quadratic one, even when the power of $f(\\Phi)$ becomes higher. The predicted spectral tilt $n_s$ and tensor-to-scalar ratio $r$ are given by,\n\\begin{align}\nn_s&\\sim1-\\frac{2n+1}{n+1}\\frac{1}{N},\\\\\nr&\\sim\\frac{8n}{n+1}\\frac{1}{N},\n\\end{align}\nwhere $N$ denotes the number of e-foldings and we have omitted smaller corrections. With a sufficiently large $n$, the scalar potential~(\\ref{Vc}) asymptotes to $\\phi^2$, which is favored by the BICEP2 data~\\cite{Ade:2014xna}. This model can be tested by forthcoming experiments.\n\\subsection{Starobinsky inflation }\\label{Starobinsky}\nNext, we choose the following function $f(\\Phi)$~\\cite{Kallosh:2013lkr},\n\\begin{align}\nf(\\Phi)=\\lambda(1-e^{-a\\Phi}),\n\\end{align}\nwhere $a$ and $\\lambda$ are real constant parameters. Then, the canonical normalized complex scalar $\\varphi$ is\n\\begin{align}\n\\varphi=4\\sqrt{C}\\lambda\\left(\\Phi+\\frac{1}{a}e^{-a\\Phi}\\right)\\sim4\\sqrt{C}\\lambda\\Phi.\n\\end{align} \nHere, we identify $\\sqrt{2}{\\rm Re}~\\varphi=\\phi$ as the inflaton. Then, the effective potential is given by\n\\begin{align}\nV\\sim\\lambda^2 (1-e^{-\\frac{\\tilde{a}}{\\sqrt{2}}\\phi})^2,\n\\end{align}\nwhere $\\tilde{a}=(4\\sqrt{C} \\lambda)^{-1}a$ and we only write down the leading terms. The resultant potential is same with one in the Starobinsky model~\\cite{Kallosh:2013lkr}, and the predicted spectral tilt~$n_s$ and tensor-to-scalar ratio~$r$ are as follows,\n\\begin{align}\nn_s&\\sim 1-\\frac{2}{N},\\label{Stilt}\\\\\nr&\\sim \\frac{16}{\\tilde{a}^2N^2} \\label{Sr},\n\\end{align} \nwhere $N$ denotes the number of e-foldings. The predicted $r$ is very small when $\\tilde{a}\\sim\\mathcal{O}(1)$. For $N\\sim 60$, the values of $(n_s,r)$ are compatible with the Planck2013 result~\\cite{Ade:2013uln}, however, this model may be excluded if the result from BICEP2~\\cite{Ade:2014xna} is confirmed by other experiments.\n\nWe also note that the mass of ${\\rm Im}\\varphi$ is much smaller than the Hubble scale during inflation although its minimum is located its origin. Then, the light direction can be a curvaton as in the case of the simplest chaotic inflation in SUGRA~\\cite{Kawasaki:2000yn,Demozzi:2010aj}, if the decay of ${\\rm Im}\\varphi$ occurs after that of the inflaton~${\\rm Re}\\varphi$. Further investigation of such a case would be interesting but it is beyond the scope of this letter.\n\\section{Conclusion}\\label{con}\nWe have proposed a new class of inflation models with a SUSY derivative term in SUGRA. We have found that the kinetic terms of chiral multiplets are included in SUSY derivative terms, even if the multiplets are absent in K\\\"ahler potential terms. By virtue of the absence of the inflaton in $K$, the $\\eta$ problem does not exist in our models. It is also remarkable that the action contains the higher order terms of $X=-\\partial_\\mu \\phi \\partial^\\mu \\phi\/2$, however, their contributions are negligible and the effective action is given by the scalar system with the standard kinetic term and the scalar potential. \n\nWe have shown two inflation models in our setup. In Sec.~\\ref{chaotic}, we discussed the chaotic inflation model. It is surprising that the effective potential is restricted from the linear to the quadratic potential regardless of the superpotential containing the arbitrary power of $\\Phi$. Therefore this model can be tested by forthcoming experiments.\n\nOn the other hand, we have also constructed the Starobinsky type inflation model in Sec.~\\ref{Starobinsky}. In contrast to the chaotic type model, it predicts the very small value of the tensor-to-scalar ratio $r$ which is compatible with the Planck2013 result. As discussed in Sec.~\\ref{Starobinsky}, this model contains the light scalar ${\\rm Im}\\varphi$ which may produce additional adiabatic perturbations. If it the case, the predicted cosmological parameters shown in Eq.(\\ref{Stilt}) and (\\ref{Sr}) can be changed. That is an interesting possibility but it requires more detailed investigation of other sectors. \n\nIn this work, we have only discussed the dynamics of the scalar sector, however we need to take fermions into account to discuss the universe after inflation. That will be our future work. \n\\section*{Acknowledgements}\nThe authors would like to thank Hiroyuki Abe for useful discussion and comments.\nY.Y. was supported by JSPS Research Fellowships for Young Scientists No. 26-4236 in Japan.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Introduction}\nEllis \\cite{ellis3} gave a description of the free crossed square of\ngroups of a CW-complex using topological methods. In contexts other\nthan groups, he has defined crossed squares, 2-crossed modules etc.\nin \\cite{ellis3}. For commutative algebras, 2-crossed modules have\nbeen defined by Grandjean and Vale \\cite{gv}.\n\nCombining earlier work \\cite{porter1} of Porter with Arvasi and Porter's joint papers\n\\cite{ap1,ap2,ap3}, one starts to see how a study\nof the links between simplicial commutative algebras and classical\nconstructions of homological algebra can be strengthened by\ninterposing crossed algebraic models for the homotopy types of\nsimplicial algebras. In this note, we continue this process using\nthese methods to give a description in terms of tensor products,\nof the top corner of a free crossed square of commutative algebras\nand the top term of the corresponding free 2-crossed module. The\nmethods are slightly different, but these results are\n2-dimensional analogues of the description of the free crossed\nmodule on a `presentation' of an algebra in terms of Kozsul-like\nterms, given in \\cite{porter1}.\n\nWe end with a section looking at possible links of this with\nAndr\\'e-Quillen homology and a squared complex form of the\ncontangent complex.\n\nThe results and general methods we use are inspired by those given for the\ncorresponding case of groups in \\cite{mp}. Some of the methods of that paper,\nof course, go across almost verbatim to this commutative algebra case,\nbut, as the audience for this paper is probably more or less disjoint from\nthat for \\cite{mp} it seems advisable to repeat arguments from that paper in\nthis algebra case even when they might be safely `left to the reader'. Of\ncourse, there are all times when the translation between the two contexts\nis less easy.\n\n\n\\textbf{Acknowledgements}. The authors wishes to thank Professor\nTimothy Porter\nfor his helpful comments.\n\n\n\\section{Preliminaries}\n\nAll algebras will be commutative and will be over the same fixed but\nunspecified ground ring.\n\n\\subsection{Simplicial algebras}\n\nDenoting the usual category of finite ordinals by $\\Delta $, we obtain for\neach $k\\geqslant 0$ a subcategory $\\Delta _{\\leqslant k}$ determined by the objects $%\n[j]$ of $\\Delta $ with $j\\leqslant k$. A simplicial algebra is a\nfunctor from the opposite category $\\Delta ^{op}$ to \\textbf{Alg};\na $k$-truncated simplicial algebra is a functor from $(\\Delta\n_{\\leqslant k}^{op})$ to \\textbf{Alg}. We denote the category of\nsimplicial algebras by \\textbf{SimpAlg} and the category of\n\\textit{k}-truncated simplicial algebras by\n$\\mathbf{Tr_{k}SimpAlg}$. By a \\textit{k-truncation of a\nsimplicial algebra}, we mean a $k$-truncated simplicial algebra\n\\textbf{tr}$_{k}\\mathbf{E}$ obtained by forgetting dimensions of\norder $>k$ in a simplicial algebra $\\mathbf{E},$ that is\nrestricting \\textbf{E} to $\\Delta _{\\leqslant k}^{op}$. This gives a\ntruncation functor\n\\begin{equation*}\n\\mathbf{tr_{k}:SimpAlg\\longrightarrow Tr_{k}SimpAlg}\n\\end{equation*}\nwhich admits a right adjoint\n\\begin{equation*}\n\\mathbf{cosk_{k}:Tr_{k}SimpAlg\\longrightarrow SimpAlg}\n\\end{equation*}\ncalled the \\textit{k-coskeleton functor}, and a left adjoint\n\\begin{equation*}\n\\mathbf{sk_{k}:Tr_{k}SimpAlg\\longrightarrow SimpAlg},\n\\end{equation*}\ncalled the \\textit{k-skeleton functor}. For explicit construction\nof these see \\cite{dusk}. We will say that a simplicial algebra\n$\\mathbf{E}$ is \\textit{\\ k-skeletal} if the natural morphism\n\\textbf{sk$_{k}\\mathbf{E}$}$ \\rightarrow \\mathbf{E}$ is an\nisomorphism.\n\nRecall that given a simplicial algebra \\textbf{E}, \\emph{the Moore\ncomplex} $(\\mathbf{NE},\\partial )$ \\emph{of} \\textbf{E} is the\nchain complex defined by\n\\begin{equation*}\n(NE)_n=\\bigcap_{i=0}^{n-1}\\mbox{\\rm Ker}d_i^n\n\\end{equation*}\nwith $\\partial _n:NE_n\\rightarrow NE_{n-1}$ induced from $d_n^n$ by\nrestriction.\n\nThe \\emph{n$^{th}$ homotopy module} $\\pi _{n}$(\\textbf{E}) of \\textbf{E} is\nthe n$^{th}$ homology of the Moore complex of \\textbf{E}, i.e.,\n\\begin{equation*}\n\\begin{array}{rcl}\n\\pi _{n}(\\mathbf{E}) & \\cong & H_{n}(\\mathbf{NE},\\partial ) \\\\\n& = & \\bigcap\\limits_{i=0}^{n}\\mbox{\\rm\nKer}d_{i}^{n}\/d_{n+1}^{n+1}(\\bigcap \\limits_{i=0}^{n}\\mbox{\\rm\nKer}d_{i}^{n+1}).\n\\end{array}\n\\end{equation*}\nWe say that the Moore complex \\textbf{NE} of a simplicial algebra is\nof \\emph{length} $k$ if $NE_{n}=0$ for all $n\\geqslant k+1$ so that a\nMoore complex is of length $k$ also of length $r$ for $r\\geqslant k.$ For\nexample, if \\textbf{E} has Moore complex of length 1, then\n$(NE_{1},NE_{0},\\partial _{1})$ is a crossed module and conversely.\nIf \\textbf{NG} is of length 2, the corresponding Moore complex gives\na 2-crossed module (cf. \\cite{ap3}).\n\n\\subsection{Free Simplicial Algebras}\n\nRecall from \\cite{ap1} the definition of free simplicial algebra\ngiven by the `step-by-step' construction of Andr\\'e \\cite{andre}.\n\nLet \\textbf{E} be a simplicial algebra and $k\\geqslant 1$, $k$-skeletal\nbe fixed. A simplicial algebra \\textbf{F} is called a\n\\textit{free} if\n\ni) $F_{n}=E_{n}$ for $nk$.\n\n\n\\textbf{Remark}: if \\textbf{A} is a simplicial algebra, then there\nexists a free simplicial algebra \\textbf{E} and an epimorphism\n$\\mathbf{E\\rightarrow A}$ which induces isomorphisms on all\nhomotopy modules. The details are omitted as they are\n'well-known'.\n\n\\subsection{Crossed Modules of Algebras}\nThroughout this paper we denote an action of $r\\in R$ on $c\\in C$\nby $r\\cdot c$.\n\nA \\emph{crossed module} is an algebra morphism $\\partial\n:C\\rightarrow R$ with an action of $R$ on $C$ satisfying (i)\n$\\partial (r\\cdot c)=r\\partial c$ and (ii) $\\partial (c)\\cdot\nc^{\\prime }=cc^{\\prime }$ for all $c,c^{\\prime }\\in M,r\\in R$. For\nthe weaker notion in which condition (ii) is not required, the\nmodels are called \\textit{pre-crossed modules}.\n\nExamples of crossed modules are:\n\ni) Any ideal, $I$, in $R$ gives an inclusion map $I\\longrightarrow\nR,$ which is a crossed module. Conversely, given any crossed\n$R$-module $\\mu :C\\longrightarrow R$, the image $I=\\mu (C)$ of $C$\nis an ideal in $R$.\n\nii) Any $R$-module $M$ can be considered as an $R$-algebra with\nzero multiplication and hence $M\\overset{0}{\\longrightarrow }R$ is\na crossed $R$-module. Conversely, if $\\mu :C\\longrightarrow R$ is\na crossed $R$-module, Ker$\\mu $ is an $R\/\\mu (C)$-module.\n\n\\section{Crossed Squares and Simplicial Algebras}\nAlthough we will be mainly concerned with crossed squares in this\npaper, some of the arguments either clearly apply or would seem to\napply in the more general case of crossed $n$-cubes and $n$-cube\ncomplexes.\n\nCrossed $n$-cubes in algebraic settings such as commutative\nalgebras, Jordan algebras, Lie algebras have been defined by\nEllis, \\cite{ellis2}.\n\nA \\emph{crossed n-cube of commutative algebras} is a family of\ncommutative algebras, $M_A$ for $A\\subseteq \\langle\nn\\rangle=\\{1,...,n\\}$ together with homomorphisms $\\mu\n_i:M_A\\rightarrow M_{A-\\{i\\}}$ for $i\\in \\langle n \\rangle$ and for\n$A,B\\subseteq \\langle n \\rangle$, functions\n\\begin{equation*}\nh:M_A\\times M_B\\longrightarrow M_{A\\cup B}\n\\end{equation*}\nsuch that for all $k\\in \\mathbf{k},\\ a,a^{\\prime }\\in M_A,\\\nb,b^{\\prime }\\in M_B,\\ c\\in M_C,$ $i,j\\in \\langle n \\rangle$ and\n$A\\subseteq B$\n\\begin{equation*}\n\\begin{array}{lrcl}\n1) & \\mu _ia & = & a\\ \\quad \\mbox{\\rm {\\rm if}}\\ i\\not \\in A \\\\\n2) & \\mu _i\\mu _ja & = & \\mu _j\\mu _ia \\\\\n3) & \\mu _ih(a,b) & = & h(\\mu _ia,\\mu _ib) \\\\\n4) & h(a,b) & = & h(\\mu _ia,b)=h(a,\\mu _ib)\\ \\quad \\mbox{\\rm {\\rm if}}\\ i\\in\nA\\cap B \\\\\n5) & h(a,a^{\\prime }) & = & aa^{\\prime } \\\\\n6) & h(a,b) & = & h(b,a) \\\\\n7) & h(a+a^{\\prime },b) & = & h(a,b)+h(a^{\\prime },b) \\\\\n8) & h(a,b+b^{\\prime }) & = & h(a,b)+h(a,b^{\\prime }) \\\\\n9) & k\\cdot h(a,b) & = & h(k\\cdot a,b)=h(a,k\\cdot b) \\\\\n10) & h(h(a,b),c) & = & h(a,h(b,c))=h(b,h(b,c)). \\\\\n& & &\n\\end{array}%\n\\end{equation*}\n\nA \\emph{morphism of crossed n-cubes} is defined in the obvious way:\nIt is a family of commutative algebra homomorphisms, for $A\\subseteq\n\\langle n \\rangle$, $\\ f_A:M_A\\longrightarrow M_A^{\\prime } $\ncommuting with the $\\mu _i^{\\prime}$s and $h^{\\prime}s$. We thus\nobtain a category of crossed $n$-cubes denoted by $\\mathbf{Crs^n} $.\n\n\\vspace{0.5cm}\n\n{\\noindent \\textbf{Examples.}} ${\\noindent(1)}$ For $n=1,$ a crossed 1-cube\nis the same as a crossed module.\n\n{\\noindent } For $n=2$ one has a crossed square:\n\\begin{equation*}\n\\xymatrix{M_{\\langle 2 \\rangle}\\ar[r]^{\\mu_2}\\ar[d]_{\\mu_1}&\nM_{\\{1\\}}\\ar[d]^{\\mu_1}\\\\\nM_{\\{2\\}}\\ar[r]_{\\mu_2}&M_{\\emptyset}. }\n\\end{equation*}\nEach $\\mu _i$ is a crossed module as is $\\mu _1\\mu _2$. The\n$h$-functions give actions and a pairing\n\\begin{equation*}\nh:M_{\\{1\\}}\\times M_{\\{2\\}}\\longrightarrow M_{\\langle 2\\rangle}.\n\\end{equation*}\nThe maps $\\mu _2$ (or $\\mu _1)$ also define a map of crossed\nmodules from $(M_{\\langle 2 \\rangle},M_{\\{2\\}},\\mu _1)$ to\n$(M_{\\langle 2 \\rangle},M_\\emptyset ,\\mu _1)$. In fact a crossed\nsquare can be thought of as a crossed module in the category of\ncrossed modules.\n\n(2) Let $I_{1}$ and $I_{2}$ be ideals of an algebra $E$. The\ncommutative square diagram of inclusions;\n\\begin{equation*}\n\\xymatrix{I_1 \\cap I_2 \\ar[r]^-{inc.}\\ar[d]_{inc.}&\nI_2\\ar[d]^{inc.}\\\\\nI_1\\ar[r]_{inc.}&E}\n\\end{equation*}\nnaturally comes together with actions of $E$ on $I_{1},I_{2}$ and\n$I_{1}\\cap I_{2}$ given by multiplication, and functions\n\\begin{equation*}\n\\begin{array}{lrll}\nh: & I_{A}\\times I_{B} & \\longrightarrow & I_{A}\\cap I_{B}=I_{A\\cup B} \\\\\n& (a,b) & \\longmapsto & ab.\n\\end{array}%\n\\end{equation*}%\nThat this is a crossed square is easily checked.\n\n(3) Let $\\mathbf{E}$ be a simplicial algebra. Let $M(\\mathbf{E},2)$\ndenote the following diagram\n\\begin{equation*}\n\\xymatrix{NE_2\/\\partial_3(NE_3)\\ar[r]^-{\\partial_2}\\ar[d]_{\\partial'_2}&NE_1\\ar[d]^{\\mu}\\\\\n\\overline{NE_1}\\ar[r]_{\\mu'}&E_1.}\n\\end{equation*}\nThen this is the underlying square of a crossed square. The extra\nstructure is given as follows: $NE_1=$\\text{Ker}$d_0^1$ and\n$\\overline{NE}_1=$\\textrm{Ker}$d_1^1$. Since $E_1$ acts on\n$NE_2\/\\partial _3NE_3,\\ \\overline{NE}_1$ and $NE_1,$ there are\nactions of $\\overline{NE}_1$ on $NE_2\/\\partial _3NE_3$ and $NE_1$\nvia $\\mu ^{\\prime },$ and $NE_1$ acts on $NE_2\/\\partial _3NE_3$ and\n$\\overline{NE}_1$ via $\\mu .$ As $\\mu $ and $\\mu ^{\\prime }$ are\ninclusions, all actions can be given by multiplication. The $h$-map\nis\n\\begin{equation*}\n\\begin{array}{rcl}\nNE_1\\times \\overline{NE}_1 & \\longrightarrow & NE_2\/\\partial _3NE_3 \\\\\n(x,\\overline{y}) & \\longmapsto & h(x,\\overline{y})=s_1x(s_1y-s_0y)+\\partial\n_3NE_3,%\n\\end{array}%\n\\end{equation*}\nwhich is bilinear. Here $x$ and $y$ are in $NE_1$ as there is a\nnatural bijection between $NE_1$ and $\\overline{NE}_1$ (by\n\\cite[Lemma 2.1]{ap2}). The element $\\bar y$ is the image of $y$\nunder this. This example effectively introduces the functor\n$$\n\\mathbf{M}(-,2):\\mathbf{SimpAlg}\\rightarrow \\mathbf{Crs^2}.\n$$\nThis is the case $n=2$ of a general construction of a crossed\n$n$-cube from a simplicial algebra given by the first author in\n\\cite{a1} where the reader may find the verification of the\naxioms. (This notational convention will be revisited at the end\nof section \\ref{sec5})\n\nNote if we consider the above crossed square as a vertical\nmorphism of crossed modules, we can take its kernel and cokernel\nwithin the category of crossed modules. In the above the morphisms\nin the top left hand corner are induced from $d_2$ so\n\\begin{equation*}\n\\text{Ker}\\left( \\partial'_2:\\frac{NE_2}{\\partial _3NE_3}\\longrightarrow \\text{Ker%\n}d_1\\right) =\\frac{NE_2\\cap \\text{Ker}d_2}{\\partial _3NE_3}\\cong \\pi _2(%\n\\mathbf{E})\n\\end{equation*}\nwhilst the other map labeled $\\mu $ is an inclusion so has trivial\nkernel, hence the kernel of this morphism of crossed modules is\n\\begin{equation*}\n\\pi _2(\\mathbf{E})\\longrightarrow 0.\n\\end{equation*}\nThe image of $\\partial_2$ (and $\\mu' $) is an ideal in both the\nalgebras on the bottom line and as Ker$d_0=NE_1$ with the\ncorresponding Im$\\mu$ being $d_2NE_2,$ the cokernel is\n$NE_1\/\\partial _2NE_2,$ whilst $E_1\/$Ker$d_0\\cong E_0,$ i.e, the\ncokernel of $\\mu $ is $M(\\mathbf{E},1)$.\n\nIn fact of course $\\mu$ is not only a morphism of crossed modules,\nit is a crossed module. This means that $\\pi\n_2(\\mathbf{E})\\rightarrow 0$ is in some sense a\n$M(\\mathbf{E},1)$-module, (cf. \\cite{ae}), and that\n$M(\\mathbf{E},2)$ can be thought of as a crossed extension of\n$M(\\mathbf{E},1)$ by $\\pi _2(\\mathbf{E})$.\n\n\\section{ Free Crossed Squares}\n\nEllis, \\cite{ellis3}, in 1993 presented the notion of a free\ncrossed square for the case of groups. In this section, we\nintroduce a commutative algebra version of this definition and\ngive a construction of a free crossed square by using the second\norder Peiffer elements and the 2-skeleton of a `step-by-step'\nconstruction of a free simplicial algebra.\n\nWe firstly adapt Ellis's definition of the free crossed square on a pair of\nfunctions $(f_2,f_3)$ to the algebra context:\n\nLet $\\mathbf{S_{1},S_{2}}$ and $\\mathbf{S_{3}}$ be sets which for\nsimplicity we assume are finite. Suppose given a function\n$f_{2}:\\mathbf{S_{2}\\rightarrow }R$ from a set $\\mathbf{S_{2}}$ to a\nfree algebra $R$ on $\\mathbf{S_{1}}$. Let $\\partial :M\\rightarrow R$\nbe the free pre-crossed module on $f_{2}$. Using the action of $R$\non $M$ we can form the semi-direct product $M\\rtimes R$. The\ninclusion $\\mu :M\\rightarrow M\\rtimes R$ given by $m\\mapsto (m,0)$\nenables us to take $M$ as an ideal of $M\\rtimes R$. (Recall from\nexamples of crossed modules that any ideal inclusion is a crossed\nmodule with action by multiplication.) There is also another ideal\nof $M\\rtimes R$ coming from $M$, namely\n\\begin{equation*}\n\\overline{M}=\\{(m,r)\\in M\\rtimes R:\\partial m=-r\\}\n\\end{equation*}\nwith inclusion denoted $\\bar{\\mu}:\\overline{M}\\rightarrow M\\rtimes R$.\n\nAssume given a function from a set $\\mathbf{S_{3}}$ to $M$,\n$f_{3}:\\mathbf{S_{3}}\\rightarrow M$, which is to satisfy $\\partial\nf_{3}=0$. Then there is a corresponding function\n$\\bar{f}_{3}:\\mathbf{S_{3}}\\rightarrow \\overline{M}$ given by\n$y\\mapsto (f_{3}(y),0)$.\n\nWe say a crossed square\n\\begin{equation*}\n\\xymatrix{L\\ar[r]^{\\partial'_2}\\ar[d]_{\\partial_2}&\\overline{M}\\ar[d]^{\\mu'}\\\\\nM\\ar[r]_-{\\mu}&M\\rtimes R}\n\\end{equation*}\nis \\emph{totaly free} on the pair of functions $(f_2,f_3)$ if\n\n{\\noindent (i) }$(M,R,\\partial )$ is the free pre-crossed module on $f_2;$\n\n{\\noindent (ii)} $\\mathbf{S_3}$ is a subset of $L$ with $f_3$ and $\\bar f_3$\nthe restrictions of $\\partial _2$ and $\\partial _2^{\\prime }$ respectively;\n\n{\\noindent (iii)} for any crossed square\n\\begin{equation*}\n\\xymatrix{L'\\ar[r]^{\\tau'}\\ar[d]_{\\tau}&\\overline{M}\\ar[d]^{\\mu'}\\\\\nM\\ar[r]_{\\mu}&M\\rtimes R}\n\\end{equation*}\nand any function $\\nu :\\mathbf{S_3}\\rightarrow L'$ satisfying $\\tau \\nu =f_3$%\n, there is a unique morphism $\\Phi =(\\phi $,id,id,id) of crossed\nsquares:\n$$\n\\xymatrix@!0{\n & L \\ar[rrrr] \\ar[dddd]\\ar[ddl]_{\\phi}\n & & & &\\overline{M} \\ar[dddd]\\ar@{=}[ddl] \\\\\n \\\\\n L' \\ar[rrrr]\\ar[dddd]\n & & & &\\overline{M} \\ar[dddd] \\\\\n \\\\\n & M \\ar[rrrr]\\ar@{=}[ddl]\n & & & &M\\rtimes R \\ar@{=}[ddl] \\\\\n \\\\\n M \\ar[rrrr]\n & & & & M\\rtimes R }\n$$\nsuch that $\\phi \\nu ^{\\prime }=\\nu ,$ where $\\nu ^{\\prime\n}:\\mathbf{S_3}\\to L $ is the inclusion.\n\nWe denote such a free crossed square of algebras by\n$(L,M,\\overline{M},M\\rtimes R). $\n\nWe know the free pre-crossed module on\n$f:\\mathbf{S_{2}}\\rightarrow R$ is $\\partial\n:R^{+}[\\mathbf{S_{2}}]\\rightarrow R$, so the function\n$$f_{3}:\\mathbf{S_{3}}\\rightarrow M,\\ \\ (\\text{with}\\ \\ M=R^{+}[\\mathbf{S_{2}}])$$\nis precisely the data $(\\mathbf{S_{3}},f_{3})$ for 2-dimensional\nconstruction data in the simplicial context. We thus need to recall\nthe 2-dimensional construction data in a free simplicial algebra\n(cf. \\cite{ap2}). This 2-dimensional form can be pictured by the\ndiagram\n\\begin{equation*}\n\\xymatrix@C=3pc{\\mathbf{E^{(2)}}:\\cdots (R[s_0 (\\mathbf{S_2}),s_1\n\\mathbf{S_2}])[\\mathbf{S_3}]\\ar@<2ex>[r]^-{d_0,d_1,d_2}\\ar@<1ex>[r]\\ar@<0pt>[r]\n&\nR[\\mathbf{S_2}]\\ar@<1ex>[r]^-{d_0,d_1}\\ar@<0ex>[r]\\ar@<1ex>[l]\\ar@<2ex>[l]^-{s_0,s_1}&R\n\\ar@<1ex>[l]^-{s_0}}\n\\end{equation*}\nwith the simplicial morphisms given as in \\cite{ap3}. Here\n$\\mathbf{S_{2}} =\\{S_{1},\\ldots ,S_{n}\\}$ and\n$\\mathbf{S_{3}}=\\{S_{1}^{\\prime },\\ldots ,S_{m}^{\\prime }\\}$ are\nfinite sets with $\\pi _{0}(\\mathbf{E}^{(2)})\\cong\nB=R\/(t_{1},\\ldots ,t_{n})$ as an $R$-algebra where $t_i=\\partial\nS_i$.\n\n\\subsection{Free crossed squares exist}\n\n\\begin{thm}\nA totally free crossed square $(L,M,\\overline{M},M\\rtimes R)$ exists\non the 2-dimensional construction data and is given by\n$\\mathbf{M(E^{(2)},2)}$ where $\\mathbf{E^{(2)}}$ is the 2-skeletal\nfree simplicial algebra defined by the construction data.\n\\end{thm}\n\n\\begin{pf}\nSuppose given the 2-dimensional construction data for a free\nsimplicial algebra, \\textbf{E}, which we will take as above as the\ndata for a totally free crossed square. We will not assume\ndetailed knowledge of \\cite{ap1} so we start with $R$ and $\\\nf_{2}:\\mathbf{S_{2}}\\rightarrow R$ and form$\\\nM=R^{+}[\\mathbf{S_{2}}]=(S_{1},\\ldots ,S_{n}).$ This gives\n$\\partial _{1}:R^{+}[\\mathbf{S_{2}}]\\rightarrow R$ as free\npre-crossed module on $f_{2}.$ The semidirect product gives us\nback\n\\begin{equation*}\nR[\\mathbf{S_{2}}]\\cong M\\rtimes R\n\\end{equation*}\nand we can identify this with $\\mathbf{E_{1}^{(2)}}$. This identification\nalso makes\n\\begin{equation*}\nM\\cong \\mbox{\\rm Ker}d_{0}^{1}\n\\end{equation*}\nfor the $d_{0}^{1}$ of $\\mathbf{E^{(2)}}$.\n\nNext form $\\overline{M}=\\{(m,r)\\in M\\rtimes R:\\partial m=-r\\}$. As\n$m\\in R^{+}[\\mathbf{S_{2}}]$, writing $m=\\sum r_{\\alpha }S^{\\alpha\n}$ for multi-indices $\\alpha $, we get $\\partial m=\\sum r_{\\alpha\n}t^{\\alpha }$ where $t_{i}=f_{2}(S_{i})$. Thus we can identify\n$\\overline{M}$ with $(S_{1}-t_{1},\\ldots ,S_{n}-t_{n})$ which is\nexactly Ker$d_{1}^{1}$ (for this see \\cite{ap1}).\n\nNow $\\ f_{3}:\\mathbf{S_{3}}\\rightarrow \\mbox{\\rm Ker}\\partial\n_{1}= \\mbox{\\rm Ker}(\\partial :NE_{1}^{(2)}\\rightarrow\nNE_{0}^{(2)})\\subset R^{+}[\\mathbf{ S_{2}}].$ We know that this\nallows us to construct $\\mathbf{E_{2}^{(2)}}$, and hence\n$\\mathbf{E_{n}^{(2)}}$ for $n\\geqslant 3$, and in addition that taking\n\\begin{equation*}\nL=NE_{2}^{(2)}\/\\partial _{3}(NE_{3}^{(2)}),\n\\end{equation*}\ngives a crossed square\n\\begin{equation*}\n\\xymatrix{L\\ar[r]^{\\partial'_2}\\ar[d]_{\\partial_2}&\\overline{M}\\ar[d]^{\\lambda'}\\\\\nM\\ar[r]_{\\lambda}&E^{(2)}_1}\n\\end{equation*}\nwhich is $M(\\mathbf{E}^{(2)},2)$. We claim this is the totally free crossed\nsquare on the construction data.\n\nAt this stage it is worth nothing that there would seem to be no simple\nadjointness statement between $\\mathbf{M(-,2)}$ and some functor that would\ngive a quick proof of freeness. The problem being that $\\mathbf{M(-,2)}$\nseems to be an adjoint only up to some sort of coherent homotopy. To avoid\nthis difficulty we use a more combinatorial approach involving the higher\norder Peiffer elements and a more concrete description of $L$.\n\nIn \\cite{ap1}, the first author and Porter analysed in general the\nstructure of algebras of boundaries such as $\\partial\n_{3}(NE_{3}^{(2)}).$ There they showed that $NE_{1}^{(2)}$ is\ngenerated as an ideal by elements of the following forms:\n\nFor all $x\\in NE_{1}^{(2)},~y\\in NE_{2}^{(2)}$,\n\\begin{align*}\nC_{(1,0)(2)}(x\\otimes y) & = (s_{1}s_{0}x-s_{2}s_{0}x)s_{2}y, \\\\\nC_{(2,0)(1)}(x\\otimes y) & = (s_{2}s_{0}x-s_{2}s_{1}x)(s_{1}y-s_{2}y), \\\\\nC_{(2,1)(0)}(x\\otimes y) & = s_{2}s_{1}x(s_{0}y-s_{1}y+s_{2}y);\\\\\n\\intertext{ whilst for all $x,~y\\in NE_{2}$,}\nC_{(1)(0)}(x\\otimes y) & = s_{1}x(s_{0}y-s_{1}y)+s_{2}(xy), \\\\\nC_{(2)(0)}(x\\otimes y) & = (s_{2}x)(s_{0}y), \\\\\nC_{(2)(1)}(x\\otimes y) & = s_{2}x(s_{1}y-s_{2}y).\n\\end{align*}\nWe know that $\\partial _{3}(NE_{3}^{(2)})$ is generated by elements\nof the forms\n\\begin{equation*}\n(s_{1}s_{0}d_{1}S_{i}-s_{0}S_{i})S_{j}^{\\prime },\\ \\\n(s_{0}S_{i}-s_{1}S_{i})(s_{1}d_{2}S_{j}^{\\prime }-S_{j}^{\\prime }),\\ \\\ns_{1}S_{i}(s_{0}d_{2}S_{j}^{\\prime }-s_{1}d_{2}S_{j}^{\\prime }+S_{j}^{\\prime\n}),\n\\end{equation*}\nand for $S_{i}^{\\prime },$ $S_{j}^{\\prime }\\in \\mathbf{S_{2}}$,\n\\begin{equation*}\nS_{i}^{\\prime }(s_{1}d_{2}S_{j}^{\\prime }-S_{j}^{\\prime }),\\ \\ S_{i}^{\\prime\n}(S_{j}^{\\prime }+s_{0}d_{2}S_{j}^{\\prime }-s_{1}d_{2}S_{j}^{\\prime }),\\ \\\n(s_{0}d_{2}S_{i}^{\\prime }-s_{1}d_{2}S_{i}^{\\prime }+S_{i}^{\\prime\n})(s_{1}d_{2}S_{j}^{\\prime }-S_{j}^{\\prime }),\n\\end{equation*}\nwhich are the second order Peiffer elements defined in \\cite{ap1},\nwhere $S_{i}\\in NE_{1}=\\mbox{\\rm Ker}d_{0}=R^{+}[\\mathbf{S_{2}}]$\nand $S_{i}^{\\prime }\\in\nNE_{2}=R[s_{0}(\\mathbf{S_{2}})]^{+}[s_{1}(\\mathbf{S_{2}}),\\mathbf{S_{3}}]\\cap\n(s_{0}(\\mathbf{S_{2}})-s_{1}(\\mathbf{S_{2}})).$\n\nThe above diagram can thus be realised as\n\\begin{equation*}\n\\xymatrix{\\dfrac{R[s_0({\\bf S_2})]^+[s_1({\\bf S_2}),{\\bf S_3}]\\cap\n(s_0({\\bf S_2})-s_1({\\bf\nS_2}))}{P_2}\\ar[r]^-{\\partial'_2}\\ar[d]_-{\\partial_2}&\\overline{R^+\n[{\\bf S_2}]}\n\\ar[d]^{\\lambda'}\\\\\nR^{+}[\\mathbf{S_2}]\\ar[r]_{\\lambda}&R[\\mathbf{S_2}]}\n\\begin{array}{cc}\n&\\\\\n&\\\\\n&(\\ast)\\\\\n\\end{array}\n\\end{equation*}\nwhere $P_{2}$ is the second order Peiffer ideal which is in fact\njust $\\partial _{3}(NE_{3}^{(2)}).$\n\nGiven any crossed square $(T,M,\\overline{M},M\\rtimes R)$ with a\nfunction $\\nu :\\mathbf{S_{3}}\\rightarrow T$, then there exists a\nmorphism\n\\begin{equation*}\n\\phi :(L,M,\\overline{M},M\\rtimes R)\\longrightarrow\n(T,M,\\overline{M},M\\rtimes R)\n\\end{equation*}\ngiven by\n\\begin{equation*}\n\\phi (S_{i}^{\\prime }+P_{2})=\\nu (S_{i}^{\\prime })\n\\end{equation*}\nsuch that $\\phi \\nu ^{\\prime }=\\nu $ where $\\nu\n:\\mathbf{S_{3}}\\rightarrow L$ is a function. The existence of\n$\\phi $ follows by using the freeness property of the algebra\n$NE_{2}^{(2)}$ and then restricting to\n$R[s_{0}(\\mathbf{S_{2}})]^{+}[s_{1}(\\mathbf{S_{2}}),\n\\mathbf{S_{3}}]\\cap\n(s_{0}(\\mathbf{S_{2}})-s_{1}(\\mathbf{S_{2}})).$ The ideal\ngenerating elements of $P_{2}$ are then easily shown to have\ntrivial image in $T$ as that algebra is part of the second crossed\nsquare.\n\nThus diagram $(\\ast )$ is the desired totally free crossed square on\nthe 2-dimensional construction data. The crossed square properties\nof $(L,M,\\overline{M},M\\rtimes R)$ may be easily verified or derived\nfrom the fact that this is exactly $\\mathbf{M(E}^{(\\mathbf{2})},2).$\n\\end{pf}\n\n\n\\vspace{0.5cm}\n\n\\textbf{Remark}:\n\n At this stage, it is important to note that nowhere\nin the argument was use made of the freeness of the 1-skeleton. If\n\\textbf{E} is any 1-skeletal simplicial algebra and we form a new\nsimplicial algebra \\textbf{F} by adding in a set $\\mathbf{S}_{3}$\nof new generators in dimension 2, so that for instance,\n$F_{2}=E_{2}^{+}[\\mathbf{S}_{3}],$ a free algebra on\n$\\mathbf{S}_{3}$, then we can use \\thinspace\n$M=NE_{1}=$Ker$d_{0}^{1}$ as before even though it need not be\nfree. The corresponding $\\overline{M}$ is then isomorphic to\nKer$d_{1}^{1}$ with the bottom right hand corner being $E_{1}$.\nThe `construction data' is now replaced by data for killing some\nelements of $\\pi _{1}(E),$ specified by\n$f_{3}:\\mathbf{S}_{3}\\rightarrow M.$ We introduce the term\n`totally free crossed square' for the type of free crossed square\nconstructed in the above theorem, using free crossed square for\nthe more general situation in which $(M,E,\\partial )$ and $f_{3}$\nare specified and no requirement $(M,E,\\partial )$ to be a free\npre-crossed module is made.\n\n\n\\subsection{ The $n$-type of the $k$-skeleton}\n\nAs in the other paper in this series, we will use the\n`step-by-step' construction of a free simplicial algebra to\nobserve the way in which the models react to the various steps of\nthe construction.\n\nBy a `step-by-step' construction of a free simplicial algebra,\nthere are simplicial inclusions\n\\begin{equation*}\n\\mathbf{E}^{(0)}\\subseteq \\mathbf{E}^{(1)}\\subseteq\n\\mathbf{E}^{(2)}\\subseteq \\cdots\n\\end{equation*}%\nThe functor, $\\mathbf{M(\\ ,\\ }n\\mathbf{)}$, from the category of\nsimplicial algebras to that of crossed $n$-cubes gives the\ncorresponding inclusions\n\\begin{equation*}\n\\mathbf{M}(\\mathbf{E}^{(0)},\\ n)\\hookrightarrow\n\\mathbf{M}(\\mathbf{E}^{(1)},\\ n)\\hookrightarrow\n\\mathbf{M}(\\mathbf{E}^{(2)},\\ n)\\hookrightarrow \\mathbf{\\cdots }\n\\end{equation*}%\nWe investigate $\\mathbf{M(E^{(i)}},n)$, for $n=0,1,2$, and varying\n$i$.\n\nFirstly look at $\\mathbf{M(E^{(0)}},n),$where the 0-skeleton\n$\\mathbf{E}^{(0)}\\,$ is\n\\begin{equation*}\n\\begin{array}{lccc}\n\\mathbf{E}^{(0)}: & \\cdots \\longrightarrow R\\longrightarrow R\\longrightarrow\nR & \\overset{f}{\\longrightarrow } & B%\n\\end{array}%\n\\end{equation*}\nwith the $d_i^n=s_j^n=\\ $identity homomorphisms.\n\nFor $n=0,\\,$there is an equality\n$\\mathbf{M(E^{(0)}},0)=E_0^{(0)}\/d_1(\\mbox{\\rm Ker}d_0)=R, $\n and so $\\mathbf{M(E^{(0)},}0)$ is just an algebra of $0$-simplices of \\textbf{E}.\n\nFor $n=1, \\mathbf{M(E^{(0)}},1)$ is $NE_1^{(0)}\/\\partial\n_2NE_2^{(0)}\\rightarrow E_0. $ It is easy to show that\n$NE_1^{(0)}\/\\partial _2NE_2^{(0)}$ is trivial in the 0-skeleton\n$\\mathbf{E}^{(0)}$ and hence\n\\begin{equation*}\n\\mathbf{M(E^{(0)}},1)\\cong (0\\longrightarrow R).\n\\end{equation*}\n\nFor $n=2$, \\ $\\mathbf{M(E^{(0)}},2)$ is the trivial crossed square\n\\begin{align*}\n\\begin{aligned}\\xymatrix{ NE_2 \/ d^{3}_{3} (NE_3 ) \\ar[d]\\ar[r]&\\text{Ker}d_0\n ^1\\ar[d]\\\\\n \\text{Ker}d_1^1\\ar[r]&E_1}\\end{aligned}&= \\begin{aligned}\\xymatrix{0 \\ar[d]\\ar[r]&0\\ar[d]\\\\\n 0\\ar[r]&R.}\\end{aligned}\n\\end{align*}\nNext take $\\mathbf{M(E^{(1)}},n)$ and recall that the 1-skeleton\n\\textbf{E}$^{(1)}$ is\n\\begin{equation*}\n\\xymatrix@C=3pc{\\mathbf{E^{(1)}}:\\cdots R[s_0 (\\mathbf{S_2}),s_1(\n\\mathbf{S_2})]\\ar@<2ex>[r]^-{d_0,d_1,d_2}\\ar@<1ex>[r]\\ar@<0pt>[r] &\nR[\\mathbf{S_2}]\\ar@<1ex>[r]^-{d_0,d_1}\\ar@<0ex>[r]\\ar@<1ex>[l]\\ar@<2ex>[l]^-{s_0,s_1}&R\n\\ar@<1ex>[l]^-{s_0}\\ar[r]^{f}&R\/I.}\n\\end{equation*}\nFor $n=0$, it follows that $\\mathbf{M(E^{(1)},} 0)$ is $\nE_0^{(1)}\/d_1(\\mbox{\\rm Ker}d_0)\\cong R\/I $ which is $\\pi\n_0(\\mathbf{E} ^{(1)})\\cong\\pi _0(\\mathbf{E})$.\n\nLet $n=1.$ We have that%\n\\begin{align*}\n\\mathbf{M(E}^{(1)}\\mathbf{,\\ }1\\mathbf{)} & = NE_{1}^{(1)}\/\\partial\n_{2}NE_{2}^{(1)}\\rightarrow E_{0}^{(1)} \\\\\n& = R^{+}[\\mathbf{S_{2}}]\/P_{1}\\rightarrow R%\n\\end{align*}\nwhich is the free crossed module. In fact this is the free crossed\nmodule on the (generalised) presentation\n$(\\mathbf{S_{1};S_{2}},f_{2})$. As pointed out in \\cite{ap1}, it is\noften convenient to generalise the notion of a presentation\n$P=(R\\,;x_{1},\\ldots ,x_{n})$ of an $R$-algebra $B$ in this way and\n$\\mathbf{E}^{(1)}$ is the 1-skeleton of the free simplicial algebra\ngenerated by this presentation, then\n\\begin{equation*}\n\\delta :NE_{1}^{(1)}\/\\partial _{2}(NE_{2}^{(1)})\\longrightarrow NE_{0}^{(1)}\n\\end{equation*}%\nis the free crossed module on $\\{S_{1},\\ldots ,S_{n}\\}\\rightarrow\nR$. This has a neat description (cf. \\cite{porter1}) as follows: The\nPeiffer ideal $P_1 =\\partial_2(NE^{(1)}_2)$ contains all such terms\nas $S_iS_j-\\delta(S_i)S_j$, so any polynomial in the $S_i$'s can be\nreduced mod $P_1$ to a linear form, hence each coset has a\nrepresentative of the form $\\sum r_iS_i$. As $S_iS_j=S_jS_i$, these\nrepresentatives are nonunique and so the free crossed $C$ is $R^n$\nfactored by all $\\delta(S_i)S_j-\\delta(S_j)S_i$ we thus have\n\\begin{equation*}\n\\pi _{1}(\\mathbf{M}(\\mathbf{E}^{(1)},1))\\cong \\mathrm{Ker}(C\\longrightarrow\nR)\\cong H_{2}(B,B)\n\\end{equation*}\nthe second Andr\\'{e}-Quillen homology group, where $C\\cong\nR^{n}\/\\mathrm{Im}d, $ for $d: \\Lambda ^{2}R^{n}\\rightarrow R$, the\nsecond Kozsul differential, see \\cite{porter1} for details. Thus\n\\begin{align*}\n\\pi _{0}(\\mathbf{M}(\\mathbf{E}^{(1)},1)) & \\cong B \\\\\n\\pi _{1}(\\mathbf{M}(\\mathbf{E}^{(1)},1)) & \\cong H_{2}(B,B)\n\\intertext{whilst} \\pi _{i}(\\mathbf{M}(\\mathbf{E}^{(1)},1))&\\cong\n0.\n\\end{align*}\n\nFor $n=2$,\n$$\n NE_{2}^{(1)}=(R[s_{0}(\\mathbf{S_{2}})]^{+}[s_{1}(\\mathbf{S_{2}})])\\cap\n(s_{0}(\\mathbf{S_{2}})-s_{1}(\\mathbf{S_{2}})),\n$$\n\n$\\mathbf{M(E}^{(1)},2)$ simplifies to give (up to isomorphism)\n\\begin{align*}\n\\begin{aligned}\n \\xymatrix{ NE_2 \/ d^{3}_{3} (NE_3 ) \\ar[d]\\ar[r]&\\text{Ker}d_0\n ^1\\ar[d]\\\\\n \\text{Ker}d_1^1\\ar[r]&E_1}\n \\end{aligned}&=\n\\begin{aligned}\n \\xymatrix{\\dfrac{(R[s_0({\\bf S_2})]^+[s_1({\\bf S_2})])\\cap\n(s_0({\\bf S_2})-s_1({\\bf S_2}))}{P_2}\n\\ar[d]\\ar[r]&\\overline{R^+[{\\bf\nS_2}]}\\ar[d]\\\\\n R^+[{\\bf S_2}]\\ar[r]&R[{\\bf S_2}]}\n \\end{aligned}\n \\end{align*}\nwhich is a crossed square.\n\nLet us next look at $\\mathbf{M(E}^{(2)},n).$ Recall the 2-skeleton\n$\\mathbf{E}^{(2)}$\n\\begin{equation*}\n\\xymatrix@C=3pc{(R[s_0 (\\mathbf{S_2}),s_1\n\\mathbf{S_2}])[\\mathbf{S_3}]\\ar@<2ex>[r]^-{d_0,d_1,d_2}\\ar@<1ex>[r]\\ar@<0pt>[r]\n&\nR[\\mathbf{S_2}]\\ar@<1ex>[r]^-{d_0,d_1}\\ar@<0ex>[r]\\ar@<1ex>[l]\\ar@<2ex>[l]^-{s_0,s_1}&R\n\\ar@<1ex>[l]^-{s_0}\\ar[r]^f&R\/I.}\n\\end{equation*}\nThe following equalities can be easily obtained by direct\ncalculation: \\thinspace for $n=0,$\n\\begin{equation*}\n\\mathbf{M(E^{(2)}},0)=E_{0}\/d_{1}(\\mbox{\\rm Ker} d_{0})\\cong \\pi\n_{0}(\\mathbf{E}^{(2)})=\\mathbf{M(E}^{(1)},0).\n\\end{equation*}\nFor $n=1,$\n\\begin{equation*}\n\\mathbf{M(E}^{(2)},1)\\cong (R^{+}[\\mathbf{S_{2}}\n]\/P_{1}\\rightarrow R)=\\mathbf{M(E}^{(1)},1).\n\\end{equation*}%\nand there is an isomorphism\n\\begin{equation*}\n\\pi _{2}(\\mathbf{E}^{(2)})\\cong \\mbox{\\rm Ker}\\left(\nNE_{2}^{(2)}\/\\partial _{3}(NE_{3}^{(2)})\\longrightarrow\nE_{1}^{(2)}\\right).\n\\end{equation*}\n Finally, let $n=2.$ Since by an earlier result of this section,\n $\\mathbf{M(E}^{(2)},2)$ corresponds to the free\ncrossed square, we obtain:\n\\begin{align*}\n\\begin{aligned}\n \\xymatrix{ NE_2^{(2)} \/ d^{3}_{3} (NE_3^{(2)} ) \\ar[d]\\ar[r]&\\text{Ker}d_0\n ^1\\ar[d]\\\\\n \\text{Ker}d_1^1\\ar[r]&E_1}\n \\end{aligned}&=\n \\begin{aligned}\n \\xymatrix{\\dfrac{R[s_0({\\bf S_2})]^+[s_1({\\bf S_2}),\\bf{S_3}]\\cap\n(s_0({\\bf S_2})-s_1({\\bf S_2}))}{P_2}\n\\ar[d]\\ar[r]&\\overline{R^+[{\\bf\nS_2}]}\\ar[d]\\\\\n R^+[{\\bf S_2}]\\ar[r]&R[{\\bf S_2}].} \\\\\n \\end{aligned}\n \\end{align*}\nThis reduces to the earlier case if $\\mathbf{S_3}$ is empty. Thus we\nhave the following relations\n$$\n\\mathbf{M(E}^{(2)},0)=\\mathbf{M(E}^{(1)},0), \\quad\n\\mathbf{M(E}^{(2)},1)=\\mathbf{M(E}^{(1)},1)\n$$\nbut $\\mathbf{M(E}^{(2)},2)$ and $\\mathbf{M(E}^{(3)},2)$ need not to\nbe the same due to the additional influence of $\\mathbf{S_3}$. Of\ncourse it is clear that, in general:\n$$\n\\mathbf{M(E}^{(i)},n)=\\mathbf{M(E}^{(i+1)},n) \\quad \\text{if}\\ i\\geqslant\nn+1.\n$$\nClearly these top left hand corner terms are unwieldy to handle\nand we will seek in section \\ref{sec5} an alternative description.\n\n\n\n\\section{Squared Complexes}\n\nThe first author and M. Ko\\c{c}ak defined $n$-crossed complexes of\nalgebras in \\cite{ak} as the analogue for commutative algebras of\nthe notion introduced by Ellis, \\cite{ellis3}, in homotopy theory.\nIn this paper we will only need the case $n=2$, which we shall\ncall a \\emph{squared complex}; it consists of a diagram of algebra\nhomomorphisms\n$$\n\\xymatrix{(*)&&\\cdots\\ar[r]&C_5\\ar[r]^{\\partial_5}&C_4\\ar[r]^{\\partial_4}&L\\ar[r]^{\\lambda'}\\ar[d]_{\\lambda}&N\n\\ar[d]^{\\mu'}\\\\\n&&&&&M\\ar[r]_{\\mu}&R}\n$$\ntogether with an action of $R$ on $L,N,M$ and $C_i$ for $i\\geqslant 4,$\nand a function $h:M\\times N\\rightarrow L.$ The following axioms need\nto be satisfied.\n\n$(i)$ The square\n\\begin{equation*}\n\\left(\n\\begin{array}{cc}\n \\xymatrix{L\\ar[r]\\ar[d]&N\\ar[d]\\\\\n M\\ar[r]&R}\n\\end{array}\n\\right)\n\\end{equation*}\nis a crossed square.\n\n$(ii)$ $C_{n}$ is an $A$-module for $n\\geqslant 4$ with $A=R\/\\left\\{ \\mu\n(M)+\\mu ^{\\prime }(N)\\right\\} .$\n\n$(iii)$ The action of $R$ on $C_{n},n\\geqslant 4,$ is such that $\\mu (M)$\nand $\\mu ^{\\prime }(N)$ operate trivially. Thus each $C_{n}$ is an\n$A$-module.\n\n$(iv)$ each $\\partial _{n}$ is $A$-module homomorphism and for\n$n\\geqslant 4$, $\\partial _{n}\\partial _{n+1}=0$.\n\nA \\emph{morphism} of square complexes\n\\begin{equation*}\n\\phi :\\left(\n\\begin{array}{cc}\nC_i, & \\left(\n\\begin{array}{cc}\nL & N \\\\\nM & R%\n\\end{array}\n\\right)%\n\\end{array}\n\\right) \\longrightarrow \\left(\n\\begin{array}{cc}\nC_i^{\\prime }, & \\left(\n\\begin{array}{cc}\nL^{\\prime } & N^{\\prime } \\\\\nM^{\\prime } & R^{\\prime }\n\\end{array}\n\\right)%\n\\end{array}\n\\right)\n\\end{equation*}\nconsists of a morphism of crossed squares $(\\phi _L,\\phi _M,\\phi\n_N,\\phi _R)$ together with a family of $\\phi _R$-equivariant\nhomomorphisms $\\phi _i$, $i\\geqslant 4$ satisfying $\\phi _L\\partial\n_4=\\partial _4^{\\prime }\\phi _4$ and $\\phi _{i-1}\\partial\n_i=\\partial _i^{\\prime }\\phi _i$ for $i\\geqslant 5.$ There is clearly a\ncategory \\textbf{SqComp} of squared complexes.\n\nBy a (totally) free squared complex, we will mean one in which the\ncrossed square is (totally) free, and in which each $C_n$ is free\nas a $\\pi _0$-module for $i\\geqslant 3.$\n\n\\begin{prop}\nThere is a functor\n\\begin{equation*}\nC(\\ ,2):\\mathbf{SimpAlg} \\longrightarrow \\mathbf{SqComp}\n\\end{equation*}\nsuch that free simplicial algebras are sent to totally free\nsquared complexes.\n\\end{prop}\n\n\\begin{pf}\nLet $\\mathbf{E}$ be a simplicial algebra. We will define a squared\ncomplex $ C(\\mathbf{E},2)$ by specifying $C(\\mathbf{E},2)_{A}$ for\neach $A\\subseteq \\langle 2\\rangle$ and for $n\\geqslant\n3,C(\\mathbf{E},2)_{n}.$ As usual, (cf. the other paper in this\nseries {\\cite{ap1, ap2, ap3}), we will denote by $D_{n}$ the ideal\nof $NE_{n}$ generated by degenerate elements. }\n\nFor $A\\subseteq \\langle 2\\rangle$, we define in particular\n\\begin{equation*}\nC(\\mathbf{E},2)_{\\langle\n2\\rangle}=\\mathbf{M}(\\mathbf{sk}_{2}\\mathbf{E},2)_{\\langle\n2\\rangle}=\\dfrac{NE_2}{\\partial_3(NE_3 \\cap D_3)}.\n\\end{equation*}%\nWe do not need to define $\\mu _{i}$ and the $h$-maps relative to\nthese algebras as they are already defined in the crossed square\n$\\mathbf{M}(\\mathbf{sk}_{2} \\mathbf{E},2)_{A}.$\n\nFor $n\\geqslant 3,$ we set\n\\begin{equation*}\nC(\\mathbf{E},2)_{n}=\\frac{NE_{n}}{(NE_{n}\\cap D_{n})+d_{n+1}(NE_{n+1}\\cap\nD_{n+1})}.\n\\end{equation*}\nAs this is part of the crossed complex associated to \\textbf{E},\nwe can take the structure maps to be those of that crossed\ncomplex, cf. \\cite{ap1}. The terms are all modules over the\ncorresponding, $\\pi _{0}$ as is easily checked. The final missing\npiece, $\\partial _{3},$ of the structure is induced by the\ndifferential $\\partial _{3}$ of $NE.$\n\nThe axioms for a squared complex can now be verified the known\nresults for crossed squares and for crossed complexes with a direct\nverificatiton of those axioms relating to the interaction of the two\nparts of the structure, much as in \\cite{ap1}.\n\nNow suppose the simplicial algebra is free. The proof of the\nfreeness of $\\mathbf{M}(sk_{2}\\mathbf{E},2)$ together with the\nfreeness of the crossed complex of a free simplicial algebra,\n\\cite{ap1}, now completes the proof.\n\\end{pf}\n\nSuppose that $\\rho $ is a general squared complex. The \\emph{homotopy modules%\n} $\\pi _{n}(\\rho ),n\\geqslant 0$ of $\\rho $ are defined in \\cite{ak} to\nbe the homology modules of the complex\n\\begin{equation*}\n\\xymatrix{\\cdots\\ar[r]^{\\partial_6}&C_5\\ar[r]^{\\partial_5}&C_4\n\\ar[r]^{\\partial_4}& L\\ar[r]^-{\\partial_3}&M\\rtimes\nN\\ar[r]^-{\\partial_2}\\ar[r]&0}\n\\end{equation*}%\nwith $\\partial _{3}(l)=(-\\lambda ^{\\prime }l,\\lambda l)$ and\n$\\partial _{2}(m,n)=\\mu (m)+\\mu ^{\\prime }(n)$. The axioms of a\ncrossed square guarantee (see \\cite{ap3}) that $\\partial _{3}$ and\n$\\partial _{2}$ are homomorphisms with $\\partial _{4}(C_{4})$ an ideal in Ker$(\\partial _{3})$, \\ $%\n\\partial _{3}(L)$ an ideal in Ker$(\\partial _{2})$, and $\\partial _{2}(M\\rtimes\nN)$ an ideal in $R$.\n\n\\begin{prop}\nThe homotopy groups of $C(\\mathbf{E},2)$ are isomorphic to those of $\\mathbf{%\nE}$ itself.\n\\end{prop}\n\n\\begin{pf}\nAgain this is a consequence of well-known results on the two parts of the\nstructure.\n\\end{pf}\n\n\n\n\\section{Alternative Description of Freeness \\label{sec5}}\n\nIn the context of CW-compexes, Ellis \\cite{ellis3} gave a neat\ndescription of the top algebra $L$ in (totally) free crossed\nsquares. A free simplicial algebra is the algebraic analogue of a\nCW-complex so one would expect a similar result to hold in this\nsetting. For this we need two constructions.\n\n\n\\subsection{Tensor Products}\n\nSuppose that $\\mu :M\\rightarrow R$ and $\\nu :N\\rightarrow R$ are crossed\nmodules of commutative algebras over $R$. The algebras $M$ and $N$ act on\neach other, and themselves, via the action of $R$. The \\emph{tensor product}\n$M\\otimes N$ is the algebra generated by the symbols $m\\otimes n$ for $m\\in\nM,\\ n\\in N$ and $r\\in R$ subsect to the relations%\n\\begin{equation*}\n\\begin{array}{rrll}\n\\text{(i)} & r\\left( m\\otimes n\\right) & = & rm\\otimes n=m\\otimes rn \\\\\n\\text{(ii)} & \\left( m+m^{\\prime }\\right) \\otimes n & = & m\\otimes\nn+m^{\\prime }\\otimes n \\\\\n& m\\otimes (n+n^{\\prime }) & = & m\\otimes n+m\\otimes n^{\\prime } \\\\\n\\text{(iii)} & \\left( m\\otimes n\\right) \\left( m^{\\prime }\\otimes n^{\\prime\n}\\right) & = & \\left( mm^{\\prime }\\otimes nn^{\\prime }\\right)%\n\\end{array}\n\\end{equation*}\nwhere $m^{\\prime }\\in M$ and $n^{\\prime }\\in N.$ There are\nmorphisms $ \\lambda :M{\\otimes }N\\rightarrow M,\\ m{\\otimes\n}n\\mapsto m\\cdot n=m\\nu (n)$ and $\\lambda ^{\\prime }:M{\\otimes\n}N\\rightarrow N,\\ m{\\otimes }n\\mapsto n\\cdot m=\\mu (m)n$. The\nalgebra $R$ acts on $M\\otimes N$ by $r\\cdot (m\\otimes n)=r\\cdot\nm\\otimes n=m\\otimes r\\cdot n,$ and there is a function $ h:M\\times\nN\\rightarrow M\\otimes N,$ \\ $(m,n)\\mapsto m{\\otimes }n$. It is\nverified in \\cite{ak} that this structure is a crossed square\n\\begin{equation*}\n\\xymatrix{M\\otimes N \\ar[r]^-{\\lambda}\\ar[d]_{\\lambda'}&N\\ar[d]^{\\nu}\\\\\nM\\ar[r]_{\\mu}&R}\n\\end{equation*}%\nwith the universal property of extending the corner\n\\begin{equation*}\n\\xymatrix{&N\\ar[d]^{\\nu}\\\\\nM\\ar[r]_{\\mu}&R.}\n\\end{equation*}\n\n\n\\subsection{Coproducts}\n\nThe following construction is due to Shammu \\cite{nizar}.\n\nLet $(M,R,\\partial _{1}),(N,R,\\partial _{2})$ be crossed\n$R$-modules. Then $N $ acts on $M$, and $M$ acts on $N$, via the\ngiven actions of $R$. Let $ M\\rtimes N$ denote the semidirect\nproduct with the multiplication given by\n\\begin{equation*}\n(m,n)(m^{\\prime },n^{\\prime })=(mm^{\\prime },n\\partial _{2}(m^{\\prime\n})+\\partial _{2}(m)n^{\\prime }+nn^{\\prime })\n\\end{equation*}\nand injections\n\\begin{equation*}\n\\begin{array}{ccccc}\ni^{\\prime }\\colon & M\\rightarrow M\\rtimes N & \\mbox{ and } & j^{\\prime\n}\\colon & N\\rightarrow M\\rtimes N \\\\\n& m\\mapsto (m,0) & & & n\\mapsto (0,n).\n\\end{array}%\n\\end{equation*}\nWe define the pre-crossed module\n\\begin{equation*}\n\\begin{array}{cccl}\n\\underline{\\delta }\\colon & M\\rtimes N & \\longrightarrow & R \\\\\n& (m,n) & \\longmapsto & \\partial _{1}(m)+\\partial _{2}(n).\n\\end{array}\n\\end{equation*}\nLet $P$ be the ideal of $M\\rtimes N$ generated by elements of the\nform\n\\begin{equation*}\n(m,n)(m^{\\prime }n^{\\prime })-\\underline{\\delta }(m,n)(m^{\\prime },n^{\\prime\n})=(-\\partial _{1}(m)n,m\\partial _{2}(n))\n\\end{equation*}\nfor all $(m,n),(m^{\\prime },n^{\\prime })\\in M\\rtimes N.$ Thus we are able to\nform the quotient algebra $M\\rtimes N\/P$ and obtain an induced morphism\n\\begin{equation*}\n\\partial \\colon M\\rtimes N\/P\\longrightarrow R\n\\end{equation*}%\ngiven by\n\\begin{equation*}\n\\partial (m,n)+P=\\partial _{1}m+\\partial _{2}n.\n\\end{equation*}%\nLet $q\\colon M\\rtimes N\\rightarrow M\\rtimes N\/P$ be projection and\nlet $ i=qi^{\\prime },j=qj^{\\prime }.$ Then $M\\sqcup N=M\\rtimes\nN\/P,$ with, the morphism $i,j,$ is the \\emph{coproduct } of $M,N$\nin the category $\\mathbf{ XMod_{k}}$. The above notation can be\nsummarised in the following diagram:\n\\begin{equation*}\n\\xymatrix{M\\ar[dr]_{i}\\ar[dd]_{\\partial_1}\\ar[rr]^{i'}&&M\\rtimes\nN\\ar[dl]^{q}\\\\\n&M\\rtimes N\/P \\ar[dl]^{\\partial}&\\\\\nR&&N.\\ar[uu]^{j'}\\ar[ll]^{\\partial_2}\\ar[ul]^{j}}\n\\end{equation*}\n\n\\begin{prop}\nLet\n\\begin{equation*}\n\\left(\n\\begin{array}{cc}\n\\xymatrix{L\\ar[r]\\ar[d]&\\overline{M}\\ar[d]\\\\\nM\\ar[r]&M\\rtimes R}\n\\end{array}%\n\\right)\n\\end{equation*}\nbe a free crossed square on the 2-dimensional construction data or\non functions $(f_{2},f_{3})$ as described above. Let $\\partial\n:C\\rightarrow M\\rtimes R$ be the free crossed module on the\nfunction \\textbf{S}$_{3}\\rightarrow M\\rtimes R$ given by $y\\mapsto\n(f_{3}y,0)$. Form the crossed module $\\partial ^{\\prime }:M\\otimes\n\\overline{M}\\rightarrow M\\rtimes R$, then\n\\begin{equation*}\nL\\cong \\{(M\\otimes \\overline{M})\\sqcup C\\}\/\\thicksim\n\\end{equation*}\nwhere $\\thicksim$ corresponds to the relations\n\\begin{equation*}\n\\begin{array}{lllll}\n1) & & i_{M\\otimes \\overline{M}}(\\partial c\\otimes \\overline{n}) & \\thicksim & j(c)-j(%\n\\overline{n}\\cdot c) \\\\\n2) & & i_{M\\otimes \\overline{M}}(m\\otimes \\partial c) & \\thicksim & j(m\\cdot c)-j(c)%\n\\end{array}%\n\\end{equation*}%\nfor $c\\in C,m\\in M$ and $\\overline{n}\\in \\overline{M.}$\n\nThe homomorphisms $L\\rightarrow M,L\\rightarrow \\overline{M}$ are\ngiven by the homomorphisms\n\\begin{equation*}\n\\lambda :M\\otimes \\overline{M}\\rightarrow M\\text{ and }\\lambda ^{\\prime\n}:M\\otimes \\overline{M}\\rightarrow \\overline{M}\n\\end{equation*}%\nand $\\partial :C\\rightarrow M\\cap \\overline{M}.$ The $h$-map of the crossed\nsquare is given by\n\\begin{equation*}\nh(m,\\overline{n})=i(m\\otimes \\overline{n})\n\\end{equation*}%\nfor $m,n\\in M.$\n\\end{prop}\n\n\\begin{pf}\nThis comes by direct verification using the universal properties of tensors\nand coproducts.\n\\end{pf}\n\n\\bigskip\n\n\\textbf{Remark}: For future applications it is again important to\nnote that the result is not dependent on the crossed square being\n\\emph{totaly} free. If $M\\rightarrow R$ is any pre-crossed module,\none can form the corner\n\\begin{equation*}\n\\xymatrix{&M\\ar[d]^{\\nu}\\\\\n\\overline{M}\\ar[r]_-{\\mu}&M\\rtimes R}\n\\end{equation*}\ncomplete it to a crossed square via $M\\otimes \\overline{M}$ and then add in $%\n\\mathbf{S_{3}}\\rightarrow M.$ Nowhere does this use freeness of\n$M\\rightarrow R$.\n\n\\begin{cor}\n\\label{c} Let \\textbf{E}$^{(1)}$ be the 1-skeleton of a free\nsimplicial algebra. Given the free crossed square\n$\\mathbf{M}(\\mathbf{E}^{(1)},2)$ described above, then\n\\begin{equation*}\nNE_{2}^{(1)}\/\\partial _{3}NE_{3}^{(1)}\\cong \\mbox{\\rm\nKer}d_{1}^{1}\\otimes _{E_{1}}\\mbox{\\rm Ker}d_{0}^{1}.\n\\end{equation*}\n\\end{cor}\n\n\\begin{pf}\nIn the 1-skeleton of a free simplicial algebra \\textbf{E}$^{(1)},$ the set\n\\textbf{S}$_{3}$ is empty. Thus this is clear from the previous proposition.\n\\end{pf}\n\n\\textbf{Remark}: If we set $M=\\ker d_{0}^{1}=NE_{1}^{(1)}$, then the\nidentification given by the Corollary gives\n\\begin{equation*}\nNE_{2}^{(1)}\/\\partial _{3}(NE_{3}^{(1)})\\cong M\\otimes \\overline{M}.\n\\end{equation*}%\nThis uses the fact that $\\ker d_{0}^{1}$ and $\\ker d_{1}^{1}$ are linked via\nthe map sending $m$ to $\\left( m-s_{0}d_{1}m\\right) $ for $m\\in \\ker\nd_{0}^{1}.$ The $h$-map\n\\begin{equation*}\nh:M\\times \\overline{M}\\longrightarrow NE_{2}^{(1)}\/\\partial _{3}NE_{3}^{(1)}\n\\end{equation*}%\ngiven by\n\\begin{equation*}\nh(x,\\overline{y})=s_{1}x(s_{1}y-s_{0}y)+\\partial _{3}NE_{3}^{(1)}.\n\\end{equation*}%\nBut this is also $h(x,\\overline{y})=x\\otimes \\overline{y}$ . Thus\n\\begin{equation*}\nx\\otimes \\overline{y}=s_{1}x(s_{1}y-s_{0}y)+\\partial\n_{3}NE_{3}^{(1)}\n\\end{equation*}%\nunder the identification via the isomorphism of the above corollary.\n\nThis explains the mysterious formula of \\cite{ap2} in the discussion\nbefore Proposition 2.6 of that paper.\n\n\\section{Applications}\n\n\\subsection{2-crossed complexes}\n\nA notion of $2$-crossed complex of commutative algebras is defined\nby Grandjean and Vale in \\cite{gv}. We have considered freeness\nconditions in \\cite{ap3} and this generalises easily to 2-crossed\ncomplexes.\n\nA \\emph{2-crossed complex} of commutative algebras is a sequence\nof $k$-algebras\n\\begin{equation*}\n\\xymatrix{C:\\cdots \\ar[r]^-{\\partial_{n+1}}&C_n\n\\ar[r]^-{\\partial_n}& C_{n-1}\\ar[r]^-{\\partial_{n-1}}&\\cdots\n\\ar[r]^{\\partial_3}&C_2 \\ar[r]^{\\partial_2 }&C_1\n\\ar[r]^{\\partial_1}&C_0}\n\\end{equation*}\ntogether with a 2-crossed module structure given by the pairing\n\\begin{equation*}\n\\left\\{ \\quad \\otimes \\quad \\right\\} :C_{1}\\otimes\n_{C_{0}}C_{1}\\longrightarrow C_{2}\n\\end{equation*}\nsuch that\n\n(i) $C_{n}$ is an $A$-module for $n\\geqslant 3$ with $A=C_{0}\/\\partial\n_{1}(C_{1});$\n\n(ii) $C_{0}$ acts on $C_{n},$ $n\\geqslant 1,$ the action of $\\partial _{1}(C_{1})$\nbeing trivial on $C_{n}$ for $n\\geqslant 3;$\n\n(iii) each $\\partial _{n}$ is an $A$-module homomorphism and $\\partial\n_{n}\\partial _{n+1}=0$ for all $n\\geqslant 1.$\n\n\\bigskip\n\nNote that $K=$ Ker $\\partial _{2}$ is an $C_{0}\/\\partial _{1}C_{1}$-module\nas $\\partial _{2}$ is a crossed module.\n\nThe notion of a morphism for 2-crossed complexes should be clear.\nSuch a morphism will be a morphism `chain complexes of algebras'\nrestricting to a morphism of 2-crossed modules on the bottom three\nterms and compatible with the action. This gives the category,\n\\textbf{2-CrsComp}, of 2-crossed complexes and morphisms between\nthem.\n\\begin{prop}\nThere is a functor%\n\\begin{equation*}\n\\mathbf{C}^{(2)}:\\mathbf{SimpAlg}\\longrightarrow\n\\mathbf{2\\text{-}CrsComp}.\n\\end{equation*}\n\\rm{(We will usually omit the superfix $(2)$ writing simply\n$\\mathbf{C}$ for this.)}\n\\end{prop}\n\n\\begin{pf}\n\\bigskip Given a simplicial algebra $\\mathbf{E},$ define\n\\begin{equation*}\nC_{n}=\n\\begin{cases}\nNE_n & \\text{for } n=0,1;\\\\\n\\dfrac{NE_{2}}{\\partial _{3}(NE_{3}\\cap D_{3})} & \\text{for } n=2; \\\\\n\\dfrac{NE_{n}}{(NE_{n}\\cap D_{n})+\\partial _{n+1}(NE_{n+1}\\cap\nD_{n+1})} & \\text{for } n\\geqslant 3,%\n\\end{cases}%\n\\end{equation*}%\nwith $\\partial _{n}$ induced by the differential of \\textbf{NE}.\nNote that the bottom three terms (for $n=0,1$ and $2$) form the\n2-crossed module considered in \\cite{ap3} and that for $n\\geqslant 3$ the algebras are all $%\nC_{0}\/\\partial _{1}(C_{1})$-modules, since in these dimensions\n$C_{n}$ is the same as the corresponding crossed complex term (cf.\n\\cite{ap1}). The only thing remaining is to check that $\\partial\n_{2}\\partial _{3}$ is trivial which is straightforward.\n\\end{pf}\n\nSince\n\\begin{equation*}\n\\mathbf{C^{(2)}}(\\mathbf{E})_{2}=NE_{2}\/\\partial _{3}(NE_{3}\\cap\nD_{3}),\n\\end{equation*}\nthe same formula as that for $\\mathbf{C(E},2)_{\\langle 2 \\rangle}$,\nwe obtain the following result.\n\n\\begin{cor}\nIf $\\mathbf{E}^{(1)}$ is the $1$-skeleton of a free simplicial\nalgebra $\\mathbf{E}$ then the $2$-crossed complex of\n$\\mathbf{E}^{(1)}$ satisfies\n\\begin{equation*}\n\\mathbf{C^{(2)}}(\\mathbf{E}^{(1)})_{2}\\cong\n\\text{Ker}d_{1}^{1}\\otimes \\text{Ker}d_{0}^{1}.\n\\end{equation*}%\nMoreover%\n\\begin{equation*}\n\\mathbf{C^{(2)}}(\\mathbf{E}^{(2)})_{2}\\cong \\left( \\left( \\text{Ker}d_{1}^{1}\\otimes \\text{%\nKer}d_{0}^{1}\\right) \\sqcup C\\right) \/\\sim\n\\end{equation*}%\nwhere as in Proposition $5.1,$ this $C$ is a free crossed module on the\n\\textquotedblleft new generators\\textquotedblright\\ in dimension $2.$\n\\end{cor}\n\n\n\n\\begin{lem}\nIf $\\mathbf{E}$ is a simplicial resolution of $B$ then, for $k\\geqslant\n3,\\mathbf{C^{(2)}}(\\mathbf{E}^{(2)})_{k}$ a free $B$-module on the\ngiven data.\n\\end{lem}\n\nTo sum up we have the following result.\n\n\\begin{thm}\nThe `step-by-step' construction of a simplicial resolution of an\nalgebra $B$ gives a `step-by-step' construction of a 2-crossed\nresolution of $B$ via the 2-crossed complex construction\n$\\mathbf{C}^{(2)}$.\n\\end{thm}\n\n\\subsection{`Quadratic' analogues of the cotangent complex?}\n\nIn this final section we take the $1$-skeleton of a simplicial\nalgebra and see how it relates to other algebraic construction,\nsuch as Andr\\'{e}-Quillen homology and a squared complex form of\nthe cotangent complex. Here our results are less conclusive than\nwe would like.\n\n\\textbf{Remark}: We will assume that rings and algebras are\nNoetherian for convenience and thus that ideals are finitely\ngenerated. This means $\\mathbf{S_2, S_3}$ etc will all be finite.\n\nFrom Proposition 4.1 we have the free crossed square\n$$\n \\left(\n\\begin{array}{cc}\n\\xymatrix{\\dfrac{(R[s_0({\\bf S_2})]^+[s_1({\\bf S_2})])\\cap (s_0({\\bf\nS_2})-s_1({\\bf S_2}))}{P_2} \\ar[d]\\ar[r]&\\overline{R^+[{\\bf\nS_2}]}\\ar[d]\\\\\n R^+[{\\bf S_2}]\\ar[r]&R[{\\bf S_2}]}\n\\end{array}\n\\right)\n$$\nso using corollary 5.2 there is the following isomorphism:\n$$\n\\dfrac{(R[s_0({\\bf S_2})]^+[s_1({\\bf S_2})])\\cap (s_0({\\bf\nS_2})-s_1({\\bf S_2}))}{P_2} \\cong R^{+}[\\mathbf{S_2}]\\otimes\n\\overline{R^{+}[\\mathbf{S_2}]}.\n$$\nThus the free crossed square becomes\n$$\n \\left(\n\\begin{array}{cc}\n\\xymatrix{ R^{+}[\\mathbf{S_2}]\\otimes\n\\overline{R^{+}[\\mathbf{S_2}]}\\ar[d]\\ar[r]&\\overline{R^+[{\\bf\nS_2}]}\\ar[d]\\\\\n R^{+}[{\\bf S_2}]\\ar[r]&R[{\\bf S_2}].}\n\\end{array}\n\\right)\n$$\nIn section 4 we saw that there is a 2-crossed module\n$$\n\\mathbb{X}:\\xymatrix{&R^{+}[\\mathbf{S_2}]\\otimes\n\\overline{R^{+}[\\mathbf{S_2}]}\\ar[r]^{\\partial_3}&R^{+}[\\mathbf{S_2}]\\rtimes\n\\overline{R^{+}[\\mathbf{S_2}]}\\ar[r]^-{\\partial_2}&R[\\mathbf{S_2}]}\n$$\nwhere\n$$\n\\partial_3(x\\otimes y)=(-\\lambda(x\\otimes y),\\lambda'(x\\otimes\ny))\n$$\nand\n$$\n\\partial_2(x,y)=\\mu(x)+\\mu'(x).\n$$\nThe axioms of a squared complex ensure that $\\partial_3$ and\n$\\partial_2$ are homomorphisms and $\\partial_3$ is a module.\n\nThe 2-crossed complex $\\mathbf{C^{(2)}(E^{(2)})}$ has a smaller\n2-crossed module at its base namely\n$$\n\\mathbb{Y}:\\xymatrix{&R^{+}[\\mathbf{S_2}]\\otimes\n\\overline{R^{+}[\\mathbf{S_2}]}\\ar[r]&R^{+}[\\mathbf{S_2}]\\ar[r]^-{\\partial_2}&R}\n$$\nand it is important to compare the two. In fact there is a split\nepimorphism from $\\mathbb{X}$ to $\\mathbb{Y}$ with kernel\n$$\n\\xymatrix{0\\ar[r]&R^{+}[\\mathbf{S_2}]\\ar[r]^{=}&R^{+}[\\mathbf{S_2}]}\n$$\nwhich has, of course, trivial homotopy. Thus $\\mathbb{X}$ and\n$\\mathbb{Y}$ encode the same information about the presentation of\n$R\/I$, $I=$Im $\\partial_2$.\n\nCrossed complexes form a category \\textbf{Crs} which can be\nconsidered as a full subcategory of both the categories of 2-crossed\ncomplexes and of squared complexes. In the case of 2-crossed\ncomplexes, any crossed complex\n$$\n\\mathbb{C}:\\xymatrix{\\cdots\\ar[r]&C\\ar[r]&M\\ar[r]&R}\n$$\nyields a 2-crossed complex with the same terms at each level and\nwith trivial Peiffer lifting $\\{\\otimes\\}:M\\otimes_{R} M\\rightarrow\nC$ whilst considered as a squared complex we get $\\mathbb{C}$ yields\n$$\n\\xymatrix{\\cdots\\ar[r]&C\\ar[r]\\ar[d]&M\\ar[d]\\\\\n&0\\ar[r]&R.}\n$$\nIn both cases higher dimensional terms are left unchanged. Both\nthese inclusions have left adjoints, i.e., the embeddings give\nreflective subcategories. The proofs are quite easily (and will be\ngiven elsewhere).\n\nThe functors from \\textbf{SimpAlg} to \\textbf{2-CrsComp} and\n\\textbf{SqComp} used above, when composed with the reflections to\n\\textbf{Crs} yield the associated crossed complex functor mentioned\nearlier (see \\cite{ap1}).\n\nFinally the category of chain complexes over $R\/I$ embeds as a\nreflexive subcategory of \\textbf{Crs} and the reflection sends a\ncrossed resolution to the (intermediate stage of the) cotangent\ncomplex (see \\cite{ap1} and \\cite{porter2}). Thus given a simplicial\nresolution of $R\/I$, constructed as in \\cite{andre} by a\nstep-by-step method, the 2-crossed and squared resolutions it gives\ncan be considered as `quadratic' analogues of the cotangent complex,\nin the same way that the crossed complex is a `linear' homotopy\nanalogue of the `homological' cotangent complex. (Here we are using\n`quadratic' and `linear' in the analogous way to that used by Baues\nin \\cite{b1} for the group based theory.)\n\nGiven this it is of interest to study the complex $\\mathbb{X}$ (or\nequivalently $\\mathbb{Y}$) and their analogues when $\\mathbf{S_3}$\ninformation is added in. Here we have no definitive results, only\nproblems.\n\nThe idea will be to try to provide algorithms for calculating and\nthus controlling, the kernel of $\\partial_3$ in $\\mathbb{X}$ (or\nequivalently $\\mathbb{Y}$). We know these give\n$\\pi_3(\\mathbf{E}^{(1)})$ and it is hoped that if these algorithms\nworked, they would allow an analysis of $\\pi_3(\\mathbf{E}^{(2)})$\nand thus to study the effect of adding in $\\mathbf{S_3}$ information\nto the higher terms of the simplicial resolution. As yet we are not\nsure if a general analysis will be possible or whether it will be\nnecessary to limit ourselves to specific classes of example, using,\nfor instance, methods from Gr\\\"{o}bner base theory.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nIn this paper, we initiate the study of the characteristic initial value problem for impulsive gravitational spacetimes in general relativity. These were considered to be solutions of the vacuum Einstein equations\n$$R_{\\mu\\nu}=0$$\nwith a delta singularity in the Riemann curvature tensor supported on a null hypersurface. Their historical origin can be traced back to the cylindrical waves of Einstein-Rosen \\cite{EinsteinRosen}, the plane waves of Brinkmann \\cite{Brinkmann} and the explicit impulsive gravitational spacetimes of Penrose \\cite{Penrose72}.\n\nImpulsive gravitational waves have often been studied within the class of plane fronted gravitational waves. This class of explicit solutions to the vacuum Einstein equations has been first studied by Brinkmann \\cite{Brinkmann} and interest in them has been revived in later decades by the work of Bondi-Pirani-Robinson \\cite{BPR}. They were later classified geometrically by Jordon-Ehlers-Kundt \\cite{JEK}. Among this class are the \\emph{pp}-waves (plane fronted waves with parallel rays) that were discovered by Brinkmann \\cite{Brinkmann}, for which the metric takes the form\n$$g=2d\\ub dr+H(\\ub,X,Y)d\\ub^2+dX^2+dY^2,$$\nand the Einstein vacuum equations imply that \n\\begin{equation}\\label{H}\n\\frac{\\partial^2 H}{\\partial X^2}+\\frac{\\partial^2 H}{\\partial Y^2}=0.\n\\end{equation}\nThese include the special case of sandwich waves, where $H$ is compactly supported in $\\ub$. Since (\\ref{H}) is linear, it follows that \\emph{pp}-waves enjoy a principle of linear superposition.\n\n\\emph{pp}-waves have a plane symmetry and originally impulsive gravitational waves have been thought of as a limiting case of the \\emph{pp}-wave with the function $H$ admitting a delta singularity in the variable $\\ub$. Precisely, explicit impulsive gravitational spacetimes were discovered and studied by Penrose \\cite{Penrose72}, who gave the metric in the following double null coordinate form:\n$$g=-2dud\\ub+(1-\\ub\\Theta(\\ub))dx^2+(1+\\ub\\Theta(\\ub))dy^2,$$\nwhere $\\Theta$ is the Heaviside step function. In the Brinkmann coordinate system, the metric has the \\emph{pp}-wave form and an obvious delta singularity:\n$$g=-2d\\ub dr-\\delta(\\ub)(X^2-Y^2)d\\ub^2+dX^2+dY^2,$$\nwhere $\\delta(\\ub)$ is the Dirac delta. Despite the presence of the delta singularity for the metric in Brinkmann coordinates, the corresponding spacetime is Lipschitz and it is only the Riemann curvature tensor (specifically, the only non-trivial $\\alpha$ component of it) that has a delta function supported on the plane null hypersurface $\\ub=0$. This spacetime turns out to possess remarkable global geometric properties \\cite{Penrose65}. In particular, it exhibits a strong focusing property such that the whole null cone emanating from a point in the past of $\\ub=0$ refocuses to a single line in the future of $\\ub=0$ (See Figure 1). In \\cite{Penrose65}, this property forms the basis of Penrose's argument that global hyperbolicity fails in this spacetime. Penrose's impulsive spacetime is plane symmetric, non-asymptotically flat and the delta curvature singularity is supported on the 3-dimensional infinite plane $\\{(u,\\ub,x,y) : \\ub=0\\}$. It has been long debated whether the strong focussing property and the resulting lack of global hyperbolicity is directly tied to the infinite extent of the impulsive gravitational wave (See Yurtsever \\cite{Yurtsever}).\n\n\\begin{figure}[htbp]\n\\begin{center}\n \n\\input{focus.pdf_t}\n \n\\caption{Focusing in Penrose's impulsive gravitational spacetime}\n\\end{center}\n\\end{figure}\n\nWhile the interest in impulsive gravitational spacetimes had been high due to the availability of explicit solutions, their global geometric properties, superposition properties and their limiting relation to general spacetimes (see Penrose \\cite{Penrose72} , Aichelburg-Sexl \\cite{AichSexl}), the first study of general spacetimes satisfying the Einstein equations and admitting possible three dimensional delta singularities was undertaken by Taub \\cite{Taub}, who derived a system of consistency relations linking the metric, curvature tensor and the geometry of the singular hypersurface. Impulsive gravitational spacetimes also arise as high-frequency limiting spacetimes considered by Choquet-Bruhat \\cite{Bruhat}. We refer the readers to \\cite{Gr}, \\cite{GrPo}, \\cite{BaHo}, \\cite{Bicak} and the references therein for a more detailed exposition on the physics literature.\n\nIn this paper, we begin the study of impulsive gravitational spacetimes viewed in the context of an (characteristic) initial value problem. We consider the data, prescribed on an outgoing null hypersurface $u=0$ and assume that the $\\alpha$ component of the curvature has a delta singularity supported on a two dimensional surface $S_{0,\\ub_s}$, which can be thought of as the intersection of the hypersurfaces $u=0$ and $\\ub=\\ub_s$. Observe that in the Penrose's explicit impulsive solution, the $\\alpha$ curvature component has precisely this type of behavior when restricted to $u=0$. Unlike the explicit impulsive spacetimes, which have only been constructed in plane symmetry and thus have infinite spatial extent, we will consider the case of $S_{0,\\ub_s}$ with compact topology, more precisely a sphere. The data on the incoming null hypersurface is prescribed to be smooth but otherwise without any smallness assumptions.\n\nWith this data, we show that a unique spacetime satisfying the vacuum Einstein equations can be constructed locally\\footnote{Notice that while the Riemann curvature tensor admits a delta function singularity, we show that the Ricci curvature tensor is well-defined in $L^2$, allowing us to make sense of the vacuum Einstein equations.}. Moreover, the delta singularity propagates along a null hypersurface emanating from the initial singularity on $S_{0,\\ub_s}$ and the spacetime is smooth away from this null hypersurface (See Figure 2). \n\n\\begin{figure}[htbp]\n\\begin{center}\n \n\\input{propagation.pdf_t}\n\n\\caption{Propagation of Singularity}\n\\end{center}\n\\end{figure}\n\nOur main result on the propagation of an impulsive gravitational wave is described by\n\\begin{theorem}\\label{giwthmv1}\nSuppose the following hold for the initial data set:\n\\begin{itemize}\n\\item The data on $\\Hb_0$ is smooth. \n\\item The data on $H_0$ is smooth except across a two sphere $S_{0,\\ub_s}$, where the traceless part of the second fundamental form $\\chih$ has a jump discontinuity.\n\\end{itemize}\nThen\n\\begin{enumerate}[(a)]\n\\item Given such initial data and $\\epsilon$ sufficiently small, there exists a unique spacetime $(\\mathcal M,g)$ endowed with a double null foliation $u$, $\\ub$ that solves the characteristic initial value problem for the vacuum Einstein equations in the region $0\\leq u\\leq u_*$, $0\\leq \\ub\\leq\\ub_*$ whenever $u_*, \\ub_* \\leq \\epsilon$.\n\\item Define $\\Hb_{\\ub_s}$ to be the incoming null hypersurface emanating from $S_{0,\\ub_s}$. Then the curvature components $\\alpha_{AB}=R(e_A,e_4,e_B,e_4)$ are measures with a singular atom supported on the hypersurface $\\Hb_{\\ub_s}$. All other components of the curvature tensor can be defined in $L^2$. Moreover, the solution is smooth away from $\\Hb_{\\ub_s}$.\n\\end{enumerate}\n\\end{theorem}\n\nFor an impulsive gravitational wave, standard local existence shows that the spacetime is smooth in $0\\leq \\ub <\\ub_s$. The problem of constructing an impulsive gravitational wave lies in making sense of the solution in the whole region with $\\ub_s\\leq \\ub\\leq \\ub_*$ and showing that the singularity propagates along the characteristic hypersurface $\\Hb_{\\ub_s}$ and that the spacetime is smooth after the impulse.\n\n\\begin{remark}[Larger class of initial data]\nThe proof introduces a new type of energy estimates for the Einstein vacuum equations which allows some components of the Riemann curvature tensor not to be in $L^2$. This discovery allows us to consider the problem of local existence and uniqueness for a larger class of non-regular data. This includes data with a Riemann curvature tensor that can only be understood as a conormal distribution. \n\\end{remark}\n\n\\begin{remark}[Uniqueness]\nPenrose's construction of explicit impulsive gravitational spacetimes is based on the gluing approach, which has later been used to generate other explicit solutions of the Einstein equations. In the general case of the characteristic initial value problem for an impulsive gravitational wave, an appropriate adaptation of the gluing philosophy would allow us to construct {\\bf weak} solutions with undetermined uniqueness. In this paper, however, we construct {\\bf strong} solutions in the sense that {\\bf uniqueness} can also be established. To prove uniqueness, we establish a priori estimates for a larger class of admissible initial data. This allows us to show that the solution is unique among all $C^0$ limits of smooth solutions to the vacuum Einstein equations.\n\\end{remark}\n\n\\begin{remark}[Propagation of singularity]\nOur theorem gives a precise description of the propagation of the initial singularity. Such problems have been extensively studied for semilinear equations (see for example \\cite{Beals}, \\cite{Metivier}). Our result can be formally compared to the works of Majda on the propagation of shocks \\cite{Maj1}, \\cite{Maj2} for systems of conservation laws and the subsequent \\cite{Alinhac}, \\cite{Metivier2}, \\cite{Metivier3}, \\cite{CS}. In these works, a short time existence, uniqueness and regularity result was established for initial data with a jump discontinuity across a surface and a precise description of the propagation of the singularity was also given. We establish an analogous result of propagation of singularity for a nonlinear system of quasilinear hyperbolic equations. However, contrary to \\cite{Maj1}, \\cite{Maj2} the singularity that is considered in the present paper is not a shock, as it propagates along the characteristics. Moreover, unlike in \\cite{Maj1}, \\cite{Maj2}, \\cite{Alinhac},\\cite{Metivier2}, \\cite{Metivier3}, \\cite{CS}, where the problem is reformulated as an initial-boundary value problem and uniqueness is known only within the class of piecewise smooth solutions, our solution is also unique among limits of smooth solutions. In order to achieve this, the special structure of the Einstein equations in the double null foliation gauge has been heavily exploited. \n\\end{remark}\n\nThe construction of spacetimes from non-regular characteristic initial data consistent with that of an impulsive gravitational wave had been known only under symmetry assumptions \\cite{ChrSph1}, \\cite{LeSm}, \\cite{LeSte2}. The work of Christodoulou, who solved the characteristic initial value problem for data with bounded variation for the spherically symmetric Einstein-scalar field system \\cite{ChrSph1}, can be thought of as a first result in that direction. In particular, in this work, the second derivatives of the scalar field, which formally is analogous to the curvature, is allowed to have a delta singularity. The study of data with bounded variation turned out to have important consequences in the global structure of spacetimes and the resolution of the cosmic censorship conjecture for the spherically symmetric Einstein-scalar field system \\cite{ChrSph2}, \\cite{ChrSph3}. The construction of distributional solutions for the vacuum Einstein equations that include \\emph{plane} impulsive gravitational wave was carried out in \\cite{LeSm}, \\cite{LeSte2}. The present paper is the first work that provides a consistent study of the initial value problem of impulsive gravitational spacetimes, including their existence, uniqueness and propagation of singularity\/regularity.\n\nGoing beyond spacetimes which represent a single impulsive gravitational wave, colliding impulsive gravitational waves had been studied by Khan-Penrose \\cite{KhanPenrose} and Szekeres \\cite{Szekeres}. In these explicit solutions, the spacetimes possess two null hypersurfaces with curvature delta singularity with a transverse intersection, representing the nonlinear interaction of two impulsive gravitational waves. The study of the characteristic initial value problem for the colliding impulsive gravitational waves will be carried out by the authors in a subsequent paper.\n\n\\subsection{First Version of the Theorem}\n\nOur general approach is based on energy estimates and transport equations in the double null foliation gauge. This general approach in the double null foliation gauge has been carried out in \\cite{KN}, \\cite{Chr} and \\cite{KlRo}.\n\nThe spacetime in question will be foliated by families of outgoing and incoming null hypersurfaces $H_u$ and $\\Hb_{\\ub}$ respectively. Their intersection is assumed to be a 2-sphere denoted by $S_{u,\\ub}$. Define a null frame $\\{e_1,e_2,e_3,e_4\\}$, where $e_3$ and $e_4$ are null, as indicated in Figure 3, and $e_1$, $e_2$ are tangent to the two spheres $S_{u,\\ub}$. $e_4$ is tangent to $H_u$ and $e_3$ is tangent to $\\Hb_{\\ub}$.\n\n\\begin{figure}[htbp]\n\\begin{center}\n \n\\input{frame.pdf_t}\n \n\\caption{The Basic Setup and the Null Frame}\n\\end{center}\n\\end{figure}\nDecompose the Riemann curvature tensor with respect to this frame:\n\\begin{equation*}\n\\begin{split}\n\\a_{AB}&=R(e_A, e_4, e_B, e_4),\\quad \\, \\,\\, \\ab_{AB}=R(e_A, e_3, e_B, e_3),\\\\\n\\b_A&= \\frac 1 2 R(e_A, e_4, e_3, e_4) ,\\quad \\bb_A =\\frac 1 2 R(e_A, e_3, e_3, e_4),\\\\\n\\rho&=\\frac 1 4 R(e_4,e_3, e_4, e_3),\\quad \\sigma=\\frac 1 4 \\,^*R(e_4,e_3, e_4, e_3)\n\\end{split}\n\\end{equation*}\nIn the context of impulsive gravitational spacetimes, the $\\alpha$ component can only be understood as a measure. \n\n\\noindent Define also the following Ricci coefficients with respect to the null frame:\n\\begin{equation*}\n\\begin{split}\n&\\chi_{AB}=g(D_A e_4,e_B),\\, \\,\\, \\quad \\chib_{AB}=g(D_A e_3,e_B),\\\\\n&\\eta_A=-\\frac 12 g(D_3 e_A,e_4),\\quad \\etab_A=-\\frac 12 g(D_4 e_A,e_3)\\\\\n&\\omega=-\\frac 14 g(D_4 e_3,e_4),\\quad\\,\\,\\, \\omegab=-\\frac 14 g(D_3 e_4,e_3),\\\\\n&\\zeta_A=\\frac 1 2 g(D_A e_4,e_3)\n\\end{split}\n\\end{equation*}\n\nDefine $\\chih$ (resp. $\\chibh$) to be the traceless part of $\\chi$ (resp. $\\chib$). For the problem of the propagation of impulsive gravitational waves, we prescribe initial data on $H_0$ such that $\\chih$ has a jump discontinuity across $S_{0,\\ub_s}$ but smooth otherwise. On $\\Hb_0$, we prescribe the initial data to be smooth but without any smallness assumptions.\n\nAs mentioned before, we will prove local existence and uniqueness for a class of data more general than that for the impulsive gravitational wave. More precisely, we require that $\\chih$ and its angular derivatives are merely bounded. However, we do not require the derivative of $\\chih$ in the $e_4$ direction even to be defined.\n\n\\begin{theorem}\\label{rdthmv1}\nSuppose the characteristic initial data are smooth on $\\Hb_0$. On $H_0$, the characteristic initial data are determined by $\\chi$ and $\\zeta_A$. Let $\\th^A$ be the coordinates on the two sphere foliating $H_0$. Suppose, in every coordinate patch\n$$\\sum_{i\\leq 3}|(\\frac{\\partial}{\\partial\\th})^i\\chih_{AB}|,\\sum_{k\\leq 1}\\sum_{i\\leq 3} |(\\frac{\\partial}{\\partial\\ub})^k(\\frac{\\partial}{\\partial\\th})^i(\\trch,\\zeta_A)|\\leq C.$$\nThen there exists $\\epsilon$ sufficiently small such that the unique solution to the vacuum Einstein equations $(\\mathcal M,g)$ endowed with a double null foliation $u$, $\\ub$ exists in $0\\leq u\\leq \\epsilon$, $0\\leq \\ub\\leq \\epsilon$. Associated to the spacetime a coordinate system $(u,\\ub,\\th^1,\\th^2)$ exists, relative to which the spacetime is in particular Lipschitz and retains higher regularity in the angular directions.\n\\end{theorem}\n\n\\begin{remark}\nWe will also explicitly construct a class of initial data that satisfy the assumptions of Theorem \\ref{giwthmv1} in Section \\ref{initialcondition}. This construction of initial data will indicate an easy modification leading to an even more general class satisfying the assumptions of Theorem \\ref{rdthmv1}.\n\\end{remark}\n\nThis Theorem in particular implies Theorem \\ref{giwthmv1}(a). An additional argument, based on the estimates derived in the proof of Theorem \\ref{rdthmv1}, will be carried out in Section \\ref{limitgiw} to prove part (b) of Theorem \\ref{giwthmv1}. The statement of Theorem \\ref{rdthmv1} will be made precise in Section \\ref{thmstatement}. In particular, the smoothness assumptions of Theorem \\ref{rdthmv1} can be weakened and we will define in what sense the vacuum Einstein equations are satisfied.\n\n\\begin{remark}[Comparison to local existence results]\nWithout symmetry assumptions, all known proofs of existence of spacetimes satisfying the Einstein equations are based on $L^2$-type estimates for the curvature tensor and its derivatives or the metric components and their derivatives. Even with the recent resolution of the $L^2$ curvature conjecture by Klainerman-Rodnianski-Szeftel \\cite{L21}, \\cite{L22}, \\cite{L23}, \\cite{L24}, \\cite{L25}, the Riemann curvature tensor has to be at least in $L^2$. (The classical local existence result of Hughes-Kato-Marsden \\cite{HKM} requires the metric to be in $H^s$, with $s> \\frac 52$.)\n\nIn this paper, we prove local existence and uniqueness under the assumption that the spacetime is merely Lipschitz, which in terms of differentiability is even one derivative weaker than the $L^2$ curvature conjecture. Of course the Lipschitz assumption refers to the worst possible behavior observed in our data and our result heavily relies on the structure of the Einstein equations which allows us to efficiently exploit the better behavior of the other components.\n\\end{remark}\n\n\\subsection{Strategy of the Proof}\n\nFor an impulsive gravitational wave, the curvature tensor can only be defined as a measure and is not in $L^2$. This is one of the main challenges of this work. Let $\\Psi$ denote the curvature components and $\\psi$ denote the Ricci coefficients. The key new observation of this paper is that the $L^2$-type energy estimates for the components of the Riemann curvature tensor\n$$\\int_{H_u} \\Psi^2+\\int_{\\Hb_{\\ub}} \\Psi^2\\leq \\int_{H_0} \\Psi^2+\\int_{\\Hb_0} \\Psi^2 +\\int_0^{\\ub} \\int_0^u\\int_{S_{u\n',\\ub'}} \\psi\\Psi\\Psi du' d\\ub'$$\ncoupled together with the null transport equations for the Ricci coefficients \n$$\\nab_3\\psi=\\Psi+\\psi\\psi,\\quad\\nab_4\\psi=\\Psi+\\psi\\psi$$\ncan be renormalized and closed avoiding the $L^2$-non-integrable components of curvature.\n\n\\subsubsection{Renormalized Energy Estimates}\\label{eeoutline}\n\nThe difficulty in carrying out the above argument is that in our setting, not all curvature components are defined in $L^2$. Even at the level of initial data, $\\alpha$ is only defined as a measure. Therefore, we need to prove $L^2$ energy estimates without involving $\\alpha$. To this end, we need to introduce and estimate the renormalized curvature components. The idea of renormalizing the curvature components has been introduced in \\cite{KlRo:causal}. Unlike this work, there the renormalization was used to obtain estimates for the Ricci coefficients.\n\nThe classical way to derive energy estimates for the Einstein equations is via the Bel-Robinson tensor. In view of our renormalization, we avoid the use of the Bel-Robinson tensor and instead prove the energy estimates directly from the Bianchi identities (\\ref{eq:null.Bianchi}). We note that the derivation of the energy estimates directly from the Bianchi equations without using the Bel Robinson tensor has also appeared in the work of Holzegel \\cite{Holzegel} in a different setting. The challenge and motivation of this method is the derivation of estimates not involving the singular component of curvature $\\alpha$.\n\nTo illustrate this, we first prove the energy estimates for $\\beta$ on $H_u$ and for $(\\rho,\\sigma)$ on $\\Hb_{\\ub}$ by considering the following set of Bianchi equations:\n$$\\nabla_4\\rho=\\div\\beta - \\frac 12 \\chibh\\cdot\\alpha+...$$\n$$\\nab_4\\sigma=-\\div ^*\\beta+\\frac 12\\chibh\\cdot\\alpha+...$$\n$$\\nab_3\\beta=\\nab\\rho+\\nab^*\\sigma+...$$\nHowever, the curvature component $\\alpha$ still appears in the nonlinear terms in these equations. In order to deal with this problem, we renormalize $\\rho$ and $\\sigma$. Define\n$$\\rhoc=\\rho-\\frac 12 \\chih\\cdot\\chibh,\\quad \\sigmac=\\sigma+\\frac 12 \\chih\\wedge\\chibh.$$\nUsing the equation\n$$\\nabla_4\\chih=-\\alpha+...,$$\nwe notice that the first two equations become\n$$\\nabla_4\\rhoc=\\div\\beta+...,\\quad\\nab_4\\sigmac=-\\div ^*\\beta+...$$\nAt the same time, the equation for $\\beta$ can be re-written in terms of of $\\rhoc$ and $\\sigmac$:\n$$\\nab_3\\beta=\\nab\\rhoc+\\nab^*\\sigmac+\\psi\\nab\\psi+...$$\nNow we have a set of renormalized Bianchi equations that does not contain $\\alpha$. Using these equations, we derive the renormalized energy estimate\n$$\\int_{H_u} \\beta^2+\\int_{\\Hb_{\\ub}} (\\rhoc,\\sigmac)^2\\leq \\int_{H_0} \\beta^2+\\int_{\\Hb_0} (\\rhoc,\\sigmac)^2 +\\int_0^{\\ub} \\int_0^u\\int_{S_{u\n',\\ub'}} \\psi\\Psi\\Psi+\\psi\\psi\\nab\\psi du' d\\ub',$$\nin which $\\alpha$ does not appear in the error term.\n\nThe same philosophy can be applied for the remaining curvature components ($\\rho,\\sigma,\\betab,\\alphab$). As a consequence, we obtain a set of $L^2$ curvature estimates which do not explicitly couple to the singular curvature component $\\alpha$. We say explicitly that there is still a remaining possibility that the Ricci coefficients $\\psi$ appearing in the nonlinear error terms for the energy estimates may depend on $\\alpha$.\n\n\\subsubsection{Estimates for the Ricci Coefficients}\\label{Riccioutline}\n\nIn order to close the estimates, it is necessary to show that all the Ricci coefficients can be estimated without \\emph{any} knowledge of $\\alpha$. The most dangerous component is $\\chih$, which naively would have to be estimated using the transport equation\n$$\\nabla_4\\chih+\\trch\\chih=-2\\omega\\chih-\\alpha.$$\nWe take an alternate route and estimate $\\chih$ from the equation\n$$\\nab_3\\chih+\\frac 1 2 \\trchb \\chih=\\nab\\widehat{\\otimes} \\eta+2\\omegab \\chih-\\frac 12 \\trch \\chibh +\\eta\\widehat{\\otimes} \\eta,$$\nor\n$$\\div\\chih=\\frac 12 \\nabla \\trch - \\frac 12 (\\eta-\\etab)\\cdot (\\chih -\\frac 1 2 \\trch) -\\beta.$$\n\nAs we shall see, the loss of information of $\\alpha$ is accompanied by the loss of information of $\\beta$ on $\\Hb_{\\ub}$. This presents yet an additional challenge in estimating the Ricci coefficients.\n\n\\subsubsection{Higher Order Energy Estimates}\\label{higherorderoutline}\n\nAnother difficulty arises considering the fact that in order to close the energy estimates, we need to prove higher derivative estimates for the curvature components. However, the derivatives of some curvature components along $e_4$ are not defined in $L^2$ initially. We will therefore only use angular covariant derivatives $\\nab$ as commutators and will prove estimates only for the $L^2$ norms of the angular covariant derivatives of the renormalized curvature components. We will show firstly that this procedure does not introduce terms that cannot be estimated in $L^2$ (in particular $\\alpha$ will not appear) and secondly that all the energy estimates can be closed only using the estimates of the angular derivatives of the renormalized curvature components.\n\n\\subsubsection{Existence and Uniqueness}\n\nSince we work with initial data with very low regularity, the a priori estimates that we have described above in Sections \\ref{eeoutline}, \\ref{Riccioutline} and \\ref{higherorderoutline} do not immediately imply the existence and uniqueness of the solutions. Instead, we need to approximate the data by a sequence of smooth data and show first that they have a common domain of existence and second that they converge in this domain. The fact that the solutions to the sequence of smooth data have a common domain of existence follows from the a priori estimates that do not involve $\\alpha$ and its derivatives as outlined above. Since the approximating data are smooth, we can conclude that the approximating solutions are also smooth and exist in a common domain.\n\nOnce we have proved the uniform a priori estimates, we proceed to prove that the sequence of solutions converges. To this end, we consider the equations for the difference of the Ricci coefficients and curvature components. We identify the spacetimes in this sequence by the value of their coordinate functions and derive equations for the difference of the Ricci coefficients and curvature components. Our a priori estimates heavily relies on the structure of the Einstein equations which allows us to eliminate any dependence of the $\\alpha$ component. A priori, there is no reason to think that this structure is preserved when considering difference of these solutions, which is necessary to show that our end result is a strong solution to the Einstein equations. It is a remarkable fact that it turns out that in the equations for the difference of the Ricci coefficients and curvature components, $\\alpha$ indeed does not appear. Thus the a priori estimates we have proved are sufficient to control the difference of the Ricci coefficients and curvature components from the difference of the initial value. This proves that the sequence of solutions converge. Even though we face the standard challenge of loss of derivative in the quasilinear equations, our estimates are still sufficient to show that the sequence converges.\n\nThe constructed limiting spacetime is not smooth. In particular, second derivative of the metric in the $e_4$ direction is not even defined. We will, however, show that the Ricci curvature tensor is better behaved than a general second derivative of the metric and that the limiting spacetime satisfies the vacuum Einstein equations $R_{\\mu\\nu}=0$ in the $L^2$ sense.\n\nThe estimates for the difference of the Ricci coefficients and curvature components imply that the constructed spacetime is the unique solution to the Einstein equations among the class of spacetimes admitting a double null foliation and satisfying strong enough a priori bounds. Moreover, we can prove uniqueness of the constructed solution among all limits of smooth spacetimes, i.e, any spacetime that arises as a $C^0$ limit of smooth solutions to the vacuum Einstein equations with initial data converging to the given initial data must coincide with the constructed spacetime.\n\nIt can be observed that the above argument for the a priori estimates, as well as that for showing existence and uniqueness, does not use the fact that initially, $\\alpha$ is a measure whose singular support is on $S_{0,\\ub_s}$. In effect, the argument avoids $\\alpha$ completely, and can be used for data such that $\\alpha$ is much rougher. Since the argument used only estimates on the Ricci coefficients and the curvature components other than $\\alpha$, it can be used to handle initial data satisfying only the assumptions of Theorem \\ref{rdthmv1}.\n\n\\subsubsection{Regularity and Propagation of Singularity}\n\nIn the setting of Theorem \\ref{giwthmv1}, i.e., that of an impulsive gravitational wave, the theorem gives a precise description of the propagation of singularity. The a priori estimates imply that all the curvature components other than $\\alpha$ are bounded. Here, we are interested in proving two additional statements: firstly, $\\alpha$ is a measure that indeed has a delta singularity on $\\Hb_{\\ub_s}$; secondly, the spacetime is smooth away from $\\Hb_{\\ub_s}$.\n\nTo show that $\\alpha$ is a measure, we approximate the data for $\\alpha$ by a sequence of smooth data $\\alpha_n$ such that $\\alpha_n$ is of size $2^n$ in a region $|\\ub-\\ub_s|\\leq 2^{-n}$. The spacetimes constructed for such data are smooth and therefore allow us to use the previously avoided $L^2$ estimate for the $\\alpha$ component of curvature. This estimate imply that in the constructed spacetime, in the region $|\\ub-\\ub_s|\\geq 2^{-n}$, the $L^2_{\\ub}$ norm of $\\alpha_n$ are uniformly bounded; while in the region $|\\ub-\\ub_s|\\leq 2^{-n}$, the $L^2_{\\ub}$ norm of $\\alpha_n$ is bounded by $2^{\\frac n2}$. By Cauchy-Schwarz, we have that the $L^1_{\\ub}$ norms of $\\alpha_n$ are uniformly bounded. This allows us to show that in the limiting spacetime, the curvature component $\\alpha$ is a measure.\n\nTo show that the singular part of $\\alpha$ is a delta function supported on the null hypersurface $\\Hb_{\\ub_s}$, we notice that\n$$\\alpha=-\\nab_4\\chih-\\trch \\chih-2 \\omega \\chih.$$\nTherefore, it suffices to show that $\\chih$ has a jump discontinuity across $\\Hb_{\\ub_s}$ and smooth everywhere else. This can be proved using the equation\n$$\\nab_3\\chih+\\frac 1 2 \\trchb \\chih=\\nab\\widehat{\\otimes} \\eta+2\\omegab \\chih-\\frac 12 \\trch \\chibh +\\eta\\widehat{\\otimes} \\eta$$\nand the fact that on the initial hypersurface $H_0$, $\\chih$ has a jump discontinuity across $S_{0,\\ub_s}$ and smooth everywhere else.\n\nIn order to show that the spacetime is smooth away from $\\Hb_{\\ub_s}$, we will estimate the higher regularity of all the curvature components in that region. We first use the Bianchi equations\n$$\\nab_3\\beta+\\trchb\\beta=\\nabla\\rho + 2\\omegab \\beta +^*\\nabla\\sigma +2\\chih\\cdot\\betab+3(\\eta\\rho+^*\\eta\\sigma).$$\nIntegrating this equation and using Gronwall's inequality, we obtain for any $\\ub\\neq\\ub_s$ that\n$$\\sup_u\\sum_{i\\leq I}||\\nab^i\\beta||_{L^2(S_{u,\\ub})}\\leq (\\sum_{i\\leq I}||\\nab^i\\beta||_{L^2(S_{0,\\ub})}+...)\\exp(\\int \\sum_{i\\leq I}||\\nab^i(\\trch,\\omegab)||_{L^\\infty(S)}).$$\nThe regularity of $\\beta$ is thus inherited from the initial data, which is smooth away from $\\ub=\\ub_s$.\nOnce we have estimates for $\\beta$, we consider the equation\n$$\\nab_3\\alpha+\\frac 12 \\trchb \\alpha=\\nabla\\hot \\beta+ 4\\omegab\\alpha-3(\\chih\\rho+^*\\chih\\sigma)+\n(\\zeta+4\\eta)\\hot\\beta.$$\nIntegrating as before, we see that $\\alpha$ also inherits the regularity from the initial data. Higher derivatives estimates for $\\alpha$ can be derived analogously by differentiating this equation. The other components of curvature can be controlled in a similar fashion. Notice that this procedure results in a loss of derivatives. In particular, in order to control the $N$-th derivative of the curvature, one needs $\\sim 2N$ derivatives initially.\n\n\\subsection{Outline of the Paper}\n\nFinally, we give the outline of the remainder of the paper. In the next Section, we give a careful introduction of the setting, describing the double null foliation, the coordinate system, the equations and relevant notations. We will state a more precise version of Theorem \\ref{rdthmv1}, which we will call Theorem \\ref{rdthmv2}. In Section \\ref{initialcondition}, we provide a construction of the initial data set satisfying the conditions in Theorem \\ref{giwthmv1} and exhibit a sequence of smooth data approximating the data with a curvature delta singularity. In Sections \\ref{estimates} and \\ref{convergence}, we prove Theorem \\ref{rdthmv2}. In Section \\ref{estimates}, we prove that for smooth initial data satisfying the assumptions of Theorem \\ref{rdthmv2}, a unique spacetime exists in a region depending only on the constants in the assumptions. This in particular implies that the approximating sequence of initial data constructed in Section \\ref{initialcondition} gives rise to spacetimes with a common and uniform region of existence (identified by a choice of a double null coordinate system). In Section \\ref{convergence}, we study the equations for the difference of two spacetimes. This allows us to show convergence of solutions with converging initial data and conclude the existence part of Theorem \\ref{rdthmv2}. In Section \\ref{limit}, we examine the regularity of the limiting spacetime and show that it is a solution to the vacuum Einstein equations. In Section \\ref{uniquenesssec}, we conclude the uniqueness part of Theorem \\ref{rdthmv2}. Finally, in Section \\ref{limitgiw}, we return to the proof of part (b) of Theorem \\ref{giwthmv1}, giving a precise description of the propagation of singularity.\\\\\n\n\\noindent{\\bf Acknowledgments:} The authors would like to thank Mihalis Dafermos for valuable discussions. We also thank Dejan Gajic, Joe Keir, Jan Sbierski, Martin Taylor, as well as an anonymous referee, for helpful comments. J. Luk is supported by the NSF Postdoctoral Fellowship DMS-1204493. I. Rodnianski is supported by the NSF grant DMS-1001500 and the FRG grant DMS-1065710.\n\n\\section{Setting and Equations}\n\n\\begin{figure}[htbp]\n\\begin{center}\n \n\\input{frame.pdf_t}\n \n\\caption{The Basic Setup and the Null Frame}\n\\end{center}\n\\end{figure}\n\nOur setting is the characteristic initial value problem with data given on the two characteristic hypersurfaces $H_0$ and $\\Hb_0$ intersecting at the sphere $S_{0,0}$ (see Figure 4). The spacetime will be a solution to the Einstein equations constructed in a neighborhood of $S_{0,0}$ bounded by the two hypersurfaces.\n\n\\subsection{Double Null Foliation}\nFor a spacetime in a neighborhood of $S_{0,0}$, we define a double null foliation as follows: Let $u$ and $\\ub$ be solutions to the eikonal equation\n$$g^{\\mu\\nu}\\partial_\\mu u\\partial_\\nu u=0,\\quad g^{\\mu\\nu}\\partial_\\mu\\ub\\partial_\\nu \\ub=0,$$\nsatisfying the initial conditions $u=0$ on $H_0$ and $\\ub=0$ on $\\Hb_0$.\nLet\n$$L'^\\mu=-2g^{\\mu\\nu}\\partial_\\nu u,\\quad \\Lb'^\\mu=-2g^{\\mu\\nu}\\partial_\\nu \\ub.$$ \nThese are null and geodesic vector fields. Define\n$$2\\Omega^{-2}=-g(L',\\Lb').$$\nDefine\n$$e_3=\\Omega\\Lb'\\mbox{, }e_4=\\Omega L'$$\nto be the normalized null pair such that \n$$g(e_3,e_4)=-2$$\nand\n$$\\Lb=\\Omega^2\\Lb'\\mbox{, }L=\\Omega^2 L'$$\nto be the so-called equivariant vector fields.\n\nWe will denote the level sets of $u$ as $H_u$ and the level sets of $\\ub$ and $\\Hb_{\\ub}$. By virtue of the eikonal equations, $H_u$ and $\\Hb_{\\ub}$ are null hypersurface. Notice that the sets defined by fixed values of $(u,\\ub)$ are 2-spheres. We denote such spheres by $S_{u,\\ub}$. They are intersections of the hypersurfaces $H_u$ and $\\Hb_{\\ub}$. The integral flows of $L$ and $\\Lb$ respect the foliation $S_{u,\\ub}$.\n\n\\subsection{The Coordinate System}\\label{coordinates}\nOn a spacetime in a neighborhood of $S_{0,0}$, we define a coordinate system $(u,\\ub,\\th^1,\\th^2)$ as follows:\nOn the sphere $S_{0,0}$, define a coordinate system $(\\th^1,\\th^2)$ for the sphere such that on each coordinate patch the metric $\\gamma$ is smooth, bounded and positive definite. Then we define the coordinates on the initial hypersurfaces by requiring \n$$\\frac{\\partial}{\\partial u}\\th^A=0\\mbox{ on $\\Hb_0$, and }\\frac{\\partial}{\\partial \\ub}\\th^A=0\\mbox{ on $H_0$}.$$\nWe now define the coordinate system in the spacetime in a neighborhood of $S_{0,0}$ by letting $u$ and $\\ub$ to be solutions to the eikonal equations:\n$$g^{\\mu\\nu}\\partial_\\mu u\\partial_\\nu u=0,\\quad g^{\\mu\\nu}\\partial_\\mu\\ub\\partial_\\nu \\ub=0,$$\nand define $\\th^1, \\th^2$ by\n$$\\Ls_L \\th^A=0,$$ \nwhere $\\Ls_L$ denote the restriction of the Lie derivative to $TS_{u,\\ub}$ (See \\cite{Chr}).\nRelative to the coordinate system, the null pair $e_3$ and $e_4$ can be expressed as\n$$e_3=\\Omega^{-1}\\left(\\frac{\\partial}{\\partial u}+b^A\\frac{\\partial}{\\partial \\th^A}\\right), e_4=\\Omega^{-1}\\frac{\\partial}{\\partial \\ub},$$\nfor some $b^A$ such that $b^A=0$ on $\\Hb_0$, while the metric $g$ takes the form\n$$g=-2\\Omega^2(du\\otimes d\\ub+d\\ub\\otimes du)+\\gamma_{AB}(d\\th^A-b^Adu)\\otimes (d\\th^B-b^Bdu).$$ \n\n\\subsection{The Equations}\\label{sec.eqns}\nWe will recast the Einstein equations as a system for Ricci coefficients and curvature components associated to a null frame $e_3$, $e_4$ defined above and an orthonormal frame ${e_1,e_2}$ tangent to the 2-spheres $S_{u,\\ub}$. Using the indices $A,B$ to denote $1,2$, we define the Ricci coefficients relative to the null fame:\n \\begin{equation}\n\\begin{split}\n&\\chi_{AB}=g(D_A e_4,e_B),\\, \\,\\, \\quad \\chib_{AB}=g(D_A e_3,e_B),\\\\\n&\\eta_A=-\\frac 12 g(D_3 e_A,e_4),\\quad \\etab_A=-\\frac 12 g(D_4 e_A,e_3)\\\\\n&\\omega=-\\frac 14 g(D_4 e_3,e_4),\\quad\\,\\,\\, \\omegab=-\\frac 14 g(D_3 e_4,e_3),\\\\\n&\\zeta_A=\\frac 1 2 g(D_A e_4,e_3)\n\\end{split}\n\\end{equation}\nwhere $D_A=D_{e_{(A)}}$. We also introduce the null curvature components,\n \\begin{equation}\n\\begin{split}\n\\a_{AB}&=R(e_A, e_4, e_B, e_4),\\quad \\, \\,\\, \\ab_{AB}=R(e_A, e_3, e_B, e_3),\\\\\n\\b_A&= \\frac 1 2 R(e_A, e_4, e_3, e_4) ,\\quad \\bb_A =\\frac 1 2 R(e_A, e_3, e_3, e_4),\\\\\n\\rho&=\\frac 1 4 R(e_4,e_3, e_4, e_3),\\quad \\sigma=\\frac 1 4 \\,^*R(e_4,e_3, e_4, e_3)\n\\end{split}\n\\end{equation}\nHere $\\, ^*R$ denotes the Hodge dual of $R$. We denote by $\\nab$ the \ninduced covariant derivative operator on $S_{u,\\ub}$ and by $\\nab_3$, $\\nab_4$\nthe projections to $S_{u,\\ub}$ of the covariant derivatives $D_3$, $D_4$, see\nprecise definitions in \\cite{KN}. \n\nObserve that,\n\\begin{equation}\n\\begin{split}\n&\\omega=-\\frac 12 \\nab_4 (\\log\\Omega),\\qquad \\omegab=-\\frac 12 \\nab_3 (\\log\\Omega),\\\\\n&\\eta_A=\\zeta_A +\\nab_A (\\log\\Omega),\\quad \\etab_A=-\\zeta_A+\\nab_A (\\log\\Omega)\n\\end{split}\n\\end{equation}\n\nLet $\\phi^{(1)}\\cdot\\phi^{(2)}$ denote an arbitrary contraction of the tensor product of $\\phi^{(1)}$ and $\\phi^{(2)}$ with respect to the metric $\\gamma$. We also define\n$$(\\phi^{(1)}\\hot\\phi^{(2)})_{AB}:=\\phi^{(1)}_A\\phi^{(2)}_B+\\phi^{(1)}_B\\phi^{(2)}_A-\\gamma_{AB}(\\phi^{(1)}\\cdot\\phi^{(2)}) \\quad\\mbox{for one forms $\\phi^{(1)}_A$, $\\phi^{(2)}_A$,}$$\n$$\\phi^{(1)}\\wedge\\phi^{(2)}:=\\eps^{AB}(\\gamma^{-1})^{CD}\\phi^{(1)}_{AC}\\phi^{(2)}_{BD}\\quad\\mbox{for symmetric two tensors $\\phi^{(1)}_{AB}$, $\\phi^{(2)}_{AB}$}.$$\nFor totally symmetric tensors, the $\\div$ and $\\curl$ operators are defined by the formulas\n$$(\\div\\phi)_{A_1...A_r}:=\\nabla^B\\phi_{BA_1...A_r},$$\n$$(\\curl\\phi)_{A_1...A_r}:=\\eps^{BC}\\nabla_B\\phi_{CA_1...A_r},$$\nwhere $\\eps$ is the volume form associated to the metric $\\gamma$.\nDefine also the trace to be\n$$(\\mbox{tr}\\phi)_{A_1...A_{r-1}}:=(\\gamma^{-1})^{BC}\\phi_{BCA_1...A_{r-1}}.$$\n\nWe separate the trace and traceless part of $\\chi$ and $\\chib$. Let $\\chih$ and $\\chibh$ be the traceless parts of $\\chi$ and $\\chib$ respectively. Then $\\chi$ and $\\chib$ satisfy the following null structure equations:\n\\begin{equation}\n\\label{null.str1}\n\\begin{split}\n\\nab_4 \\trch+\\frac 12 (\\trch)^2&=-|\\chih|^2-2\\omega \\trch\\\\\n\\nab_4\\chih+\\trch \\chih&=-2 \\omega \\chih-\\alpha\\\\\n\\nab_3 \\trchb+\\frac 12 (\\trchb)^2&=-2\\omegab \\trchb-|\\chibh|^2\\\\\n\\nab_3\\chibh + \\trchb\\, \\chibh&= -2\\omegab \\chibh -\\alphab\\\\\n\\nab_4 \\trchb+\\frac1 2 \\trch \\trchb &=2\\omega \\trchb +2\\rho- \\chih\\cdot\\chibh +2\\div \\etab +2|\\etab|^2\\\\\n\\nab_4\\chibh +\\frac 1 2 \\trch \\chibh&=\\nab\\widehat{\\otimes} \\etab+2\\omega \\chibh-\\frac 12 \\trchb \\chih +\\etab\\widehat{\\otimes} \\etab\\\\\n\\nab_3 \\trch+\\frac1 2 \\trchb \\trch &=2\\omegab \\trch+2\\rho- \\chih\\cdot\\chibh+2\\div \\eta+2|\\eta|^2\\\\\n\\nab_3\\chih+\\frac 1 2 \\trchb \\chih&=\\nab\\widehat{\\otimes} \\eta+2\\omegab \\chih-\\frac 12 \\trch \\chibh +\\eta\\widehat{\\otimes} \\eta\n\\end{split}\n\\end{equation}\nThe other Ricci coefficients satisfy the following null structure equations:\n\\begin{equation}\n\\label{null.str2}\n\\begin{split}\n\\nabla_4\\eta&=-\\chi\\cdot(\\eta-\\etab)-\\b\\\\\n\\nabla_3\\etab &=-\\chib\\cdot (\\etab-\\eta)+\\bb\\\\\n\\nabla_4\\omegab&=2\\omega\\omegab+\\frac 34 |\\eta-\\etab|^2-\\frac 14 (\\eta-\\etab)\\cdot (\\eta+\\etab)-\n\\frac 18 |\\eta+\\etab|^2+\\frac 12 \\rho\\\\\n\\nabla_3\\omega&=2\\omega\\omegab+\\frac 34 |\\eta-\\etab|^2+\\frac 14 (\\eta-\\etab)\\cdot (\\eta+\\etab)- \\frac 18 |\\eta+\\etab|^2+\\frac 12 \\rho\\\\\n\\end{split}\n\\end{equation}\nThe Ricci coefficients also satisfy the following constraint equations\n\\begin{equation}\n\\label{null.str3}\n\\begin{split}\n\\div\\chih&=\\frac 12 \\nabla \\trch - \\frac 12 (\\eta-\\etab)\\cdot (\\chih -\\frac 1 2 \\trch) -\\beta,\\\\\n\\div\\chibh&=\\frac 12 \\nabla \\trchb + \\frac 12 (\\eta-\\etab)\\cdot (\\chibh-\\frac 1 2 \\trchb) +\\betab\\\\\n\\curl\\eta &=-\\curl\\etab=\\sigma +\\frac 1 2\\chibh \\wedge\\chih\\\\\nK&=-\\rho+\\frac 1 2 \\chih\\cdot\\chibh-\\frac 1 4 \\trch \\trchb\n\\end{split}\n\\end{equation}\nwith $K$ the Gauss curvature of the surfaces $S$.\nThe null curvature components satisfy the following null Bianchi equations:\n\\begin{equation}\n\\label{eq:null.Bianchi}\n\\begin{split}\n&\\nab_3\\alpha+\\frac 12 \\trchb \\alpha=\\nabla\\hot \\beta+ 4\\omegab\\alpha-3(\\chih\\rho+^*\\chih\\sigma)+\n(\\zeta+4\\eta)\\hot\\beta,\\\\\n&\\nab_4\\beta+2\\trch\\beta = \\div\\alpha - 2\\omega\\beta + \\eta \\alpha,\\\\\n&\\nab_3\\beta+\\trchb\\beta=\\nabla\\rho + 2\\omegab \\beta +^*\\nabla\\sigma +2\\chih\\cdot\\betab+3(\\eta\\rho+^*\\eta\\sigma),\\\\\n&\\nab_4\\sigma+\\frac 32\\trch\\sigma=-\\div^*\\beta+\\frac 12\\chibh\\cdot ^*\\alpha-\\zeta\\cdot^*\\beta-2\\etab\\cdot\n^*\\beta,\\\\\n&\\nab_3\\sigma+\\frac 32\\trchb\\sigma=-\\div ^*\\betab+\\frac 12\\chih\\cdot ^*\\alphab-\\zeta\\cdot ^*\\betab-2\\eta\\cdot \n^*\\betab,\\\\\n&\\nab_4\\rho+\\frac 32\\trch\\rho=\\div\\beta-\\frac 12\\chibh\\cdot\\alpha+\\zeta\\cdot\\beta+2\\etab\\cdot\\beta,\\\\\n&\\nab_3\\rho+\\frac 32\\trchb\\rho=-\\div\\betab- \\frac 12\\chih\\cdot\\alphab+\\zeta\\cdot\\betab-2\\eta\\cdot\\betab,\\\\\n&\\nab_4\\betab+\\trch\\betab=-\\nabla\\rho +^*\\nabla\\sigma+ 2\\omega\\betab +2\\chibh\\cdot\\beta-3(\\etab\\rho-^*\\etab\\sigma),\\\\\n&\\nab_3\\betab+2\\trchb\\betab=-\\div\\alphab-2\\omegab\\betab+\\etab \\cdot\\alphab,\\\\\n&\\nab_4\\alphab+\\frac 12 \\trch\\alphab=-\\nabla\\hot \\betab+ 4\\omega\\alphab-3(\\chibh\\rho-^*\\chibh\\sigma)+\n(\\zeta-4\\etab)\\hot \\betab\n\\end{split}\n\\end{equation}\nwhere $^*$ denotes the Hodge dual on $S_{u,\\ub}$.\n\nIn the sequel, we will use capital Latin letters $A\\in \\{1,2\\}$ for indices on the spheres $S_{u,\\ub}$ and Greek letters $\\mu\\in\\{1,2,3,4\\}$ for indices in the whole spacetime.\n\nIn the following it will be useful to apply a schematic notation. We will let $\\phi$ denote an arbitrary tensorfield, $\\psi$ a Ricci coefficient and $\\Psi$ a null curvature component different from $\\alpha$. We will simply write $\\psi\\psi$ or $\\psi^2$ (or $\\psi\\Psi$, etc.) to denote an arbitrary contraction. Moreover, we will denote by $\\nab^i\\psi^j$ the sum of all terms which are products of $j$ factors, with each factor being $\\nab^{i_k}\\psi$ and that the sum of all $i_k$'s being $i$, i.e., \n$$\\nab^i\\psi^j=\\displaystyle\\sum_{i_1+i_2+...+i_j}\\underbrace{\\nab^{i_1}\\psi\\nab^{i_2}\\psi...\\nab^{i_j}\\psi}_\\text{j factors}.$$\nThe use of the schematic notation is reserved for the cases when the precise nature of the contraction is not important to the argument. In particular, when using this schematic notation, we will neglect all constant factors.\n\n\\subsection{Integration and Norms}\n\nLet $U$ be a coordinate patch on $S_{0,0}$ and $p_U$ be a partition of unity in $D_U$ such that $p_U$ is supported in $D_U$. Given a function $\\phi$, the integration on $S_{u,\\ub}$ is given by the formula:\n$$\\int_{S_{u,\\ub}} \\phi :=\\sum_U \\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}\\phi p_U\\sqrt{\\det\\gamma}d\\th^1 d\\th^2.$$\nLet $D_{u_*,\\ub_*}$ by the region $0\\leq u\\leq u_*$, $0\\leq \\ub\\leq \\ub_*$. The integration on $D_{u,\\ub}$ is given by the formula\n\\begin{equation*}\n\\begin{split}\n\\int_{D_{u,\\ub}} \\phi :=&\\sum_U \\int_0^u\\int_0^{\\ub}\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}\\phi p_U\\sqrt{-\\det g}d\\th^1 d\\th^2d\\ub du\\\\\n=&2\\sum_U \\int_0^u\\int_0^{\\ub}\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}\\phi p_U\\Omega^2\\sqrt{-\\det \\gamma}d\\th^1 d\\th^2d\\ub du.\n\\end{split}\n\\end{equation*}\nSince there are no canonical volume forms on $H_u$ and $\\Hb_{\\ub}$, we define integration by\n$$\\int_{H_{u}} \\phi :=\\sum_U \\int_0^{\\epsilon}\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}\\phi2 p_U\\Omega\\sqrt{\\det\\gamma}d\\th^1 d\\th^2d\\ub,$$\nand\n$$\\int_{H_{\\ub}} \\phi :=\\sum_U \\int_0^\\epsilon\\int_{-\\infty}^{\\infty}\\int_{-\\infty}^{\\infty}\\phi2p_U\\Omega\\sqrt{\\det\\gamma}d\\th^1 d\\th^2du.$$\n\nWith these definitions of integration, we can define the norms that we will use. Let $\\phi$ be a tensorfield. For $1\\leq p<\\infty$, define\n$$||\\phi||_{L^p(S_{u,\\ub})}^p:=\\int_{S_{u,\\ub}} <\\phi,\\phi>_\\gamma^{p\/2},$$\n$$||\\phi||_{L^p(H_u)}^p:=\\int_{H_{u}} <\\phi,\\phi>_\\gamma^{p\/2},$$\n$$||\\phi||_{L^p(\\Hb_{\\ub})}^p:=\\int_{\\Hb_{\\ub}} <\\phi,\\phi>_\\gamma^{p\/2}.$$\nDefine also the $L^\\infty$ norm by\n$$||\\phi||_{L^\\infty(S_{u,\\ub})}:=\\sup_{\\th\\in S_{u,\\ub}} <\\phi,\\phi>_\\gamma^{1\/2}(\\th).$$\n\n\\subsection{Precise Statement of the Main Theorem}\\label{thmstatement}\nWith the notations introduced in this Section, we give a precise version of the statement of Theorem \\ref{rdthmv1}:\n\\begin{theorem}\\label{rdthmv2}\nSuppose the initial data set for the characteristic initial value problem is given on $H_0$ for $0\\leq \\ub\\leq \\ub_*$ and on $\\Hb_0$ for $0\\leq u\\leq u_*$ such that\n$$c\\leq |\\det\\gamma \\restriction_{S_{u,0}} |, |\\det\\gamma \\restriction_{S_{0,\\ub}} |\\leq C,$$\n$$\\sum_{i\\leq 3}\\left(|(\\frac{\\partial}{\\partial\\th})^i\\gamma \\restriction_{S_{u,0}}|+|(\\frac{\\partial}{\\partial\\th})^i\\gamma \\restriction_{S_{0,\\ub}}|\\right)\\leq C,$$\n$$\\mathcal O_0:= \\sum_{i\\leq 3} \\sup_{\\ub}||\\nabla^i\\psi||_{L^2(S_{0,\\ub})}+\\sum_{i\\leq 3} \\sup_{u}||\\nabla^i\\psi||_{L^2(S_{u,0})}\\leq C,$$\n$$\\mathcal R_0:=\\sum_{i\\leq 2}\\left(\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}\\sup_{\\ub}||\\nab^i\\Psi||_{L^2(S_{0,\\ub})}+\\sum_{\\Psi\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}}\\sup_u||\\nab^i\\Psi||_{L^2(S_{u,0})}\\right)\\leq C.$$\nThen for $\\epsilon$ sufficiently small depending only on $C$ and $c$, there exists a spacetime $(\\mathcal M,g)$ endowed with a double null foliation $u$, $\\ub$ that solves the characteristic initial value problem to the vacuum Einstein equations in $0\\leq u\\leq u_*$, $0\\leq \\ub\\leq \\ub_*$ for $u_*,\\ub_*\\leq \\epsilon$. The metric is continuous and takes the form\n$$g=-2\\Omega^2(du\\otimes d\\ub+d\\ub\\otimes du)+ \\gamma_{AB}(d\\th^A-b^A du)\\otimes(d\\th^B-b^B du).$$\n$(\\mathcal M,g)$ is a $C^0$ limit of smooth solutions to the vacuum Einstein equations and is the unique spacetime solving the characteristic initial value problem among all $C^0$ limits of smooth solutions. Moreover, \n$$\\frac{\\partial}{\\partial \\th}g,\\frac{\\partial}{\\partial u}g\\in C^0_u C^0_{\\ub} L^4(S),$$\n$$\\frac{\\partial^2}{\\partial \\th^2}g,\\frac{\\partial^2}{\\partial u\\partial\\th}g,\\frac{\\partial^2}{\\partial u^2}g\\in C^0_u C^0_{\\ub} L^2(S),$$\n$$\\frac{\\partial}{\\partial \\ub}g, \\frac{\\partial}{\\partial\\ub}((\\gamma^{-1})^{AB}\\frac{\\partial}{\\partial\\ub}(\\gamma)_{AB}) \\in L^\\infty_u L^\\infty_{\\ub} L^\\infty(S),$$\n$$\\frac{\\partial^2}{\\partial \\th \\partial \\ub}g,\\frac{\\partial^2}{\\partial u\\partial\\ub}g,\\frac{\\partial^2}{\\partial \\ub^2}b^A\\in L^\\infty_u L^\\infty_{\\ub} L^4(S).$$\nIn the $(u,\\ub,\\th^1,\\th^2)$ coordinates, the Einstein equations are satisfied in $L^\\infty_uL^\\infty_{\\ub}L^2(S)$. Furthermore, the higher angular differentiability in the data results in higher angular differentiability.\n\\end{theorem}\nWe prove this Theorem in two steps. First, we show that if the initial data are smooth and the bounds for the initial data in the assumption of Theorem \\ref{rdthmv2} hold, then an $\\epsilon$ can be chosen depending only on the bounds in the assumption of Theorem \\ref{rdthmv2} such that a smooth spacetime solving the Einstein equations exists in $0\\leq u\\leq\\epsilon$ and $0\\leq \\ub\\leq \\epsilon$. This is formulated as Theorem \\ref{timeofexistence}, and is proved in Section \\ref{estimates}. Then, we show that a sequence of solutions to the Einstein equations with a converging sequence of smooth initial data, satisfying the assumptions of Theorem \\ref{rdthmv2} with uniform constants $C$ and $c$, converges. The limit spacetime satisfies the Einstein equations and has the properties stated in Theorem \\ref{rdthmv2}. This is formulated as Theorem \\ref{convergencethm2} and is proved in Section \\ref{convergence}. We furthermore show that the solution is unique and that the regularity in the angular directions persists in Section \\ref{convergence}.\n\n\\section{The Initial Data}\\label{initialcondition}\n\nIn this Section, we construct data satisfying the constraint equations such that $\\chih$ has a jump discontinuity across a two sphere. We also construct a sequence of smooth data satisfying the constraint equations approaching the data with discontinuous $\\chih$. We derive precise bounds for the Ricci coefficients and the curvature components for the initial data in this approximating sequence.\n\nWe fix\n$$\\Omega=1$$\nidentically on the initial hypersurfaces $H_0$ and $\\Hb_0$. \nOn $H_0$, $\\gamma$ and $\\chi$ have to satisfy the equations\n\\begin{equation}\\label{con1}\n\\Ls_L \\gamma=2\\chi,\\quad \\Ls_L \\trch= -\\frac 12 (\\trch)^2-|\\chih|_\\gamma^2,\n\\end{equation}\nwhile on $\\Hb_0$, $\\gamma$ and $\\chib$ have to satisfy the equations\n\\begin{equation}\\label{con2}\n\\Ls_{\\Lb}\\gamma=2\\chib,\\quad\\Ls_{\\Lb} \\trchb= -\\frac 12 (\\trchb)^2-|\\chibh|_\\gamma^2\n\\end{equation}\nHere $\\Ls$ denotes the restriction of the Lie derivative to $TS_{u,\\ub}$.\n\nFollowing \\cite{Chr}, we can obtain initial data satisfying the above constraint equations by prescribing $\\gamma_{AB}$, $\\zeta_A$, $\\trch$ and $\\trchb$ on the two sphere $S_{0,0}$ and prescribing the conformal class of the metric $\\hat{\\gamma}_{AB}$ satisfying $\\sqrt{\\det \\hat{\\gamma}_{AB}}=1$ on each of the initial hypersurfaces. Relative to the coordinate system $(\\th^1,\\th^2)$, we require that on $S_{0,0}$,\n$$\\sum_{i\\leq I+3}|(\\frac{\\partial}{\\partial\\th})^i(\\gamma_{AB}, \\zeta_A, tr\\chi, tr\\chib)|\\leq C.$$\nOn the initial incoming hypersurface $\\Hb_0$, in the coordinate system $(u,\\th^1,\\th^2)$ as in Section \\ref{coordinates}, we require the conformal class of the metric $\\hat{\\gamma}_{AB}$ to be smooth, satisfying $\\sqrt{\\det \\hat{\\gamma}_{AB}}=1$ and obeying the estimates\n$$\\sum_{j\\leq J+1}\\sum_{i\\leq I+3}|(\\frac{\\partial}{\\partial u})^j(\\frac{\\partial}{\\partial\\th})^i\\hat{\\gamma}_{AB}|\\leq C.$$\n\nOn the initial outgoing hypersurface $H_0$, in the coordinate system $(u,\\th^1,\\th^2)$ as in Section \\ref{coordinates}, we now prescribe the conformal class of the metric $\\hat{\\gamma}_{AB}$ satisfying $\\sqrt{\\det \\hat{\\gamma}_{AB}}=1$ such that its $\\ub$ derivative has a jump discontinuity. To this end we define smooth matrices $((\\hat{\\gamma})_1)_{AB}$ and $((\\hat{\\gamma})_2)_{AB}$, such that $((\\hat{\\gamma})_1)_{AB}$ is positive definite with determinant equals to $1$ and both $((\\hat{\\gamma})_1)_{AB}$ and $((\\hat{\\gamma})_2)_{AB}$ satisfy\n$$\\sum_{k\\leq K+1}\\sum_{i\\leq I+3}|(\\frac{\\partial}{\\partial\\ub})^k(\\frac{\\partial}{\\partial\\th})^i((\\hat{\\gamma})_j)_{AB}|\\leq C.$$\nFix $\\ub_s\\leq\\frac{\\epsilon}{2}$, where $\\epsilon>0$ is a small parameter depending on the constants $C$ above and will be determine later. \nLet\n$$\\underline{\\hat{\\gamma}}_{AB}=((\\hat{\\gamma})_1)_{AB}+(\\ub-\\ub_s)((\\hat{\\gamma})_2)_{AB}\\mathbbm 1_{\\{\\ub\\geq\\ub_s\\}}.$$\nFor $\\epsilon$ sufficiently small depending on $C$, $\\underline{\\hat{\\gamma}}_{AB}$ is positive definite for $0\\leq\\ub\\leq\\epsilon$. We then define \n$$\\hat{\\gamma}_{AB}=\\frac{1}{\\sqrt{\\det \\underline{\\hat{\\gamma}}}}\\underline{\\hat{\\gamma}}_{AB}.$$\n\nAccording to the procedure in \\cite{Chr}, there exists $\\epsilon$ sufficiently small depending only on $C$ such that there exists initial data for $0\\leq u\\leq \\epsilon$ on $\\Hb_0$ satisfying \\eqref{con2} and initial data for $0\\leq \\ub\\leq \\epsilon$ on $H_0$ obeying \\eqref{con1}. We refer the readers to \\cite{Chr} for details.\n\nWe note that according to \\cite{Chr}, in order to obtain the initial data set on $H_0$, we need to solve for the conformal factor $\\Phi$ defined by \n$$\\gamma_{AB}=\\Phi^2 \\hat{\\gamma}_{AB},$$\nwhich obeys the ODE\n\\begin{equation}\\label{PhiODE}\n\\frac{\\partial^2\\Phi}{\\partial \\ub^2}+\\frac 18 ({\\hat\\gamma}^{-1})^{AC}({\\hat\\gamma}^{-1})^{BD}\\frac{\\partial}{\\partial \\ub}\\hat\\gamma_{AB}\\frac{\\partial}{\\partial \\ub}\\hat\\gamma_{CD}\\Phi=0,\n\\end{equation}\nwith initial data\n$$\\Phi |_{S_{0,0}}=1.$$\nThe time of existence $\\epsilon$ for this ODE depends only on the size of $({\\hat\\gamma}^{-1})^{AC}({\\hat\\gamma}^{-1})^{BD}\\frac{\\partial}{\\partial \\ub}\\hat\\gamma_{AB}\\frac{\\partial}{\\partial \\ub}\\hat\\gamma_{CD}$. Therefore, even though $\\frac{\\partial}{\\partial \\ub}\\hat\\gamma_{AB}$ is discontinuous, we can prescribe the discontinuity at $\\ub=\\ub_s$ such that $\\ub_s=\\frac{\\epsilon}{2}$.\n\nGiven the conformal part of the metric $\\hat{\\gamma}$ and the conformal factor $\\Phi$, we can identify\n\\begin{equation}\\label{chihPhi}\n\\chih_{AB}=\\frac{1}{2}\\Phi^2\\frac{\\partial}{\\partial \\ub}\\hat\\gamma_{AB},\n\\end{equation}\nand\n\\begin{equation}\\label{trchPhi}\n\\trch=\\frac{2}{\\Phi}\\frac{\\partial\\Phi}{\\partial\\ub}.\n\\end{equation}\nWith this identification, $\\chih$ has a jump discontinuity at $\\ub=\\ub_s$ while $\\trch$ is continuous.\n\n\\subsection{Approximation Procedure}\n\nWe now introduce an approximation procedure to construct a sequence of smooth initial data approaching the data described above. Consider a $C^\\infty_0(\\mathbb R)$ function $\\tilde{h}_0$ that is supported in $[-1,1]$ and is identically 1 in $[-\\frac{1}{2},\\frac{1}{2}]$. Let\n$$\\tilde{h}(x)=\\left\\{\\begin{array}{clcr}\\tilde{h}_0(x)-\\tilde{h}_0(2x)&x\\ge 0\\\\0&x< 0\\end{array}\\right.$$\nNote that $\\tilde{h}$ is smooth. Now let\n$$h_n(x)=\\mathbbm 1_{\\{x\\geq 0\\}}+\\sum_{j=-\\infty}^{n}\\tilde{h}(2^j x).$$\nWe note that $h_n$ is supported in $\\{x\\geq 2^{-(n+1)}\\}$ and $h'_n:=h_n-h_{n-1}$ is supported in $\\{2^{-(n+1)}\\leq x\\leq 2^{-n}\\}$. Moreover, $h_{n}\\to\\mathbbm 1_{\\{x\\geq 0\\}}$ in $L^p$ for every $p<\\infty$ as $n\\to\\infty$ and $x h_n\\to x\\mathbbm 1_{\\{x\\geq 0\\}}$ in $L^p$ for every $p\\leq\\infty$ as $n\\to\\infty$. At the $n$-th step, we define \n$$(\\gamma_n,\\zeta_n,\\trch_n)=(\\gamma,\\zeta,\\trch)\\mbox{ on $S_{0,0}$}$$\nand\n$$(\\chibh_n,\\trchb_n)=(\\chibh,\\trchb)\\mbox{ on $\\Hb_{0}$}$$\nas before. Define\n$$(\\underline{\\hat{\\gamma}}_n)_{AB}=((\\hat{\\gamma})_1)_{AB}+(\\ub-\\ub_s)h_n(\\ub-\\ub_s)((\\hat{\\gamma})_2)_{AB}\\mbox{ on $H_{0}$,}$$\nand\n$$(\\hat{\\gamma}_n)_{AB}=\\frac{1}{\\sqrt{\\det (\\underline{\\hat{\\gamma}})_n}}(\\underline{\\hat{\\gamma}}_n)_{AB}\\mbox{ on $H_{0}$}.$$\nNotice that for $\\epsilon$ sufficiently small, we have uniform upper bounds for each component of $\\hat{\\underline{\\gamma}}$ and their angular derivatives and a uniform lower bound for $\\det\\hat{\\underline{\\gamma}}$. Moreover, it is easy to see that for \n$${\\hat{\\gamma}}'_n:={\\hat{\\gamma}}_n-{\\hat{\\gamma}}_{n-1},$$\nwe have\n$$\\sum_{i\\leq I+3}|(\\frac{\\partial}{\\partial \\ub})^k(\\frac{\\partial}{\\partial\\th})^i{\\hat{\\gamma}}'_n|\\leq C2^{(-1+k)n},\\quad\\mbox{ for }\\ub_s\\leq \\ub\\leq\\ub_s+2^{-n}, k\\leq K+1, $$\n$$(\\frac{\\partial}{\\partial \\ub})^k(\\frac{\\partial}{\\partial\\th})^i{\\hat{\\gamma}}'_n=0,\\quad\\mbox{ for }\\ub\\geq\\ub_s+2^{-n},\\mbox{ for $i\\leq I+3$, $k\\leq K+1$}, $$\nThus\n$$(\\int_0^\\epsilon|(\\frac{\\partial}{\\partial \\ub})^k(\\frac{\\partial}{\\partial\\th})^i{\\hat{\\gamma}}'_n|^p d\\ub)^{\\frac 1p}\\leq C2^{(-1+k-\\frac 1p) n}\\mbox{ for }k\\leq K+1.$$\nMoreover, we know that ${\\hat{\\gamma}}_n\\to {\\hat{\\gamma}}$ in the sense that\n$$(\\int_0^\\epsilon|(\\frac{\\partial}{\\partial \\ub})^k(\\frac{\\partial}{\\partial\\th})^i({\\hat{\\gamma}}_n-{\\hat{\\gamma}})|^p d\\ub)^{\\frac 1p}\\leq C2^{(-1+k-\\frac 1p) n}\\mbox{ for }k\\leq K+1.$$\n\n\nWith the definition of $(\\hat{\\gamma}_n)_{AB}$, we can solve (\\ref{PhiODE}) for $\\Phi_n$ and use (\\ref{chihPhi}) and (\\ref{trchPhi}) to solve for $\\chih_n$ and $(\\trch)_n$. This then allows us to use the null structure equations and the Bianchi identities to solve for all Ricci coefficients and curvature components, as well as their derivatives. In the following we will use the notation that a subscript $n$ denotes that metric component, Ricci coefficient or curvature component associated to $\\hat{\\gamma}_n$. Moreover, we will use the notation\n$$\\Phi'_n=\\Phi_n-\\Phi_{n-1}.$$\nWe now derive the bounds for each of the $\\Phi_n$'s and $\\Phi'_n$'s in the initial data. By uniqueness of the constraint ODEs, all \n$$\\Phi'_n=0 \\mbox{ for }\\ub\\leq\\ub_s.$$\nWe therefore only derive bounds for $\\ub_s\\leq\\ub\\leq \\epsilon$.\n\nFrom (\\ref{PhiODE}), we have\n$$c\\leq \\Phi_n\\leq C,\\quad \\sum_{k\\leq K+2}\\sum_{i\\leq I+3}|(\\frac{\\partial}{\\partial \\ub})^k(\\frac{\\partial}{\\partial\\th})^i\\Phi_n|\\leq C$$\nthus\n$$\\sum_{k\\leq K}\\sum_{i\\leq I+3}|(\\frac{\\partial}{\\partial \\th})^i(\\chih,\\trch)_n|\\leq C.$$\nThen using the equation (\\ref{PhiODE}) again, \n$$\\sum_{k\\leq K+2}\\sum_{i\\leq I+3}|(\\frac{\\partial}{\\partial \\ub})^k(\\frac{\\partial}{\\partial \\th})^i\\Phi'_n|\\leq C2^{(-2+k)n},\\quad\\mbox{ for }\\ub_s\\leq\\ub\\leq\\ub_s+2^{-n},$$\nand\n$$(\\frac{\\partial}{\\partial \\ub})^k(\\frac{\\partial}{\\partial \\th})^i\\Phi'_n=0,\\quad\\mbox{ for }\\ub\\geq\\ub_s+2^{-n}\\mbox{ and }2\\leq k\\leq K+2,\\mbox{ for all }i\\leq I+3.$$\nThen by (\\ref{chihPhi}) and (\\ref{trchPhi}), we have for $\\chih'_n$,\n$$\\sum_{k\\leq K}\\sum_{i\\leq I+3}|(\\frac{\\partial}{\\partial \\ub})^k(\\frac{\\partial}{\\partial \\th})^i(\\chih'_n)_{AB}|\\leq C2^{kn},\\quad\\mbox{ for }\\ub_s\\leq\\ub\\leq\\ub_s+2^{-n},$$\n$$\\sum_{k\\leq K}\\sum_{i\\leq I+3}|(\\frac{\\partial}{\\partial \\ub})^k(\\frac{\\partial}{\\partial \\th})^i(\\chih'_n)_{AB}|\\leq C2^{-n},\\quad\\mbox{ for }\\ub\\geq\\ub_s+2^{-n};$$\nand for $\\trch'_n$:\n$$\\sum_{k\\leq K}\\sum_{i\\leq I+3}|(\\frac{\\partial}{\\partial \\ub})^k(\\frac{\\partial}{\\partial \\th})^i\\trch'_n|\\leq C2^{(-1+k)n},\\quad\\mbox{ for }\\ub_s\\leq\\ub\\leq\\ub_s+2^{-n}$$\n$$\\sum_{k\\leq K}\\sum_{i\\leq I+3}|(\\frac{\\partial}{\\partial \\ub})^k(\\frac{\\partial}{\\partial \\th})^i\\trch'_n|\\leq C2^{-n},\\quad\\mbox{ for }\\ub\\geq\\ub_s+2^{-n}.$$\nFrom the equation for $\\zeta$ on $H_0$ together with the Codazzi equation, we have\n$$\\nabla_4\\zeta=\\div\\chih-\\frac 12 \\nab tr\\chi-\\chih\\zeta-\\frac 32 \\trch\\zeta.$$\nNotice that since $\\Omega=1$ on $H_0$, we can write\n$$\\nab_4\\zeta=\\frac{\\partial}{\\partial \\ub}\\zeta+\\chi\\zeta.$$\nThus, we can integrate to get the estimate\n$$\\sum_{k\\leq K+1}\\sum_{i\\leq I+2}|(\\frac{\\partial}{\\partial \\ub})^k(\\frac{\\partial}{\\partial\\th})^i\\zeta_n|\\leq C.$$\nFor the difference $\\zeta'_n=\\zeta_n-\\zeta_{n-1}$,\n$$\\sum_{k\\leq K+1}\\sum_{i\\leq I+2}|(\\frac{\\partial}{\\partial \\ub})^k(\\frac{\\partial}{\\partial\\th})^i\\zeta'_n|\\leq C2^{-n}.$$\nNotice that we have used the control on the metric to relate $\\div$ to the coordinates derivatives. Since on $H_0$,\n$$\\zeta=\\eta=-\\etab,$$\nwe have\n$$\\sum_{k\\leq K+1}\\sum_{i\\leq I+2}|(\\frac{\\partial}{\\partial \\ub})^k(\\frac{\\partial}{\\partial\\th})^i(\\eta_n,\\etab_n)|\\leq C.$$\nand\n$$\\sum_{k\\leq K+1}\\sum_{i\\leq I+2}|(\\frac{\\partial}{\\partial \\ub})^k(\\frac{\\partial}{\\partial\\th})^i(\\eta'_n,\\etab'_n)|\\leq C2^{-n}.$$\nBy the equation of $\\trchb$ on $H_0$\n$$\\nabla_4(\\trchb)+\\frac 12\\trch \\trchb=2\\rho-\\chih\\cdot\\chibh-2\\div\\zeta+|\\zeta|^2_\\gamma,$$\nand the Gauss equation\n$$K=-\\rho+\\frac 12 \\chih\\cdot\\chibh-\\frac 14 \\trch\\trchb,$$\nwe have\n$$\\nab_4(\\trchb)+\\trch \\trchb=-2K-2\\div\\zeta+|\\zeta|^2_\\gamma.$$\nSince the estimates for the Gauss curvature $K$ can be derived from the estimates for $\\gamma$ and its derivatives alone, by the estimates derived above and the initial estimate for $\\trchb$ on $S_{0,0}$,\n$$\\sum_{k\\leq K+1}\\sum_{i\\leq I+1}|(\\frac{\\partial}{\\partial\\ub})^k(\\frac{\\partial}{\\partial\\th})^i\\trchb_n|\\leq C.$$\nand\n$$\\sum_{k\\leq K+1}\\sum_{i\\leq I+1}|(\\frac{\\partial}{\\partial\\ub})^k(\\frac{\\partial}{\\partial\\th})^i \\trchb'_n|\\leq C2^{-n}.$$\nBy the equation of $\\chibh$ on $H_0$\n$$\\nabla_4 \\chibh=-\\frac{1}{2}\\trch\\chibh-\\nabla\\hot\\zeta+\\zeta\\hot\\zeta-\\frac{1}{2}\\trchb\\chih,$$\nwe have\n$$\\sum_{k\\leq K+1}\\sum_{i\\leq I+1}|(\\frac{\\partial}{\\partial\\ub})^k(\\frac{\\partial}{\\partial\\th})^i\\chibh_n|\\leq C.$$\nand\n$$\\sum_{k\\leq K+1}\\sum_{i\\leq I+1}|(\\frac{\\partial}{\\partial\\ub})^k(\\frac{\\partial}{\\partial\\th})^i\\chibh'_n|\\leq C2^{-n}.$$\nBy the following equation on $H_0$\n$$\\nabla_4\\chih=-\\trch \\chih-\\alpha,$$\nwe have\n$$\\sum_{k\\leq K-1}\\sum_{i\\leq I+3}|(\\frac{\\partial}{\\partial\\ub})^k(\\frac{\\partial}{\\partial\\th})^i\\alpha_n|\\leq C2^{(1+k)n},\\quad\\mbox{ for }\\ub_s\\leq \\ub\\leq \\ub_s+2^{-n},$$\n$$\\sum_{k\\leq K-1}\\sum_{i\\leq I+3}|(\\frac{\\partial}{\\partial\\ub})^k(\\frac{\\partial}{\\partial\\th})^i\\alpha_n|\\leq C2^{-n},\\quad\\mbox{ for }\\ub\\geq \\ub_s+2^{-n},$$\nBy the Codazzi equation on $H_0$\n$$\\div\\chi=\\frac{1}{2}\\nabla \\trch-\\zeta\\cdot(\\chih-\\frac{1}{2}\\trch)-\\beta,$$\nwe have\n$$\\sum_{k\\leq K}\\sum_{i\\leq I+2}|(\\frac{\\partial}{\\partial\\ub})^k(\\frac{\\partial}{\\partial\\th})^i\\beta_n|\\leq C \\mbox{ for }\\ub_s\\leq \\ub\\leq \\ub_s+2^{-n},$$\n$$\\sum_{k\\leq K}\\sum_{i \\leq I+2}|(\\frac{\\partial}{\\partial\\ub})^k(\\frac{\\partial}{\\partial\\th})^i\\beta'_n|\\leq C2^{-n} \\mbox{ for }\\ub\\geq \\ub_s+2^{-n}.$$\nBy the Gauss equation on $H_0$\n$$K=-\\rho+\\frac{1}{2}\\chih\\cdot\\chibh-\\frac{1}{4}tr\\chi tr\\chib,$$\nwe have\n$$\\sum_{k\\leq K}\\sum_{i\\leq I+1}|(\\frac{\\partial}{\\partial\\ub})^k(\\frac{\\partial}{\\partial\\th})^i\\rho_n|\\leq C \\mbox{ for }\\ub_s\\leq \\ub\\leq \\ub_s+2^{-n},$$\n$$\\sum_{k\\leq K}\\sum_{i\\leq I+1}|(\\frac{\\partial}{\\partial\\ub})^k(\\frac{\\partial}{\\partial\\th})^i\\rho'_n|\\leq C2^{-n} \\mbox{ for }\\ub\\geq \\ub_s+2^{-n}.$$\nBy the equation on $H_0$\n$$\\curl \\zeta=\\sigma+\\frac 12\\chibh\\wedge\\chih,$$\nwe have\n$$\\sum_{k\\leq K}\\sum_{i\\leq I+1}|(\\frac{\\partial}{\\partial\\ub})^k(\\frac{\\partial}{\\partial\\th})^i\\sigma_n|\\leq C \\mbox{ for }\\ub_s\\leq \\ub\\leq \\ub_s+2^{-n},$$\n$$\\sum_{k\\leq K}\\sum_{i\\leq I+1}|(\\frac{\\partial}{\\partial\\th})^i\\sigma'_n|\\leq C2^{-n} \\mbox{ for }\\ub\\geq \\ub_s+2^{-n}.$$\nOn $S_{0,0}$, using the Codazzi equation\n$$\\div\\chibh=\\frac{1}{2}\\nabla \\trchb+\\zeta\\cdot(\\chih-\\frac{1}{2}\\trch)+\\beta,$$\nwe have\n$$\\sum_{i\\leq I+2}|(\\frac{\\partial}{\\partial\\th})^i\\betab_n|\\leq C \\mbox{ on $S_{0,0}$}.$$\nBy the following Bianchi equation on $H_0$\n$$\\nabla_4\\betab=-\\nab\\rho+^*\\nab\\sigma-tr\\chi\\betab+2\\chibh\\cdot\\beta-3(^*\\zeta\\sigma-\\zeta\\rho),$$\nwe have\n$$\\sum_{k\\leq K+1}\\sum_{i\\leq I}|(\\frac{\\partial}{\\partial\\ub})^k(\\frac{\\partial}{\\partial\\th})^i\\betab_n|\\leq C,$$\n$$\\sum_{k\\leq K+1}\\sum_{i\\leq I}|(\\frac{\\partial}{\\partial\\ub})^k(\\frac{\\partial}{\\partial\\th})^i\\betab'_n|\\leq C2^{-n}.$$\nBy the null structure equation on $H_0$\n$$\\nabla_4\\omegab=3|\\zeta|^2+\\frac 12\\rho,$$\nwe have \n$$\\sum_{k\\leq K+1}\\sum_{i=0}^{I+1}|(\\frac{\\partial}{\\partial\\ub})^k(\\frac{\\partial}{\\partial\\th})^i\\omegab_n|\\leq C.$$\n$$\\sum_{k\\leq K+1}\\sum_{i=0}^{I+1}|(\\frac{\\partial}{\\partial\\ub})^k(\\frac{\\partial}{\\partial\\th})^i\\omegab'_n|\\leq C2^{-n}.$$\nTogether with $\\omega=0$ which follows from $\\omega=-\\frac{1}{2}\\nabla_4(\\log\\Omega)$, we have the bounds in $L^\\infty$ for all the Ricci coefficients and null curvature components on $H_0$.\n\nA similar procedure allows us to solve for the Ricci coefficients and the null curvature components on $\\Hb_0$. Recall that for an impulsive gravitational wave, we take the data on $\\Hb_0$ to be fixed, i.e., $\\psi_n=\\psi$ and $\\Psi_n=\\Psi$ on $\\Hb_0$.\n\nFinally, we see that on $H_0$, we can define $\\nab_3^j\\psi_n$ and $\\nab_3^j\\Psi_n$ for $j\\leq J$ by taking the values of $\\nab_3^j\\psi_n$ and $\\nab_3^j\\Psi_n$ restricted to $S_{0,0}$ and solve the ODEs as above. Notice that they can be solved as long as $I$ is sufficiently large depending on $J$.\n\nThis allows us to conclude\n\\begin{proposition}\\label{dataprop}\nFor every $I\\geq 2$, $J,K\\geq 0$, there exists characteristic initial data for an impulsive gravitational wave such that on $\\Hb_0$:\n\\begin{equation}\\label{data1}\n\\sum_{j\\leq J}\\sum_{k\\leq K}\\sum_{i\\leq I+1}\\sup_{\\ub}||\\nabla_3^j\\nab_4^k\\nabla^i\\psi||_{L^2(S_{0,\\ub})}\\leq C_{I,J,K},\n\\end{equation}\n\\begin{equation}\\label{data2}\n\\sum_{j\\leq J}\\sum_{k\\leq K}\\sum_{i\\leq I}\\sum_{\\Psi\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}}\\sup_u||\\nabla_3^j\\nab_4^k\\nabla^i\\Psi||_{L^2(S_{u,0})}\\leq C_{I,J,K},\\end{equation}\nOn $H_0$, $\\chih$ has a jump discontinuity across $S_{0,\\ub_s}$, and\n\\begin{equation}\\label{data3}\n\\sum_{j\\leq J}\\sum_{i\\leq I+1}\\sup_{\\ub}||\\nab_3^j\\nabla^i\\psi||_{L^2(S_{0,\\ub})}\\leq C_{I,J},\n\\end{equation}\n\\begin{equation}\\label{data4}\n\\sum_{j\\leq J}\\sum_{i\\leq I}\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}\\sup_{\\ub}||\\nab_3^j\\nabla^i\\Psi||_{L^2(S_{0,\\ub})}\\leq C_{I,J},\n\\end{equation}\nand for $H_0\\setminus\\{|\\ub-\\ub_s|\\geq\\eta\\}$ for every $\\eta>0$,\n$$\\sum_{j\\leq J}\\sum_{k\\leq K}\\sum_{i\\leq I+1}\\sup_{\\ub}||\\nab_3^j\\nab_4^k\\nabla^i\\psi||_{L^2(S_{0,\\ub})}\\leq C_{I,J,K},$$\n$$\\sum_{j\\leq J}\\sum_{k\\leq K}\\sum_{i\\leq I}\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}\\sup_{\\ub}||\\nab_3^j\\nab_4^k\\nabla^i\\Psi||_{L^2(S_{0,\\ub})}\\leq C_{I,J,K},$$\nand for $\\epsilon$ sufficiently small, $\\gamma$ is positive definite on $H_0$ and $\\Hb_0$ with bounded angular derivatives up to the $(I+2)$-nd derivative. \n\nThere exists a sequence of of smooth initial data approaching the characteristic initial data described above such that (\\ref{data1})-(\\ref{data4}) hold for $\\psi_n$ and $\\Psi_n$, \nand for $H_0\\setminus\\{\\ub_s\\leq\\ub\\leq \\ub_s+2^{-n}\\}$,\n\\begin{equation}\\label{data5}\n\\sum_{j\\leq J}\\sum_{k\\leq K}\\sum_{i\\leq I+1}\\sup_{\\ub}||\\nab_3^j\\nab_4^k\\nabla^i\\psi_n||_{L^2(S_{0,\\ub})}\\leq C_{I,J,K},\n\\end{equation}\n\\begin{equation}\\label{data6}\n\\sum_{j\\leq J}\\sum_{k\\leq K}\\sum_{i\\leq I}\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}\\sup_{\\ub}||\\nab_3^j\\nab_4^k\\nabla^i\\Psi_n||_{L^2(S_{0,\\ub})}\\leq C_{I,J,K},\n\\end{equation}\nand for $H_0\\cap\\{\\ub_s\\leq\\ub\\leq \\ub_s+2^{-n}\\}$,\n\\begin{equation}\\label{data7}\n\\sum_{j\\leq J}\\sum_{i\\leq I}\\sup_{\\ub}||\\nab_3^j\\nabla^i\\alpha_n||_{L^2(S_{0,\\ub})}\\leq C_{I,J,K}2^n.\n\\end{equation}\nThis sequence of data converges in the sense that on $\\Hb_0$, the data is fixed; and on $H_0$, for $\\psi'_n=\\psi_n-\\psi_{n-1}$ and $\\Psi'_n=\\Psi_n-\\Psi_{n-1}$,\n\\begin{equation}\\label{data8}\n\\sum_{j\\leq J}\\sum_{i\\leq I+1}\\sup_{\\ub}||\\nab_3^j\\nabla^i\\psi'_n||_{L^2(S_{0,\\ub})}\\leq C_{I,J}2^{-n},\n\\end{equation}\n\\begin{equation}\\label{data9}\n\\sum_{j\\leq J}\\sum_{i\\leq I}\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}\\sup_{\\ub}||\\nab_3^j\\nabla^i\\Psi'_n||_{L^2(S_{0,\\ub})}\\leq C_{I,J}2^{-n},\n\\end{equation}\non $H_0\\setminus\\{\\ub_s\\leq\\ub\\leq \\ub_s+2^{-n}\\}$,\n\\begin{equation}\\label{data10}\n\\sum_{j\\leq J}\\sum_{k\\leq K}\\sum_{i\\leq I+1}\\sup_{\\ub}||\\nab_3^j\\nab_4^k\\nabla^i\\psi'_n||_{L^2(S_{0,\\ub})}\\leq C_{I,J,K}2^{-n},\n\\end{equation}\n\\begin{equation}\\label{data11}\n\\sum_{j\\leq J}\\sum_{k\\leq K}\\sum_{i\\leq I}\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}\\sup_{\\ub}||\\nab_3^j\\nab_4^k\\nabla^i\\Psi'_n||_{L^2(S_{0,\\ub})}\\leq C_{I,J,K}2^{-n}.\n\\end{equation}\n\\end{proposition}\n\\begin{remark}\nIn the context of Theorem \\ref{rdthmv2}, corresponding sequences of $\\psi_n$ and $\\Psi_n$ can be constructed. As opposed to the case of an impulsive gravitational wave, such sequences will satisfy (\\ref{data1})-(\\ref{data4}) and (\\ref{data8})-(\\ref{data9}), but they will not obey the improved estimates of Proposition \\ref{dataprop} which holds on $H_0\\setminus\\{\\ub_s\\leq\\ub\\leq\\ub_s+2^{-n}\\}$. This additional regularity is only relevant for the argument that the impulsive gravitational wave is smooth away from $\\Hb_{\\ub_s}$.\n\\end{remark}\nProposition \\ref{dataprop} in particular constructs the set of initial data satisfying the assumptions of Theorem \\ref{giwthmv1}. In the following, we will focus on the case $I=2$, $J=K=0$. This level of regularity is sufficient to prove all the estimates and to construct a spacetime metric satisfying the Einstein equations.\n\n\\section{A Priori Estimates}\\label{estimates}\n\nIn this section, we begin the proof of Theorem \\ref{rdthmv2}. We show that for any smooth characteristic initial data satisfying the estimates in the assumption of Theorem \\ref{rdthmv2}, there is an $\\epsilon$ depending only on the constants in the estimates in the assumptions such that the unique resulting spacetime admits a double null foliation $(u, \\ub)$ and remains smooth with precise estimates on its Ricci coefficients and curvature components in $0\\leq u\\leq u_*$ and $0\\leq \\ub\\leq\\ub_*$ for $u, \\ub\\leq \\epsilon$. In particular, no assumptions have to be made on the size of $\\alpha$.\n\nIn the context of Theorem \\ref{rdthmv2}, our Theorem in this Section asserts that a region of existence can be found independent of $n$ for any smooth approximating sequence of data satisfying the bounds in the assumptions of Theorem \\ref{rdthmv2}.\n\nWe first define the norms that we will work with. Let\n$$\\mathcal R=\\sum_{i\\leq 2}\\left(\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}\\sup_u||\\nabla^i\\Psi||_{L^2(H_u)} +\\sum_{\\Psi\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}}\\sup_{\\ub}||\\nabla^i\\Psi||_{L^2(\\underline{H}_{\\ub})}\\right),$$\n$$\\mathcal O_{i,p}=\\sup_{u,\\ub}||\\nabla^i\\psi||_{L^p(S_{u,\\ub})},$$\nand\n$$\\tilde{\\mathcal O}_{i,2}=\\sum_{\\psi_H\\in\\{\\trch,\\chih,\\eta,\\etab,\\omega,\\trchb\\}}\\sup_u||\\nabla^i\\psi_H||_{L^2(H_u)}+ \\sum_{\\psi_{\\Hb}\\in\\{\\trch,\\eta,\\etab,\\omegab,\\trchb,\\chibh\\}}\\sup_{\\ub}||\\nabla^i\\psi_{\\Hb}||_{L^2(\\Hb_{\\ub})}.$$\nWe write\n$$\\mathcal O=\\sum_{i=0}^{2} \\mathcal O_{i,2}+\\sum_{i=0}^{1}\\mathcal O_{i,4}+\\mathcal O_{0,\\infty}.$$\nThe following is the main theorem in this section:\n\\begin{theorem}\\label{timeofexistence}\nSuppose the initial data set for the characteristic initial value problem is smooth and satisfies\n$$c\\leq |\\det\\gamma \\restriction_{S_{u,0}} |\\leq C,\\quad \\sum_{i\\leq 3}|(\\frac{\\partial}{\\partial\\th})^i\\gamma \\restriction_{S_{u,0}}|\\leq C,$$\n$$\\mathcal O_0:= \\sum_{i\\leq 2} \\sup_{\\ub}||\\nabla^i\\psi||_{L^2(S_{0,\\ub})}+\\sum_{i\\leq 2} \\sup_{u}||\\nabla^i\\psi||_{L^2(S_{u,0})}+||\\nabla^3\\psi||_{L^2(H_0)}+||\\nabla^3\\psi||_{L^2(\\Hb_0)}\\leq C,$$\n$$\\mathcal R_0:=\\sum_{i\\leq 2}\\left(\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}||\\nab^i\\Psi||_{L^2(H_0)}+\\sum_{\\Psi\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}}||\\nab^i\\Psi||_{L^2(\\Hb_0)}\\right)\\leq C.$$\nThen for $\\epsilon$ sufficiently small depending only on $C$ and $c$, there exists a unique spacetime $(\\mathcal M, g)$ endowed with a double null foliation $(u,\\ub)$ that solves the Einstein equations in the region $0\\leq u\\leq u_*$, $0\\leq \\ub\\leq \\ub_*$, whenever $u_*, \\ub_*\\leq \\epsilon$. Moreover, in this region, the following norms are bounded above by a constant $C'$ depending only on $C$ and $c$:\n$$\\mathcal O, \\tilde{\\mathcal O}_{3,2}, \\mathcal R < C'.$$\n\\end{theorem}\nIt follows by \\cite{Rendall} that for every smooth initial data set, a unique smooth spacetime satisfying the Einstein equations exists in a small neighborhood of $S_{0,0}$. In order to show that the size of this region only depends on the constants in the assumption of Theorem \\ref{timeofexistence}, it suffices to establish a priori control in the region $0\\leq u\\leq\\epsilon$, $0\\leq \\ub\\leq \\epsilon$. The Theorem then follows from a standard last slice argument (See \\cite{L}).\n\nIn this section, all estimates will be proved under the following bootstrap assumption:\n\\begin{equation}\\tag{A}\\label{BA1}\n\\mathcal O_{0,\\infty}\\leq \\Delta_0.\n\\end{equation}\n\nWe now outline the steps in proving a priori estimates. In Section \\ref{metric}, we will first estimate the metric components under the bootstrap assumption (\\ref{BA1}). In Section \\ref{transportsec}-\\ref{elliptic}, we will derive some preliminary estimates and formulae. In Section \\ref{transportsec}, we provide Propositions which gives $L^p$ estimates for general quantities satisfying transport equations. They will be used to control the Ricci coefficients and curvature components. In Section \\ref{Embedding}, we establish Sobolev Embedding Theorems in our setting. In Section \\ref{commutation}, we state some formulae for the commutators $[\\nabla_4,\\nabla]$ and $[\\nabla_3,\\nabla]$. They will then be used to obtain higher order estimates for general quantities satisfying transport equations. In Section \\ref{elliptic}, we prove elliptic estimates for general quantities obeying Hodge systems. \n\nAfter these preliminary estimates, we estimate the Ricci coefficients and the curvature components. This proceeds in three steps:\\\\\n\n\\noindent {\\bf STEP 1}: In Section \\ref{Riccisec}, we prove that $\\mathcal R<\\infty$ and $\\tilde{\\mathcal O}_{3,2}<\\infty$ together imply that $\\mathcal O\\leq C(\\mathcal O_0)$, where $C(\\mathcal O_0)$ is a constant depending only on the initial norm $\\mathcal O_0$. These estimates are proved via the null structure equations.\\\\\n\n\\noindent {\\bf STEP 2}: In Section \\ref{Ricciellipticsec}, we use elliptic estimates to prove that $\\mathcal R<\\infty$ implies $\\tilde{\\mathcal O}_{3,2}\\leq C(\\mathcal O_0,\\mathcal R_0,\\mathcal R)$, where $C(\\mathcal O_0,\\mathcal R_0,\\mathcal R)$ is a constant depending both on the initial norm and $\\mathcal R$. \\\\\n\n\\noindent {\\bf STEP 3}: In Section \\ref{energyestimatessec}, we derive the energy estimates and prove the boundedness of $\\mathcal R$ by the initial data.\n\\\\\n\nThese steps provide bounds for the Ricci coefficients up to three angular derivatives and the curvature components up to two angular derivatives depending only on the constants in the assumptions of Theorem \\ref{timeofexistence}. In Section \\ref{Propregsec}, we show that this implies that higher regularity propagates and that the spacetime remains smooth in a region depending only on the constants in the assumptions of Theorem \\ref{timeofexistence}. This allows us to conclude the proof of Theorem \\ref{timeofexistence} in Section \\ref{EndofProof}.\n\nFinally, in Section \\ref{AddEst}, we return to the sequence of smooth spacetimes arising from the sequence of data approximating the data of an impulsive gravitational wave as described in Section \\ref{initialcondition}. We show that these spacetimes obey estimates in addition to those given in Theorem \\ref{timeofexistence}. These bounds will be applied in Section \\ref{limitgiw} to establish the regularity and singularity properties of an impulsive gravitational spacetime.\n\n\\subsection{Estimates for Metric Components}\\label{metric}\nWe first show that we can control $\\Omega$ under the bootstrap assumption (\\ref{BA1}):\n\\begin{proposition}\\label{Omega}\nThere exists $\\epsilon_0=\\epsilon_0(\\Delta_0)$ such that for every $\\epsilon\\leq\\epsilon_0$,\n$$\\frac 12\\leq \\Omega\\leq 2.$$\n\\end{proposition}\n\\begin{proof}\nConsider the equation\n\\begin{equation}\\label{Omegatransport}\n \\omega=-\\frac{1}{2}\\nabla_4\\log\\Omega=\\frac{1}{2}\\Omega\\nabla_4\\Omega^{-1}=\\frac{1}{2}\\frac{\\partial}{\\partial \\ub}\\Omega^{-1}.\n\\end{equation}\nFix $\\ub$. Notice that both $\\omega$ and $\\Omega$ are scalars and therefore the $L^\\infty$ norm is independent of the metric. We can integrate equation (\\ref{Omegatransport}) using the fact that $\\Omega^{-1}=1$ on $H_0$ to obtain\n$$||\\Omega^{-1}-1||_{L^\\infty(S_{u,\\ub})}\\leq C\\int_0^{u}||\\omega||_{L^\\infty(S_{u',\\ub})}du\\leq C\\Delta_0\\epsilon.$$\nThis implies both the upper and lower bounds for $\\Omega$ for sufficiently small $\\epsilon$.\n\\end{proof}\n\n\nWe then show that we can control $\\gamma$ under the bootstrap assumption (\\ref{BA1}):\n\\begin{proposition}\\label{gamma}\nConsider a coordinate patch $U$ on $S_{0,0}$ and define $U_{u,0}$ to be a coordinate patch on $S_{u,0}$ given by the one-parameter diffeomorphism generated by $\\Lb$. Define $U_{u,\\ub}$ to be the image of $U_{u,0}$ under the one-parameter diffeomorphism generated by $L$. Define also $D_U=\\bigcup_{0\\leq u\\leq \\epsilon ,0\\leq \\ub\\leq \\epsilon} U_{u,\\ub}$. For $\\epsilon$ small enough depending on initial data and $\\Delta_0$, there exists $C$ and $c$ depending only on initial data such that the following pointwise bounds hold for $\\gamma$ in $\\mathcal D_U$:\n$$c\\leq \\det\\gamma\\leq C. $$\nMoreover, in $D_U$,\n$$|\\gamma_{AB}|,|(\\gamma^{-1})^{AB}|\\leq C.$$\n\\end{proposition}\n\\begin{proof}\nThe first variation formula states that\n$$\\Ls_L\\gamma=2\\Omega\\chi.$$\nIn coordinates, this means\n$$\\frac{\\partial}{\\partial \\ub}\\gamma_{AB}=2\\Omega\\chi_{AB}.$$\nFrom this we derive that \n$$\\frac{\\partial}{\\partial \\ub}\\log(\\det\\gamma)=\\Omega\\trch.$$\nDefine $\\gamma_0(u,\\ub,\\th^1,\\th^2)=\\gamma(0,\\ub,\\th^1,\\th^2)$. \n$$|\\det\\gamma-\\det(\\gamma_0)|\\leq C\\int_0^{\\ub}|\\trch|d\\ub'\\leq C\\Delta_0\\epsilon.$$\nThis implies that the $\\det \\gamma$ is bounded above and below. Let $\\Lambda$ be the larger eigenvalue of $\\gamma$. Clearly,\n\\begin{equation}\\label{La}\n\\Lambda\\leq C\\sup_{A,B=1,2}\\gamma,\n\\end{equation}\nand \n$$\\sum_{A,B=1,2}|\\chi_{AB}|^2\\leq C\\Lambda ||\\chi||_{L^\\infty(S_{u,\\ub})}.$$\nThen\n$$|\\gamma_{AB}-(\\gamma_0)_{AB}|\\leq C\\int_0^{\\ub}|\\chi_{AB}|d\\ub'\\leq C\\Lambda\\Delta_0\\epsilon.$$\nUsing the upper bound (\\ref{La}), we thus obtain the upper bound for $|\\gamma_{AB}|$. The upper bound for $|(\\gamma^{-1})^{AB}|$ follows from the upper bound for $|\\gamma_{AB}|$ and the lower bound for $\\det\\gamma$.\n\\end{proof}\n\nA consequence of the previous Proposition is an estimate on the surface area of each two sphere $S_{u,\\ub}$.\n\\begin{proposition}\\label{area}\n$$\\sup_{u,\\ub}|\\mbox{Area}(S_{u,\\ub})-\\mbox{Area}(S_{u,0})|\\leq C\\Delta_0\\epsilon.$$\n\\end{proposition}\n\\begin{proof}\nThis follows from the fact that $\\sqrt{\\det\\gamma}$ is pointwise only slightly perturbed if $\\epsilon$ is chosen to be appropriately small.\n\\end{proof}\nWith the estimate on the volume form, we can now show that the $L^p$ norms defined with respect to the metric and the $L^p$ norms defined with respect to the coordinate system are equivalent.\n\\begin{proposition}\\label{eqnorm}\nGiven a covariant tensor $\\phi_{A_1...A_r}$ on $S_{u,\\ub}$, we have\n$$\\int_{S_{u,\\ub}} <\\phi,\\phi>_{\\gamma}^{p\/2} \\sim \\sum_{i=1}^r\\sum_{A_i=1,2}\\iint |\\phi_{A_1...A_r}|^p \\sqrt{\\det\\gamma} d\\th^1 d\\th^2.$$\n\\end{proposition}\nWe can also control $b$ under the bootstrap assumption, thus controlling the full spacetime metric: \n\\begin{proposition}\\label{b}\nIn the coordinate system $(u,\\ub,\\th^1,\\th^2)$,\n$$|b^A|\\leq C\\Delta_0\\epsilon.$$\n\\end{proposition}\n\\begin{proof}\n$b^A$ satisfies the equation\n$$\\frac{\\partial b^A}{\\partial \\ub}=-4\\Omega^2\\zeta^A.$$\nThis can be derived from \n$$[L,\\Lb]=\\frac{\\partial b^A}{\\partial \\ub}\\frac{\\partial}{\\partial \\th^A}.$$\nNow, integrating and using Proposition \\ref{eqnorm} gives the result.\n\\end{proof}\n\n\\subsection{Estimates for Transport Equations}\\label{transportsec}\nThe estimates for the Ricci coefficients and the null curvature components are derived from the null structure equations and the null Bianchi equations respectively. In order to use the equations, we need a way to obtain estimates from the null transport type equations. Such estimates require the boundedness of $\\trch$ and $\\trchb$, which is consistent with our bootstrap assumption (\\ref{BA1}). Below, we state two Propositions which provide $L^p$ estimates for general quantities satisfying transport equations either in the $e_3$ or $e_4$ direction.\n\\begin{proposition}\\label{transport}\nThere exists $\\epsilon_0=\\epsilon_0(\\Delta_0)$ such that for all $\\epsilon \\leq \\epsilon_0$ and for every $2\\leq p<\\infty$, we have\n\\[\n ||\\phi||_{L^p(S_{u,\\ub})}\\leq C(||\\phi||_{L^p(S_{u,\\ub'})}+\\int_{\\ub'}^{\\ub} ||\\nabla_4\\phi||_{L^p(S_{u,\\ub''})}d{\\ub''}),\n\\]\n\\[\n ||\\phi||_{L^p(S_{u,\\ub})}\\leq C(||\\phi||_{L^p(S_{u',\\ub})}+\\int_{u'}^{u} ||\\nabla_3\\phi||_{L^p(S_{u'',\\ub})}d{u''}).\n\\]\n\\end{proposition}\n\n\\begin{proof}\n\nThe following identity holds for any scalar $f$:\n\\[\n \\frac{d}{d\\ub}\\int_{\\mathcal S_{u,\\ub}} f=\\int_{\\mathcal S_{u,\\ub}} \\left(\\frac{df}{d\\ub}+\\Omega \\trch f\\right)=\\int_{\\mathcal S_{u,\\ub}} \\Omega\\left(e_4(f)+ \\trch f\\right).\n\\]\nSimilarly, we have\n\\[\n \\frac{d}{du}\\int_{\\mathcal S_{u,\\ub}} f=\\int_{\\mathcal S_{u,\\ub}} \\Omega\\left(e_3(f)+ \\trchb f\\right).\n\\]\nHence, taking $f=|\\phi|_{\\gamma}^p$, we have\n\\begin{equation}\\label{Lptransport}\n\\begin{split}\n ||\\phi||^p_{L^p(S_{u,\\ub})}=&||\\phi||^p_{L^p(S_{u,\\ub'})}+\\int_{\\ub'}^{\\ub}\\int_{S_{u,\\ub''}} p|\\phi|^{p-2}\\Omega\\left(<\\phi,\\nabla_4\\phi>_\\gamma+ \\frac{1}{p}\\trch |\\phi|^2_{\\gamma}\\right)d{\\ub''}\\\\\n ||\\phi||^p_{L^p(S_{u,\\ub})}=&||\\phi||^p_{L^p(S_{u',\\ub})}+\\int_{u'}^{u}\\int_{S_{u'',\\ub}} p|\\phi|^{p-2}\\Omega\\left(<\\phi,\\nabla_3\\phi>_\\gamma+ \\frac{1}{p}\\trchb |\\phi|^2_{\\gamma}\\right)d{u''}\n\\end{split}\n\\end{equation}\nThe Proposition is proved using Cauchy-Schwarz on the sphere and the $L^\\infty$ bounds for $\\Omega$ and $\\trch$ ($\\trchb$) which are provided by Proposition \\ref{Omega} and the bootstrap assumption (\\ref{BA1}) respectively.\n\\end{proof}\nThe above estimates also hold for $p=\\infty$:\n\\begin{proposition}\\label{transportinfty}\nThere exists $\\epsilon_0=\\epsilon_0(\\Delta_0)$ such that for all $\\epsilon \\leq \\epsilon_0$, we have\n\\[\n ||\\phi||_{L^\\infty(S_{u,\\ub})}\\leq C\\left(||\\phi||_{L^\\infty(S_{u,\\ub'})}+\\int_{\\ub'}^{\\ub} ||\\nabla_4\\phi||_{L^\\infty(S_{u,\\ub''})}d{\\ub''}\\right)\n\\]\n\\[\n ||\\phi||_{L^\\infty(S_{u,\\ub})}\\leq C\\left(||\\phi||_{L^\\infty(S_{u',\\ub})}+\\int_{u'}^{u} ||\\nabla_3\\phi||_{L^\\infty(S_{u'',\\ub})}d{u''}\\right).\n\\]\n\\end{proposition}\n\\begin{proof}\nThis follows simply from integrating along the integral curves of $L$ and $\\Lb$, and the estimate on $\\Omega$ in Proposition \\ref{Omega}.\n\\end{proof}\n\n\\subsection{Sobolev Embedding}\\label{Embedding}\nUnder the bootstrap assumption (\\ref{BA1}), the Sobolev Embedding hold on a 2-sphere $S_{u,\\ub}$. \n\\begin{proposition}\\label{L4}\nThere exists $\\epsilon_0=\\epsilon_0(\\Delta_0)$ such that as long as $\\epsilon\\leq \\epsilon_0$, we have\n$$||\\phi||_{L^4(S_{u,\\ub})}\\leq C\\sum_{i=0}^1||\\nabla^i\\phi||_{L^2(S_{u,\\ub})}. $$\n\\end{proposition}\n\\begin{proof}\nWe first prove this for scalars. Given our coordinate system, the desired Sobolev Embedding follows from standard Sobolev Embedding Theorems and the lower and upper bound of the volume form. Thus, the proposition holds for scalars. Now, for $\\phi$ being a tensor, let $f=\\sqrt{|\\phi|_{\\gamma}^2+\\delta^2}$. Then\n$$||f||_{L^4(S_{u,\\ub})}\\leq C\\left(||f||_{L^2(S_{u,\\ub})}+||\\frac{<\\phi,\\nabla\\phi>_{\\gamma}}{\\sqrt{|\\phi|_{\\gamma}^2+\\delta^2}}||_{L^2(S_{u,\\ub})}\\right)\\leq C\\left(||f||_{L^2(S_{u,\\ub})}+||\\nabla\\phi||_{L^2(S_{u,\\ub})}\\right).$$\nThe Proposition can be achieved by sending $\\delta\\to 0$.\n\\end{proof}\nWe can also prove the Sobolev Embedding Theorem for the $L^\\infty$ norm: \n\\begin{proposition}\\label{Linfty}\nThere exists $\\epsilon_0=\\epsilon_0(\\Delta_0)$ such that as long as $\\epsilon\\leq \\epsilon_0$, we have\n$$||\\phi||_{L^\\infty(S_{u,\\ub})}\\leq C\\left(||\\phi||_{L^2(S_{u,\\ub})}+||\\nabla\\phi||_{L^4(S_{u,\\ub})}\\right). $$\nAs a consequence,\n$$||\\phi||_{L^\\infty(S_{u,\\ub})}\\leq C\\sum_{i=0}^2||\\nabla^i\\phi||_{L^2(S_{u,\\ub})}. $$\n\\end{proposition}\n\\begin{proof}\nThe first statement follows from coordinate considerations as in Proposition \\ref{L4}. The second statement follows from applying the first and Proposition \\ref{L4}.\n\\end{proof}\n\n\\subsection{Commutation Formulae}\\label{commutation}\nWe have the following formula from \\cite{KN}:\n\\begin{proposition}\nThe commutator $[\\nabla_4,\\nabla]$ acting on an $(0,r)$ S-tensor is given by\n\\begin{equation*}\n \\begin{split}\n[\\nabla_4,\\nabla_B]\\phi_{A_1...A_r}=&[D_4,D_B]\\phi_{A_1...A_r}+(\\nabla_B\\log\\Omega)\\nabla_4\\phi_{A_1...A_r}-(\\gamma^{-1})^{CD}\\chi_{BD}\\nabla_C\\phi_{A_1...A_r} \\\\\n&-\\sum_{i=1}^r (\\gamma^{-1})^{CD}\\chi_{BD}\\etab_{A_i}\\phi_{A_1...\\hat{A_i}C...A_r}+\\sum_{i=1}^r (\\gamma^{-1})^{CD}\\chi_{A_iB}\\etab_{D}\\phi_{A_1...\\hat{A_i}C...A_r}.\n \\end{split}\n\\end{equation*}\nSimilarly, the commutator $[\\nabla_3,\\nabla]$ acting on an $(0,r)$ S-tensor is given by\n\\begin{equation*}\n \\begin{split}\n[\\nabla_3,\\nabla_B]\\phi_{A_1...A_r}=&[D_3,D_B]\\phi_{A_1...A_r}+(\\nabla_B\\log\\Omega)\\nabla_3\\phi_{A_1...A_r}-(\\gamma^{-1})^{CD}\\chib_{BD}\\nabla_C\\phi_{A_1...A_r} \\\\\n&-\\sum_{i=1}^r (\\gamma^{-1})^{CD}\\chib_{BD}\\eta_{A_i}\\phi_{A_1...\\hat{A_i}C...A_r}+\\sum_{i=1}^r (\\gamma^{-1})^{CD}\\chib_{A_iB}\\eta_{D}\\phi_{A_1...\\hat{A_i}C...A_r}.\n \\end{split}\n\\end{equation*}\n\\end{proposition}\n\nBy induction, we get the following schematic formula for repeated commutations:\n\\begin{proposition}\\label{commuteeqn}\nSuppose $\\nabla_4\\phi=F_0$. Let $\\nabla_4\\nabla^i\\phi=F_i$.\nThen\n\\begin{equation*}\n\\begin{split}\nF_i\\sim &\\sum_{i_1+i_2+i_3=i}\\nabla^{i_1}(\\eta,\\underline{\\eta})^{i_2}\\nabla^{i_3} F_0+\\sum_{i_1+i_2+i_3+i_4=i}\\nabla^{i_1}(\\eta,\\underline{\\eta})^{i_2}\\nabla^{i_3}\\chi\\nabla^{i_4} \\phi\\\\\n&+\\sum_{i_1+i_2+i_3+i_4=i-1} \\nabla^{i_1}(\\eta,\\underline{\\eta})^{i_2}\\nabla^{i_3}\\beta\\nabla^{i_4} \\phi.\n\\end{split}\n\\end{equation*}\nwhere we have applied our schematic notation and by $\\nabla^{i_1}(\\eta,\\underline{\\eta})^{i_2}$ we mean the sum of all terms which is a product of $i_2$ factors, each factor being $\\nabla^j \\eta$ or $\\nabla^j\\underline{\\eta}$ for some $j$ and that the sum of all $j$'s is $i_1$, i.e., $$\\nabla^{i_1}(\\eta,\\underline{\\eta})^{i_2}=\\displaystyle\\sum_{j_1+...+j_{i_2}=i_1}\\sum_{\\psi_1,...,\\psi_{i_2}\\in\\{\\eta,\\etab\\}}\\nabla^{j_1}\\psi_1...\\nabla^{j_{i_2}}\\psi_{i_2}.$$ Similarly, suppose $\\nabla_3\\phi=G_{0}$. Let $\\nabla_3\\nabla^i\\phi=G_{i}$.\nThen\n\\begin{equation*}\n\\begin{split}\nG_{i}\\sim &\\sum_{i_1+i_2+i_3=i}\\nabla^{i_1}(\\eta,\\underline{\\eta})^{i_2}\\nabla^{i_3} G_{0}+\\sum_{i_1+i_2+i_3+i_4=i}\\nabla^{i_1}(\\eta,\\underline{\\eta})^{i_2}\\nabla^{i_3}\\underline{\\chi}\\nabla^{i_4} \\phi\\\\\n&+\\sum_{i_1+i_2+i_3+i_4=i-1} \\nabla^{i_1}(\\eta,\\underline{\\eta})^{i_2}\\nabla^{i_3}\\underline{\\beta}\\nabla^{i_4} \\phi.\n\\end{split}\n\\end{equation*}\n\n\\end{proposition}\n\\begin{proof}\nThe proof is by induction. We will prove it for the first statement. The proof of second one is analogous. The formula obviously holds for $i=0$. Assume that the statement is true for $i_\\gamma =&\\int <\\beta,\\nabla\\rhoc+^*\\nabla\\sigmac>_\\gamma+<\\beta,\\psi\\Psi>_\\gamma \\\\\n=&\\int_{D_{u,\\ub}} -<\\div\\beta,\\rhoc>_\\gamma+<\\div ^*\\beta,\\sigmac>_\\gamma +<\\beta,\\psi\\Psi+\\psi\\nabla\\psi>_\\gamma\\\\\n=&\\int_{D_{u,\\ub}} -<\\nabla_4\\rhoc,\\rhoc>_\\gamma-<\\nabla_4\\sigmac,\\sigmac>_\\gamma +<\\Psi,\\psi\\Psi+\\psi\\nabla\\psi+\\psi\\psi\\psi>_\\gamma.\n\\end{split}\n\\end{equation*}\nApplying Proposition \\ref{intbyparts34} then yields the energy estimates for the $0$-th derivative of the curvature:\n\\begin{equation*}\n\\begin{split}\n&\\int_{H_u} |\\beta|^2_\\gamma+\\int_{\\Hb_{\\ub}} \\rhoc^2+\\sigmac^2 \\\\\n\\leq &\\int_{H_{u'}} |\\beta|^2_\\gamma+\\int_{\\Hb_{\\ub'}} \\rhoc^2+\\sigmac^2\n+\\int_{D_{u,\\ub}}\\Psi(\\psi\\Psi+\\psi\\nabla\\psi+\\psi\\psi\\psi).\n\\end{split}\n\\end{equation*}\nWe now commute equations (\\ref{rcsc}) and (\\ref{bc}) with $\\nabla^i$ to get\n\\begin{equation*}\n\\begin{split}\n&\\nabla_3\\nabla^i\\beta- \\nabla \\nabla^i\\rhoc-\\nabla\\nabla^i\\sigmac \\\\\n\\sim&\\sum_{i_1+i_2+i_3+i_4=i+1}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\psi\\nabla^{i_4}\\psi \\\\\n&+\\sum_{i_1+i_2+i_3+i_4+i_5=i-1}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}K^{i_4}\\nabla^{i_5}(\\rhoc,\\sigmac)+\\sum_{i_1+i_2+i_3=i}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}(\\psi\\Psi),\\\\\n\\end{split}\n\\end{equation*}\nand\n\\begin{equation*}\n\\begin{split}\n&\\nabla_4\\nabla^i\\rhoc- \\div\\nabla^i\\beta\\\\\n\\sim&\\sum_{i_1+i_2+i_3+i_4=i+1}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\psi\\nabla^{i_4}\\psi \\\\\n&+\\sum_{i_1+i_2+i_3+i_4+i_5=i-1}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}K^{i_4}\\nabla^{i_5}\\beta+\\sum_{i_1+i_2+i_3=i}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}(\\psi\\Psi),\\\\\n\\end{split}\n\\end{equation*}\nand \n\\begin{equation*}\n\\begin{split}\n&\\nabla_4\\nabla^i\\sigmac- \\div^*\\nabla^i\\beta\\\\\n\\sim&\\sum_{i_1+i_2+i_3+i_4=i+1}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\psi\\nabla^{i_4}\\psi \\\\\n&+\\sum_{i_1+i_2+i_3+i_4+i_5=i-1}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}K^{i_4}\\nabla^{i_5}\\beta+\\sum_{i_1+i_2+i_3=i}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}(\\psi\\Psi),\\\\\n\\end{split}\n\\end{equation*}\nwhere \n$$(\\div\\nabla^i\\beta)_{A_1...A_i}=(\\nabla^{i+1}\\beta)^B{ }_{A_1...A_iB}.$$\nWe integrate by parts using Proposition \\ref{intbyparts34} and \\ref{intbypartssph} as in the $0$-th derivative case to get\n\\begin{equation*}\n\\begin{split}\n&\\int_{H_u} |\\nabla^i\\beta|^2_\\gamma+\\int_{\\Hb_{\\ub}} |\\nabla^i(\\rhoc,\\sigmac)|^2_\\gamma \\\\\n\\leq &\\int_{H_{u'}} |\\nabla^i\\beta|^2_\\gamma+\\int_{\\Hb_{\\ub'}} |\\nabla^i(\\rhoc,\\sigmac)|^2_\\gamma+\\int_{D_{u,\\ub}}\\nabla^i\\Psi\\sum_{i_1+i_2+i_3+i_4=i}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\psi\\nabla^{i_4}\\Psi\\\\\n&+\\int_{D_{u,\\ub}}\\nabla^i\\Psi\\sum_{i_1+i_2+i_3+i_4=i-1}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}K\\nabla^{i_4}\\Psi\\\\\n&+\\int_{D_{u,\\ub}}\\nabla^i\\Psi\\sum_{i_1+i_2+i_3+i_4=i+1}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\psi\\nabla^{i_4}\\psi.\n\\end{split}\n\\end{equation*}\nWe then consider the following set of Bianchi equations. Notice that $\\alpha$ does not appear in any equations in this set.\n\\begin{equation*}\n\\begin{split}\n&\\nabla_3\\sigma+\\frac 32tr\\chib\\sigma=-\\div ^*\\betab+\\frac 12\\chih\\cdot ^*\\alphab-\\zeta\\cdot ^*\\betab-2\\eta\\cdot \n^*\\betab,\\\\\n&\\nabla_3\\rho+\\frac 32tr\\chib\\rho=-\\div\\betab- \\frac 12\\chih\\cdot\\alphab+\\zeta\\cdot\\betab-2\\eta\\cdot\\betab,\\\\\n&\\nabla_4\\betab+tr\\chi\\betab=-\\nabla\\rho +^*\\nabla\\sigma+ 2\\omega\\betab +2\\chibh\\cdot\\beta-3(\\etab\\rho-^*\\etab\\sigma),\\\\\n\\end{split}\n\\end{equation*}\nFrom these we can derive the following estimates:\n\\begin{equation*}\n\\begin{split}\n&\\int_{H_u} |\\nabla^i(\\rho,\\sigma)|^2_\\gamma+\\int_{\\Hb_{\\ub}} |\\nabla^i\\betab|^2_\\gamma \\\\\n\\leq &\\int_{H_{u'}} |\\nabla^i(\\rho,\\sigma)|^2_\\gamma+\\int_{\\Hb_{\\ub'}} |\\nabla^i\\betab|^2_\\gamma+\\int_{D_{u,\\ub}}\\nabla^i\\Psi\\sum_{i_1+i_2+i_3+i_4=i}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\psi\\nabla^{i_4}\\Psi\\\\\n&+\\int_{D_{u,\\ub}}\\nabla^i\\Psi\\sum_{i_1+i_2+i_3+i_4=i-1}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}K\\nabla^{i_4}\\Psi.\n\\end{split}\n\\end{equation*}\nFinally, we look at the following set of Bianchi equations:\n\\begin{equation*}\n\\begin{split}\n&\\nabla_3\\betab+2tr\\chib\\betab=-\\div\\alphab-2\\omegab\\betab+\\etab \\cdot\\alphab,\\\\\n&\\nabla_4\\alphab+\\frac 12 tr\\chi\\alphab=-\\nabla\\hot \\betab+ 4\\omega\\alphab-3(\\chibh\\rho-^*\\chibh\\sigma)+\n(\\zeta-4\\etab)\\hot \\betab\n\\end{split}\n\\end{equation*}\nAgain, we notice the absence of the $\\alpha$ terms. Hence we have the estimate\n\\begin{equation*}\n\\begin{split}\n&\\int_{H_u} |\\nabla^i\\betab|^2_\\gamma+\\int_{\\Hb_{\\ub}} |\\nabla^i\\alphab|^2_\\gamma \\\\\n\\leq &\\int_{H_{u'}} |\\nabla^i\\betab|^2_\\gamma+\\int_{\\Hb_{\\ub'}} |\\nabla^i\\alphab|^2_\\gamma+\\int_{D_{u,\\ub}}\\nabla^i\\Psi\\sum_{i_1+i_2+i_3+i_4=i}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\psi\\nabla^{i_4}\\Psi\\\\\n&+\\int_{D_{u,\\ub}}\\nabla^i\\Psi\\sum_{i_1+i_2+i_3+i_4=i-1}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}K\\nabla^{i_4}\\Psi.\n\\end{split}\n\\end{equation*}\nWe have thus established that \n$$\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}\\int_{H_u} |\\nabla^i\\Psi|^2_\\gamma+\\sum_{\\Psic\\in\\{\\rhoc,\\sigmac,\\betab,\\alphab\\}}\\int_{\\Hb_{\\ub}} |\\nabla^i\\Psic|^2_\\gamma$$\ncan be bounded by the right hand side in the statement of the Proposition. To conclude, we use the fact that\n$$\\Psic=\\Psi+\\psi\\psi,$$\nand thus\n\\begin{equation*}\n\\begin{split}\n&\\sum_{\\Psi\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}}\\int_{\\Hb_{\\ub}} |\\nabla^i\\Psi|^2_\\gamma\\\\\n\\leq &\\sum_{\\Psic\\in\\{\\rhoc,\\sigmac,\\betab,\\alphab\\}}\\int_{\\Hb_{\\ub}} |\\nabla^i\\Psic|^2_\\gamma+\\sum_{i_1+i_2\\leq 2}||\\nabla^{i_1}\\psi\\nabla^{i_2}\\psi||_{L^2(\\Hb_{\\ub})}\\\\\n\\leq& \\sum_{\\Psic\\in\\{\\rhoc,\\sigmac,\\betab,\\alphab\\}}\\int_{\\Hb_{\\ub}} |\\nabla^i\\Psic|^2_\\gamma+\\epsilon^{\\frac{1}{2}}C(\\mathcal O_0).\n\\end{split}\n\\end{equation*}\n\\end{proof}\nFinally, we prove the boundedness of the norm $\\mathcal R$:\n\\begin{proposition}\\label{curvature}\nThere exists $\\epsilon_0=\\epsilon_0(\\mathcal O_0,\\mathcal R_0)$ such that for every $\\epsilon\\leq\\epsilon_0$,\n\\[\n \\mathcal R\\leq C(\\mathcal O_0, \\mathcal R_0).\n\\]\n\\end{proposition}\n\\begin{proof}\nAssume as a bootstrap assumption:\n\\begin{equation}\\label{BA3}\n\\mathcal R\\leq \\Delta_2.\n\\end{equation}\nThus we can apply Propositions \\ref{Ricci}, \\ref{Kest} and \\ref{Riccielliptic}.\nBy Proposition \\ref{ee}, we have\n\\begin{equation}\\label{eenonlinear}\n\\begin{split}\n&\\sum_{i=0}^{2}(\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}\\int_{H_u} (\\nabla^i\\Psi)^2+\\sum_{\\Psi\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}}\\int_{\\underline{H}_{\\underline{u}}} (\\nabla^i\\Psi)^2)\\\\\n\\leq& C\\mathcal R_0^2+\\epsilon^{\\frac{1}{2}}C(\\mathcal O_0)+\\int_{\\mathcal D_{u,\\underline{u}}}\\sum_{i\\leq 2}\\nabla^i\\Psi\\sum_{i_1+i_2+i_3+i_4\\leq 2}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\psi\\nabla^{i_4}\\Psi \\\\\n&+\\int_{\\mathcal D_{u,\\underline{u}}}\\sum_{i\\leq 2}\\nabla^i\\Psi\\sum_{i_1+i_2+i_3+i_4\\leq 1}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}K\\nabla^{i_4}\\Psi \\\\\n&+\\int_{\\mathcal D_{u,\\underline{u}}}\\sum_{i\\leq 2}\\nabla^i\\Psi\\sum_{i_1+i_2+i_3+i_4\\leq 3}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\psi\\nabla^{i_4}\\psi \\\\\n\\leq &C\\mathcal R_0^2+C\\sum_{i\\leq 2}\\int_{D_{u,\\ub}} (\\nab^i\\Psi)^2 +\\sum_{i_1+i_2+i_3+i_4\\leq 2}\\int_{\\mathcal D_{u,\\underline{u}}}\\left(\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\psi\\nabla^{i_4}\\Psi\\right)^2\\\\\n&+\\sum_{i_1+i_2+i_3+i_4\\leq 1}\\int_{\\mathcal D_{u,\\underline{u}}}\\left(\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3} K\\nabla^{i_4}\\Psi\\right)^2 +\\sum_{i_1+i_2+i_3+i_4\\leq 3}\\int_{\\mathcal D_{u,\\underline{u}}}\\left(\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3} \\psi\\nabla^{i_4}\\psi\\right)^2.\n\\end{split}\n\\end{equation}\nIt is helpful to first note that \n$$\\sum_{i\\leq 2}||\\nabla^i\\Psi||_{L^2(D_{u,\\ub})}\\leq C\\epsilon^{\\frac 12}\\mathcal R,$$\nsince $\\nab^i\\Psi$ can be estimated either in $L^2(H_u)$ or $L^2(\\Hb_{\\ub})$.\nBy Sobolev Embedding in Propositions \\ref{L4} and \\ref{Linfty}, we thus have\n$$\\sum_{i\\leq 1}||\\nabla^i\\Psi||_{L^2_uL^2_{\\ub}L^4(S)}+||\\Psi||_{L^2_uL^2_{\\ub}L^\\infty(S)}\\leq C\\epsilon^{\\frac 12}\\mathcal R,$$\nA similar argument shows that\n$$\\sum_{i\\leq 3}||\\nabla^i\\psi||_{L^2(D_{u,\\ub})}+\\sum_{i\\leq 2}||\\nabla^i\\psi||_{L^2_uL^2_{\\ub}L^4(S)}+\\sum_{i\\leq 1}||\\nab^i\\psi||_{L^2_uL^2_{\\ub}L^\\infty(S)}\\leq C\\epsilon^{\\frac 12}\\tilde{\\mathcal O}_{3,2},$$\nThus by Proposition \\ref{Riccielliptic}, we have\n$$\\sum_{i\\leq 3}||\\nabla^i\\psi||_{L^2(D_{u,\\ub})}+\\sum_{i\\leq 2}||\\nabla^i\\psi||_{L^2_uL^2_{\\ub}L^4(S)}+\\sum_{i\\leq 1}||\\nab^i\\psi||_{L^2_uL^2_{\\ub}L^\\infty(S)}\\leq C(\\mathcal O_0,\\mathcal R_0)\\epsilon^{\\frac 12}\\mathcal R.$$\nNow we estimate the first nonlinear term in (\\ref{eenonlinear}):\n\\begin{equation*}\n\\begin{split}\n\\sum_{i_1+i_2+i_3+i_4\\leq 2}\\int_{\\mathcal D_{u,\\underline{u}}}\\left(\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\psi\\nabla^{i_4}\\Psi\\right)^2. \\\\\n\\end{split}\n\\end{equation*}\nWe can estimate, by Proposition \\ref{Ricci},\n\\begin{equation*}\n\\begin{split}\n&\\sum_{i_1,i_2\\leq 1,i_3\\leq 1}||\\psi^{i_1}\\nabla^{i_2}\\psi\\nabla^{i_3}\\Psi||_{L^2(D_{u,\\ub})}\\\\\n\\leq &(\\sum_{i_1\\leq 2}\\sup_{u,\\ub}||\\psi||_{L^\\infty(S_{u,\\ub})}^{i_1})(\\sum_{i_2\\leq 2}\\sup_{u,\\ub}||\\nabla^{i_2}\\psi||_{L^2(S_{u,\\ub})})(\\sum_{i_3\\leq 2}||\\nabla^{i_3}\\Psi||_{L^2(D_{u,\\ub})})\\\\\n\\leq &\\epsilon^{\\frac 12} C(\\mathcal O_0)R.\n\\end{split}\n\\end{equation*}\nWe then estimate the term\n$$\\sum_{i_1+i_2+i_3\\leq 1}\\int_{\\mathcal D_{u,\\underline{u}}}\\left(\\psi^{i_1}\\nabla^{i_2} K\\nabla^{i_3}\\Psi\\right)^2.$$\nWe have, by Propositions \\ref{Ricci} and \\ref{Kest},\n\\begin{equation*}\n\\begin{split}\n&\\sum_{i_1+i_2+i_3\\leq 1}||\\psi^{i_1}\\nabla^{i_2} K\\nabla^{i_3}\\Psi||_{L^2(D_{u,\\ub})}\\\\\n\\leq &(\\sum_{i_1\\leq 2}\\sup_{u,\\ub}||\\psi||_{L^\\infty(S_{u,\\ub})}^{i_1})(\\sum_{i_2\\leq 1}\\sup_{u,\\ub}||\\nabla^{i_2}K||_{L^2(S_{u,\\ub})})(\\sum_{i_3\\leq 2}||\\nabla^{i_3}\\Psi||_{L^2(D_{u,\\ub})})\\\\\n\\leq &\\epsilon^{\\frac 12} C(\\mathcal O_0,\\mathcal R_0)R.\n\\end{split}\n\\end{equation*}\nFinally, we estimate\n$$\\sum_{i_1+i_2+i_3+i_4\\leq 3}\\int_{\\mathcal D_{u,\\underline{u}}}\\left(\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3} \\psi\\nabla^{i_4}\\psi\\right)^2.$$\nBy Propositions \\ref{Ricci} and \\ref{Riccielliptic},\n\\begin{equation*}\n\\begin{split}\n&\\sum_{i_1+i_2+i_3+i_4\\leq 3}||\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3} \\psi\\nabla^{i_4}\\psi||_{L^2(D_{u,\\ub})}\\\\\n\\leq &(\\sum_{i_1\\leq 3}\\sup_{u,\\ub}||\\psi||_{L^\\infty(S_{u,\\ub})}^{i_1})(\\sum_{i_2\\leq 2}\\sup_{u,\\ub}||\\nabla^{i_2}\\psi||_{L^2(S_{u,\\ub})})(\\sum_{i_3\\leq 3}||\\nabla^{i_3}\\psi||_{L^2(D_{u,\\ub})})\\\\\n\\leq &\\epsilon^{\\frac 12} C(\\mathcal O_0,\\mathcal R_0)R.\n\\end{split}\n\\end{equation*}\nThus\n$$\\sum_{i=0}^{2}(\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}\\int_{H_u} (\\nabla^i\\Psi)^2+\\sum_{\\Psi\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}}\\int_{\\underline{H}_{\\underline{u}}} (\\nabla^i\\Psi)^2)\n\\leq C\\mathcal R_0^2+\\epsilon^{\\frac{1}{2}}C(\\mathcal O_0)+\\epsilon^{\\frac 12} C(\\mathcal O_0,\\mathcal R_0)R.$$\nBy choosing $\\epsilon$ sufficiently small we have\n$$\\mathcal R\\leq C(\\mathcal O_0,\\mathcal R_0).$$\nThis improves (\\ref{BA3}) for $\\Delta_2$ sufficiently large. Moreover, $\\Delta_2$ can be chosen to depend only on $\\mathcal O_0$ and $\\mathcal R_0$. Thus the choice of $\\epsilon$ depends only on $\\mathcal O_0$ and $\\mathcal R_0$.\n\\end{proof}\n\\subsection{Propagation of Regularity}\\label{Propregsec}\nUp to this point, we have been considering spacetimes with smooth characteristic initial data. In the context of an impulsive gravitational wave, our estimates in this Section apply to spacetimes arising from the smooth data approximating the given singular data. For these spacetimes, we can also prove that estimates for the higher order derivatives of the curvature. In view of the bounds that we have already obtained, this follows from standard arguments. In particular, energy estimates for the higher order derivatives can be derived as in Section \\ref{energyestimatessec}. In this case, since the equations are linear in the highest order derivatives, one can use the bounds from the previous Sections to obtain the desired estimates. More precisely, we have\n\\begin{proposition}\\label{propagationregularity}\nSuppose, in addition to the assumptions of Theorem \\ref{timeofexistence}, the bound \n$$\\sum_{j\\leq J}\\sum_{k\\leq K}\\sum_{i\\leq I}(\\sum_{\\Psi\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}}||\\nab_3^j\\nab_4^k \\nab^i\\Psi||_{L^2(\\Hb_0)}+\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}||\\nab_3^i \\nab_4^k\\nab^j\\Psi||_{L^2(H_0)})\\leq D_{I,J,K}. $$\nholds initially for some $I,J,K$. Then, the following bounds hold for $0\\leq u\\leq u_*$, $0\\leq \\ub\\leq \\ub_*$,\n$$\\sum_{j\\leq J}\\sum_{k\\leq K}\\sum_{i\\leq I}(\\sum_{\\Psi\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}}||\\nab_3^j\\nab_4^k \\nab^i\\Psi||_{L^2(\\Hb_{\\ub})}+\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}||\\nab_3^j \\nab_4^k\\nab^i\\Psi||_{L^2(H_u)})\\leq D'_{I,J,K}. $$\nfor $u_*,\\ub_*\\leq \\epsilon$, where $\\epsilon$ can be chosen as in Theorem \\ref{timeofexistence} and the constant $D'_{I,J,K}$ depends only on the size of the initial data norm in the assumption of Theorem \\ref{timeofexistence} and $D_{I,J,K}$.\n\\end{proposition}\n\\begin{proof}\nSince the arguments are similar to those in Section \\ref{energyestimatessec}, we will only provide a sketch. We will proceed with an induction argument in $i$, $j $ and $k$ and we will only consider the highest order terms, assuming appropriate control on the lower order terms. From \\cite{KN}, we have the commutation formula\n$$[\\nab_3,\\nab_4]f=-2\\omega \\nab_3 f+2\\omegab\\nab_4 f+4\\zeta\\cdot\\nab f$$\nfor all scalar functions $f$. Therefore, by commuting $\\nab_3^j\\nab_4^k\\nab^i$ with the equations \\eqref{rcsc} and \\eqref{bc}, we have\n\\begin{equation*}\n\\begin{split}\n&\\nabla_3\\nab_3^j\\nab_4^k\\nabla^i\\beta- \\nabla \\nab_3^j\\nab_4^k\\nabla^i\\rhoc-\\nabla\\nab_3^j\\nab_4^k\\nabla^i\\sigmac \\\\\n\\sim&\\psi\\nab_3^j\\nab_4^k\\nab^i(\\beta,\\rhoc,\\sigmac,\\betab)+\\psi\\psi\\nab_3^j\\nab_4^k\\nab^{i+1}\\psi+\\mbox{terms with fewer derivatives},\\\\\n\\end{split}\n\\end{equation*}\nand\n\\begin{equation*}\n\\begin{split}\n&\\nabla_4\\nab_3^j\\nab_4^k\\nabla^i\\rhoc- \\div\\nab_3^j\\nab_4^k\\nabla^i\\beta\\\\\n\\sim&\\psi\\nab_3^j\\nab_4^k\\nab^i(\\beta,\\rhoc,\\sigmac,\\betab)+\\psi\\psi\\nab_3^j\\nab_4^k\\nab^{i+1}\\psi+\\mbox{terms with fewer derivatives},\\\\\n\\end{split}\n\\end{equation*}\nand \n\\begin{equation*}\n\\begin{split}\n&\\nabla_4\\nab_3^j\\nab_4^k\\nabla^i\\sigmac- \\div^*\\nab_3^j\\nab_4^k\\nabla^i\\beta\\\\\n\\sim&\\psi\\nab_3^j\\nab_4^k\\nab^i(\\beta,\\rhoc,\\sigmac,\\betab)+\\psi\\psi\\nab_3^j\\nab_4^k\\nab^{i+1}\\psi+\\mbox{terms with fewer derivatives}.\n\\end{split}\n\\end{equation*}\nNotice that since $\\alpha$ does not appear in \\eqref{rcsc} and \\eqref{bc}, we do not have the term $\\nab_3^j\\nab_4^k\\nab^i\\alpha$ in these commuted equations. Therefore, integrating by parts as in Section \\ref{energyestimatessec}, we have\n\\begin{equation*}\n\\begin{split}\n&||\\nab_3^j\\nab_4^k \\nab^i(\\rhoc,\\sigmac)||_{L^2(\\Hb_{\\ub})}^2+||\\nab_3^j \\nab_4^k\\nab^i\\beta||_{L^2(H_u)}^2\\\\\n\\leq &\\mbox{Data}+\\|\\nab_3^j\\nab_4^k\\nab^i\\Psi\\psi\\nab_3^j\\nab_4^k\\nab^i(\\beta,\\rhoc,\\sigmac,\\betab)\\|_{L^1(\\mathcal D_{u,\\ub})}+\\|\\nab_3^j\\nab_4^k\\nab^i\\Psi\\psi\\psi\\nab_3^j\\nab_4^k\\nab^{i+1}\\psi\\|_{L^1(\\mathcal D_{u,\\ub})}\\\\\n&+\\|\\nab_3^j\\nab_4^k\\nab^i\\Psi\\times(\\mbox{terms with fewer derivative})\\|_{L^1(\\mathcal D_{u,\\ub})}.\n\\end{split}\n\\end{equation*}\nSimilarly, using the other null Bianchi equations in \\eqref{eq:null.Bianchi}, we also have\n\\begin{equation}\\label{hr.1}\n\\begin{split}\n&\\sum_{\\Psic\\in\\{\\rhoc,\\sigmac,\\betab,\\alphab\\}}||\\nab_3^j\\nab_4^k \\nab^i\\Psic||_{L^2(\\Hb_{\\ub})}^2+\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}||\\nab_3^j \\nab_4^k\\nab^i\\Psi||_{L^2(H_u)}^2\\\\\n\\leq &\\mbox{Data}+\\|\\nab_3^j\\nab_4^k\\nab^i\\Psi\\psi\\nab_3^j\\nab_4^k\\nab^i(\\beta,\\rhoc,\\sigmac,\\betab,\\alphab)\\|_{L^1(\\mathcal D_{u,\\ub})}+\\|\\nab_3^j\\nab_4^k\\nab^i\\Psi\\psi\\psi\\nab_3^j\\nab_4^k\\nab^{i+1}\\psi\\|_{L^1(\\mathcal D_{u,\\ub})}\\\\\n&+\\|\\nab_3^j\\nab_4^k\\nab^i\\Psi\\times(\\mbox{terms with fewer derivative})\\|_{L^1(\\mathcal D_{u,\\ub})}.\n\\end{split}\n\\end{equation}\nAs in Section \\ref{Ricciellipticsec}, the highest order derivatives of the Ricci coefficients $\\nab_3^j\\nab_4^k\\nab^{i+1}\\psi$ can be controlled by the highest order derivatives of the curvature components using elliptic estimates:\n\\begin{equation}\\label{hr.2}\n\\begin{split}\n&\\sum_{\\psi\\in\\{\\trch,\\chih,\\omega,\\trchb,\\eta,\\etab\\}}\\|\\nab_3^j\\nab_4^k\\nab^{i+1}\\psi\\|_{L^2(H_u)}+\\sum_{\\psi\\in\\{\\trch,\\chih,\\omegab,\\trchb,\\eta,\\etab\\}}\\|\\nab_3^j\\nab_4^k\\nab^{i+1}\\psi\\|_{L^2(\\Hb_{\\ub})}\\\\\n\\leq &\\sum_{\\Psic\\in\\{\\rhoc,\\sigmac,\\betab,\\alphab\\}}||\\nab_3^j\\nab_4^k \\nab^i\\Psic||_{L^2(\\Hb_{\\ub})}+\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}||\\nab_3^j \\nab_4^k\\nab^i\\Psi||_{L^2(H_u)}\\\\\n&+\\mbox{terms under control}.\n\\end{split}\n\\end{equation}\nUsing the boundedness of the $L^\\infty$ norm of $\\psi$ and the control of the terms with fewer derivatives that we have assumed as the induction hypothesis, \\eqref{hr.1} and \\eqref{hr.2} imply\n\\begin{equation}\\label{hr.3}\n\\begin{split}\n&\\sum_{\\Psic\\in\\{\\rhoc,\\sigmac,\\betab,\\alphab\\}}||\\nab_3^j\\nab_4^k \\nab^i\\Psic||_{L^2(\\Hb_{\\ub})}^2+\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}||\\nab_3^j \\nab_4^k\\nab^i\\Psi||_{L^2(H_u)}^2\\\\\n\\leq &\\mbox{Data term}+\\mbox{Bounded terms}+\\int_0^{\\ub} \\sum_{\\Psic\\in\\{\\rhoc,\\sigmac,\\betab,\\alphab\\}}||\\nab_3^j\\nab_4^k \\nab^i\\Psic||_{L^2(\\Hb_{\\ub_*})}^2 d\\ub_*\\\\\n&+\\int_0^u\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}||\\nab_3^j \\nab_4^k\\nab^i\\Psi||_{L^2(H_{u_*})}^2 du_*\\\\\n&+\\int_0^{\\ub}\\int_0^u\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}||\\nab_3^j \\nab_4^k\\nab^i\\Psi||_{L^2(H_{u_*})}\\sum_{\\Psic\\in\\{\\rhoc,\\sigmac,\\betab,\\alphab\\}}||\\nab_3^j\\nab_4^k \\nab^i\\Psic||_{L^2(\\Hb_{\\ub_*})} du_*d\\ub_*.\n\\end{split}\n\\end{equation}\nTo obtain the desired conclusion, introduce the bootstrap assumption\n$$\\sum_{\\Psic\\in\\{\\rhoc,\\sigmac,\\betab,\\alphab\\}}||\\nab_3^j\\nab_4^k \\nab^i\\Psic||_{L^2(\\Hb_{\\ub}(0,u))}^2+\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}||\\nab_3^j \\nab_4^k\\nab^i\\Psi||_{L^2(H_u(0,\\ub))}^2\\leq A e^{2A(u+\\ub)}$$\nfor $A$ sufficiently large. Then \\eqref{hr.3} implies that we have\n$$\\sum_{\\Psic\\in\\{\\rhoc,\\sigmac,\\betab,\\alphab\\}}||\\nab_3^j\\nab_4^k \\nab^i\\Psic||_{L^2(\\Hb_{\\ub}(0,u))}^2+\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}||\\nab_3^j \\nab_4^k\\nab^i\\Psi||_{L^2(H_u(0,\\ub))}^2\\leq C e^{2A(u+\\ub)}$$\nfor some constant $C$ depending on the data term and the bounded terms. Therefore, this improves the bootstrap assumption for $A$ sufficiently large. Finally, notice that since we can control the derivatives of the Ricci coefficients by the derivatives of the renormalized curvature components, we in fact have the bound for the curvature components themselves. This concludes the proof of the proposition.\n\\end{proof}\n\\begin{remark}\nNotice that for approximating solutions for an impulsive gravitational wave, the constants in the initial norms $D_{I,J,K}$ depend on $n$. However, the initial data satisfy uniform estimates for each $I\\geq 2,J\\geq 0$ with $K=0$. Therefore, the corresponding uniform estimates will hold in the region $0\\leq u\\leq u_*$ and $0\\leq\\ub\\leq \\ub_*$.\n\\end{remark}\n\n\n\\subsection{End of Proof of Theorem \\ref{timeofexistence}}\\label{EndofProof}\nOnce we have the estimates for all the higher derivatives, the proof of Theorem \\ref{timeofexistence} is standard. We refer, for example, to Section 6 of \\cite{L} for details.\n\n\\subsection{Additional Estimates for an Impulsive Gravitational Wave}\\label{AddEst}\nIn this Subsection, we focus our attention on the sequence of spacetimes $(\\mathcal M_n, g_n)$ with characteristic initial data converging to that of an impulsive gravitational wave as constructed in Section \\ref{initialcondition}. The conclusion of Theorem \\ref{timeofexistence} can be applied to $(\\mathcal M_n, g_n)$. The initial data of the sequence of spacetimes, however, possess an additional property: $\\alpha_n$ and its angular derivatives are bounded uniformly in $L^1_{\\ub}$. In this Subsection, we prove that in the constructed spacetimes $(\\mathcal M_n, g_n)$, the same estimates hold for all $u\\leq u_*$ with $u_*\\leq \\epsilon$. Moreover, we show that $\\alpha_n$ concentrates around $\\ub_s$ and if $\\alpha_n$ is initially more regular uniformly in $n$ away from $\\ub_s$, it is also uniformly more regular away from $\\ub_s$ for $0\\leq u\\leq u_*\\leq \\epsilon$.\n\nWe begin by proving energy estimates for $\\alpha_n$ in $L^2(H_u)$ and $\\beta_n$ in $L^2(\\Hb_{\\ub})$ using the pair of Bianchi equations for $(\\nab_3\\alpha_n,\\nab_4\\beta_n)$ in (\\ref{eq:null.Bianchi}).\n\\begin{proposition}\\label{alphaenergy}\nThe following estimate holds:\n\\begin{equation*}\n\\begin{split}\n&\\int_{H_u} |\\nabla^i\\alpha_n|^2_\\gamma+\\int_{\\Hb_{\\ub}} |\\nabla^i\\beta_n|^2_\\gamma \\\\\n\\leq &\\int_{H_{u'}} |\\nabla^i\\alpha_n|^2_\\gamma+\\int_{\\Hb_{\\ub'}} |\\nabla^i\\beta|^2_\\gamma+\\int_{D_{u,\\ub}}\\nabla^i(\\alpha_n,\\beta_n)\\sum_{i_1+i_2+i_3+i_4=i}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\psi\\nabla^{i_4}\\Psi\\\\\n&+\\int_{D_{u,\\ub}}\\nabla^i(\\alpha_n,\\beta_n)\\sum_{i_1+i_2+i_3+i_4=i}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}K\\nabla^{i_4}\\Psi,\n\\end{split}\n\\end{equation*}\nwhere here, unlike in other places, we use $\\Psi$ to denote all possible curvature components, including $\\alpha_n$.\n\\end{proposition}\n\\begin{proof}\nConsider the following Bianchi equations:\n\\begin{equation*}\n\\begin{split}\n&\\nabla_3\\alpha+\\frac 12 tr\\chib \\alpha=\\nabla\\hot \\beta+ 4\\omegab\\alpha-3(\\chih\\rho+^*\\chih\\sigma)+\n(\\zeta+4\\eta)\\hot\\beta,\\\\\n&\\nabla_4\\beta+2tr\\chi\\beta = \\div\\alpha - 2\\omega\\beta + \\eta \\alpha,\\\\\n\\end{split}\n\\end{equation*}\nUsing Proposition \\ref{intbypartssph},\n\\begin{equation*}\n\\begin{split}\n\\int <\\alpha_n,\\nabla_3\\alpha_n>_\\gamma =&\\int <\\alpha_n,\\nabla\\hot\\beta_n>_\\gamma+<\\alpha_n,\\psi\\Psi>_\\gamma \\\\\n=&\\int -<\\div\\alpha_n,\\beta_n>_\\gamma+<\\alpha_n,\\psi\\Psi>_\\gamma \\\\\n=&\\int -<\\nabla_4\\beta_n,\\beta_n>_\\gamma+<\\alpha_n,\\psi\\Psi>_\\gamma +<\\beta_n,\\psi\\Psi>_\\gamma\\\\\n\\end{split}\n\\end{equation*}\nIntegrate by parts using Proposition \\ref{intbyparts34} to get that for $u\\geq u'$, $\\ub\\geq \\ub'$,\n\\begin{equation*}\n\\begin{split}\n\\int_{H_u} |\\alpha_n|^2_\\gamma+\\int_{\\Hb_{\\ub}} |\\beta_n|^2_\\gamma \\leq \\int_{H_{u'}} |\\alpha_n|^2_\\gamma+\\int_{\\Hb_{\\ub'}} |\\beta_n|^2_\\gamma+\\int_{D_{u,\\ub}}<(\\alpha_n,\\beta_n),\\psi\\Psi>_\\gamma\\\\\n\\end{split}\n\\end{equation*}\nWe use the commutation formula, and note that the special structure is preserved in the highest order:\n\\begin{equation*}\n\\begin{split}\n&\\nabla_3\\nabla^i\\alpha- \\nabla\\hot \\nabla^i\\beta \\\\\n\\sim&\\sum_{i_1+i_2+i_3+i_4=i}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\psi\\nabla^{i_4}\\alpha \\\\\n&+\\sum_{i_1+i_2+i_3+i_4+i_5=i-1}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}K^{i_4}\\nabla^{i_5}\\beta+\\sum_{i_1+i_2+i_3=i}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}(\\psi\\Psi),\\\\\n&\\nabla_4\\nabla^i\\beta- \\div\\nabla^i\\alpha\\\\\n\\sim&\\sum_{i_1+i_2+i_3+i_4=i}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\psi\\nabla^{i_4}\\beta\\\\\n&+\\sum_{i_1+i_2+i_3+i_4+i_5=i-1}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}K^{i_4}\\nabla^{i_5}\\alpha+\\sum_{i_1+i_2+i_3=i}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}(\\psi\\Psi),\\\\\n\\end{split}\n\\end{equation*}\nPerforming the integration by parts as before using Propositions \\ref{intbyparts34} and \\ref{intbypartssph}, we have\n\\begin{equation*}\n\\begin{split}\n&\\int_{H_u} |\\nabla^i\\alpha_n|^2_\\gamma+\\int_{\\Hb_{\\ub}} |\\nabla^i\\beta_n|^2_\\gamma \\\\\n\\leq &\\int_{H_{u'}} |\\nabla^i\\alpha_n|^2_\\gamma+\\int_{\\Hb_{\\ub'}} |\\nabla^i\\beta_n|^2_\\gamma+\\int_{D_{u,\\ub}}\\nabla^i(\\alpha_n,\\beta_n)\\sum_{i_1+i_2+i_3+i_4=i}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}\\psi\\nabla^{i_4}\\Psi\\\\\n&+\\int_{D_{u,\\ub}}\\nabla^i(\\alpha_n,\\beta_n)\\sum_{i_1+i_2+i_3+i_4=i}\\nabla^{i_1}\\psi^{i_2}\\nabla^{i_3}K\\nabla^{i_4}\\Psi.\n\\end{split}\n\\end{equation*}\n\\end{proof}\n\n\\subsubsection{Before the Impulse}\n\nWe first apply Proposition \\ref{alphaenergy} for the region $0\\leq \\ub\\leq \\ub_s$. Notice that the initial norm\n$$\\sum_{i\\leq 2}\\int_0^{\\ub_s} ||\\nabla^i\\alpha_n||_{L^2(S_{0,\\ub})}^2 d\\ub\\leq C$$\nis bounded independent of $n$. Therefore,\n\\begin{proposition}\\label{beforeshock}\n$$\\sum_{i\\leq 2}(\\sup_{u\\leq \\epsilon}||\\nabla^i\\alpha_n||_{L^2(H_u(0,\\ub_s))}^2+\\sup_{\\ub\\leq \\ub_s}||\\nabla^i\\beta_n||^2_{L^2(\\Hb_{\\ub})}) \\leq C(\\mathcal O_0,\\mathcal R_0),$$\nindependent of $n$.\n\\end{proposition}\n\\begin{proof}\nBy Proposition \\ref{alphaenergy} and all the estimates in the previous section, we have\n\\begin{equation*}\n\\begin{split}\n&\\sum_{i\\leq 2}(||\\nabla^i\\alpha_n||_{L^2(H_u(0,\\ub_s))}^2+\\sup_{\\ub\\leq \\ub_s}||\\nabla^i\\beta_n||^2_{L^2(\\Hb_{\\ub})})\\\\\n \\leq &C(\\mathcal O_0,\\mathcal R_0)(1+||\\nabla^i\\alpha_n||_{L^2(H)}+\\int_0^u ||\\nabla^i\\alpha_n||^2_{L^2(H_{u'})} du').\n\\end{split}\n\\end{equation*}\nBy Gronwall's inequality, we have\n$$\\sum_{i\\leq 2}(\\sup_{u\\leq \\epsilon}||\\nabla^i\\alpha_n||_{L^2(H_u(0,\\ub_s))}^2+\\sup_{\\ub\\leq \\ub_s}||\\nabla^i\\beta_n||^2_{L^2(\\Hb_{\\ub})}) \\leq C(\\mathcal O_0,\\mathcal R_0),$$\nas claimed.\n\\end{proof} \n\n\\subsubsection{The Impulse Region}\n\nFor an impulsive gravitational wave, the initial data represented by the $\\alpha$ component of curvature and its derivative are not in $L^2$. By the construction of Section \\ref{initialcondition}, the approximating sequence $\\alpha_n$ satisfies initially \n$$\\sum_{i\\leq 2}\\int_{\\ub_s}^{\\ub_s+2^{-n}} ||\\nabla^i\\alpha_n||_{L^2(S_{0,\\ub})}^2 d\\ub\\leq C2^{n}.$$\nIn the following Proposition, we show that this bound is propagated.\n\\begin{proposition}\\label{duringshock}\n$$\\sum_{i\\leq 2}(\\sup_{u\\leq \\epsilon}||\\nabla^i\\alpha_n||_{L^2(H_u(\\ub_s,\\ub_s+2^{-n}))}^2+\\sup_{\\ub_s\\leq \\ub\\leq \\ub_s+2^{-n}}||\\nabla^i\\beta_n||^2_{L^2(\\Hb_{\\ub})}) \\leq C(\\mathcal O_0,\\mathcal R_0)2^{n},$$\nwhere $C(\\mathcal O_0,\\mathcal R_0)$ is independent of $n$.\n\\end{proposition}\n\\begin{proof}\nThe proof is the same as Proposition \\ref{beforeshock}, except that the initial data for $\\nabla^i\\alpha$ can only be bounded by $2^{\\frac n2}$ in $L^2(H_0(\\ub_s,\\ub_s+2^{-n}))$.\n\\end{proof}\n\n\\subsubsection{After the Impulse}\n\nFor $\\ub_s+2^{-n}\\leq \\ub\\leq \\ub_*$, since the initial data satisfy $$\\sum_{i\\leq 2}\\int_{\\ub_s+2^{-n}}^{\\ub_*} ||\\nabla^i\\alpha_n||_{L^2(S_{0,\\ub})}^2 d\\ub\\leq C,$$\nindependent of $n$, we again have a better estimate. An extra challenge arises from the fact that the estimates derived in Proposition \\ref{duringshock} depends on $n$. This can be overcome if we allow a loss in derivative:\n\\begin{proposition}\\label{aftershock}\n$$\\sum_{i\\leq 1}(\\sup_{u\\leq u_*}||\\nabla^i\\alpha_n||_{L^2(H_u(\\ub_s+2^{-n},\\ub_*))}^2+\\sup_{\\ub_s+2^{-n}\\leq\\ub\\leq \\ub_*}||\\nabla^i\\beta_n||^2_{L^2(\\Hb_{\\ub})}) \\leq C(\\mathcal O_0,\\mathcal R_0),$$\nindependent of $n$.\n\\end{proposition}\n\\begin{proof}\nTo bound the left hand side, we can apply the argument as in Proposition \\ref{beforeshock} to get\n$$\\sum_{i\\leq 1}(\\sup_{u\\leq u_*}||\\nabla^i\\alpha_n||_{L^2(H_u(\\ub_s+2^{-n},\\ub_*))}^2+\\sup_{\\ub_s+2^{-n}\\leq\\ub\\leq \\ub_*}||\\nabla^i\\beta_n||^2_{L^2(\\Hb_{\\ub})}) \\leq C(\\mathcal O_0,\\mathcal R_0)2^{\\frac n2}.$$\nThe problem in proving Proposition \\ref{aftershock} with a bound independent of $n$ is that according to Proposition \\ref{duringshock}, $||\\nab^i\\beta_n||_{L^2(\\Hb_{\\ub_s})}$ may grow in $n$. However, we can still obtain the desired bounds for $\\alpha_n$ and $\\nab\\alpha_n$ (note that the argument below will not apply to $\\nab^2\\alpha_n$ unless extra assumptions on the initial data are used) by using Proposition \\ref{RS}, which implies that\n$$\\sum_{i\\leq 1}\\sup_{u,\\ub}||\\nab^i\\beta_n||_{L^2(S_{u,\\ub})}\\leq C(\\mathcal O_0,\\mathcal R_0).$$\nTaking $\\ub=\\ub_s+2^{-n}$ and integrating, we thus have\n$$\\sum_{i\\leq 1}||\\nabla^i\\beta_n||^2_{L^2(\\Hb_{\\ub_s+2^{-n}})}\\leq C(\\mathcal O_0,\\mathcal R_0).$$\n\\end{proof}\n\nWe can also prove estimates for higher derivatives away from $\\ub_s$. The influence of a propagating curvature impulse results in a loss of derivatives in such estimates compared to the assumed regularity of the initial data. By the remark after Proposition \\ref{propagationregularity}, we have estimates, uniform in $n$, for the $\\nab_3$ and $\\nab$ derivatives for $\\psi$ and $\\Psi$. We now show that we can also have uniform in $n$ estimates for the $\\nab_4$ derivatives for $\\psi$ and $\\Psi$ for $\\ub\\geq\\ub_s+2^{-n}$. By the Bianchi equations (\\ref{eq:null.Bianchi}), this in turn can be reduced to showing estimates for $\\nab_4$ derivatives of $\\alpha$. This is proved in the following Proposition:\n\\begin{proposition}\\label{alphaapriori}\nFor $\\ub_s+2^{-n}\\leq\\ub\\leq\\epsilon$, if the initial data set satisfies\n$$\\sum_{k\\leq K}\\sum_{i\\leq I}\\sum_{\\Psi\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}}||\\nab_4^k \\nab^i\\Psi_n||_{L^2(\\Hb_0)}\\leq C_{I,K}. $$\n$$\\sum_{k\\leq K}\\sum_{i\\leq I}\\sup_{\\ub}\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}|| \\nab_4^k\\nab^i\\Psi_n||_{L^2(S_{0,\\ub})}\\leq C_{I,K}.$$\nthen\n$$\\sum_{i\\leq I-1}\\sum_{k\\leq \\min\\{K,\\lfloor\\frac{I-i-1}{2}\\rfloor\\}}(\\sup_{\\ub}||\\nab_4^k\\nabla^i\\alpha_n||_{L^2(H_u(\\ub_s+2^{-n},\\epsilon))}+\\sup_{\\ub_s+2^{-n}\\leq \\ub\\leq \\epsilon}||\\nab_4^k\\nabla^i\\beta_n||_{L^2(\\Hb_{\\ub})})\\leq C_{I,K}',$$\nwhere $C_{I,K}$ is independent of $n$.\n\\end{proposition}\n\\begin{proof}\nThe main step is to prove that on $\\Hb_{\\ub_s+2^{-n}}$, we have the estimate\n$$||\\nab_4^k\\nab^i\\beta_n||_{L^2(\\Hb_{\\ub_s+2^{-n}})}\\leq C$$\nindependent of $n$. For $k=0$, we can commute with angular derivatives and integrate along the $u$ direction using the Bianchi equation\n$$\\nab_3\\beta+\\trchb\\beta=\\nab\\rho+2\\omegab\\beta+^*\\nab\\sigma+2\\chih\\cdot\\betab+3(\\eta\\rho+^*\\eta\\sigma).$$\nSince the right hand side has one more angular derivative, this integration loses a derivative, allowing us only to prove\n$$\\sum_{i\\leq I-1}||\\nab^i\\beta_n||_{L^2(\\Hb_{\\ub_s+2^{-n}})}\\leq C$$\n\nNow we consider also the $\\nab_4$ derivatives. Differentiate the equation by $\\nab_4^k\\nab_i$ and commute $[\\nab_3,\\nab_4^k\\nab_i]$ on the left hand side. Moreover, except for the component $\\alpha$, whenever we see $\\nab_4^{k_1}\\Psi$, we substitute an appropriate Bianchi equation from (\\ref{eq:null.Bianchi}). Then we get an equation\n$$\\nab_3\\nab_4^k\\nab^i\\beta=\\nab_4^{k-1}\\nab^{i+2}\\beta+...$$\nwhere $...$ denote terms that are lower order in terms of derivatives. We then integrate this along the $u$ direction. Thus\n$$||\\nab_4^k\\nab^i\\beta_n||_{L^2(\\Hb_{\\ub_s+2^{-n}})}\\leq C||\\nab_4^{k-1}\\nab^{i+2}\\beta_n||_{L^2(\\Hb_{\\ub_s+2^{-n}})}+...$$\nInducting in $k$, we get that\n$$||\\nab_4^k\\nab^i\\beta_n||_{L^2(\\Hb_{\\ub_s+2^{-n}})}\\leq C||\\nab^{i+2k}\\beta_n||_{L^2(\\Hb_{\\ub_s+2^{-n}})}+... $$\nNow we have reduced to the case where there are only angular derivatives falling on $\\beta_n$. By the above, we need $i+2k+1\\leq I$. Thus we can prove \n\\begin{equation}\\label{smoothnessdata}\n\\sum_{i\\leq I-1}\\sum_{k\\leq \\min\\{K,\\lfloor\\frac{I-i-1}{2}\\rfloor\\}}||\\nab_4^k\\nab^i\\beta_n||_{L^2(\\Hb_{\\ub_s+2^{-n}})}\\leq C.\n\\end{equation}\nThe conclusion thus follows from a standard energy estimate type argument as in Proposition \\ref{alphaenergy} and using (\\ref{smoothnessdata}) as the initial data on $\\Hb_{\\ub_s+2^{-n}}$\n\\end{proof}\n\nCombining the estimates in Propositions \\ref{beforeshock}, \\ref{duringshock}, \\ref{alphaapriori} and Cauchy-Schwarz, we have the following uniform $L^1_{\\ub}$ estimate.\n\\begin{proposition}\\label{totalvariation}\n$$\\sup_{u\\leq u_*}\\sum_{i\\leq I}\\int_0^{\\ub_*} ||\\nabla^i\\alpha_n||_{L^2(S_{u,\\ub})}d\\ub\\leq C_I,$$\nwhere $C_I$ is independent of $n$.\n\\end{proposition}\nProposition \\ref{totalvariation} is crucial in showing that the limiting spacetime will have $\\alpha$ defined as a finite measure with a singular atom at $\\ub_s$.\n\n\\section{Convergence}\\label{convergence}\nIn this Section, we show that a sequence of initial data satisfying uniform estimates of Theorem \\ref{rdthmv2} with converging initial data gives rise to a sequence of converging spacetimes. The convergence will be understood as follows: the spacetime will be identified in the system of double null coordinates $(u,\\ub,\\th^1,\\th^2)$ and convergence will be established for the sequence of the corresponding spacetime metrics. In view of the quasilinear nature of the Einstein equations, we can only hope to prove convergence of our approximating spacetimes in a norm with one derivative fewer than the a priori estimates that we established for them. \n\nTo get estimates of metric, we use the fact that the metric components satisfy inhomogeneous transport equations with right hand side expressed as Ricci coefficients. The estimates for the difference of Ricci coefficients and curvature components are derived by considering the system of difference equations obtained from the original system of transport, elliptic and Bianchi equations. As was the case for the a priori estimates, the challenge for the difference system is a lack of any a priori control of the difference of the singular curvature components $\\alpha$. For the approximating sequence of initial data, $\\alpha_n$ is not a Cauchy sequence in $L^2(H_0)$. In the proof of the a priori estimates, we handled the lack of information of the $\\alpha$ component of curvature via a renormalization procedure and the observation that the system satisfied by the Ricci coefficients and the renormalized curvature components can be closed without any reference to the $\\alpha$ component of curvature. Potentially, this property may fail when we consider the difference system. It is however a remarkable fact as we will show below that the difference equations still possess the same structure.\n\nThe following is the main Theorem in which we estimate the difference of the metrics, Ricci coefficients and curvature components of two spacetimes:\n\n\\begin{theorem}\\label{convergencethm}\nSuppose we have two sets of initial data $(1)$ and $(2)$ satisfying the conditions in Theorem \\ref{timeofexistence} with the same constants $C$ and $c$. By Theorem \\ref{timeofexistence}, we can solve for vacuum spacetimes $(\\mathcal M^{(1)}, g^{(1)})$ and $(\\mathcal M^{(2)}, g^{(2)})$ corresponding to the initial data sets $(1)$ and $(2)$ in the region $0\\leq u\\leq u_*$ and $0\\leq \\ub\\leq\\ub_*$ for $u_*,\\ub_*\\leq \\epsilon$ . Let $(u,\\ub,\\th^1,\\th^2)$ be the coordinate system introduced in Section \\ref{coordinates} such that the metrics take the form\n$$g^{(i)}=-2(\\Omega^{(i)})^2(du\\otimes d\\ub+d\\ub\\otimes du)+(\\gamma^{(i)})_{AB}(d\\th^A-(b^{(i)})^Adu)\\otimes (d\\th^B-(b^{(i)})^Bdu),$$ \nwhere $\\Omega=1$ and $b^{A}=0$ on $H_0$ and $\\Hb_0$. We can now identify the two spacetimes by identifying points with the same value of coordinate functions. Define $g'=g^{(1)}-g^{(2)}$, $\\psi'=\\psi^{(1)}-\\psi^{(2)}$ and $\\Psi'=\\Psi^{(1)}-\\Psi^{(2)}$ to be the difference of the metric, the difference of the Ricci coefficients and the difference of the curvature components respectively. If the data satisfy \n$$\\sup_u|(\\frac{\\partial}{\\partial\\th})^i\\gamma_{AB}'(\\ub=0)|+\\sup_{\\ub}|(\\frac{\\partial}{\\partial\\th})^i\\gamma_{AB}'(u=0)|\\leq a,$$\n$$\\sum_{\\psi\\neq\\chih,\\omega}(\\sum_{i\\leq 1} \\sup_{u}||\\nabla^i\\psi'||_{L^2(S_{u,0})}+\\sum_{i\\leq 1} \\sup_{\\ub}||\\nabla^i\\psi'||_{L^2(S_{0,\\ub})})\\leq a,$$\n$$\\sum_{i\\leq 1} ||\\nabla^i(\\chih',\\omega')||_{L^{p_0}_{\\ub}L^2(S_{0,\\ub})}\\leq a\\quad\\mbox{for some fixed }2\\leq p_0<\\infty,$$\n$$||\\nabla^2(\\chih',\\omega',\\eta',\\etab')||_{L^2(H_0)}+||\\nabla^2(\\chibh',\\omegab',\\eta',\\etab')||_{L^2(\\Hb_{0})}\\leq a,$$\n$$\\sup_{u} ||\\nabla^2(\\trch',\\trchb')||_{L^2(S_{u,0})}+\\sup_{\\ub} ||\\nabla^2(\\trch',\\trchb')||_{L^2(S_{0,\\ub})}\\leq a,$$\n$$\\sum_{i\\leq 1}\\left(\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}} ||\\nabla^i\\Psi'||_{L^{2}_{\\ub}L^2(S_{0,\\ub})}+\\sum_{\\Psi\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}} ||\\nabla^i\\Psi'||_{L^{2}_{u}L^2(S_{u,0})}\\right)\\leq a,$$\nwhere the angular covariant derivative and all the norms are defined with respect to the spacetime $(1)$. Then the following estimates hold in $\\{0\\leq u\\leq u_*\\}\\cap\\{0\\leq\\ub\\leq\\ub_*\\}$:\n$$\\sup_{u,\\ub}|(\\frac{\\partial}{\\partial\\th})^i\\gamma_{AB}'|\\leq C'a,$$\n$$\\sum_{\\psi\\neq\\chih,\\omega}\\sum_{i\\leq 1} \\sup_{u,\\ub}||\\nabla^i\\psi'||_{L^2(S_{u,\\ub})}\\leq C'a,$$\n$$\\sum_{i\\leq 1} \\sup_u ||\\nabla^i(\\chih',\\omega')||_{L^{p_0}_{\\ub}L^2(S_{u,\\ub})}\\leq C'a,$$\n$$\\sup_u ||\\nabla^2(\\chih',\\omega',\\eta',\\etab')||_{L^2(H_u)}+\\sup_{\\ub} ||\\nabla^2(\\chibh',\\omegab',\\eta',\\etab')||_{L^2(\\Hb_{\\ub})}+\\sup_{u,\\ub} ||\\nabla^2(\\trch',\\trchb')||_{L^2(S_{u,\\ub})}\\leq C'a,$$\n$$\\sum_{i\\leq 1}\\left(\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}} \\sup_u||\\nabla^i\\Psi'||_{L^2_{\\ub}L^2(S_{u,\\ub})}+\\sum_{\\Psi\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}} \\sup_{\\ub}||\\nabla^i\\Psi'||_{L^2_{u}L^2(S_{u,\\ub})}\\right)\\leq C'a,$$\nfor some constant $C'$ depending only on $C$ and $c$ and independent of $a$.\n\\end{theorem}\nBy iterating Theorem \\ref{convergencethm}, we can reduce its proof to the region $0\\leq u\\leq \\delta$, $0\\leq \\ub\\leq \\delta$ for some sufficiently small $\\delta$ independent of $a$. Most of this Section will be devoted to a proof of Theorem \\ref{convergencethm} for $0\\leq u\\leq\\delta$ and $0\\leq \\ub\\leq\\delta$. The proof of Theorem \\ref{convergencethm} will be carried out in Sections \\ref{convsec1}, \\ref{convsec2}, \\ref{convsec3}, \\ref{convsec4} and \\ref{convsec5}.\n\nTheorem \\ref{convergencethm} implies the following convergence result:\n\\begin{theorem}\\label{convergencethm2}\nSuppose we have a sequence of initial data that coincide on $\\Hb_0$ and satisfy the assumptions of Theorem \\ref{timeofexistence} with uniform constants $C$ and $c$ on $H_0$ and $\\Hb_0$. By Theorem \\ref{timeofexistence}, for every initial data in the sequence, a unique smooth solution to the vacuum Einstein equations $(\\mathcal M_n, g_n)$ exists in $0\\leq u\\leq u_*$, $0\\leq \\ub\\leq \\ub_*$ for $u_*, \\ub_*\\leq \\epsilon$ and the metric takes the following form in the coordinate system $(u,\\ub,\\th^1,\\th^2)$:\n$$g_n=-2(\\Omega_n)^2(du\\otimes d\\ub+d\\ub\\otimes du)+(\\gamma_n)_{AB}(d\\th^A-(b_n)^Adu)\\otimes (d\\th^B-(b_n)^Bdu),$$ \nwhere $\\Omega=1$ and $b^A=0$ on $H_0$ and $\\Hb_0$. Denote the Ricci coefficients and curvature components by $\\psi_n$ and $\\Psi_n$ respectively. Identify the spacetimes in the sequence by the value of the coordinate functions $(u,\\ub,\\th^1,\\th^2)$. Define also $g_n'=g_n-g_{n-1}$, $\\psi'_n=\\psi_n-\\psi_{n-1}$ and $\\Psi'_n=\\Psi_n-\\Psi_{n-1}$. If\n$$\\sup_u|(\\frac{\\partial}{\\partial\\th})^i(\\gamma_{AB})'_n(\\ub=0)|\\leq a_n,$$\n$$\\sum_{\\psi\\neq\\chih,\\omega}\\sum_{i\\leq 1} \\sup_{\\ub}||\\nabla^i\\psi'_n||_{L^2(S_{0,\\ub})}\\leq a_n,$$\n$$\\sum_{i\\leq 1} ||\\nabla^i(\\chih_n',\\omega_n')||_{L^{p_0}_{\\ub}L^2(S_{0,\\ub})}\\leq a_n\\quad\\mbox{for some fixed }2\\leq p_0<\\infty,$$\n$$||\\nabla^2(\\chih_n',\\omega_n',\\eta_n',\\etab_n')||_{L^2(H_0)}+\\sup_{\\ub} ||\\nabla^2(\\trch_n',\\trchb_n')||_{L^2(S_{0,\\ub})}\\leq a_n,$$\n$$\\sum_{i\\leq 1}\\left(\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}} ||\\nabla^i\\Psi_n'||_{L^{2}_{\\ub}L^2(S_{0,\\ub})}+\\sum_{\\Psi\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}} ||\\nabla^i\\Psi_n'||_{L^{2}_{u}L^2(S_{u,0})}\\right)\\leq a_n,$$\nfor some $a_n$ such that $\\displaystyle\\sum a_n<\\infty$,\nthen the spacetime metrics converge uniformly to a continuous limiting spacetime metric $g_\\infty$\n$$g_\\infty=-2(\\Omega_\\infty)^2(du\\otimes d\\ub+d\\ub\\otimes du)+ (\\gamma_{\\infty})_{AB}(d\\th^A-(b_\\infty)^Adu)\\otimes(d\\th^B-(b_\\infty)^Bdu)$$\nin the region $0\\leq u\\leq u_*$, $0\\leq \\ub\\leq \\ub_*$.\nMoreover, \n$$(\\frac{\\partial}{\\partial \\th}g_n,\\frac{\\partial}{\\partial u}g_n)\\mbox{ converge to }(\\frac{\\partial}{\\partial \\th}g_\\infty,\\frac{\\partial}{\\partial u}g_\\infty)\\mbox{ in }L^\\infty_u L^\\infty_{\\ub} L^4(S),$$\n$$(\\frac{\\partial^2}{\\partial \\th^2}g_n,\\frac{\\partial^2}{\\partial u\\partial\\th}g_n,\\frac{\\partial^2}{\\partial u^2}g_n)\\mbox{ converge to }(\\frac{\\partial^2}{\\partial \\th^2}g_\\infty,\\frac{\\partial^2}{\\partial u\\partial\\th}g_\\infty,\\frac{\\partial^2}{\\partial u^2}g_{\\infty})\\mbox{ in }L^\\infty_u L^\\infty_{\\ub} L^2(S),$$\n$$(\\frac{\\partial}{\\partial \\ub}g_n, \\frac{\\partial}{\\partial\\ub}((\\gamma_n^{-1})^{AB}\\frac{\\partial}{\\partial\\ub}(\\gamma_n)_{AB}))\\mbox{ converge to }(\\frac{\\partial}{\\partial \\ub}g_{\\infty}, \\frac{\\partial}{\\partial\\ub}((\\gamma_{\\infty}^{-1})^{AB}\\frac{\\partial}{\\partial\\ub}(\\gamma_{\\infty})_{AB}))\\mbox{ in }L^\\infty_u L^{p_0}_{\\ub} L^\\infty(S),$$\n$$(\\frac{\\partial^2}{\\partial \\th \\partial \\ub}g_n,\\frac{\\partial^2}{\\partial u\\partial\\ub}g_n,\\frac{\\partial^2}{\\partial\\ub^2}(b^A)_n)\\mbox{ converge to }(\\frac{\\partial^2}{\\partial \\th \\partial \\ub}g_\\infty,\\frac{\\partial^2}{\\partial u\\partial\\ub}g_\\infty,\\frac{\\partial^2}{\\partial\\ub^2}(b^A)_\\infty)\\mbox{ in }L^\\infty_u L^{p_0}_{\\ub} L^4(S).$$\nAs a consequence, in the limiting spacetime,\n$$\\frac{\\partial}{\\partial \\th}g_\\infty,\\frac{\\partial}{\\partial u}g_\\infty\\in C^0_u C^0_{\\ub} L^4(S),$$\n$$\\frac{\\partial^2}{\\partial \\th^2}g_\\infty,\\frac{\\partial^2}{\\partial u\\partial\\th}g_\\infty,\\frac{\\partial^2}{\\partial u^2}g_\\infty\\in C^0_u C^0_{\\ub} L^2(S),$$\n$$\\frac{\\partial}{\\partial \\ub}g_\\infty, \\frac{\\partial}{\\partial\\ub}((\\gamma_\\infty^{-1})^{AB}\\frac{\\partial}{\\partial\\ub}(\\gamma_\\infty)_{AB}) \\in L^\\infty_u L^\\infty_{\\ub} L^\\infty(S),$$\n$$\\frac{\\partial^2}{\\partial \\th \\partial \\ub}g_\\infty,\\frac{\\partial^2}{\\partial u\\partial\\ub}g_\\infty,\\frac{\\partial^2}{\\partial \\ub^2}(b^A)_\\infty\\in L^\\infty_u L^\\infty_{\\ub} L^4(S).$$\n\\end{theorem}\n\n\\begin{remark}\nNotice that in the case of initial data of the impulsive gravitational wave, we have $\\nabla^i\\chih_n$ converging in $L^p_{\\ub} L^2(S)$ for any $2\\leq p <\\infty$, but not for $p=\\infty$. We thus take a sequence of initial data that converges in a topology which is consistent with $\\nab^i\\chih_n$ converging in $L^p_{\\ub}L^2(S)$.\n\\end{remark}\n\nThe convergence of the Ricci coefficients and the acceptable curvature components follows from Theorem \\ref{convergencethm}. In Section \\ref{limit}, we will show that the convergence of the Ricci coefficients and the acceptable curvature components imply the asserted convergence of the metric and the regularity property of the limiting spacetime. The above convergence theorem is strong enough to show that the limiting spacetime is a solution to the Einstein equations:\n\\begin{theorem}\\label{Einstein}\nSuppose all the assumptions of Theorem \\ref{convergencethm2} hold. Then the limiting spacetime metric satisfies the Einstein equations in $L^\\infty_u L^\\infty_{\\ub}L^2(S)$.\n\\end{theorem}\nThis will also be proved in Section \\ref{limit}. Moreover, this limiting spacetime solution is the unique solution to the vacuum Einstein equations.\n\\begin{theorem}\\label{uniquenessthm}\nThe solution to the Einstein equations given by Theorems \\ref{convergencethm2} and \\ref{Einstein} is unique among spacetimes that arise as $C^0$ limit of smooth solutions to the vacuum Einstein equations.\n\\end{theorem}\nA more precise version of the uniqueness theorem is formulated as Proposition \\ref{uniquenessprop} and will be proved in Section \\ref{uniquenesssec}. Moreover, if the initial data is assumed to be more regular, the limit spacetime metric is more regular:\n\\begin{theorem}\\label{regularitythm}\nSuppose, in addition to the assumptions of Theorem \\ref{convergencethm2}, the bounds\n$$\\sum_{j\\leq J}\\sum_{i\\leq I}(\\sum_{\\Psi\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}}||\\nab_3^j\\nab^i\\Psi_n||_{L^2(\\Hb_0)}+\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}||\\nab_3^i \\nab^j\\Psi_n||_{L^2(H_0)})\\leq C $$\nhold uniformly independent of $n$.\nThen \n$$\\sum_{j\\leq J+2}\\sum_{i\\leq \\min\\{I,j-2\\}}(\\frac{\\partial}{\\partial u})^j(\\frac{\\partial}{\\partial \\th})^i g_\\infty\\in L^\\infty_u L^\\infty_{\\ub} L^2(S),$$\n$$\\sum_{j\\leq J+2}\\sum_{i\\leq \\min\\{I,j-2\\}}(\\frac{\\partial}{\\partial u})^j(\\frac{\\partial}{\\partial \\th})^i\\frac{\\partial}{ \\partial \\ub}g_\\infty\\in L^\\infty_u L^{p_0}_{\\ub} L^2(S).$$\n\\end{theorem}\nThis will be proved in Section \\ref{regularityp}. \n\nFor general initial data satisfying the assumptions of Theorem \\ref{rdthmv2}, using the construction in Section \\ref{initialcondition}, there exists an approximating sequence of smooth initial data satisfying the assumption of Theorem \\ref{convergencethm2}. Thus, the combination of Theorems \\ref{timeofexistence}, \\ref{convergencethm2}, \\ref{Einstein}, \\ref{regularitythm} and \\ref{uniquenessthm} together imply Theorem \\ref{rdthmv2}.\n\nThe remainder of this Section will be organized as follows: Theorem \\ref{convergencethm} is proved in Sections \\ref{convsec1}-\\ref{convsec5}. After the definition of the norms in Section \\ref{convsec1}, the proof is carried out in three steps:\\\\\n\n\\noindent{\\bf STEP 1} (Section \\ref{convsec2}): The difference of the metric components are estimated assuming the bounds for the difference of the Ricci coefficients.\\\\\n\n\\noindent{\\bf STEP 2} (Section \\ref{convsec4}): The difference of the Ricci coefficients are controlled assuming the estimates of the difference of the curvature components. This relies on the transport equations for the difference quantities derived in Section \\ref{convsec3}.\\\\\n\n\\noindent{\\bf STEP 3} (Section \\ref{convsec5}): Finally, the bounds for the difference of curvature components are obtained, closing all the estimates for Theorem \\ref{convergencethm}.\\\\\n\nIn Section \\ref{AddEstMetric}, additional estimates are derived for the metric components. These additional estimates will be used together with Theorem \\ref{convergencethm} to construct a limiting spacetime and to obtain Theorem \\ref{convergencethm2} in Section \\ref{limit}. In this Section, Theorem \\ref{Einstein} is also proved, showing that the limiting spacetime satisfies the Einstein equations. In Section \\ref{uniquenesssec}, we formulate and prove a precise version of Theorem \\ref{uniquenessthm}, establishing uniqueness of the limiting spacetime. In Section \\ref{regularityp}, Theorem \\ref{regularitythm} is proved, showing additional regularity in the spacetime with more regular initial data. Finally, in Section \\ref{limitgiw}, we return to the case of an impulsive gravitational wave and obtain extra regularity properties for these spacetimes.\n\n\\subsection{Norms}\\label{convsec1}\nWe begin the proof of Theorem \\ref{convergencethm}.\nDefine the following $L^2$ norms for the difference of the null curvature components or their renormalized versions:\n$$\\mathcal R'=\\sum_{i\\leq 1}\\sup_u||\\nabla^i(\\beta',\\rho',\\sigma',\\betab')||_{L^2(H_u)}+\\sum_{i\\leq 1}\\sup_{\\ub}||\\nabla^i(\\rhoc',\\sigmac',\\betab',\\alphab')||_{L^2(\\Hb_{\\ub})}$$\nDefine the following norm for difference of the the Ricci coefficients:\n$$\\mathcal O'=\\sum_{i\\leq 1}\\sup_{u}||\\nabla^i(\\hat{\\chi}',\\omega')||_{L^{p_0}_{\\ub}L^2(S_{u,\\ub})}+\\sum_{i\\leq 1}\\sup_{u,\\ub}||(\\nabla^i(tr\\chi',\\eta',\\underline{\\eta}',\\underline{\\omega}',\\underline{\\hat{\\chi}}',tr\\underline{\\chi}')||_{L^2(S_{u,\\ub})}.$$\nThese norms will be used to estimate the difference of the null Ricci coefficients except for those involving the highest derivatives, for which the estimates are weaker. We therefore also introduce the norms:\n\\begin{equation*}\n\\begin{split}\n\\tilde{\\mathcal O}'=\\sup_u ||\\nabla^2(\\chih',\\omega',\\eta',\\etab')||_{L^2(H_u)}+\\sup_{\\ub} ||\\nabla^2(\\chibh',\\omegab',\\eta',\\etab')||_{L^2(\\Hb_{\\ub})}+\\sup_{u,\\ub} ||\\nabla^2(\\trch',\\trchb')||_{L^2(S_{u,\\ub})},\n\\end{split}\n\\end{equation*}\nNotice that the norms $\\mathcal R'$, $\\mathcal O'$ and $\\tilde{\\mathcal O}'$ are difference counterparts of the norms $\\mathcal R$, $\\mathcal O$ and $\\tilde{\\mathcal O}$ for the difference quantities. Note, however, that the former provides control of one fewer derivatives than the latter. This is due to the fact that convergence will be proved in a norm which is one derivative weaker than the corresponding norms for the a priori estimates. Define also the following norms:\n\\begin{equation*}\n\\begin{split}\n\\mathcal O''=\\sup_u(||(\\chih,\\omega,\\eta,\\underline{\\eta},\\trch,\\trchb)'||_{L^2_{\\ub}L^\\infty(S_{u,\\ub})}+||\\nabla(\\chih,\\omega,\\eta,\\underline{\\eta},\\trch,\\trchb)'||_{L^2_{\\ub}L^4(S_{u,\\ub})}).\n\\end{split}\n\\end{equation*}\nIt follows by Sobolev Embedding in Propositions \\ref{L4} and \\ref{Linfty} that\n\\begin{proposition}\\label{OpSobolev}\n$$\\mathcal O''\\leq C(\\tilde{\\mathcal O}'+\\mathcal O').$$\n\\end{proposition}\n\n\\subsection{Estimates for the Difference of the Metrics}\\label{convsec2}\nIn this section, we show that the difference of the metrics and their coordinate angular derivatives can be controlled by the $\\mathcal O'$ and $\\tilde{\\mathcal O}'$ norms. Recall that the metrics take the form\n$$g^{(i)}=-2(\\Omega^{(i)})^2(du\\otimes d\\ub+d\\ub\\otimes du)+(\\gamma^{(i)})_{AB}(d\\th^A-(b^{(i)})^Adu)\\otimes (d\\th^B-(b^{(i)})^Bdu),$$ \nfor $i=1,2$.\n\\begin{proposition}\\label{Omegap}\n\\[\n\\sup_{u,\\ub}|(\\Omega',(\\Omega^{-1})')(u,\\ub)|\\leq C\\delta^{\\frac 12}\\mathcal O''.\n\\]\n\\end{proposition}\n\\begin{proof}\nRecall that\n\\[\n \\omega=-\\frac{1}{2}\\nabla_4\\log\\Omega=\\frac{1}{2}\\Omega\\nabla_4\\Omega^{-1}=\\frac{1}{2}\\frac{\\partial}{\\partial \\ub}\\Omega^{-1}.\n\\]\nHence \n\\[\n \\frac{1}{2}\\frac{\\partial}{\\partial\\ub}(\\Omega^{-1})'=\\omega'.\n\\]\nBy integrating along the $\\ub$ direction, noticing that $\\Omega'=0$ on $\\Hb_0$, and using Cauchy-Schwarz we get\n$$|(\\Omega^{-1})'|\\leq C\\delta^{\\frac 12}\\mathcal O''.$$\nIn order to get the estimate for $\\Omega'$, we note that\n\\[\n(\\Omega'+\\Omega^{(2)})^{-1}=(\\Omega^{(1)})^{-1}=(\\Omega^{-1})^{(2)}+(\\Omega^{-1})'.\n\\]\nTherefore,\n\\[\n \\Omega'=((\\Omega^{(2)})^{-1}+(\\Omega)^{-1})')^{-1}-\\Omega^{(2)}=\\frac{(\\Omega^{-1})'}{1+\\frac{(\\Omega^{-1})'}{(\\Omega^{-1})^{(2)}}}.\n\\]\nIn view of the upper and lower bounds of $\\Omega$ in Proposition \\ref{Omega},\n$$|\\Omega'|\\leq C\\delta^{\\frac 12}\\mathcal O''.$$\n\n\\end{proof}\nUsing the estimates for $\\Omega'$, we also have estimates for $\\gamma'$.\n\\begin{proposition}\\label{gammap}\n$\\gamma'$ satisfies the following pointwise bounds:\n$$\\sup_{u,\\ub}|(\\gamma_{AB}',((\\gamma^{-1})^{AB})')(u,\\ub)|\\leq a+C\\delta^{\\frac 12}\\mathcal O''_u.$$\n\\end{proposition}\n\\begin{proof}\nThe components of $\\gamma$ solve the following ODE:\n$$\\frac{\\partial}{\\partial \\ub}\\gamma_{AB}=2\\Omega\\chi_{AB}.$$\nThis implies\n$$\\frac{\\partial}{\\partial \\ub}\\log(\\det\\gamma)=\\Omega\\trch.$$\nFrom this we can derive an equation for $(\\det\\gamma)'$:\n$$\\frac{\\partial}{\\partial \\ub}(\\det\\gamma)'=\\frac{1}{(\\det\\gamma)^{(2)}}\\left(-(\\det\\gamma)'\\frac{\\partial}{\\partial \\ub}(\\det\\gamma)^{(1)}+(\\det\\gamma)^{(1)}(\\det\\gamma)^{(2)}(\\Omega\\trch)'\\right). $$\nBy Proposition \\ref{gamma}, we have uniform upper and lower bounds on $\\det\\gamma$ and uniform estimates for $\\Omega$ and $\\frac{\\partial}{\\partial \\ub}(\\det\\gamma)$. The previous Proposition gives $|\\Omega'|\\leq C\\delta^{\\frac 12}\\mathcal O''$. Moreover, $\\int |\\trch'| d\\ub'\\leq \\delta\\mathcal O''$ by definition. Thus,\n\\begin{equation}\\label{volumeformd}\n|(\\det\\gamma)'|\\leq C\\delta^{\\frac 12}\\mathcal O''.\n\\end{equation}\nWe can also derive an equation for $(\\gamma_{AB})'$:\n$$\\frac{\\partial}{\\partial \\ub}(\\gamma_{AB})'=2(\\Omega\\chi_{AB})'.$$\nThus\n$$|(\\gamma_{AB})'(u,\\ub)|\\leq |(\\gamma_{AB})'(u,0)|+C\\int_0^{\\ub}|(\\chi_{AB})'|d\\ub'+C\\delta^{\\frac 12}\\mathcal O'' |\\chi_{AB}|.$$\nBy assumption, $|(\\gamma_{AB})'(u,0)|\\leq a$. By Proposition \\ref{gamma}, matrices $\\gamma^{(1)}$ and $\\gamma^{(2)}$ are uniformly non-degenerate matrices. Hence, regardless of whether we define the $L^\\infty$ norm with respect to $\\gamma^{(1)}$ or $\\gamma^{(2)}$, we have\n$$\\sum_{A,B=1,2}|(\\chih_{AB})'|\\leq C\\sup_{u,\\ub}||\\chi'||_{L^\\infty(S_{u,\\ub})}.$$\nTherefore, \n\\begin{equation}\\label{metricd}\n|(\\gamma_{AB})'|\\leq a+C\\delta^{\\frac 12}\\mathcal O''.\n\\end{equation}\nNow, (\\ref{volumeformd}) and (\\ref{metricd}) together also imply the pointwise bound in coordinates for \n$$|((\\gamma^{-1})^{AB})'|\\leq a+C\\delta^{\\frac 12}\\mathcal O''.$$\n\\end{proof}\nThis estimate allows us to conclude that the $L^p$ norms defined with respect to either metric $(1)$ or $(2)$ differ only by $a+C\\delta^{\\frac 12}\\mathcal O''$.\n\\begin{proposition}\\label{normscomparable}\nGiven any tensor, we can define its $L^p(S)$ norms $||\\phi||^{(1)}_{L^p(S_{u,\\ub})}$ and $||\\phi||^{(2)}_{L^p(S_{u,\\ub})}$ with respect to the first and second metric. Suppose\n$$||\\phi||^{(1)}_{L^p(S_{u,\\ub})}<\\infty.$$\nThen\n$$|||\\phi||^{(1)}_{L^p(S_{u,\\ub})}-||\\phi||^{(2)}_{L^p(S_{u,\\ub})}|\\leq C(a+\\delta^{\\frac 12}\\mathcal O'')||\\phi||^{(1)}_{L^p(S_{u,\\ub})}.$$\nIn particular, $\\phi$ is also in $L^p$ with respect to the second metric.\n\\end{proposition}\n\\begin{proof}\nThis follows from the pointwise control of $(\\gamma_{AB})'$, $((\\gamma^{-1})^{AB})'$ and $(\\det\\gamma)'$ in coordinates.\n\\end{proof}\nIn a similar manner, we can control the $(\\frac{\\partial}{\\partial\\th^A})$ derivatives of $\\gamma'$.\n\\begin{proposition}\\label{dgammap}\n$$\\sup_{u,\\ub}||(\\frac{\\partial}{\\partial\\th^C})\\gamma_{AB}'||_{L^4(S_{u,\\ub})}\\leq Ca+C\\delta^{\\frac 12}\\mathcal O'',$$\nwhere $L^4(S)$ is understood as the $L^4$ norm for a scalar function and by Proposition \\ref{normscomparable} can be defined with respect either to the metric $\\gamma^{(1)}$ or $\\gamma^{(2)}$.\n\\end{proposition}\n\\begin{proof}\nBy the equation \n$$\\frac{\\partial}{\\partial \\ub}\\gamma_{AB}=2\\Omega\\chi_{AB}$$\nand the assumption, we have\n\\begin{equation}\\label{gammadd}\n\\begin{split}\n||(\\frac{\\partial}{\\partial\\th^C})^i\\gamma_{AB}'(u,\\ub)||_{L^4(S_{u,\\ub'})}\\leq& Ca+C\\int_0^{\\ub}\\left(\\|(\\frac{\\partial}{\\partial\\th^C})^i\\Omega'||_{L^4(S_{u,\\ub'})}+\\|(\\frac{\\partial}{\\partial\\th^C})^i\\chi'||_{L^4(S_{u,\\ub'})}\\right) d\\ub'.\n\\end{split}\n\\end{equation}\nSince\n$$\\nabla(\\log \\Omega)=\\frac 12 (\\eta+\\etab),$$\nwe have\n$$|(\\frac{\\partial}{\\partial\\th^C})\\Omega'|\\leq C|(\\eta,\\etab)'|+C|\\Gamma'||\\Omega|+C|\\Gamma||\\Omega'| \\leq C|(\\eta,\\etab)'|+C|\\frac{\\partial}{\\partial\\th^D}\\gamma_{AB}'|+C|\\Omega'|,$$\nwhere $\\Gamma$ is the connection coefficients on the spheres with respect to $\\gamma$.\nMoreover, by Cauchy-Schwarz,\n$$\\int ||(\\frac{\\partial}{\\partial\\th^C})\\chi'||_{L^4(S_{u,\\ub'})} d\\ub'\\leq C\\delta^{\\frac 12}\\mathcal O''.$$\nThus by (\\ref{gammadd}), we have\n\\begin{equation*}\n\\begin{split}\n&||(\\frac{\\partial}{\\partial\\th^C})\\gamma_{AB}'(u,\\ub)||_{L^4(S_{u,\\ub})}\\\\\n\\leq&Ca+C\\delta^{\\frac 12}\\mathcal O''+C\\int_0^{\\ub} ||\\eta',\\etab',\\chih',\\frac{\\partial}{\\partial\\th^D}\\gamma_{AB}',\\Omega'||_{L^4(S_{u,\\ub'})}d\\ub'.\n\\end{split}\n\\end{equation*}\nUsing already established estimates, Gronwall's inequality and Cauchy-Schwarz, we thus have\n$$||(\\frac{\\partial}{\\partial\\th^C})\\gamma_{AB}'(u,\\ub)||_{L^4(S_{u,\\ub})}\\leq Ca+C\\delta^{\\frac 12}\\mathcal O''.$$\n\\end{proof}\nA consequence of the above Proposition is the estimates on the difference of the connection coefficients:\n\\begin{proposition}\\label{Gammap}\n$$\\sup_{u,\\ub}||\\Gamma'||_{L^4(S_{u,\\ub})}\\leq Ca+C\\delta^{\\frac 12}\\mathcal O''.$$\n\\end{proposition}\n\\begin{proof}\nThis follows from the fact that $\\Gamma'$ can be expressed as a linear combination $\\gamma'$ and its first angular derivative.\n\\end{proof}\nThis implies that $\\nabla^{(1)}$ and $\\nabla^{(2)}$ are comparable in the following sense:\n\\begin{proposition}\\label{connectionpL4}\nLet $p\\leq 4$. Suppose $\\phi$ is a tensor. \nThen\n$$||(\\nabla^{(1)})\\phi-(\\nabla^{(2)})\\phi||_{L^p(S_{u,\\ub})} \\leq C(a+\\delta^{\\frac 12}\\mathcal O'')||\\phi||_{L^\\infty(S_{u,\\ub})}.$$\n\\end{proposition}\n\\begin{proof}\nSince we have estimates for $\\gamma_{AB}$ and $(\\gamma^{-1})^{AB}$. It suffice to estimate \n$$(\\nabla^{(1)})_{\\frac{\\partial}{\\partial\\th^A}}\\phi-(\\nabla^{(2)})_{\\frac{\\partial}{\\partial\\th^A}}\\phi.$$\nUsing Proposition \\ref{Gammap}, we have\n\\begin{equation*}\n\\begin{split}\n&||(\\nabla^{(1)})_{\\frac{\\partial}{\\partial\\th^A}}\\phi-(\\nabla^{(2)})_{\\frac{\\partial}{\\partial\\th^A}}\\phi||_{L^p(S_{u,\\ub})}\\\\\n\\leq &||\\Gamma^{(1)}\\phi-\\Gamma^{(2)}\\phi||_{L^p(S_{u,\\ub})}\\\\\n\\leq &C||\\Gamma'||_{L^4(S_{u,\\ub})}||\\phi||_{L^\\infty(S_{u,\\ub})}\\\\\n\\leq &C(a+\\delta^{\\frac 12}\\mathcal O'')||\\phi||_{L^\\infty(S_{u,\\ub})}.\n\\end{split}\n\\end{equation*}\n\\end{proof}\nWe now show that the difference between the angular covariant derivatives of two tensors is comparable to the angular covariant derivative of the difference:\n\\begin{proposition}\\label{angularpL4}\nLet $p\\leq 4$. Suppose $\\phi^{(1)}$ and $\\phi^{(2)}$ are tensors defined on spacetimes $(1)$ and $(2)$ respectively.\nThen\n$$||\\nabla^{(1)}(\\phi')-(\\nabla\\phi)'||_{L^p(S_{u,\\ub})}\\leq C(a+\\delta^{\\frac 12}\\mathcal O'')||\\phi||_{L^\\infty(S_{u,\\ub})}.$$\n\\end{proposition}\n\\begin{proof}\nIt is easy to see that\n$$\\nabla^{(1)}(\\phi')-(\\nabla\\phi)'=-\\nabla^{(1)}\\phi^{(2)}+\\nabla^{(2)}\\phi^{(2)}.$$\nThe Proposition thus follows from Proposition \\ref{connectionpL4}.\n\\end{proof}\nWe now estimate the coordinate angular derivative of $\\Omega'$ in $L^4(S)$:\n\\begin{proposition}\\label{dOmegap}\nWe have\n\\[\n\\sup_{u,\\ub}||\\frac{\\partial}{\\partial \\th^A}\\Omega'||_{L^4(S_{u,\\ub})}\\leq Ca+C\\delta^{\\frac 12}\\mathcal O''.\n\\]\n\\end{proposition}\n\\begin{proof}\nRecall that\n\\[\n \\omega=-\\frac{1}{2}\\nabla_4\\log\\Omega=\\frac{1}{2}\\Omega\\nabla_4\\Omega^{-1}=\\frac{1}{2}\\frac{\\partial}{\\partial \\ub}\\Omega^{-1}.\n\\]\nHence \n\\[\n \\frac{1}{2}\\frac{\\partial}{\\partial\\ub}(\\Omega^{-1})'=\\omega'.\n\\]\nThus, in order to estimate $\\frac{\\partial}{\\partial \\th^A}\\Omega'$, we need to estimate $\\frac{\\partial}{\\partial \\th^A}\\omega'$ in $L^1_{\\ub}L^4(S)$. Since by the a priori estimates in the previous section we have $||\\nabla\\omega||_{L^\\infty(S)}\\leq C$, it suffices by Proposition \\ref{angularpL4} to estimate \n$$\\sum_{i\\leq 1}\\int_0^{\\ub}||\\nabla^i\\omega'||_{L^4(S_{u,\\ub'})}d\\ub'$$\nBy the definition of the norm and Cauchy-Schwarz\n$$\\sum_{i\\leq 1}\\int_0^{\\ub}||\\nabla^i\\omega'||_{L^4(S_{u,\\ub'})}d\\ub'\\leq C\\delta^{\\frac 12}\\mathcal O''.$$\nThe conclusion thus follows.\n\\end{proof}\nThe above Proposition allows us to estimate the second coordinate angular derivatives of $\\gamma'$ in $L^2(S)$.\n\\begin{proposition}\\label{ddgammap}\n$$\\sup_{u,\\ub}||(\\frac{\\partial^2}{\\partial\\th^C \\partial\\th^D})\\gamma_{AB}'||_{L^2(S_{u,\\ub})}\\leq Ca+C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}'),$$\nwhere $L^2(S)$ is understood as the $L^2$ norms for scalar functions.\n\\end{proposition}\n\\begin{proof}\nBy the equation \n$$\\frac{\\partial}{\\partial \\ub}\\gamma_{AB}=2\\Omega\\chi_{AB},$$\nand the a priori estimates in the previous section, we have\n\\begin{equation}\\label{gamma2dd}\n\\begin{split}\n&||(\\frac{\\partial^2}{\\partial\\th^C\\partial\\th^D})\\gamma_{AB}'(u,\\ub)||_{L^2(S_{u,\\ub})}\\\\\n\\leq& Ca+C\\int_0^{\\ub}\\left(\\|(\\frac{\\partial^2}{\\partial\\th^C\\partial\\th^D})\\Omega'||_{L^2(S_{u,\\ub'})}+\\|(\\frac{\\partial^2}{\\partial\\th^C\\partial\\th^D})\\chi'||_{L^2(S_{u,\\ub'})}\\right) d\\ub'\\\\\n&+ C\\int_0^{\\ub}\\left(\\|(\\frac{\\partial}{\\partial\\th^C})\\Omega'||_{L^2(S_{u,\\ub'})}+\\|(\\frac{\\partial}{\\partial\\th^C})\\chi'||_{L^2(S_{u,\\ub'})}\\right) d\\ub'.\n\\end{split}\n\\end{equation}\nThe last term has already been estimated in the proof of Proposition \\ref{dgammap}:\n\\begin{equation*}\n\\begin{split}\n&\\int_0^{\\ub}\\left(\\|(\\frac{\\partial}{\\partial\\th^C})\\Omega'||_{L^2(S_{u,\\ub'})}+\\|(\\frac{\\partial}{\\partial\\th^C})\\chi'||_{L^2(S_{u,\\ub'})}\\right) d\\ub'\\\\\n\\leq& Ca+C\\delta^{\\frac 12}\\mathcal O''.\n\\end{split}\n\\end{equation*}\nSince\n$$\\nabla(\\log \\Omega)=\\frac 12 (\\eta+\\etab),$$\nwe have\n\\begin{equation*}\n\\begin{split}\n&||(\\frac{\\partial^2}{\\partial\\th^C\\partial\\th^D})\\Omega'||_{L^2(S_{u,\\ub'})}\\\\\n\\leq& C||\\frac{\\partial}{\\partial\\th}(\\eta,\\etab)'||_{L^2(S_{u,\\ub'})}+C||\\frac{\\partial}{\\partial\\th}\\Gamma'||_{L^2(S_{u,\\ub'})}||\\Omega||_{L^\\infty(S_{u,\\ub'})}+C||\\frac{\\partial}{\\partial\\th}\\Gamma||_{L^2(S_{u,\\ub'})}||\\Omega'|||_{L^\\infty(S_{u,\\ub'})}\\\\\n& +C||\\Gamma'||_{L^4(S_{u,\\ub'})}||\\frac{\\partial}{\\partial\\th}\\Omega||_{L^4(S_{u,\\ub'})}+C||\\Gamma||_{L^\\infty(S_{u,\\ub'})}||\\frac{\\partial}{\\partial\\th}\\Omega'||_{L^2(S_{u,\\ub'})}\\\\\n\\leq &C\\sum_{i\\leq 1}||\\nabla^i(\\eta,\\etab)'||_{L^2(S_{u,\\ub'})}+C\\sum_{i\\leq 1}||(\\frac{\\partial}{\\partial\\th})^i\\Gamma'||_{L^2(S_{u,\\ub'})}+C\\sum_{i\\leq 1}||(\\frac{\\partial}{\\partial\\th})^i\\Omega'||_{L^2(S_{u,\\ub'})}.\n\\end{split}\n\\end{equation*}\nUsing Propositions \\ref{dOmegap} and the definition of the norms $\\mathcal O''$ and $\\tilde{\\mathcal O}'$, we have\n\\begin{equation*}\n\\begin{split}\n&\\sum_{i\\leq 1}\\int_0^{\\ub}(||\\nabla^i(\\eta,\\etab)'||_{L^2(S_{u,\\ub'})}+||(\\frac{\\partial}{\\partial\\th})^i\\Gamma'||_{L^2(S_{u,\\ub'})}+||(\\frac{\\partial}{\\partial\\th})^i\\Omega'||_{L^2(S_{u,\\ub'})})d\\ub'\\\\\n\\leq &C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}'_u)+C\\int_0^{\\ub}||(\\frac{\\partial^2}{\\partial\\th^E\\partial\\th^F})\\gamma_{AB}'(u,\\ub)||_{L^2(S_{u,\\ub'})}d\\ub').\n\\end{split}\n\\end{equation*}\nMoreover,\n$$\\int ||(\\frac{\\partial^2}{\\partial\\th^C\\partial\\th^D})\\chi'||_{L^2(S_{u,\\ub'})} d\\ub'\\leq C(\\mathcal O''+\\tilde{\\mathcal O}').$$\nThus by (\\ref{gamma2dd}), we have\n\\begin{equation*}\n\\begin{split}\n&||(\\frac{\\partial^2}{\\partial\\th^C\\partial\\th^D})\\gamma_{AB}'(u,\\ub)||_{L^2(S_{u,\\ub})}\\\\\n\\leq&Ca+C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}')+C\\int_0^{\\ub}||(\\frac{\\partial^2}{\\partial\\th^E\\partial\\th^F})\\gamma_{AB}'(u,\\ub)||_{L^2(S_{u,\\ub'})}d\\ub').\n\\end{split}\n\\end{equation*}\nBy Gronwall's inequality, we thus have\n$$||(\\frac{\\partial^2}{\\partial\\th^C \\partial\\th^D})\\gamma_{AB}'||_{L^2(S_{u,\\ub})}\\leq Ca+C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}').$$\n\\end{proof}\nThe proof of the previous Proposition also shows that\n\\begin{proposition}\\label{ddOmegap}\n$$\\sup_{u,\\ub}||(\\frac{\\partial^2}{\\partial\\th^C \\partial\\th^D})\\Omega'||_{L^2(S_{u,\\ub})}\\leq Ca+C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}'),$$\n\\end{proposition}\nProposition \\ref{ddgammap} implies the following estimates of $K'$, the difference of the Gauss curvatures:\n\\begin{proposition}\\label{Kp}\n$K'$ satisfies the following bounds:\n$$\\sup_{u,\\ub}||K'||_{L^2(S_{u,\\ub})}\\leq Ca+C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}').$$\n\\end{proposition}\n\\begin{proof}\nThis follows from the estimates of the derivatives of $\\gamma'$ in Propositions \\ref{dgammap} and \\ref{ddgammap}.\n\\end{proof}\nMoreover, Proposition \\ref{ddgammap} allows us to conclude that the covariant second derivative with respect to connection $(1)$ and connection $(2)$ are comparable:\n\\begin{proposition}\\label{connectionpL2}\nLet $p\\leq 2$. Suppose $\\phi$ is a tensor. \nThen\n$$||(\\nabla^{(1)})^2\\phi-(\\nabla^{(2)})^2\\phi||_{L^p(S_{u,\\ub})} \\leq C(a+\\delta^{\\frac 12}\\mathcal O'')\\sum_{i\\leq 1}||\\nabla^i\\phi||_{L^4(S_{u,\\ub})}.$$\n\\end{proposition}\n\\begin{proof}\nBy Propositions \\ref{Gammap}, \\ref{ddgammap} and \\ref{Kp}.\n\\end{proof}\nThis implies that taking the covariant second derivatives of the difference is comparable to taking the difference of the covariant second derivatives:\n\\begin{proposition}\\label{angularpL2}\nLet $p\\leq 2$. Suppose $\\phi^{(1)}$ and $\\phi^{(2)}$ are tensors defined on spacetimes $(1)$ and $(2)$ respectively.\nThen\n$$||(\\nabla^{(1)})^2(\\phi')-(\\nabla^2\\phi)'||_{L^p(S_{u,\\ub})}\\leq C(a+\\delta^{\\frac 12}\\mathcal O'')\\sum_{i\\leq 1}||\\nabla^i\\phi||_{L^4(S_{u,\\ub})}.$$\n\\end{proposition}\n\\begin{proof}\nThe proof follows as that of Proposition \\ref{angularpL4}, using Proposition \\ref{connectionpL2} instead of Proposition \\ref{connectionpL4}.\n\\end{proof}\nWe now prove the estimates for $b'$:\n\\begin{proposition}\\label{bp}\n\\[\n\\sup_{u,\\ub}||(b^A)'||_{L^\\infty(S_{u,\\ub})}, \\sup_{u,\\ub}||\\frac{\\partial}{\\partial\\th^B}(b^A)'||_{L^4(S_{u,\\ub})},\\sup_{u,\\ub}||\\frac{\\partial^2}{\\partial \\th^C\\partial\\th^B}(b^A)'||_{L^2(S_{u,\\ub})}\\leq C(\\mathcal O''+\\tilde{\\mathcal O}').\n\\]\n\\end{proposition}\n\\begin{proof}\nRecall that $b$ satisfies the transport equation \n$$\\frac{\\partial b^A}{\\partial \\ub}=-4\\Omega^2\\zeta^A.$$\nThe Proposition thus follows from differentiating the equation by $\\frac{\\partial}{\\partial\\th}$, integrating in $\\ub$ and using the the bounds in Propositions \\ref{Omegap}, \\ref{dOmegap} and \\ref{ddOmegap} and the definitions of the norms.\n\\end{proof}\n\n\n\n\\subsection{Transport Equations for the Difference Quantities}\\label{convsec3}\nIn this Subsection, we show how to derive estimates for the difference $\\phi'$ of the quantities $\\phi^{(1)}$ and $\\phi^{(2)}$ satisfying transport equations in the $\\nab_4^{(1)}$, $\\nab_4^{(2)}$ or $\\nab_3^{(1)}$, $\\nab_3^{(2)}$ directions. This should be compared with the previous Subsection where we considered difference quantities satisfying transport equations, but unlike in this Subsection, the corresponding transport equations held with respect to the $\\frac{\\partial}{\\partial \\ub}$, $\\frac{\\partial}{\\partial u}$ derivatives rather than $\\nab_4$, $\\nab_3$, which themselves depend on the spacetimes under consideration.\n\nIn order to estimate the quantities $\\phi'$ associated to the difference between that of different spacetimes, we need to derive a transport equation for $\\phi'$ from the transport equation for $\\phi$. \n\\begin{proposition}\\label{peqn}\nConsider spacetimes that satisfy the hypotheses of Theorem \\ref{timeofexistence} and \\ref{convergencethm}. Let $\\phi$ be a $(0,r)$ S-tensor. Suppose $\\nabla_4\\phi=F$. Then\n\\begin{equation*}\n\\begin{split}\n\\nabla_4^{(1)}\\phi'\\sim & F'+\\frac{(\\Omega^{-1})'}{\\Omega^{-1}}\\nabla_4\\phi+\\frac{(\\Omega^{-1})'}{\\Omega^{-1}}\\gamma^{-1}\\chi\\phi+(\\gamma^{-1}\\chi)'\\phi.\n\\end{split}\n\\end{equation*}\nSimilarly, suppose $\\nabla_3\\phi=G$. Then\n\\begin{equation*}\n\\begin{split}\n\\nabla_3^{(1)}\\phi'\\sim & G'+\\frac{(\\Omega^{-1})'}{\\Omega^{-1}}\\nabla_3\\phi+\\Omega^{-1}(b^A)'\\nabla_{\\frac{\\partial}{\\partial \\th^A}}\\phi+\\frac{(\\Omega^{-1})'}{\\Omega^{-1}}\\gamma^{-1}\\chib\\phi+(\\gamma^{-1}\\chib)'\\phi+\\Omega^{-1}\\frac{\\partial b'}{\\partial\\th}\\phi.\n\\end{split}\n\\end{equation*}\n\\end{proposition}\n\\begin{proof}\nWe write in coordinates\n$$\\nabla_4\\phi_{A_1...A_r}=\\Omega^{-1}\\frac{\\partial}{\\partial \\ub}\\phi_{A_1...A_r}-(\\gamma^{-1})^{CD}\\chi_{A_iD}\\phi_{A_1...\\hat{A_i}C...A_r},$$\nwhere $\\hat{A_i}$ denotes that the original $A_i$ in the $i$-th slot of the tensor is removed. This equation holds in both spacetimes $(1)$ and $(2)$.\nThen, we derive the equations for $\\phi'$. We will write schematically without the exact constant depending only on $r$:\n\\begin{equation*}\n\\begin{split}\n&(\\nabla_4)^{(1)}\\phi' \\\\\n=&\\left((\\Omega^{-1})^{(1)}\\frac{\\partial}{\\partial \\ub}-(\\gamma^{-1})^{(1)}\\chi^{(1)}\\right)\\left(\\phi^{(1)}-\\phi^{(2)}\\right) \\\\\n=&\\left((\\Omega^{-1})^{(1)}\\frac{\\partial}{\\partial \\ub}-(\\gamma^{-1})^{(1)}\\chi^{(1)}\\right)\\phi^{(1)}-\\left((\\Omega^{-1})^{(2)}\\frac{\\partial}{\\partial \\ub}-(\\gamma^{-1})^{(2)}\\chi^{(2)}\\right)\\phi^{(2)}\\\\\n& -(\\Omega^{-1})'\\frac{\\partial}{\\partial \\ub}\\phi^{(2)}+(\\gamma^{-1}\\chi)'\\phi^{(2)}\\\\\n=&F'-(\\Omega^{-1})'\\frac{\\partial}{\\partial \\ub}\\phi^{(2)}+(\\gamma^{-1}\\chi)'\\phi^{(2)}\\\\\n=&F'-\\frac{(\\Omega^{-1})'}{(\\Omega^{-1})^{(1)}}(\\nabla_4)^{(1)}\\phi^{(2)}+\\frac{(\\Omega^{-1})'}{(\\Omega^{-1})^{(1)}}(\\gamma^{-1}\\chi)^{(1)}\\phi^{(2)}+(\\gamma^{-1}\\chi)'\\phi^{(2)}.\n\\end{split}\n\\end{equation*}\nFor the $\\nabla_3$ equations, we write in coordinates\n\\begin{equation*}\n\\begin{split}\n\\nabla_3\\phi_{A_1...A_r}=&\\Omega^{-1}\\frac{\\partial}{\\partial u}\\phi_{A_1...A_r}+\\Omega^{-1}b^B\\frac{\\partial}{\\partial \\th^B}\\phi_{A_1...A_r} \\\\\n&-(\\gamma^{-1})^{CD}\\chib_{A_i D}\\phi_{A_1...\\hat{A_i}C...A_r}+\\Omega^{-1}\\frac{\\partial b^B}{\\partial\\th^{A_i}}\\phi_{A_1...\\hat{A_i}C...A_r}.\n\\end{split}\n\\end{equation*}\nWe derive the difference equations as before. However, $e_3$ in the respective spacetimes are not parallel to each other and we therefore have extra terms.\n\\begin{equation*}\n\\begin{split}\n&(\\nabla_3)^{(1)}\\phi' \\\\\n=&\\left((\\Omega^{-1})^{(1)}\\frac{\\partial}{\\partial u}+(\\Omega^{-1})^{(1)}b^{(1)}\\frac{\\partial}{\\partial \\th} -(\\gamma^{-1})^{(1)}\\chib^{(1)}+(\\Omega^{-1})^{(1)}\\frac{\\partial b^{(1)}}{\\partial \\th}\\right)\\left(\\phi^{(1)}-\\phi^{(2)}\\right) \\\\\n=&\\left((\\Omega^{-1})^{(1)}\\frac{\\partial}{\\partial u}+(\\Omega^{-1})^{(1)}b^{(1)}\\frac{\\partial}{\\partial \\th} -(\\gamma^{-1})^{(1)}\\chib^{(1)}+(\\Omega^{-1})^{(1)}\\frac{\\partial b^{(1)}}{\\partial \\th}\\right)\\phi^{(1)} \\\\\n&-\\left((\\Omega^{-1})^{(2)}\\frac{\\partial}{\\partial u}+(\\Omega^{-1})^{(2)}b^{(2)}\\frac{\\partial}{\\partial \\th} -(\\gamma^{-1})^{(2)}\\chib^{(2)}+(\\Omega^{-1})^{(2)}\\frac{\\partial b^{(2)}}{\\partial \\th}\\right)\\phi^{(2)}\\\\\n& -(\\Omega^{-1})'\\frac{\\partial}{\\partial u}\\phi^{(2)}+(\\gamma^{-1}\\chib)'\\phi^{(2)}-(\\Omega^{-1}b)'\\frac{\\partial}{\\partial \\th}\\phi^{(2)}-(\\Omega^{-1}\\frac{\\partial b}{\\partial \\th})'\\phi^{(2)}\\\\\n=&G'-\\frac{(\\Omega^{-1})'}{(\\Omega^{-1})^{(1)}}\\nabla_3^{(1)}\\phi^{(1)}-(\\Omega^{-1})^{(2)}b'\\frac{\\partial}{\\partial\\th}\\phi^{(2)}-(\\Omega^{-1})^{(2)}\\frac{\\partial b'}{\\partial \\th}\\phi^{(2)}\\\\\n&+\\frac{(\\Omega^{-1})'}{(\\Omega^{-1})^{(1)}}(\\gamma^{-1}\\chib)^{(1)}\\phi^{(1)}+(\\gamma^{-1}\\chib)'\\phi^{(2)}.\n\\end{split}\n\\end{equation*}\n\\end{proof}\nThis, together with the commutation estimates in Proposition \\ref{commuteeqn}, gives the following estimates for $(\\nabla^i\\phi)'$.\n\\begin{proposition}\\label{pcommuteeqn}\nSuppose $\\nabla_4\\phi=F$. Then\n\\begin{equation*}\n\\begin{split}\n\\nabla_4(\\nabla^i\\phi)'\\sim &\\sum_{i_1+i_2+i_3=i}(\\nabla^{i_1}(\\eta,\\etab)^{i_2}\\nabla^{i_3} F)'+\\sum_{i_1+i_2+i_3+i_4=i}(\\nabla^{i_1}(\\eta,\\etab)^{i_2}\\nabla^{i_3}\\chi\\nabla^{i_4} \\phi)' \\\\\n&+(\\Omega^{-1})'\\left(\\sum_{i_1+i_2+i_3=i}\\nabla^{i_1}(\\eta,\\etab)^{i_2}\\nabla^{i_3} F+\\sum_{i_1+i_2+i_3+i_4=i}\\nabla^{i_1}(\\eta,\\etab)^{i_2}\\nabla^{i_3}\\chi\\nabla^{i_4} \\phi\\right)\\\\\n&+(\\gamma^{-1}\\chi)'\\nabla^i\\phi.\n\\end{split}\n\\end{equation*}\nSimilarly, suppose $\\nabla_3\\phi=G$. Then\n\\begin{equation*}\n\\begin{split}\n\\nabla_3(\\nabla^i\\phi)'\\sim &\\sum_{i_1+i_2+i_3=i}(\\nabla^{i_1}(\\eta,\\etab\n)^{i_2}\\nabla^{i_3} G)'+\\sum_{i_1+i_2+i_3+i_4=i}(\\nabla^{i_1}(\\eta,\\etab)^{i_2}\\nabla^{i_3}\\chib\\nabla^{i_4} \\phi)' \\\\\n&+(\\Omega^{-1})'\\left(\\sum_{i_1+i_2+i_3=i}\\nabla^{i_1}(\\eta,\\etab)^{i_2}\\nabla^{i_3} G+\\sum_{i_1+i_2+i_3+i_4=i}\\nabla^{i_1}(\\eta,\\etab)^{i_2}\\nabla^{i_3}\\chib\\nabla^{i_4} \\phi\\right)\\\\\n&+b'\\nabla^{i+1}\\phi+((\\gamma^{-1}\\chi)'+\\frac{\\partial b'}{\\partial\\th})\\nabla^i\\phi.\n\\end{split}\n\\end{equation*}\n\\end{proposition}\n\\begin{proof}\nAccording to the already established estimates, we have $\\frac{1}{2}\\leq \\Omega\\leq 2$. The result now follows directly from Propositions \\ref{commuteeqn} and \\ref{peqn}.\n\\end{proof}\nWe now use Propositions \\ref{transport} and \\ref{pcommuteeqn} to obtain estimates from a transport equation.\n\\begin{proposition}\\label{transportp}\nSuppose $\\nabla_4\\phi=F$. Then\n\\begin{equation*}\n\\begin{split}\n&\\sum_{i\\leq 1}||\\nabla^{i}\\phi'||_{L^2(S_{u,\\ub})} \\\\\n\\leq &\\sum_{i\\leq 1}||\\nabla^{i}\\phi'||_{L^2(S_{u,0})}+C\\int_0^{\\ub} \\sum_{i\\leq 1}||(\\nabla^iF)'||_{L^2(S_{u,\\ub'})} d\\ub'\\\\\n&+C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}'+\\mathcal O' )(\\sum_{i\\leq 1}\\sup_{\\ub'\\leq \\ub}(||\\nabla^iF||_{L^2(S_{u,\\ub'})}+||\\nabla^i\\phi||_{L^2(S_{u,\\ub'})})).\n\\end{split}\n\\end{equation*}\n\nSimilarly, suppose $\\nabla_3\\phi=G$. Then\n\\begin{equation*}\n\\begin{split}\n&\\sum_{i\\leq 1}||\\nabla^{i}\\phi'||_{L^2(S_{u,\\ub})} \\\\\n\\leq &\\sum_{i\\leq 1}||\\nabla^{i}\\phi'||_{L^2(S_{0,\\ub})}+C\\int_0^{\\ub} \\sum_{i\\leq 1}||(\\nabla^iG)'||_{L^2(S_{u,\\ub'})} d\\ub'\\\\\n&+C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}'+\\mathcal O' )(\\sum_{i\\leq 1}\\sup_{\\ub'\\leq \\ub}(||\\nabla^iG||_{L^2(S_{u,\\ub'})}+||\\nabla^i\\phi||_{L^2(S_{u,\\ub'})})).\n\\end{split}\n\\end{equation*}\n\\end{proposition}\n\\begin{proof}\nWe use the pointwise estimates for $\\gamma'$, $\\Omega'$ and $b'$ proved in the last section.\n\nBy Propositions \\ref{angularpL4}, it suffices to estimate \n$$\\sum_{i\\leq 1}||(\\nabla^i\\phi)'||_{L^2(S_{u,\\ub})}$$\nsince the difference can be estimated by\n$$\\sum_{i\\leq 1}||(\\nabla^i\\phi)'-\\nabla(\\phi')||_{L^2(S_{u,\\ub})}\\leq C(\\mathcal O''+\\tilde{\\mathcal O}')(||\\nabla\\phi||_{L^4(S_{u,\\ub})}+||\\phi||_{L^\\infty(S_{u,\\ub})}).$$\nWe can estimate $||(\\nabla^i\\phi)'||_{L^2(S_{u,\\ub})}$ using Proposition \\ref{transport} and the equation in Proposition \\ref{pcommuteeqn}. Recall the formula in Proposition \\ref{pcommuteeqn}\n\\begin{equation*}\n\\begin{split}\n\\nabla_4(\\nabla^i\\phi)'\\sim &\\sum_{i_1+i_2+i_3=i}(\\nabla^{i_1}(\\eta,\\etab)^{i_2}\\nabla^{i_3} F)'+\\sum_{i_1+i_2+i_3+i_4=i}(\\nabla^{i_1}(\\eta,\\etab)^{i_2}\\nabla^{i_3}\\chi\\nabla^{i_4} \\phi)' \\\\\n&+(\\Omega^{-1})'\\left(\\sum_{i_1+i_2+i_3=i}\\nabla^{i_1}(\\eta,\\etab)^{i_2}\\nabla^{i_3} F+\\sum_{i_1+i_2+i_3+i_4=i}\\nabla^{i_1}(\\eta,\\etab)^{i_2}\\nabla^{i_3}\\chi\\nabla^{i_4} \\phi\\right)\\\\\n&+(\\gamma^{-1}\\chi)'\\nabla^i\\phi.\n\\end{split}\n\\end{equation*}\nWe estimate term by term:\n\\begin{equation*}\n\\begin{split}\n&\\int_0^{\\ub}\\sum_{i_1+i_2+i_3\\leq 1}||(\\nabla^{i_1}(\\eta,\\etab)^{i_2}\\nabla^{i_3} F)'||_{L^2(S_{u,\\ub'})}d\\ub'\\\\\n\\leq &C\\int_0^{\\ub}(\\sum_{i_1\\leq 1}||(\\eta,\\etab)||^{i_1}_{L^\\infty(S_{u,\\ub'})})\\sum_{i_2\\leq 1}||(\\nabla^{i_2} F)'||_{L^2(S_{u,\\ub'})}d\\ub'\\\\\n\\leq &C\\sum_{i\\leq 1}||(\\nabla^{i} F)'||_{L^1_{\\ub}L^2(S_{u,\\ub})}.\n\\end{split}\n\\end{equation*}\nThe second term can be estimated by\n\\begin{equation*}\n\\begin{split}\n&\\int_0^{\\ub}\\sum_{i_1+i_2+i_3\\leq 1}||(\\nabla^{i_1}(\\eta,\\etab)^{i_2}\\nabla^{i_3}\\chi\\nabla^{i_4} \\phi)'||_{L^2(S_{u,\\ub'})}d\\ub'\\\\\n\\leq &C\\int_0^{\\ub}(\\sum_{i_1\\leq 1}\\sum_{i_2\\leq 2}||\\nabla^{i_1}(\\eta,\\etab,\\trch,\\chih)||^{i_2}_{L^\\infty(S_{u,\\ub'})})\\sum_{i_3\\leq 1}||(\\nabla^{i_3} \\phi)'||_{L^2(S_{u,\\ub'})}d\\ub'\\\\\n&+C\\int_0^{\\ub}(\\sum_{i_1\\leq 1}||\\nab^{i_1}(\\eta',\\etab',\\trch',\\chih')||_{L^2(S_{u,\\ub'})})\\\\\n&\\quad\\quad\\quad\\quad\\times(\\sum_{i_2\\leq 1}\\sum_{i_3\\leq 1}||\\nabla^{i_1}(\\eta,\\etab,\\trch,\\chih)||^{i_3}_{L^\\infty(S_{u,\\ub'})})(\\sum_{i_4\\leq 1}||\\nabla^{i_4} \\phi||_{L^2(S_{u,\\ub'})})d\\ub'\\\\\n\\leq &C\\sum_{i\\leq 1}||(\\nabla^{i} \\phi)'||_{L^1_{\\ub}L^2(S_{u,\\ub})}+C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}'+\\mathcal O')(||\\nabla\\phi||_{L^2(S_{u,\\ub})}+||\\phi||_{L^\\infty(S_{u,\\ub})}).\\\\\n\\end{split}\n\\end{equation*}\nThe third term is easier to estimate since by Proposition \\ref{Omegap}, $(\\Omega^{-1})'$ can be estimated in $L^\\infty$:\n\\begin{equation*}\n\\begin{split}\n&\\int_0^{\\ub}\\sum_{i_1+i_2+i_3\\leq 1}||(\\Omega^{-1})'\\left(\\nabla^{i_1}(\\eta,\\etab)^{i_2}\\nabla^{i_3} F+\\nabla^{i_1}(\\eta,\\etab)^{i_2}\\nabla^{i_3}\\chi\\nabla^{i_4} \\phi\\right)||_{L^2(S_{u,\\ub'})}d\\ub'\\\\\n\\leq &C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}')(\\sum_{i\\leq 1}\\sup_{\\ub'\\leq \\ub}||\\nabla^iF||_{L^2(S_{u,\\ub'})}+\\sum_{i\\leq 1}\\sup_{\\ub'\\leq \\ub}||\\nabla^i\\phi||_{L^2(S_{u,\\ub'})}).\n\\end{split}\n\\end{equation*}\nThe fourth term can be controlled in the same way as the third term, since $(\\gamma^{-1}\\chi)'$ can be estimated in $L^2_{\\ub}L^\\infty$:\n\\begin{equation*}\n\\begin{split}\n&\\int_0^{\\ub}\\sum_{i\\leq 1}||(\\gamma^{-1}\\chi)'\\nabla^i\\phi||_{L^2(S_{u,\\ub'})}d\\ub'\\\\\n\\leq &C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}')(\\sum_{i\\leq 1}\\sup_{\\ub'\\leq \\ub}||\\nabla^i\\phi||_{L^2(S_{u,\\ub'})}).\n\\end{split}\n\\end{equation*}\nPutting all these estimates together using Proposition \\ref{transport}, we have\n\\begin{equation*}\n\\begin{split}\n&\\sum_{i\\leq 1}||\\nabla^{i}\\phi'||_{L^2(S_{u,\\ub})} \\\\\n\\leq &\\sum_{i\\leq 1}||\\nabla^{i}\\phi'||_{L^2(S_{0,\\ub})}+C\\sum_{i\\leq 1}||(\\nabla^iF)'||_{L^1_{\\ub}L^2(S_{u,\\ub})}+C\\delta||\\nabla^{i}\\phi'||_{L^2_{\\ub}L^2(S_{u,\\ub})}\\\\\n&+C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}'+\\mathcal O' )(\\sum_{i\\leq 2}\\sup_{\\ub'\\leq \\ub}(||\\nabla^iF||_{L^2(S_{u,\\ub'})}+||\\nabla^i\\phi||_{L^2(S_{u,\\ub'})})).\n\\end{split}\n\\end{equation*}\nTake $\\delta$ to be sufficiently small depending only on $C$ but not $a$, we can absorb the third term to the left hand side to get\n\\begin{equation*}\n\\begin{split}\n&\\sum_{i\\leq 1}||\\nabla^{i}\\phi'||_{L^2(S_{u,\\ub})} \\\\\n\\leq &\\sum_{i\\leq 1}||\\nabla^{i}\\phi'||_{L^2(S_{0,\\ub})}+C\\sum_{i\\leq 1}||(\\nabla^iF)'||_{L^1_{\\ub}L^2(S_{u,\\ub})}\\\\\n&+C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}'+\\mathcal O' )(\\sum_{i\\leq 2}\\sup_{\\ub'\\leq \\ub}(||\\nabla^iF||_{L^2(S_{u,\\ub'})}+||\\nabla^i\\phi||_{L^2(S_{u,\\ub'})})).\n\\end{split}\n\\end{equation*}\nThe proof for the $\\nabla_3$ equation is analogous.\n\\end{proof}\n\\subsection{Estimates for the Ricci Coefficient Difference}\\label{convsec4}\nThe results of the previous Subsection can now be used to estimate the norm ${\\mathcal O}'$ in terms of $\\mathcal R'$. \n\nIn this Subsection and the subsequent Section \\ref{convsec5}, in order to simplify notation, we will omit the superindices $(1)$ and $(2)$ in all the background quantities, for example, $\\nab_3, \\nab, \\Omega, \\beta...$\n\n\\begin{proposition}\\label{Riccip}\nThere exists $\\delta$ and $C$ depending only on the a priori estimates in Theorem \\ref{timeofexistence} (but independent of $a$) such that\n$$\\mathcal O'\\leq Ca+C\\delta^{\\frac 12}\\mathcal R'+C\\delta^{\\frac 12}\\tilde{\\mathcal O}'.$$\n\\end{proposition}\n\\begin{proof}\nWe control $\\mathcal O'$ using the null structure equations.\nFirst, we consider the Ricci coefficients $\\trchb, \\chibh, \\etab$. They all satisfy $\\nabla_3$ equations of the form\n$$\\nabla_3\\psi=\\psi\\psi+\\Psi$$\nsuch that $\\chih, \\omega$ do not appear in the $\\psi$ terms on the right hand side; and that $\\beta$ does not appear as $\\Psi$. This is important because $\\chih'$ and $\\omega'$ cannot be controlled by $C\\mathcal O'$ in the $L^2(S)$ norms and $\\beta'$ cannot be controlled by $C\\mathcal R'$ in $L^2(\\Hb)$. In order to estimate $\\psi$ using this equation, we need to estimate the curvature term:\n\\begin{equation*}\n\\begin{split}\n\\sum_{i\\leq 1}||(\\nabla^i\\Psi)'||_{L^1_uL^2(S)}\\leq C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}')+C\\delta^{\\frac 12}\\mathcal R'.\n\\end{split}\n\\end{equation*}\nand the nonlinear terms:\n\\begin{equation*}\n\\begin{split}\n&\\sum_{i\\leq 1}||(\\nabla^i(\\psi\\psi))'||_{L^1_u L^2(S_{u,\\ub})}\\\\\n\\leq &C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}')+C \\delta^{\\frac 12}\\sum_{i_1+i_2\\leq 1}\\sup_{u'\\leq u}||\\gamma'\\nabla^{i_1}\\psi\\nabla^{i_2}\\psi||_{ L^2(S_{u',\\ub})}\\\\\n&+C\\delta^{\\frac 12}\\sum_{i_1+i_2\\leq 1}\\sup_{u'\\leq u}||\\nabla^{i_1}\\psi\\nabla^{i_2}\\psi'||_{L^2(S_{u',\\ub})}\\\\\n\\leq &C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}'+\\mathcal O').\n\\end{split}\n\\end{equation*}\nBy Proposition \\ref{transportp}, we thus have\n\\begin{equation}\\label{Ricci31}\n\\begin{split}\n&\\sum_{i\\leq 1}\\sup_{u}||\\nabla^i(\\trchb',\\chibh',\\etab')||_{L^2(S_{u,\\ub})}\\\\\n\\leq &\\sum_{i\\leq 1}||\\nabla^i(\\trchb',\\chibh',\\etab')||_{L^2(S_{0,\\ub})}+ C\\delta^{\\frac 12}\\sup_{u}(\\mathcal O''_{u}+\\tilde{\\mathcal O}'_{u}+\\mathcal O')+C\\delta^{\\frac 12}\\mathcal R'\\\\\n\\leq & a+ C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}'+\\mathcal O')+C\\delta^{\\frac 12}\\mathcal R'.\n\\end{split}\n\\end{equation}\nWe now consider the equation for $\\omega$, which schematically looks like\n$$\\nabla_3\\omega=\\omega\\psi+\\chih\\psi+\\psi\\psi+\\rhoc,$$\nwhere $\\psi\\neq \\chih,\\omega$ are the good components as above. The term $\\psi'\\omega$ or $\\psi'\\chih$ can be estimated since $\\psi'$ can be put in an appropriate $L^p(S)$ norm and be controlled by $C\\mathcal O'$. However, the most difficult terms are $\\omega'\\psi$ and $\\chih'\\psi$ since $\\omega'$ and $\\chih'$ cannot be bounded in $L^1_uL^p(S)$ by $C\\mathcal O'$. Thus, by Proposition \\ref{transportp}, we have\n\\begin{equation*}\n\\begin{split}\n&\\sum_{i\\leq 1}||\\nabla^i\\omega'||_{L^2(S_{u,\\ub})}\\\\\n\\leq &C\\sum_{i\\leq 1}||\\nabla^i\\omega'||_{L^2(S_{0,\\ub})}+ C\\sum_{i_1+i_2\\leq 1}||\\nabla^{i_1}\\psi\\nabla^{i_2}(\\chih',\\omega')||_{L^1_u L^p(S_{u,\\ub})}+C\\delta^{\\frac 12}\\mathcal R'+C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}'+\\mathcal O').\n\\end{split}\n\\end{equation*}\nSince this holds for every $\\ub$, we can integrate in $L^2$ in $\\ub$ to get\n\\begin{equation}\\label{Ricci32}\n\\begin{split}\n&\\sum_{i\\leq 1}||\\nabla^i\\omega'||_{L^2_{\\ub}L^2(S_{u,\\ub})}\\\\\n\\leq &C\\sum_{i\\leq 1}||\\nabla^i\\omega'||_{L^2_{\\ub}L^2(S_{0,\\ub})}+ \\int_0^u||\\nabla^{i_1}\\psi\\nabla^{i_2}(\\chih',\\omega')||_{L^2_{\\ub} L^2(S_{u',\\ub})} du'+C\\mathcal R'+C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}'+\\mathcal O')\\\\\n\\leq & a+C\\mathcal R'+C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}'+\\mathcal O'),\n\\end{split}\n\\end{equation}\nsince we can control $\\nabla^i\\omega'$ by $C\\mathcal O'$ after integrating along the $\\ub$ direction.\nWe can estimate $\\chih$ in a similar manner as $\\omega$. $\\chih$ satisfies\n$$\\nabla_3\\chih=\\psi\\psi+\\psi\\chih+\\nabla\\eta,$$\nwhere $\\psi\\neq \\chih,\\omega$. Putting $\\nabla\\eta$ in the $\\tilde{\\mathcal O}'$ norm and using already obtained estimates, we have\n\\begin{equation*}\n\\begin{split}\n&\\sum_{i\\leq 1}||\\nabla^i\\chih'||_{L^2(S_{u,\\ub})}\\\\\n\\leq &C\\sum_{i\\leq 1}||\\nabla^i\\chih'||_{L^2(S_{0,\\ub})}+ C\\sum_{i_1+i_2\\leq 1}||\\nabla^{i_1}\\psi \\nabla^{i_2}\\chih'||_{L^1_u L^2(S_{u,\\ub})}+C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}'+\\mathcal O').\n\\end{split}\n\\end{equation*}\nAs for $\\omega'$, we now integrate in $L^2$ in $\\ub$ to get\n\\begin{equation}\\label{Ricci33}\n\\begin{split}\n&\\sum_{i\\leq 1}||\\nabla^i\\chih'||_{L^2_{\\ub}L^2(S_{u,\\ub})}\\\\\n\\leq &\\sum_{i\\leq 1}||\\nabla^i\\chih'||_{L^2_{\\ub}L^2(S_{0,\\ub})}+ C \\int_0^u||\\nabla^{i_1}\\psi \\nabla^{i_2}\\chih'||_{L^2_{\\ub} L^2(S_{u',\\ub})}du'+C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}'+\\mathcal O')\\\\\n\\leq &a+C\\delta^{\\frac 12}(\\mathcal O''+\\tilde{\\mathcal O}'+\\mathcal O').\n\\end{split}\n\\end{equation}\nIt remains to consider $\\trch,\\eta,\\omegab$. They satisfy $\\nabla_4$ equations of the form:\n$$\\nabla_4\\psi=\\psi\\psi+\\Psi,$$\nwhere $\\psi$ can be any Ricci coefficients and $\\Psi$ can be any curvature components $\\neq\\alpha,\\alphab$, i.e., all the $\\Psi$ terms can be controlled in $L^2(H)$ by $\\mathcal R'$. Moreover, since we are now integrating in $\\ub$, all terms $\\psi'$ can be estimated by $C\\mathcal O'$. By Proposition \\ref{transportp}, we have\n$$\\sum_{i\\leq 1}||\\nabla^i(\\trch',\\eta',\\omegab')||_{L^2(S_{u,\\ub})}\\leq \\sum_{i\\leq 1}||\\nabla^i(\\trch',\\eta',\\omegab')||_{L^2(S_{0,\\ub})}+ C\\mathcal R'+C\\int_0^{\\ub}\\mathcal O' d\\ub'.$$\nTherefore,\n$$\\sum_{i\\leq 1}||\\nabla^i(\\trch',\\eta',\\omegab')||_{L^2(S_{u,\\ub})}\\leq a+C\\delta^{\\frac 12}\\mathcal R'+C\\delta^{\\frac 12}(\\sup_u\\mathcal O''+\\tilde{\\mathcal O}'+\\mathcal O').$$\nUsing $\\mathcal O''\\leq C(\\tilde{\\mathcal O}'+\\mathcal O')$ and all the above estimates, we have\n$$\\mathcal O'\\leq Ca+C\\delta^{\\frac 12}\\mathcal R'+C\\delta^{\\frac 12}(\\tilde{\\mathcal O}'+\\mathcal O').$$\nBy choosing $\\delta$ sufficiently small so that $C\\delta^{\\frac 12}\\leq \\frac 12$, we have\n$$\\mathcal O'\\leq Ca+C\\delta^{\\frac 12}\\mathcal R'+C\\delta^{\\frac 12}\\tilde{\\mathcal O}'.$$\nNotice that in particular $\\delta$ and $C$ depend only on the a priori estimates from Theorem \\ref{timeofexistence} and are independent of $a$.\n\\end{proof}\nWe now move on to estimate $\\tilde{\\mathcal O}'$ by $\\mathcal R'$. Recall from Propositions \\ref{Theta} and \\ref{ellipticTheta} that $\\tilde{\\mathcal O}$ was controlled using a combination of transport equations for $\\Theta$ and Hodge systems. The norm $\\tilde{\\mathcal O}'$ can be dealt with in a similar fashion. In order to perform this scheme, we need show that as was with the case of the norm $\\tilde{\\mathcal O}$ where we controlled $\\nab\\psi$ from $\\Theta$, the difference $\\psi'$ satisfies elliptic equations with $\\Theta'$ as a source. This is given by the following Proposition:\n\\begin{proposition}\\label{commuteelliptic}\nLet $\\phi$ be a $(0,r)$-tensorfield. Suppose \n$$\\div\\phi=f,\\quad\\curl\\phi=g,\\quad\\tr\\phi=h$$\non each of the spacetimes. Then\n$$\\div\\phi'\\sim f'+ (\\gamma^{-1})'\\nabla\\phi+\\Gamma'\\phi,$$\n$$\\curl\\phi'\\sim g'+ (\\eps)'\\nabla\\phi,$$\n$$\\tr\\phi' \\sim h' + (\\gamma^{-1})'\\phi.$$\n\\end{proposition}\n\\begin{proof}\nThis is a straightforward coordinate computation.\n\\end{proof}\nBased on the estimates we have on $\\gamma'$, $(\\gamma^{-1})'$ and $\\Gamma'$, and using the estimates in Proposition \\ref{ellipticthm}, we have\n\\begin{proposition}\\label{ellipticthmp}\nLet $\\phi^{(1)}$ and $\\phi^{(2)}$ be totally symmetric $r+1$ covariant tensorfields on 2-spheres $(\\mathbb S^2,\\gamma^{(1)})$, $(\\mathbb S^2,\\gamma^{(2)})$ respectively satisfying\n$$\\div\\phi^{(i)}=f^{(i)},\\quad \\curl\\phi^{(i)}=g^{(i)},\\quad \\mbox{tr}\\phi^{(i)}=h^{(i)},$$\nfor $i=1,2$. \nThen\n\\begin{equation*}\n\\begin{split}\n||\\nabla^{2}\\phi'||_{L^2(S)}\\leq &Ca\\sum_{i\\leq 1}(||\\nabla^{i}f||_{L^2(S)}+||\\nabla^{i}g||_{L^2(S)}+||\\nabla^{i}h||_{L^2(S)}+||\\phi||_{L^2(S)})\\\\\n&+C\\sum_{i\\leq 1}(||\\nabla^{i}f'||_{L^2(S)}+||\\nabla^{i}g'||_{L^2(S)}+||\\nabla^{i}h'||_{L^2(S)}+||\\phi'||_{L^2(S)})\n\\end{split}\n\\end{equation*}\n\n\\end{proposition}\n\\begin{proof}\nThis is a direct consequence of Propositions \\ref{commuteelliptic} and \\ref{ellipticthm}.\n\\end{proof}\n\n\n\nWe can now begin to estimate $\\tilde{\\mathcal O}'$. First, we define $\\Theta'$ and derive estimates for $\\Theta'$ using transport equations. Let $\\Theta'$ denote $(\\nabla\\trch)',(\\nabla\\trchb)',\\mu',\\mub',\\kappa',\\kappab'$. We have the following estimates:\n\\begin{proposition}\\label{Thetap}\n$$\\sum_{i\\leq 1}||\\nabla^i(\\nabla\\trch',\\nabla\\trchb',\\mu',\\mub')||_{L^2(S_{u,\\ub})}\\leq Ca+C\\delta^{\\frac 12}(\\tilde{\\mathcal O}'+\\mathcal R').$$\n\\end{proposition}\n\\begin{proof}\nFrom the proof of Proposition \\ref{ellipticTheta}, we know that each of $\\nabla\\trch,\\nabla\\trchb,\\mu,\\mub$ satisfies either\n$$\\nab_3\\Theta=\\psi\\nab\\psi_{\\Hb}+\\psi\\psi\\psi+\\psi\\Psi_{\\Hb},$$\nor\n$$\\nab_4\\Theta=\\psi\\nab\\psi_{H}+\\psi\\psi\\psi+\\psi\\Psi_H,$$\nwhere as in Section \\ref{Ricciellipticsec}, we use the notation $\\psi_H\\in\\{\\trch,\\chih,\\eta,\\etab,\\omega,\\trchb\\}$, $\\psi_{\\Hb}\\in\\{\\trch,\\eta,\\etab,\\omegab,\\trchb,\\chibh\\}$, $\\Psi_H\\in\\{\\beta,\\rho,\\sigma,\\betab\\}$ and $\\Psi_{\\Hb}\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}$.\n\nWe focus on those $\\Theta$'s satisfying the $\\nab_4$ equation. The other case can be treated analogously. By Proposition \\ref{transportp}, we need to control\n$$\\sum_{i\\leq 1}||(\\nabla^i(\\psi\\nab\\psi_{H}+\\psi\\psi\\psi+\\psi\\Psi_{H}))'||_{L^1_{\\ub}L^2(S_{u,\\ub})},$$\n$$\\sum_{i\\leq 1}||\\nabla^i(\\psi\\nab\\psi_{H}+\\psi\\psi\\psi+\\psi\\Psi_{H})||_{L^2(S_{u,\\ub})},$$\nand \n$$\\sum_{i\\leq 1}||\\nabla^i\\Theta||_{L^2(S_{u,\\ub})}.$$\nWe first estimate the $'$ terms. Among those we first look at the term with two derivatives on the Ricci coefficient. We have, using Proposition \\ref{Riccip}, \n$$||(\\psi\\nabla^2\\psi_H)'||_{L^1_{\\ub}L^2(S_{u,\\ub})}\\leq C\\tilde{\\mathcal O}'+ C||\\psi'||_{L^1_{\\ub}L^\\infty(S_{u,\\ub})}+C\\delta^{\\frac 12}||\\nabla^2\\psi_H'||_{L^2_{\\ub}L^2(S_{u,\\ub})}\\leq C(a+\\tilde{\\mathcal O}'+\\delta^{\\frac 12}\\mathcal R').$$\nThe other term with Ricci coefficients can be estimated by $\\mathcal O'$ and thus by Proposition \\ref{Riccip}:\n$$\\sum_{i_1+i_2\\leq 1} ||(\\psi^{i_1}\\nabla^{i_2}\\psi\\nabla\\psi)'||_{L^1_{\\ub}L^2(S_{u,\\ub})}\\leq C(a+\\delta^{\\frac 12}\\mathcal R'+\\delta^{\\frac 12}\\tilde{\\mathcal O}').$$\nWe then move to the term with the curvature components:\n\\begin{equation*}\n\\begin{split}\n&\\sum_{i_1+i_2+i_3\\leq 1} ||(\\psi^{i_1}\\nabla^{i_2}\\psi\\nabla^{i_3}\\Psi_{H})'||_{L^1_{\\ub}L^2(S_{u,\\ub})}\\\\\n\\leq &C(a+\\delta^{\\frac 12}\\mathcal R'+\\delta^{\\frac 12}\\tilde{\\mathcal O}')+C\\delta^{\\frac 12}\\sum_{i\\leq 1}||\\nab^i\\Psi_{H}'||_{L^2_{\\ub}L^2(S_{u,\\ub})}\\\\\n\\leq &C(a+\\delta^{\\frac 12}\\mathcal R'+\\delta^{\\frac 12}\\tilde{\\mathcal O}').\n\\end{split}\n\\end{equation*}\nThe terms without $'$ can be estimated using the a priori estimates in Theorem \\ref{timeofexistence}:\n$$\\sum_{i\\leq 1}(||\\nabla^i(\\psi\\nab\\psi_{H}+\\psi\\psi\\psi+\\psi\\Psi_{H})||_{L^2(S_{u,\\ub})}+\n||\\nabla^i\\Theta||_{L^2(S_{u,\\ub})})\\leq C.$$\nThus, by Proposition \\ref{transportp}, we have\n$$\\sum_{i\\leq 1}||\\nabla^i(\\nabla\\trch',\\nabla\\trchb',\\mu',\\mub')||_{L^2(S_{u,\\ub})}\\leq Ca+C\\delta^{\\frac 12}(\\tilde{\\mathcal O}'+\\mathcal R').$$\n\\end{proof}\nWe now consider the difference quantities $\\kappa'$ and $\\kappab'$. As in Proposition \\ref{omegaelliptic}, they will only satisfy $L^2(H_u)$ and $L^2(\\Hb_{\\ub})$ estimates. As in the proof of Proposition \\ref{omegaelliptic}, in what follows, we allow $\\psi$ also to be $\\omega^\\dagger$ and $\\omegab^\\dagger$.\n\\begin{proposition}\\label{ellipticomegap}\n$$\\sum_{i\\leq 1}||\\nabla^i\\kappa'||_{L^2(H_u)}\\leq C\\delta^{\\frac 12}(a+\\tilde{\\mathcal O}'+\\mathcal R'),$$\n$$\\sum_{i\\leq 1}||\\nabla^i\\kappab'||_{L^2(\\Hb_{\\ub})}\\leq C\\delta^{\\frac 12}(a+\\tilde{\\mathcal O}'+\\mathcal R').$$\n\\end{proposition}\n\\begin{proof}\nRecall equations (\\ref{kappaeqn}) and (\\ref{kappabeqn}):\n$$\\nab_3\\kappa=\\psi\\nab\\psi+\\psi\\psi\\psi+\\psi\\Psi,$$\nor\n$$\\nab_4\\kappab=\\psi\\nab\\psi+\\psi\\psi\\psi+\\psi\\Psi.$$\nWe focus on $\\kappa$. As $\\kappab$ is easier. The only terms that are new compared to the proof of Proposition \\ref{Thetap} \n$$\\sum_{i\\leq 1}||\\nab^i(\\psi\\beta)'||_{L^1_{u}L^2(S_{u,\\ub})}\\mbox{ and }\\sum_{i\\leq 2}||\\nab^i(\\chih',\\omega')||_{L^1_{u}L^2(S_{u,\\ub})}.$$\nThus using the estimates in the proof of Proposition \\ref{Thetap}, we have\n$$\\sum_{i\\leq 1}||\\nabla^i\\kappa'||_{L^2(S_{u,\\ub})}\\leq Ca+C\\delta^{\\frac 12}(\\tilde{\\mathcal O}'+\\mathcal R)+C\\delta^{\\frac 12}(\\sum_{i\\leq 1}||\\nab^i\\beta'||_{L^2_uL^2(S_{u,\\ub})}+\\sum_{i\\leq 2}||\\nab^i(\\chih',\\omega')||_{L^2_uL^2(S_{u,\\ub})}).$$\nIntegrating over $\\ub$ in $L^2$, we get\n\\begin{equation*}\n\\begin{split}\n&\\sum_{i\\leq 1}||\\nabla^i\\kappa'||_{L^2(H_u)}\\\\\n\\leq &C\\delta^{\\frac 12}a+C\\delta(\\tilde{\\mathcal O}'+\\mathcal R')+C\\delta^{\\frac 12}(\\sum_{i\\leq 1}||\\nab^i\\beta'||_{L^2_{\\ub}L^2_uL^2(S_{u,\\ub})}+\\sum_{i\\leq 2}||\\nab^i(\\chih',\\omega')||_{L^2_{\\ub}L^2_uL^2(S_{u,\\ub})}).\n\\end{split}\n\\end{equation*}\nThus, using the definition of the norms $\\tilde{\\mathcal O}'$ and $\\mathcal R'$,\n$$\\sum_{i\\leq 1}||\\nabla^i\\kappa'||_{L^2(H_u)}\\leq C\\delta^{\\frac 12}a+C\\delta(\\tilde{\\mathcal O}'+\\mathcal R').$$\nSimilarly\n$$\\sum_{i\\leq 1}||\\nabla^i\\kappab'||_{L^2(H_u)}\\leq C\\delta^{\\frac 12}a+C\\delta(\\tilde{\\mathcal O}'+\\mathcal R').$$\n\\end{proof}\nWe use Propositions \\ref{Thetap} and \\ref{ellipticomegap} and elliptic estimates to derive all the estimates for $\\nabla^{2}\\psi'$.\n\\begin{proposition}\\label{Ricciellipticp}\n$$\\tilde{\\mathcal O}'\\leq Ca+C\\mathcal R'.$$\n\\end{proposition}\n\\begin{proof}\nWe will first prove\n$$\\sup_u||\\nabla^{2}(\\chih',\\omega',\\eta',\\etab')||_{L^2(H_u)}+\\sup_{\\ub}||\\nabla^2(\\chibh',\\omegab',\\eta',\\etab')||_{L^2(\\Hb_{\\ub})}\\leq Ca+C\\delta^{\\frac 12}\\tilde{\\mathcal O}'+C\\mathcal R'.$$\nWe have the following div-curl systems:\n$$\\div(\\chih,\\eta,\\etab)=(\\nab\\trch,\\mu)+\\psi\\psi+(\\beta,\\rho,\\sigma),$$\n$$\\curl(\\chih,\\eta,\\etab)=(\\nab\\trch,\\mu)+\\psi\\psi+(\\beta,\\rho,\\sigma),$$\nand\n$$\\div(\\chibh,\\eta,\\etab)=(\\nab\\trchb,\\mub)+\\sum_{\\psi\\neq \\chih,\\omega}\\psi\\psi+(\\rhoc,\\sigmac,\\betab),$$\n$$\\curl(\\chibh,\\eta,\\etab)=(\\nab\\trchb,\\mub)+\\sum_{\\psi\\neq \\chih,\\omega}\\psi\\psi+(\\rhoc,\\sigmac,\\betab);$$\nas well as the div-curl system\n$$\\div(\\nab\\omega,\\nab\\omega^\\dagger)=\\nab\\kappa+\\psi\\psi+(\\beta,\\rho,\\sigma),$$\n$$\\curl(\\nab\\omega,\\nab\\omega^\\dagger)=\\nab\\kappa+\\psi\\psi+(\\beta,\\rho,\\sigma),$$\nand\n$$\\div(\\nab\\omegab,\\nab\\omegab^\\dagger)=\\nab\\kappab+\\sum_{\\psi\\neq \\chih,\\omega}\\psi\\psi+(\\rhoc,\\sigmac,\\betab),$$\n$$\\curl(\\nab\\omegab,\\nab\\omegab^\\dagger)=\\nab\\kappab+\\sum_{\\psi\\neq \\chih,\\omega}\\psi\\psi+(\\rhoc,\\sigmac,\\betab).$$\nBy Proposition \\ref{ellipticthmp}, we need to control the terms involving $\\Theta$\n$$\\sup_{u}\\sum_{i\\leq 1}||\\nabla^i(\\nab\\trch',\\mu',\\kappa')||_{L^2(H_u)}+\\sup_{\\ub}\\sum_{i\\leq 1}||\\nabla^i(\\nab\\trchb',\\mub',\\kappab')||_{L^2(\\Hb_{\\ub})},$$\nthe curvature terms\n$$\\sup_u\\sum_{i\\leq 1}||\\nabla^i(\\beta',\\rho',\\sigma')||_{L^2(H_u)}+\\sup_{\\ub}||\\nabla^i(\\rhoc',\\sigmac',\\betab')||_{L^2(\\Hb_{\\ub})}$$\nand the lower order terms involving the Ricci coefficients\n$$\\sum_{i\\leq 1}||\\nabla^i(\\psi'\\psi)||_{L^2(H_u)}+\\sum_{i\\leq 1}\\sum_{\\psi\\neq \\chih,\\omega}||\\nabla^i(\\psi'\\psi)||_{L^2(\\Hb_{\\ub})}.$$\nThe terms involving $\\Theta$ are controlled by the estimates in Proposition \\ref{Thetap} by\n$$C\\delta^{\\frac 12}(a+\\tilde{\\mathcal O}'+\\mathcal R').$$\nThe curvature terms can be estimated using the definition of $\\mathcal R'$ by\n$$C\\mathcal R'.$$\nThe remaining terms can be bounded by\n\\begin{equation*}\n\\begin{split}\n&\\sum_{i\\leq 1}||\\nabla^i(\\psi'\\psi)||_{L^2(H_u)}\\\\\n\\leq &C(||\\psi'||_{L^\\infty(S_{u,\\ub})}\\sum_{i\\leq 1}||\\nabla^{i}\\psi||_{L^2(S_{u,\\ub})}+\\sum_{i\\leq 1}||\\nabla^{i}\\psi'||_{L^2(S_{u,\\ub})}||\\nabla^{i_2}\\psi||_{L^\\infty(S_{u,\\ub})})\\\\\n\\leq &Ca+C\\delta^{\\frac 12}\\tilde{\\mathcal O}'+C\\delta^{\\frac 12}\\mathcal R'.\n\\end{split}\n\\end{equation*}\nThus we have\n$$\\sup_u||\\nabla^{2}(\\chih',\\omega',\\eta',\\etab')||_{L^2(H_u)}+\\sup_{\\ub}||\\nabla^2(\\chibh',\\omegab',\\eta',\\etab')||_{L^2(\\Hb_{ub})}\\leq Ca+C\\delta^{\\frac 12}\\tilde{\\mathcal O}'+C\\mathcal R'.$$\nTherefore, together with the estimates for $\\nab^2\\trch'$ and $\\nab^2\\trchb'$ in Proposition \\ref{Thetap}, we have\n$$\\tilde{\\mathcal O}\\leq Ca+C\\delta^{\\frac 12}\\tilde{\\mathcal O}'+C\\mathcal R'.$$\nBy choosing $\\delta$ sufficiently small depending on $C$ so that $C\\delta^{\\frac 12}$, we have\n$$\\tilde{\\mathcal O}\\leq Ca+C\\mathcal R'.$$\nNotice that $C$ and $\\delta$ depend only on the a priori estimates in Theorem \\ref{timeofexistence} and are both independent of $a$.\n\\end{proof}\n\\subsection{Estimates for the Curvature Difference}\\label{convsec5}\nIn order to finish the proof of Theorem \\ref{convergencethm}, we need to estimate $\\mathcal R'$ by $Ca$. This will be proved using an energy-type estimate. As in the bounds for the curvature components themselves, we will derive the energy estimates directly from the Bianchi equations without using the Bel Robinson tensor. This method is especially useful in controlling the difference of the curvature components since these difference quantities do not arise from a Weyl field with respect to either of the background spacetime metrics.\n\\begin{proposition}\n\\begin{equation*}\n\\begin{split}\n&\\sum_{i\\leq 1}\\left(\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}\\int_{H_u} (\\nabla^i\\Psi')^2+\\sum_{\\Psic\\in\\{\\rhoc,\\sigmac,\\betab,\\alphab\\}}\\int_{\\underline{H}_{\\underline{u}}} (\\nabla^i\\Psi')^2\\right)\\\\\n\\leq& Ca^2+C\\delta(\\mathcal R')^2+C||(K\\Psi)'||_{L^2_u L^2_{\\ub}L^2(S)}^2\\\\\n&+C\\sum_{i_1+i_2+i_3\\leq 1}||(\\psi^{i_1}\\nabla^{i_2}\\psi\\nabla^{i_3}\\Psi)'||_{L^2_u L^2_{\\ub}L^2(S)}^2+\\sum_{i_1+i_2+i_3\\leq 2}||(\\psi^{i_1}\\nabla^{i_2}\\psi\\nabla^{i_3}\\psi)'||_{L^2_u L^2_{\\ub}L^2(S)}^2.\n\\end{split}\n\\end{equation*}\n\\end{proposition}\n\\begin{proof}\nThe energy estimates for the curvature difference follows from the difference Bianchi equations. Once these are derived, the energy estimates will be obtained as in Section \\ref{estimates} by integration by parts in appropriate subsets of the difference Bianchi system. In the proof below, we concentrate on the most difficult case which involves renormalized curvature components. The estimates for the other components can be derived in a similar fashion.\n\nRecall that\n\\begin{equation*}\n\\begin{split}\n&\\nabla_3\\beta+tr\\chib\\beta=\\nabla\\rho + 2\\omegab \\beta +^*\\nabla\\sigma +2\\chih\\cdot\\betab+3(\\eta\\rho+^*\\eta\\sigma),\\\\\n&\\nabla_4\\sigma+\\frac 32tr\\chi\\sigma=-\\div^*\\beta+\\frac 12\\chibh\\cdot ^*\\alpha-\\zeta\\cdot^*\\beta-2\\etab\\cdot\n^*\\beta,\\\\\n&\\nabla_4\\rho+\\frac 32tr\\chi\\rho=\\div\\beta-\\frac 12\\chibh\\cdot\\alpha+\\zeta\\cdot\\beta+2\\etab\\cdot\\beta,\\\\\n\\end{split}\n\\end{equation*}\nAs before, we renormalize the equations. Define\n$$\\sigmac:=\\sigma+\\frac{1}{2}\\chibh\\wedge\\chih,$$\n$$\\rhoc:=\\rho-\\frac{1}{2}\\chibh\\cdot\\chih.$$\nSchematically, we have\n\\begin{equation*}\n\\begin{split}\n&\\nabla_3\\beta=\\nabla\\rhoc+^*\\nabla\\sigmac +\\psi\\nabla\\psi+\\psi\\Psi+\\psi\\psi\\psi,\\\\\n&\\nabla_4\\sigmac=-\\div^*\\beta+\\psi\\nabla\\psi+\\psi\\Psi+\\psi\\psi\\psi,\\\\\n&\\nabla_4\\rhoc=\\div\\beta+\\psi\\nabla\\psi+\\psi\\Psi+\\psi\\psi\\psi.\n\\end{split}\n\\end{equation*}\nFrom these we can derive equations for $\\beta'$, $\\sigma'$ and $\\rhoc'$. By Proposition \\ref{peqn}, \n\\begin{equation*}\n\\begin{split}\n&\\nabla_3\\beta'=(\\nabla_3\\beta)'+\\frac{(\\Omega^{-1})'}{\\Omega^{-1}}\\nabla_3\\beta+\\Omega^{-1}(b^A)'\\nabla_{\\frac{\\partial}{\\partial\\th^A}}\\beta+\\Omega^{-1}(\\gamma^{-1}\\chi)'\\beta+\\Omega^{-1}b\\Gamma'\\beta,\\\\\n&\\nabla_4\\sigmac=(\\nabla_4\\sigmac)'+\\frac{(\\Omega^{-1})'}{\\Omega^{-1}}\\nabla_4\\sigmac+\\Omega^{-1}(\\gamma^{-1}\\chi)'\\sigmac,\\\\\n&\\nabla_4\\rhoc=(\\nabla_4\\rhoc)'+\\frac{(\\Omega^{-1})'}{\\Omega^{-1}}\\nabla_4\\rhoc+\\Omega^{-1}(\\gamma^{-1}\\chi)'\\rhoc.\n\\end{split}\n\\end{equation*}\nMoreover, we can write schematically\n\\begin{equation*}\n\\begin{split}\n&(\\nab(\\rhoc,\\sigmac))'=\\nabla(\\rhoc',\\sigmac')+\\Gamma'(\\rhoc,\\sigmac)\\\\\n&(\\div\\beta)'=\\div\\beta'+\\gamma'\\nabla\\beta+\\Gamma'\\beta.\n\\end{split}\n\\end{equation*}\nTherefore,\n\\begin{equation*}\n\\begin{split}\n&\\nabla_3\\beta'=\\nabla\\rhoc'+^*\\nabla\\sigmac' +(\\psi\\nabla\\psi+\\psi\\Psi+\\psi\\psi\\psi)'+F_{\\beta},\\\\\n&\\nabla_4\\sigmac'=-\\div^*\\beta'+(\\psi\\nabla\\psi+\\psi\\Psi+\\psi\\psi\\psi)'+F_{\\sigmac},\\\\\n&\\nabla_4\\rhoc'=\\div\\beta'+(\\psi\\nabla\\psi+\\psi\\Psi+\\psi\\psi\\psi)'+F_{\\rhoc},\n\\end{split}\n\\end{equation*}\nwhere $F_{\\beta}$, $F_{\\sigmac}$ and $F_{\\rhoc}$ satisfy the bound\n$$\\sum_{i\\leq 1}||\\nab^iF||_{L^2_uL^2_{\\ub}L^2(S)}\\leq C\\delta^{\\frac 12}(a+\\mathcal R').$$\nTherefore, multiplying the first equation by $\\beta'$, integrating by parts and using the other equations, we have\n\\begin{equation*}\n\\begin{split}\n&\\int_{H_u} |\\beta'|^2+\\int_{\\Hb_{\\ub}} (|\\rhoc'|^2+|\\sigmac'|^2)\\\\\n\\leq& Ca^2 +||(\\beta',\\rhoc',\\sigma')(\\psi\\nabla\\psi+\\psi\\Psi+\\psi\\psi\\psi)'||_{L^1_u L^1_{\\ub}L^1(S)}+||(\\beta',\\rhoc',\\sigma')F_{\\beta,\\rhoc,\\sigmac}||_{L^1_u L^1_{\\ub}L^1(S)}\\\\\n\\leq &Ca^2 +||(\\beta',\\rhoc',\\sigma')||_{L^2_u L^2_{\\ub}L^2(S)}+||(\\psi\\nabla\\psi+\\psi\\Psi+\\psi\\psi\\psi)'||_{L^2_u L^2_{\\ub}L^2(S)}^2++||F||_{L^2_u L^2_{\\ub}L^2(S)}^2\\\\\n\\leq &Ca^2 +C\\delta(\\mathcal R')^2+||(\\psi\\nabla\\psi+\\psi\\Psi+\\psi\\psi\\psi)'||_{L^2_u L^2_{\\ub}L^2(S)}^2.\n\\end{split}\n\\end{equation*}\nWe now look at the equation the first covariant angular derivative for the \ncurvature components:\n\\begin{equation*}\n\\begin{split}\n&\\nabla_3\\nabla\\beta=\\nabla^2\\rhoc+^*\\nabla^2\\sigmac+K(\\rhoc,\\sigmac) +\\psi^2\\nabla\\Psi+\\nabla\\psi\\Psi+\\psi\\nabla\\Psi+\\psi\\nabla^2\\psi+\\psi^2\\nabla\\psi+\\psi^4,\\\\\n&\\nabla_4\\nabla\\sigmac=-\\div^*\\nabla\\beta+K\\beta+\\psi^2\\nabla\\Psi+\\nabla\\psi\\Psi+\\psi\\nabla\\Psi+\\psi\\nabla^2\\psi+\\psi^2\\nabla\\psi+\\psi^4,\\\\\n&\\nabla_4\\nabla\\rhoc=\\div\\nabla\\beta+K\\beta+\\psi^2\\nabla\\Psi+\\nabla\\psi\\Psi+\\psi\\nabla\\Psi+\\psi\\nabla^2\\psi+\\psi^2\\nabla\\psi+\\psi^4.\n\\end{split}\n\\end{equation*}\nA similar argument as before involving writing the equations for $\\nabla(\\beta',\\rhoc',\\sigmac')$ and integrating by parts yield\n\\begin{equation*}\n\\begin{split}\n&\\int_{H_u} |\\nabla\\beta'|^2+\\int_{\\Hb_{\\ub}} (|\\nabla\\rhoc'|^2+|\\nabla\\sigmac'|^2)\\\\\n\\leq &Ca^2 +C\\delta(\\mathcal R')^2\\\\\n&+||(K\\Psi+\\psi^2\\nabla\\Psi+\\nabla\\psi\\Psi+\\psi\\nabla\\Psi+\\psi\\nabla^2\\psi+(\\nabla\\psi)^2+\\psi^2\\nabla\\psi+\\psi^4)'||_{L^2_u L^2_{\\ub}L^2(S)}^2.\n\\end{split}\n\\end{equation*}\n\\end{proof}\n\nFrom this one can conclude\n\\begin{proposition}\\label{curvaturep}\n\\begin{equation*}\n\\begin{split}\n\\mathcal R'\\leq Ca.\n\\end{split}\n\\end{equation*}\n\\end{proposition}\n\\begin{proof}\nBy the previous Proposition, we need to estimate\n$$||(K\\Psi)'||_{L^2_u L^2_{\\ub}L^2(S)}^2,$$\n$$\\sum_{i_1+i_2+i_3\\leq 1}||(\\psi^{i_1}\\nabla^{i_2}\\psi\\nabla^{i_3}\\Psi)'||_{L^2_u L^2_{\\ub}L^2(S)}^2,$$\n$$\\sum_{i_1+i_2+i_3\\leq 2}||(\\psi^{i_1}\\nabla^{i_2}\\psi\\nabla^{i_3}\\psi)'||_{L^2_u L^2_{\\ub}L^2(S)}^2$$\nBy Proposition \\ref{Kp} and the definition of the norm $\\mathcal R'$ and the a priori estimates from Theorem \\ref{timeofexistence}, we have\n$$||(K\\Psi)'||_{L^2_u L^2_{\\ub}L^2(S)}^2\\leq ||K'\\Psi||_{L^2_u L^2_{\\ub}L^2(S)}^2+||K \\Psi'||_{L^2_u L^2_{\\ub}L^2(S)}^2\\leq C\\delta^2(\\mathcal R')^2.$$\nTo estimate the remaining terms, we will repeatedly invoke Proposition \\ref{angularpL4} to exchange $(\\nabla\\Psi)'$ and $\\nabla(\\Psi')$ (or $(\\nabla\\psi)'$ and $\\nabla(\\psi')$ etc.), at the expense of an error term $\\delta^2(\\mathcal R')^2$.\n\\begin{equation*}\n\\begin{split}\n&\\sum_{i_1+i_2+i_3\\leq 1}||(\\psi^{i_1}\\nabla^{i_2}\\psi\\nabla^{i_3}\\Psi)'||_{L^2_u L^2_{\\ub}L^2(S)}^2\\\\\n\\leq &C\\delta^2(\\mathcal R')^2+C(\\sum_{i_1\\leq 1}\\sum_{i_2\\leq 2}||\\nabla^{i_1}\\psi||_{L^\\infty_u L^\\infty_{\\ub}L^\\infty(S)}^{i_2})^2(\\sum_{i_3\\leq 1}||\\nabla^{i_3}\\Psi'||_{L^2_u L^2_{\\ub}L^2(S)}^2)\\\\\n&+C(\\sum_{i_1\\leq 1}||\\nabla^{i_1}\\psi||_{L^\\infty_u L^\\infty_{\\ub}L^\\infty(S)}^2)(\\sum_{i_2\\leq 1}||\\nabla^{i_3}\\Psi||_{L^2_u L^2_{\\ub}L^4(S)}^2)(\\sum_{i_3\\leq 1}||\\nabla^{i_1}\\psi'||_{L^\\infty_u L^\\infty_{\\ub}L^4(S)}^2)\\\\\n\\leq &C\\delta(a^2+(\\mathcal R')^2),\n\\end{split}\n\\end{equation*}\nwhere the second term is estimated by Theorem \\ref{timeofexistence} and the definition of $\\mathcal R'$; and the last term is estimated by Theorem \\ref{timeofexistence} and Proposition \\ref{Riccip}. Finally, we estimate using Theorem \\ref{timeofexistence} and Proposition \\ref{Riccip}\n\\begin{equation*}\n\\begin{split}\n&\\sum_{i_1+i_2+i_3\\leq 1}||(\\psi^{i_1}\\nabla^{i_2}\\psi\\nabla^{i_3}\\psi)'||_{L^2_u L^2_{\\ub}L^2(S)}^2\\\\\n\\leq &C\\delta^2(\\mathcal R')^2+C(\\sum_{i_1\\leq 1}\\sum_{i_2\\leq 2}||\\nabla^{i_1}\\psi||_{L^\\infty_u L^\\infty_{\\ub}L^\\infty(S)}^{i_2})^2(\\sum_{i_3\\leq 2}||\\nabla^{i_3}\\psi'||_{L^2_u L^2_{\\ub}L^2(S)}^2)\\\\\n\\leq &C\\delta(a^2+(\\mathcal R')^2),\n\\end{split}\n\\end{equation*}\nPutting these together, we have\n$$(\\mathcal R')^2\\leq Ca^2+C\\delta(\\mathcal R')^2.$$\nThe conclusion follows by choosing $\\delta$ sufficiently small depending only on $C$.\n\\end{proof}\nThis implies all the quantities that we have estimated in this section can be estimated by $Ca$:\n\\begin{proposition}\\label{p}\n$$||g'||_{L^\\infty_u L^\\infty_{\\ub} L^\\infty(S)},||\\frac{\\partial}{\\partial\\th}g'||_{L^\\infty_u L^\\infty_{\\ub} L^4(S)},||\\frac{\\partial^2}{\\partial\\th^2}g'||_{L^\\infty_u L^\\infty_{\\ub} L^2(S)},\\mathcal O',\\tilde{\\mathcal O'},\\mathcal R'\\leq Ca.$$\n\\end{proposition}\n\\begin{proof}\nThis is a direct consequence of Propositions \\ref{Omegap}, \\ref{gammap}, \\ref{dgammap}, \\ref{dOmegap}, \\ref{ddgammap}, \\ref{ddOmegap}, \\ref{bp}, \\ref{Riccip}, \\ref{Thetap}, \\ref{Ricciellipticp} and \\ref{curvaturep}.\n\\end{proof}\nThis concludes the proof of Theorem \\ref{convergencethm}.\n\n\\subsection{Additional Estimates on the Difference of the Metrics}\\label{AddEstMetric}\n\nIn this Subsection, we derive some additional estimates on the difference of the metrics. They follow directly from the bounds already derived. This will be used in Section \\ref{limit} to show that the limiting spacetime metric is in the space asserted in Theorem \\ref{convergencethm2} and satisfies the Einstein equations.\n\n\\begin{proposition}\\label{ugp}\n$$||\\frac{\\partial}{\\partial u}(\\gamma',\\Omega',b')||_{L^4(S_{u,\\ub})}\\leq Ca.$$\n\\end{proposition}\n\\begin{proof}\n$\\gamma$ satisfies a transport equation in the $\\Lb$ direction:\n$$\\Ls_{\\Lb}\\gamma=2\\Omega\\chib.$$\nFrom this we can derive an equation for $\\gamma'$:\n$$\\frac{\\partial}{\\partial u}\\gamma'=-b^A\\frac{\\partial}{\\partial\\th^A}\\gamma'-(b^A)'\\frac{\\partial}{\\partial\\th^A}\\gamma+2(\\Omega\\chi)'.$$\nEstimating the right hand side using Proposition \\ref{p}, we have\n$$||\\frac{\\partial}{\\partial u}\\gamma'||_{L^4(S_{u,\\ub})}\\leq Ca.$$\n$\\Omega$ also satisfies a transport equation in the $\\Lb$ direction:\n$$\\omegab=-\\frac 12\\nabla_3\\log\\Omega=\\frac 12\\Omega\\nab_3\\Omega^{-1}=\\frac 12(\\frac{\\partial}{\\partial u}+b^A\\frac{\\partial}{\\partial\\th^A})\\Omega^{-1}.$$\nAs for $\\gamma'$, we can derive an equation for $\\frac{\\partial}{\\partial u}(\\Omega^{-1})'$. By direct estimates using Proposition \\ref{p}, we have\n$$||\\frac{\\partial}{\\partial u}\\Omega'||_{L^4(S_{u,\\ub})}\\leq Ca.$$\nFinally, we move to $\\frac{\\partial}{\\partial u}b'$. $b$ does not satisfy a transport equation. We therefore resort to the equation\n$$\\frac{\\partial b^A}{\\partial \\ub}=-4\\Omega^2\\zeta^A.$$\nApplying $\\frac{\\partial}{\\partial u}$, we get\n$$\\frac{\\partial^2 b^A}{\\partial \\ub\\partial u}=-4\\frac{\\partial}{\\partial u}(\\Omega^2\\zeta^A).$$\nSince $(b')^A=0$ on $\\Hb_0$, $\\frac{\\partial b'}{\\partial u}=0$ on $\\Hb_0$.\nThus\n$$||\\frac{\\partial}{\\partial u}b'||_{L^4(S_{u,\\ub})}\\leq C\\int_0^{\\ub} ||\\frac{\\partial}{\\partial u}(\\Omega^2\\zeta^A)'||_{L^4(S_{u,\\ub'})}d\\ub'.$$\nBy Proposition \\ref{p}, the right hand side $\\leq Ca$.\n\\end{proof}\n\n\\begin{proposition}\\label{u2gp}\n$$||\\frac{\\partial^2}{\\partial u^2}(\\gamma',\\Omega',b')||_{L^2(S_{u,\\ub})}\\leq Ca.$$\n\\end{proposition}\n\\begin{proof}\nWe prove this Proposition by taking an extra $\\frac{\\partial}{\\partial u}$ derivative of the equations in the proof of Proposition \\ref{ugp}. By\n$$\\Ls_{\\Lb}\\gamma=2\\Omega\\chib,$$\nwe have\n$$\\frac{\\partial^2}{\\partial u^2}\\gamma=2\\frac{\\partial}{\\partial u}(\\Omega\\chib)-\\frac{\\partial}{\\partial u}(b^A\\frac{\\partial}{\\partial\\th^A}\\gamma).$$\nNotice that \n$$\\Omega\\frac{\\partial}{\\partial u}\\chib=\\nab_3\\chib-\\chib\\chib.$$\nThus using the null structure equation for $\\nab_3\\chib$, and estimating directly, we get\n$$||\\frac{\\partial^2}{\\partial u^2}\\gamma'||_{L^2(S_{u,\\ub})}\\leq Ca.$$\nTo estimate $\\frac{\\partial^2}{\\partial u^2}\\Omega$, we use\n$$\\omegab=\\frac 12(\\frac{\\partial}{\\partial u}+b^A\\frac{\\partial}{\\partial\\th^A})\\Omega^{-1}.$$\nDifferentiating with respect to $\\frac{\\partial}{\\partial u}$, using the estimates for $\\frac{\\partial}{\\partial u}b^A$ and the null structure equation for $\\nab_3\\omega$, we get\n$$||\\frac{\\partial^2}{\\partial u^2}\\Omega'||_{L^2(S_{u,\\ub})}\\leq Ca.$$\nFinally, differentiating \n$$\\frac{\\partial b^A}{\\partial \\ub}=-4\\Omega^2\\zeta^A$$\ntwice in the $u$ direction, we get\n$$\\frac{\\partial^3 b^A}{\\partial \\ub\\partial u^2}=-4\\frac{\\partial^2}{\\partial u^2}(\\Omega^2\\zeta^A).$$\nUsing the null structure equations as well as the Bianchi equation for $\\nab_3\\betab$, we see that all the terms on the right hand side can be bounded in $L^2(S_{u,\\ub})$. Thus we can integrate to get\n$$||\\frac{\\partial^2}{\\partial u^2}b'||_{L^2(S_{u,\\ub})}\\leq Ca.$$\n\\end{proof}\nIt is also possible to prove estimates for the mixed second derivatives\n\\begin{proposition}\\label{utgp}\n$$||\\frac{\\partial^2}{\\partial u\\partial\\th^A}(\\gamma',\\Omega',b')||_{L^2(S_{u,\\ub})}\\leq Ca.$$\n\\end{proposition}\n\\begin{proof}\nThis can be proved in a similar way as Proposition \\ref{ugp}, taking $\\frac{\\partial}{\\partial\\th^A}$ instead of $\\frac{\\partial}{\\partial u}$ derivative.\n\\end{proof}\nOn the other hand, $\\ub$ derivatives can only be taken once and can only be estimated after taking the $L^{p_0}_{\\ub}$ norm:\n\\begin{proposition}\\label{ubgp}\n$$||\\frac{\\partial}{\\partial \\ub}(\\gamma',\\Omega',b')||_{L^{p_0}_{\\ub}L^\\infty(S_{u,\\ub})}\\leq Ca.$$\n\\end{proposition}\n\\begin{proof}\nWe can directly estimate the right hand side of the equations\n$$\\frac{\\partial}{\\partial \\ub}\\gamma'=2(\\Omega\\chi)',$$\n$$\\frac 12\\frac{\\partial}{\\partial \\ub}(\\Omega^{-1})'=\\omega',$$\n$$\\frac{\\partial}{\\partial \\ub}(b^A)'=-4(\\Omega^2\\zeta^A)'.$$\n\\end{proof}\nThe mixed $\\ub$ and $\\th$ derivatives can also be estimates:\n\\begin{proposition}\\label{ubtgp}\n$$||\\frac{\\partial^2}{\\partial \\ub \\partial \\th^A}(\\gamma',\\Omega',b')||_{L^{p_0}_{\\ub}L^4(S_{u,\\ub})}\\leq Ca.$$\n\\end{proposition}\\label{ubugp}\n\\begin{proof}\nTake $\\frac{\\partial}{\\partial \\th^A}$ derivatives of the equations\n$$\\frac{\\partial}{\\partial \\ub}\\gamma'=2(\\Omega\\chi)',$$\n$$\\frac 12\\frac{\\partial}{\\partial \\ub}(\\Omega^{-1})'=\\omega',$$\n$$\\frac{\\partial}{\\partial \\ub}(b^A)'=-4(\\Omega^2\\zeta^A)',$$\nand apply the estimates in Proposition \\ref{p}.\n\\end{proof}\nSimilarly for the mixed $\\ub$ and $u$ derivatives:\n\\begin{proposition}\n$$||\\frac{\\partial^2}{\\partial \\ub \\partial u}(\\gamma',\\Omega',b')||_{L^{p_0}_{\\ub}L^2(S_{u,\\ub})}\\leq Ca.$$\n\\end{proposition}\n\\begin{proof}\nTake $\\frac{\\partial}{\\partial u}$ derivatives of the equations\n$$\\frac{\\partial}{\\partial \\ub}\\gamma'=2(\\Omega\\chi)',$$\n$$\\frac 12\\frac{\\partial}{\\partial \\ub}(\\Omega^{-1})'=\\omega',$$\n$$\\frac{\\partial}{\\partial \\ub}(b^A)'=-4(\\Omega^2\\zeta^A)',$$\nand use the null structure equations.\n\\end{proof}\nWe do not have estimates for the second $\\ub$ derivatives of the metric, since they can only be understood as measures even for the initial data. However, the following substitute is useful in showing that the limit spacetime that we construct satisfies the Einstein equations, in particular, $R_{u\\ub}=R_{A\\ub}=R_{\\ub\\ub}=0$.\n\\begin{proposition}\\label{trchp}\n$$||\\frac{\\partial^2}{\\partial \\ub^2}(b^A)'||_{L^{p_0}_{\\ub}L^\\infty(S_{u,\\ub})}\\leq Ca$$\nand\n$$||\\frac{\\partial}{\\partial \\ub}((\\gamma^{-1})^{AB}\\frac{\\partial}{\\partial \\ub}\\gamma_{AB})'||_{L^{p_0}_{\\ub}L^4(S_{u,\\ub})}\\leq Ca.$$\n\\end{proposition}\n\\begin{proof}\nThe first estimate follows directly from \n$$\\frac{\\partial}{\\partial \\ub}(b^A)'=-4(\\Omega^2\\zeta^A)'$$\nand the null structure equations.\n\nFor the second estimate, notice that in coordinates,\n$$\\trch=(\\gamma^{-1})^{AB}\\frac{\\partial}{\\partial \\ub}\\gamma_{AB}.$$\nThus we want to estimate $\\frac{\\partial}{\\partial\\ub}\\trch'$. Using the null structure equation\n$$\\nab_4\\trch+\\frac 12(\\trch)^2=-|\\chih|^2-2\\omega\\trch,$$\nwe have\n$$\\frac{\\partial}{\\partial\\ub}\\trch'=(-\\frac \\Omega 2(\\trch)^2-\\Omega|\\chih|^2-2\\Omega\\omega\\trch)'.$$\nThe conclusion follows from estimating the right hand side directly using Proposition \\ref{p}.\n\\end{proof}\n\n\n\\subsection{The Limiting Spacetime Metric}\\label{limit}\n\nIn this Subsection, we prove Theorems \\ref{convergencethm2} and \\ref{Einstein}. We therefore assume that the conditions of Theorem \\ref{convergencethm2} hold. Consider the sequence of metric\n$$g_n=-2(\\Omega_n)^2(du\\otimes d\\ub+d\\ub\\otimes du)+(\\gamma_n)_{AB}(d\\th^A-(b_n)^Adu)\\otimes (d\\th^B-(b_n)^Bdu).$$ \nBy Proposition \\ref{p}, $\\gamma_n$, $b_n$ and $\\Omega_n$ converge uniformly to their limiting values $\\gamma_{\\infty}$, $b_\\infty$ and $\\Omega_\\infty$ and are therefore continuous functions of $(u,\\ub,\\th^1,\\th^2)$ and define a continuous limiting spacetime metric\n\\begin{equation}\\label{limitmetric}\ng_\\infty=-2(\\Omega_\\infty)^2(du\\otimes d\\ub+d\\ub\\otimes du)+ (\\gamma_{\\infty})_{AB}(d\\th^A-(b_\\infty)^Adu)\\otimes(d\\th^B-(b_\\infty)^Bdu).\n\\end{equation}\nTo complete the proof of Theorem \\ref{convergencethm2}, we need to demonstrate the desired regularity of the limiting spacetime. This follows easily from the estimates in the previous Subsections.\n\\begin{proposition}\n$$(\\frac{\\partial}{\\partial \\th}g_n,\\frac{\\partial}{\\partial u}g_n)\\mbox{ converge to }(\\frac{\\partial}{\\partial \\th}g_\\infty,\\frac{\\partial}{\\partial u}g_\\infty)\\mbox{ in }L^\\infty_u L^\\infty_{\\ub} L^4(S),$$\n\\begin{equation*}\n\\begin{split}\n(\\frac{\\partial^2}{\\partial \\th^2}g_n,\\frac{\\partial^2}{\\partial u\\partial\\th}g_n,\\frac{\\partial^2}{\\partial u^2}g_n)\\mbox{ converge to }\\\\\n(\\frac{\\partial^2}{\\partial \\th^2}g_\\infty,\\frac{\\partial^2}{\\partial u\\partial\\th}g_\\infty,\\frac{\\partial^2}{\\partial u^2}g_{\\infty})\\mbox{ in }L^\\infty_u L^\\infty_{\\ub} L^2(S),\n\\end{split}\n\\end{equation*}\n\\begin{equation*}\n\\begin{split}\n(\\frac{\\partial}{\\partial \\ub}g_n, \\frac{\\partial}{\\partial\\ub}((\\gamma_n^{-1})^{AB}\\frac{\\partial}{\\partial\\ub}(\\gamma_n)_{AB}))\\mbox{ converge to }\\\\\n(\\frac{\\partial}{\\partial \\ub}g_{\\infty}, \\frac{\\partial}{\\partial\\ub}((\\gamma_{\\infty}^{-1})^{AB}\\frac{\\partial}{\\partial\\ub}(\\gamma_{\\infty})_{AB}))\\mbox{ in }L^\\infty_u L^{p_0}_{\\ub} L^\\infty(S),\n\\end{split}\n\\end{equation*}\n\\begin{equation*}\n\\begin{split}\n(\\frac{\\partial^2}{\\partial \\th \\partial \\ub}g_n,\\frac{\\partial^2}{\\partial u\\partial\\ub}g_n,\\frac{\\partial^2}{\\partial \\ub^2} (b^A)_n)\\mbox{ converge to }\\\\\n(\\frac{\\partial^2}{\\partial \\th \\partial \\ub}g_\\infty,\\frac{\\partial^2}{\\partial u\\partial\\ub}g_\\infty,\\frac{\\partial^2}{\\partial \\ub^2} (b^A)_{\\infty})\\mbox{ in }L^\\infty_u L^{p_0}_{\\ub} L^4(S).\n\\end{split}\n\\end{equation*}\n\\end{proposition}\n\\begin{proof}\nThat\n$$\\frac{\\partial}{\\partial \\th}g_n,\\frac{\\partial}{\\partial u}g_n\\mbox{ converge in }L^\\infty_u L^\\infty_{\\ub} L^4(S)$$\nfollows from Propositions \\ref{p} and \\ref{ugp} respectively.\n\nThat\n$$\\frac{\\partial^2}{\\partial \\th^2}g_n,\\frac{\\partial^2}{\\partial u^2}g_n,\\frac{\\partial^2}{\\partial u\\partial\\th}g_n\\mbox{ converge in } L^\\infty_u L^\\infty_{\\ub} L^2(S)$$\nfollows from Propositions \\ref{p}, \\ref{u2gp} and \\ref{utgp} respectively.\n\nThat\n$$\\frac{\\partial}{\\partial \\ub}g_n, \\frac{\\partial}{\\partial\\ub}((\\gamma_n^{-1})^{AB}\\frac{\\partial}{\\partial\\ub}(\\gamma_n)_{AB})\\mbox{ converge in }L^\\infty_u L^{p_0}_{\\ub} L^\\infty(S)$$\nfollows from Propositions \\ref{ubgp} and \\ref{trchp} respectively.\n\nThat\n$$\\frac{\\partial^2}{\\partial \\th \\partial \\ub}g_n,\\frac{\\partial^2}{\\partial u\\partial\\ub}g_n,\\frac{\\partial^2}{\\partial \\ub^2} (b^A)_n\\mbox{ converge in }L^\\infty_u L^{p_0}_{\\ub} L^4(S)$$\nfollows from Propositions \\ref{utgp}, \\ref{ubugp} and \\ref{trchp} respectively.\n\\end{proof}\n\\begin{proposition}\\label{gspace}\n$$\\frac{\\partial}{\\partial \\th}g_\\infty,\\frac{\\partial}{\\partial u}g_\\infty\\in C^0_u C^0_{\\ub} L^4(S),$$\n$$\\frac{\\partial^2}{\\partial \\th^2}g_\\infty,\\frac{\\partial^2}{\\partial u\\partial\\th}g_\\infty,\\frac{\\partial^2}{\\partial u^2}g_\\infty\\in C^0_u C^0_{\\ub} L^2(S),$$\n$$\\frac{\\partial}{\\partial \\ub}g_\\infty, \\frac{\\partial}{\\partial\\ub}((\\gamma_\\infty^{-1})^{AB}\\frac{\\partial}{\\partial\\ub}(\\gamma_\\infty)_{AB}) \\in L^\\infty_u L^\\infty_{\\ub} L^\\infty(S),$$\n$$\\frac{\\partial^2}{\\partial \\th \\partial \\ub}g_\\infty,\\frac{\\partial^2}{\\partial u\\partial\\ub}g_\\infty,\\frac{\\partial^2}{\\partial \\ub^2} (b^A)_{\\infty}\\in L^\\infty_u L^\\infty_{\\ub} L^4(S).$$\n\\end{proposition}\n\\begin{proof}\nTo prove the first two statements, notice that since $g_n$ are smooth, the convergence in $L^\\infty_u L^\\infty_{\\ub}L^p(S)$ implies that the limit is in $C^0_u C^0_{\\ub} L^p(S)$.\n\nFor the latter two statements, we use the fact that for $p_0<\\infty$, if $f_n\\to f$ in $L^{p_0}$ and $f_n$ is uniformly bounded in $L^\\infty$, then $f\\in L^\\infty$.\n\\end{proof}\n\n\nProposition \\ref{gspace} allows us to conclude that all first derivatives of $g_{\\infty}$ can be defined and belong to an appropriate space. These norms also allow us to conclude that the product of any two first derivatives of the metric belongs to $L^\\infty_u L^\\infty_{\\ub}L^2(S)$. For the second derivatives, Proposition \\ref{gspace} shows that all second derivatives of the metric except $\\frac{\\partial^2}{\\partial \\ub^2}(\\gamma,\\Omega)$ are defined as functions in $L^\\infty L^{p_0}_{\\ub}L^2(S)$. The second derivative $\\frac{\\partial^2}{\\partial \\ub^2}(\\gamma,\\Omega)$ is merely a distribution. As we will see below, its definition is not necessary to make sense of the Einstein equations. \n\nWe now show that all components of the curvature for the limiting spacetime, except for $R_{\\ub A\\ub B}$, $R_{\\ub u \\ub A}$ and $R_{\\ub u \\ub u}$, can be defined at worst as functions in $L^\\infty_uL^\\infty_{\\ub}L^2(S)$. For this we use the coordinate definition of the Riemann curvature tensor:\n\\begin{equation}\\label{curvdef}\nR_{\\delta\\sigma\\mu\\nu}=g_{\\delta\\rho}(\\frac{\\partial}{\\partial x^{\\mu}}\\Gamma^\\rho_{\\nu\\sigma}-\\frac{\\partial}{\\partial x^{\\nu}}\\Gamma^\\rho_{\\mu\\sigma}+\\Gamma^{\\rho}_{\\mu\\lambda}\\Gamma^\\lambda_{\\nu\\sigma}-\\Gamma^\\rho_{\\nu\\lambda}\\Gamma^\\lambda_{\\mu\\sigma}).\n\\end{equation}\n\\begin{proposition}\nIn the limiting spacetime, all components of the Riemann curvature tensor, defined by the coordinate expression (\\ref{curvdef}) expect for $R_{\\ub A\\ub B}$, $R_{\\ub u \\ub A}$ and $R_{\\ub u \\ub u}$, can be defined as functions in $L^\\infty_uL^\\infty_{\\ub}L^2(S)$. Moreover, $R(L,\\Lb,L,e_A)$ and $R(L,\\Lb,L,\\Lb)$ can be defined as functions in $L^\\infty_uL^\\infty_{\\ub}L^2(S)$.\n\\end{proposition}\n\\begin{proof}\nUsing (\\ref{curvdef}), it is easy to see that if $\\delta,\\sigma,\\mu,\\nu\\neq \\ub$, the expression has at most one $\\ub$ derivative. Thus, by the regularity properties of $g_\\infty$, these components of the Riemann curvature tensor are well-defined as $L^\\infty_uL^\\infty_{\\ub}L^2(S)$ functions. \n\nIn the case that exactly one of $\\delta,\\sigma,\\mu,\\nu$ is equal to $\\ub$, we can assume, by the symmetry properties of the Riemann curvature tensor, that $\\delta=\\ub$. Then, the formula for the component of the Riemann curvature tensor has at most one $\\ub$ derivative and by the regularity properties, it is well-defined as $L^\\infty_uL^\\infty_{\\ub}L^2(S)$ functions. Notice, however, that strictly speaking, at the level of regularity that we have, the symmetry properties of the Riemann curvature tensor may not hold. Consider, for example, the component $R_{12\\ub 1}$. Using the formula above, we would have a term $g_{1A}(g^{A\\ub}g_{12,\\ub})_{,\\ub}$, which is not defined. However, the sum of \\emph{all} terms containing two $\\ub$ derivatives is equal to the expression\n$$g_{1\\alpha}\\frac{\\partial}{\\partial \\ub}(g^{\\ub\\alpha}\\frac{\\partial}{\\partial \\ub}g_{12}).$$\nwhich can be rearranged, up to terms with at most one $\\ub$ derivative, in the form\n$$\\frac{\\partial}{\\partial \\ub}(g_{1\\alpha}g^{\\ub\\alpha}\\frac{\\partial}{\\partial \\ub}g_{12}).$$\nThis expression vanishes since $g_{1\\alpha}g^{\\ub\\alpha}=0$. The above calculation is a reflection of the symmetry properties of the Riemann curvature tensor and provides an appropriate distributional definition of $R_{\\sigma\\delta\\mu\\nu}$ for which the symmetry properties formally hold.\n\nWe have therefore shown that all components of curvature with the exception of $R_{\\ub A\\ub B}$, $R_{\\ub u \\ub A}$ and $R_{\\ub u \\ub u}$ are defined as $L^\\infty_uL^\\infty_{\\ub}L^2(S)$ functions. However, replacing the double null coordinate system $(u,\\ub,\\th^1,\\th^2)$ by the null frame $(L,\\Lb,e^1,e^2)$, we can show that additional components of curvature are defined as functions in $L^\\infty_uL^\\infty_{\\ub}L^2(S)$. To see this, we can write\n$$R(L,\\Lb,L,e_A)=R_{\\ub u \\ub A}+ b^B R_{\\ub B\\ub A},$$\nand\n$$R(L,\\Lb,L,\\Lb)=R_{\\ub u \\ub u}+ b^B R_{\\ub u \\ub B}+b^A R_{\\ub A \\ub u}+b^A b^B R_{\\ub A\\ub B}=R_{\\ub u \\ub u}+2b^A R_{\\ub A \\ub u}+b^A b^B R_{\\ub A\\ub B}.$$\nNote that\n$$R_{\\ub u \\ub u}=-\\frac 12 g_{uu,\\ub\\ub}+\\mbox{good terms}=-\\frac 12 b^A b^B (\\frac{\\partial}{\\partial \\ub})^2 g_{AB}+\\mbox{good terms},$$\n$$R_{\\ub u \\ub A}=-\\frac 12 g_{Au,\\ub\\ub}+\\mbox{good terms}=\\frac 12 b^B (\\frac{\\partial}{\\partial \\ub})^2 g_{AB}+\\mbox{good terms},$$\n$$R_{\\ub B\\ub A}=-\\frac 12 g_{AB,\\ub\\ub}+\\mbox{good terms},$$\nwhere good terms denote terms with at most one $\\ub$ derivative. Therefore, the terms with two $\\ub$ derivatives cancel in $R(L,\\Lb,L,A)$ and $R(L,\\Lb,L,\\Lb)$ and they can be defined as $L^\\infty_uL^\\infty_{\\ub}L^2(S)$ functions.\n\\end{proof}\nWe now verify that the limiting spacetime that we have constructed satisfies the vacuum Einstein equations in the sense that relative to the system of double null coordinates $(u,\\ub,\\th^1,\\th^2)$,\n$$R_{\\mu\\nu}=0\\quad\\mbox{in }L^\\infty_uL^\\infty_{\\ub}L^2(S).$$\nFor this, we use the coordinate definition of the Ricci curvature:\n\\begin{equation*}\nR_{\\mu\\nu}=\\frac{\\partial}{\\partial x^{\\rho}}\\Gamma^\\rho_{\\mu\\nu}-\\frac{\\partial}{\\partial x^{\\nu}}\\Gamma^\\rho_{\\rho\\mu}+\\Gamma^\\rho_{\\rho\\lambda}\\Gamma^{\\lambda}_{\\mu\\nu}-\\Gamma^\\rho_{\\mu\\lambda}\\Gamma^{\\lambda}_{\\rho\\nu}.\n\\end{equation*}\n\\begin{proposition}\nThe limiting spacetime satisfies the vacuum Einstein equations in the sense that relative to the system of double null coordinates $(u,\\ub,\\th^1,\\th^2)$,\n$$R_{\\mu\\nu}=0\\quad\\mbox{in }L^\\infty_uL^\\infty_{\\ub}L^2(S).$$.\n\\end{proposition}\n\\begin{proof}\nNow, notice that with the metric given by (\\ref{limitmetric}), we have\n$$(g_{\\infty}^{-1})^{\\ub\\ub}=(g_{\\infty}^{-1})^{uu}=(g_{\\infty}^{-1})^{u1}=(g_{\\infty}^{-1})^{u2}=0.$$\nTherefore,\n$\\Gamma^{\\ub}_{\\mu\\nu}$ does not contain a term with $g_{\\mu\\nu,\\ub}$. In other words, the expressions for \n$$R_{uu}, R_{uA}, R_{AB}\\quad\\mbox{for $A,B=1,2$}$$\ndo not contain two $\\ub$ derivatives. Therefore, we have \n$$R_{uu}= R_{uA}= R_{AB}= 0$$\nas functions in $L^\\infty_uL^\\infty_{\\ub}L^2(S)$. For the component $R_{u\\ub}$, there are terms involving two $\\ub$ derivatives of the metric. In particular, we have\n\\begin{equation*}\n\\begin{split}\nR_{u\\ub}=&\\frac{\\partial}{\\partial \\ub}\\Gamma^{\\ub}_{u\\ub}-\\frac{\\partial}{\\partial \\ub}\\Gamma^{\\ub}_{\\ub u}-\\frac{\\partial}{\\partial\\ub}\\Gamma^A_{Au}-\\frac{\\partial}{\\partial\\ub}\\Gamma^u_{uu}\\\\\n&+\\mbox{terms involving at most one $\\ub$ derivative of the metric}\\\\\n=&\\frac{-1}{4\\Omega^2}\\gamma_{AB}b^A\\frac{\\partial^2}{\\partial\\ub^2} b^B+\\mbox{terms involving at most one $\\ub$ derivative of the metric}.\n\\end{split}\n\\end{equation*}\nFrom this expression, we see that except for the second $\\ub$ derivatives of $b$, the terms involving two $\\ub$ derivatives of the metric cancel. Recall from Proposition \\ref{gspace} that $\\frac{\\partial^2}{\\partial\\ub^2} b^A$ is in $L^\\infty_uL^\\infty_{\\ub}L^2(S)$. Thus we also have\n$$R_{u\\ub}=0$$\nas functions in $L^\\infty_uL^\\infty_{\\ub}L^2(S)$. Similarly, for $R_{A\\ub}$, we have\n\\begin{equation*}\n\\begin{split}\nR_{A\\ub}=&\\frac{\\partial}{\\partial \\ub}\\Gamma^{\\ub}_{A\\ub}-\\frac{\\partial}{\\partial \\ub}\\Gamma^{\\ub}_{\\ub A}-\\frac{\\partial}{\\partial\\ub}\\Gamma^B_{BA}-\\frac{\\partial}{\\partial\\ub}\\Gamma^u_{uA}\\\\\n&+\\mbox{terms involving at most one $\\ub$ derivative of the metric}\\\\\n=&\\frac{1}{4\\Omega^2}\\gamma_{AB}\\frac{\\partial^2}{\\partial\\ub^2} b^B+\\mbox{terms involving at most one $\\ub$ derivative of the metric}.\n\\end{split}\n\\end{equation*}\nAs for $R_{u\\ub}$, because of the cancellation, we have\n$$R_{\\ub A}=0$$\nas functions in $L^\\infty_{u}L^\\infty_{\\ub}L^2(S)$.\n\nIt now remains to study $R_{\\ub\\ub}$. Indeed this term involves two $\\ub$ derivatives of the metric and there are no cancellations to remove this term. We have\n\\begin{equation*}\n\\begin{split}\nR_{\\ub\\ub}=&\\frac{\\partial}{\\partial \\ub}\\Gamma^{\\ub}_{\\ub\\ub}-\\frac{\\partial}{\\partial \\ub}\\Gamma^{\\ub}_{\\ub\\ub}+\\frac{\\partial}{\\partial \\th^A}\\Gamma^{A}_{\\ub\\ub}+\\frac{\\partial}{\\partial \\ub}\\Gamma^A_{A\\ub}\\\\\n&+\\mbox{terms involving at most one $\\ub$ derivative of the metric}.\n\\end{split}\n\\end{equation*}\nThe first two terms cancel each other. Note that\n$$\\Gamma^A_{\\ub\\ub}=0$$\nsince\n$$g_{\\ub\\ub}=g_{\\ub B}=0,\\quad g^{Au}=0.$$\nWe are thus left with\n$$\\frac{\\partial}{\\partial \\ub}\\Gamma^A_{A\\ub}=\\frac{\\partial}{\\partial \\ub}(g^{AB}\\frac{\\partial}{\\partial \\ub}g_{AB}).$$\nIt might seem that this term behaves like $\\frac{\\partial^2}{\\partial \\ub^2}\\gamma$ which we cannot control. However, this particular combination of derivatives, by Proposition \\ref{gspace}, is in fact in $L^\\infty_uL^\\infty_{\\ub}L^4(S)$. Thus $R_{\\ub\\ub}=0$ in $L^\\infty_uL^\\infty_{\\ub}L^2(S)$.\n\\end{proof}\n\n\\subsection{Uniqueness}\\label{uniquenesssec}\n\nIn the Subsection, we show that the spacetime constructed in Theorem \\ref{convergencethm2} is the unique spacetime among the class of continuous spacetimes which arise as $C^0$ limits of smooth spacetime solutions to the vacuum Einstein equations. In view of Theorem \\ref{convergencethm}, uniqueness follows easily if the spacetime solution is assumed to be in the class of spacetimes satisfying the conclusions of \\ref{convergencethm}. Nevertheless, we prove uniqueness within a larger class of spacetimes for which it is not a priori assumed that the spacetime is Lipschitz, or that it is more regular in the $u$ and the angular directions.\n\nThe following is a precise formulation of the uniqueness theorem (Theorem \\ref{uniquenessthm}) and is the main result in this Subsection:\n\\begin{proposition}\\label{uniquenessprop}\nLet $(\\mathcal M^{(1)}, g^{(1)})$ and $(\\mathcal M^{(2)}, g^{(2)})$ be two $C^0$ Lorentzian spacetimes in double null coordinates, defined in $0\\leq u\\leq u_*, 0\\leq \\ub\\leq \\ub_*$ and taking the form\n$$g^{(i)}=-2(\\Omega^{(i)})^2(du\\otimes d\\ub+d\\ub\\otimes du)+(\\gamma^{(i)})_{AB}(d\\th^A-(b^{(i)})^Adu)\\otimes (d\\th^B-(b^{(i)})^Bdu)$$ \nsuch that\n\\begin{itemize}\n\\item $\\Omega=1$ and $b^A=0$ on $H_0$ and $\\Hb_0$ and $\\gamma^{(1)}|_{H_0}=\\gamma^{(2)}|_{H_0}$ and $\\gamma^{(1)}|_{\\Hb_0}=\\gamma^{(2)}|_{\\Hb_0}$\n\\item for $i=1,2$, there exists a sequence of smooth spacetimes $(\\mathcal M_n^{(i)}, g^{(i)}_n)$ such that for every $n$, $g^{(i)}_n$ is defined in $0\\leq u\\leq u_*$ and $0\\leq \\ub\\leq \\ub_*$ and takes the following form in double null coordinates:\n$$g^{(i)}_n=-2(\\Omega^{(i)}_n)^2(du\\otimes d\\ub+d\\ub\\otimes du)+(\\gamma^{(i)}_n)_{AB}(d\\th^A-(b^{(i)}_n)^Adu)\\otimes (d\\th^B-(b^{(i)}_N)^Bdu)$$ \nwith the property that\n\\begin{enumerate}\n\\item $$g^{(i)}_n \\to g^{(i)} \\quad\\mbox{in }C^0\\mbox{ in }0\\leq u\\leq u_*, 0\\leq \\ub\\leq \\ub_*,$$\n\n\\item The initial data for $g^{(i)}_n$ satisfy the assumptions of Theorem \\ref{timeofexistence} uniformly,\n\n\\item The initial data for $g^{(i)}_n$ converges to the initial data for $g^{(i)}$, i.e.,\n$$g^{(i)}_n|_{H_0} \\to g^{(i)}|_{H_0} \\quad\\mbox{and }g^{(i)}_n|_{\\Hb_0} \\to g^{(i)}|_{\\Hb_0}$$\nin the norms in the assumptions of Theorem \\ref{convergencethm}.\n\\end{enumerate}\n\\end{itemize}\nThen, if $u_*, \\ub_*\\leq \\epsilon$, where $\\epsilon$ is as given in Theorem \\ref{timeofexistence},\n$$g^{(1)}=g^{(2)}\\quad\\mbox{in }0\\leq u\\leq u_*, 0\\leq \\ub\\leq \\ub_* .$$\n\\end{proposition}\n\\begin{proof}\nBy Property $(3)$ in the assumptions, for every $i$, there exists $n_i$ such that \n$$\\sup_u|(\\frac{\\partial}{\\partial\\th})^i(\\gamma_{AB})_{n_i}'(\\ub=0)|+\\sup_{\\ub}||(\\frac{\\partial}{\\partial\\th})^i(\\gamma_{AB})_{n_i}'(u=0)|\\leq 2^{-i},$$\n$$\\sum_{\\psi\\neq\\chih,\\omega}(\\sum_{i\\leq 1} \\sup_{u}||\\nabla^i\\psi_{n_i}'||_{L^2(S_{u,0})}+\\sum_{i\\leq 1} \\sup_{\\ub}||\\nabla^i\\psi_{n_i}'||_{L^2(S_{0,\\ub})})\\leq 2^{-i},$$\n$$\\sum_{i\\leq 1} ||\\nabla^i(\\chih_{n_i}',\\omega_{n_i}')||_{L^{p_0}_{\\ub}L^2(S_{0,\\ub})}\\leq 2^{-i}\\quad\\mbox{for some fixed }2\\leq p_0<\\infty,$$\n$$||\\nabla^2(\\chih_{n_i}',\\omega_{n_i}',\\eta_{n_i}',\\etab_{n_i}')||_{L^2(H_0)}+\\sup_{\\ub} ||\\nabla^2(\\chibh_{n_i}',\\omegab_{n_i}',\\eta_{n_i}',\\etab_{n_i}')||_{L^2(\\Hb_{0})}\\leq 2^{-i},$$\n$$\\sup_{u} ||\\nabla^2(\\trch_{n_i}',\\trchb_{n_i}')||_{L^2(S_{u,0})}+\\sup_{\\ub} ||\\nabla^2(\\trch_{n_i}',\\trchb_{n_i}')||_{L^2(S_{0,\\ub})}\\leq 2^{-i},$$\n$$\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}\\sum_{i\\leq 1} ||\\nabla^i\\Psi_{n_i}'||_{L^{2}_{\\ub}L^2(S_{0,\\ub})}\\leq 2^{-i}.$$\nBy Theorem \\ref{convergencethm}, there exists $C$ depending only on the uniform bounds in the assumption $(2)$ such that\n$$|g^{(1)}_{n_i}-g^{(2)}_{n_i}|\\leq C2^{-i}.$$\nTherefore, in $C^0$ norm,\n$$|g^{(1)}-g^{(2)}|\\leq C|g^{(1)}_{n_i}-g^{(2)}_{n_i}|\\leq C2^{-i}.$$\nSince this holds for every $i$, we have\n$$g^{(1)}=g^{(2)}.$$\n\\end{proof}\n\n\\subsection{Propagation of Regularity}\\label{regularityp}\nUsing Proposition \\ref{propagationregularity}, we show that if the initial data is more regular in the $\\nab_3$ and $\\nab$ directions, then so is the limiting spacetime.\n\\begin{proposition}\\label{regularityprop}\nSuppose, in addition to the assumptions of Theorem \\ref{convergencethm}, we have the following estimates for the initial data:\n$$\\sum_{j\\leq J}\\sum_{i\\leq I}(\\sum_{\\Psi\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}}||\\nab_3^j \\nab^i\\Psi||_{L^2(\\Hb_0)}+\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}||\\nab_3^i \\nab^j\\Psi||_{L^2(H_0)})\\leq C. $$\nThen\n$$\\sum_{j\\leq J}\\sum_{i\\leq I}(\\sum_{\\Psi\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}}\\sup_{\\ub}||\\nab_3^j \\nab^i\\Psi||_{L^2(\\Hb_{\\ub})}+\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}\\sup_u||\\nab_3^i \\nab^j\\Psi||_{L^2(H_u)})\\leq C'.$$\n\\end{proposition}\n\\begin{proof}\nBy Proposition \\ref{propagationregularity}, for an approximating sequence of spacetimes, the following bound holds independent of $n$:\n$$\\sum_{j\\leq J}\\sum_{i\\leq I}(\\sum_{\\Psi\\in\\{\\rho,\\sigma,\\betab,\\alphab\\}}\\sup_{\\ub}||\\nab_3^j \\nab^i\\Psi_n||_{L^2(\\Hb_{\\ub})}+\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}\\sup_u||\\nab_3^i \\nab^j\\Psi_n||_{L^2(H_u)})\\leq C'.$$\nThe conclusion thus follows.\n\\end{proof}\nUsing the equations\n$$\\frac{\\partial}{\\partial \\ub}\\gamma=2\\Omega\\chi,$$\n$$\\frac 12\\frac{\\partial}{\\partial \\ub}\\Omega^{-1}=\\omega,$$\n$$\\frac{\\partial}{\\partial \\ub}b^A=-4\\Omega^2\\zeta^A,$$\ntogether with Proposition \\ref{regularityprop}, we can show that $g$ is in the desired space as indicated in Theorem \\ref{regularitythm}. The details are straightforward and will be omitted.\n\n\\subsection{The Limiting Spacetime for an Impulsive Gravitational Wave}\\label{limitgiw}\nIn the Subsection, we show that for the case of an impulsive gravitational wave, the spacetime is more regular than a general spacetime with merely bounded $\\chih$, thus concluding the proof of Theorem \\ref{giwthmv1}. In particular, we show that $\\alpha_\\infty$ is well defined as a measure which is the weak limit of $\\alpha_n$. All other curvature components can be defined in $L^\\infty$. Moreover, the singularity of $\\alpha_\\infty$ is supported on a null hypersurface and the constructed spacetime is smooth away from this null hypersurface.\n\n\\subsubsection{Limit of the Curvature Component $\\alpha$}\n\nWe first show that $\\alpha_n$ has a well defined limit as a measure.\n\\begin{proposition}\\label{alphalimit}\nIn the case of an impulsive gravitational wave, for every $u$ and every $\\vartheta=(\\th^1,\\th^2)\\in\\mathbb S^2$, $\\alpha_\\infty(u,\\vartheta)$ is a measure on $[0,\\ub_*]$, defined as the limit of $\\alpha_n(u,\\vartheta)$\n\\end{proposition}\n\\begin{proof}\nBy Proposition \\ref{totalvariation}, for every $u$ and $\\vartheta$, $\\alpha_n$ has uniformly bounded $L^1_{\\ub}$ norms. Thus to show convergence of $\\alpha_n$, we only need to show that for every $u$ and every $\\vartheta=(\\th^1,\\th^2)\\in\\mathbb S^2$, \n$$\\int_0^{\\ub} \\alpha_n(u,\\ub',\\vartheta) d\\ub' $$\nconverges to a limit $\\alpha_{\\infty}(u,\\vartheta)([0,\\ub))$ as $n\\to\\infty$ for all $\\ub$ such that $\\alpha_{\\infty}(u,\\vartheta)([0,\\ub))$ is continuous.\nBy the equation\n$$\\nab_4\\chih+\\trch\\chih=-2\\omega\\chih-\\alpha,$$\nwe have\n$$\\int_0^{\\ub} \\alpha_n(u,\\ub',\\vartheta) d\\ub'=\\int_0^{\\ub} (\\Omega^{-1}_n\\frac{\\partial}{\\partial\\ub}\\chih_n+\\trch_n\\chih_n+2\\omega_n\\chih_n)(u,\\ub',\\vartheta) d\\ub'.$$\nIntegrating by parts and using \n$$\\frac{\\partial}{\\partial\\ub}\\Omega^{-1}_n=2\\omega_n,$$\nwe derive\n\\begin{equation}\\label{intalpha}\n\\int_0^{\\ub} \\alpha_n(u,\\vartheta) d\\ub=(\\Omega^{-1}_n\\chih_n)(u,\\ub,\\vartheta)-(\\Omega^{-1}_n\\chih_n)(u,\\ub=0,\\vartheta)+\\int_0^{\\ub} (\\trch_n\\chih_n)(u,\\ub',\\vartheta) d\\ub'.\n\\end{equation}\nBy Proposition \\ref{p}, \n$$\\int_0^{\\ub} (\\trch_n\\chih_n)(u,\\ub',\\vartheta) d\\ub'$$\nconverges. Thus, in order to show weak convergence of $\\alpha_n$, we need to show pointwise convergence for $\\Omega^{-1}_n\\chih_n$ for all $\\ub$ such that $\\alpha_{\\infty}(u,\\vartheta)([0,\\ub))$ is continuous. By the construction of the data for an impulsive gravitational wave in Section \\ref{initialcondition}, we know that for the initial data\n$$(\\chih'_n)_{AB}(u=0,\\ub,\\vartheta)\\leq C2^{-n}\\quad\\mbox{for }\\ub\\leq\\ub_s\\mbox{ or }\\ub\\geq\\ub_s+2^{-n}.$$\nUsing the equation\n$$\\nabla_3\\chih+\\frac 12\\trchb\\chih=\\nabla\\hot\\eta+2\\omegab\\chih-\\frac 12\\trch\\chibh+\\eta\\hot\\eta,$$\none sees that \n$$(\\chih'_n)_{AB}(u,\\ub,\\vartheta)\\leq C2^{-n}\\quad\\mbox{for }\\ub\\leq\\ub_s\\mbox{ or }\\ub\\geq\\ub_s+2^{-n}.$$\nTherefore, $\\chih_n$ converges for all $\\ub\\neq\\ub_s$. By Proposition \\ref{convergencethm2}, $\\Omega^{-1}_n\\chih_n$ also converges for all $\\ub\\neq\\ub_s$.\n\nIt remains to show that the limit $\\alpha_{\\infty}(u,\\vartheta)([0,\\ub))$ is discontinuous at $\\ub=\\ub_s$. In view of (\\ref{intalpha}), it suffices to show that $\\chih$ has a jump discontinuity across $\\Hb_{\\ub_s}$. This follows from the equation\n$$\\nabla_3\\chih+\\frac 12\\trchb\\chih=\\nabla\\hot\\eta+2\\omegab\\chih-\\frac 12\\trch\\chibh+\\eta\\hot\\eta.$$\nRewriting in coordinates and expressing $\\chih$, $\\chibh$, $\\eta$, $\\etab$ in terms of the coordinate vector fields $\\frac{\\partial}{\\partial\\th^1}$ and $\\frac{\\partial}{\\partial\\th^2}$ on the spheres, we have\n\\begin{equation*}\n\\begin{split}\n&\\Omega^{-1}(\\frac{\\partial}{\\partial u}+b^A\\frac{\\partial}{\\partial\\th^A})\\chih-\\gamma^{-1}\\chib\\chih+\\frac 1 2 \\trchb \\chih\\\\\n=&\\nab\\widehat{\\otimes} \\eta+2\\omegab \\chih-\\frac 12 \\trch \\chibh +\\eta\\widehat{\\otimes} \\eta.\n\\end{split}\n\\end{equation*}\nConsider the coordinate system $(u,\\ub,\\tilde{\\th}^1,\\tilde{\\th}^2)$ such that \n$$\\frac{d}{d\\ub}\\tilde{\\th}^A(\\ub;u,\\th)=b^A(u,\\ub,\\tilde\\th^1,\\tilde\\th^2),$$\nwith initial data\n$$\\tilde{\\th}^A(0;u,\\th)=\\th^A.$$\nAccording to the proven estimates for $b$, this change of variable is $W^{2,\\infty}$. Therefore, the vector fields $\\frac{\\partial}{\\partial\\tilde{\\th}^1}$ and $\\frac{\\partial}{\\partial\\tilde{\\th}^2}$ associated to the new coordinate system are $W^{1,\\infty}$ with respect to the differentiable structure given by the original coordinate system $(u,\\ub,\\th^1,\\th^2)$. In the new coordinate system, we can rewrite the transport equation for $\\chih$ as\n$$\\Omega^{-1}\\frac{\\partial}{\\partial u}\\chih-\\gamma^{-1}\\chib\\chih+\\frac 1 2 \\trchb \\chih\n=\\nab\\widehat{\\otimes} \\eta+2\\omegab \\chih-\\frac 12 \\trch \\chibh +\\eta\\widehat{\\otimes} \\eta,$$\nwhere $\\chih$, $\\chibh$, $\\eta$, $\\etab$ are now expressed in terms of the new coordinate vector fields $\\frac{\\partial}{\\partial\\tilde{\\th}^1}$ and $\\frac{\\partial}{\\partial\\tilde{\\th}^2}$.\nNotice that $\\chib$ is continuous and the expression\n$$\\nab\\widehat{\\otimes} \\eta+2\\omegab \\chih-\\frac 12 \\trch \\chibh +\\eta\\widehat{\\otimes} \\eta$$\nis also continuous. Since for the initial data, $\\chih(\\tilde{u}_0,\\ub,\\theta)$ has a jump discontinuity for $\\ub=\\ub_s$, $\\chih_{\\tilde{A}\\tilde{B}}$ also has a jump discontinuity across $\\ub=\\ub_s$. As noted before, the the vector fields $\\frac{\\partial}{\\partial\\tilde{\\th}^1}$ and $\\frac{\\partial}{\\partial\\tilde{\\th}^2}$ associated to the new coordinate system are $W^{1,\\infty}$ with respect to the differentiable structure given by the original coordinate system $(u,\\ub,\\th^1,\\th^2)$. Therefore, $\\chih_{AB}$ also has a jump discontinuity.\n\\end{proof}\n\n\\subsubsection{Control of the Curvature Components $\\beta,\\rho,\\sigma,\\betab,\\alphab$}\n\nExcept for the curvature component $\\alpha$ which can only be defined as a measure, all other curvature components can be defined as $L^\\infty_uL^\\infty_{\\ub}L^\\infty(S)$ functions:\n\n\\begin{proposition}\n$\\beta,\\rho,\\sigma,\\betab,\\alphab$ can be defined as functions in $L^\\infty_uL^\\infty_{\\ub}L^\\infty(S)$.\n\\end{proposition}\n\\begin{proof}\nThat $\\rho,\\sigma,\\betab,\\alphab$ can be defined as functions in $L^\\infty_uL^\\infty_{\\ub}L^\\infty(S)$ follows from Proposition \\ref{regularityprop} and the Sobolev Embedding Theorem. To show that $\\beta$ can be defined as a $L^\\infty_uL^\\infty_{\\ub}L^\\infty(S)$ function, we consider the following Bianchi equation:\n$$\\nabla_3\\beta=\\sum_{\\Psi\\in\\{\\rho,\\sigma\\}}\\nabla\\Psi+\\psi\\sum_{\\Psi\\in\\{\\beta,\\rho,\\sigma,\\betab\\}}\\Psi.$$\nSince $\\nab\\rho, \\psi, \\rho, \\sigma, \\betab$ can be bounded in $L^\\infty_uL^\\infty_{\\ub}L^\\infty(S)$, it follows from Proposition \\ref{transport} and Gronwall's inequality that $\\beta$ can be estimated in $L^\\infty_uL^\\infty_{\\ub}L^\\infty(S)$, as desired.\n\\end{proof}\n\n\n\\subsubsection{Smoothness of Spacetime away from $\\Hb_{\\ub_s}$}\n\nWe now show that the spacetime is smooth away from the null hypersurface $\\ub=\\ub_s$. In the region $\\ub<\\ub_s$ before the impulse, this follows from standard theory of local-wellposedness. We focus on the region after the impulse, where $\\ub>\\ub_s$. In the following Proposition, we prove uniform estimates for the $\\nab_4$ and $\\nab$ derivatives of $\\alpha_\\infty$ in the region $\\ub>\\ub_s$. The bounds for all derivatives of the curvature components follow from a combination of the estimates on $\\alpha_\\infty$, Theorem \\ref{regularitythm} and the Bianchi equations.\n\\begin{proposition}\\label{giwsmoothness}\nSuppose the data for the impulsive gravitational wave satisfy the estimates as given in Proposition \\ref{dataprop} for some $I\\geq 2$, $K\\geq 0$. Then\n$$\\sum_{i\\leq I-1}\\sum_{k\\leq \\min\\{K,\\lfloor\\frac{I-i-1}{2}\\rfloor\\}} \\limsup_{\\tilde{\\ub}\\to \\ub_s+}||\\nab_4^k\\nab^i\\alpha_{\\infty}||_{L^2(H_u(\\tilde{\\ub},\\ub_*))}\\leq C_{I,K}'.$$\n\\end{proposition}\n\\begin{proof}\nBy Proposition \\ref{alphaapriori},\n\\begin{equation}\\label{alphaaprioricon}\n\\sum_{i\\leq I-1}\\sum_{k\\leq \\min\\{K,\\lfloor\\frac{I-i-1}{2}\\rfloor\\}}\\sup_{\\ub}||\\nab_4^k\\nabla^i\\alpha_n||_{L^2(H_u(\\ub_s+2^{-n},\\epsilon))}\\leq C_{I,K}'.\n\\end{equation}\nFix $\\delta>0$. Take $n$ large enough such that $2^{-n}< \\delta$. Then (\\ref{alphaaprioricon}) implies that \n$$\\sum_{i\\leq I-1}\\sum_{k\\leq \\min\\{K,\\lfloor\\frac{I-i-1}{2}\\rfloor\\}}\\sup_{\\ub}||\\nab_4^k\\nabla^i\\alpha_n||_{L^2(H_u(\\ub_s+\\delta,\\epsilon))}\\leq C_{I,K}'.$$\nThe conclusion follows from the fact that this bound is uniform in $\\delta$.\n\\end{proof}\nAs a consequence, we show\n\\begin{proposition}\nThe limiting spacetime is smooth away from the the null hypersurface $\\Hb_{\\ub_s}$.\n\\end{proposition}\n\n\\begin{remark}\nIn Proposition \\ref{giwsmoothness}, we have only used the fact that for $\\ub>\\ub_s$, the initial data set is smooth. Therefore, suppose we have a initial data set satisfying the assumptions of Theorem \\ref{rdthmv2}, with the additional assumption that it is smooth for $\\ub>\\tilde{\\ub}$ for some $\\tilde{\\ub}$, we can also prove that the spacetime is smooth in $0\\leq u\\leq \\epsilon$, $\\tilde{\\ub} <\\ub\\leq\\epsilon$.\n\\end{remark}\n\n\\subsubsection{Decomposition of $\\alpha_\\infty$ into Singular and Regular Parts}\n\nFinally, we show that $\\alpha_\\infty$ has a delta singularity on the incoming null hypersurface $\\Hb_{\\ub_s}$.\n\\begin{proposition}\n$\\alpha_{\\infty}$ can be decomposed as\n$$\\alpha_{\\infty}=\\delta(\\ub_s)\\alpha_s+\\alpha_r,$$\nwhere $\\delta(\\ub_s)$ is the scalar delta function supported on the null hypersurface $\\Hb_{\\ub_s}$, $\\alpha_s=\\alpha_s(u,\\vartheta)\\neq 0$ belongs to $L^\\infty_uL^\\infty(S)$ and $\\alpha_r$ belongs to $L^\\infty_uL^\\infty_{\\ub}L^\\infty(S)$.\n\\end{proposition}\n\\begin{proof}\nDefine \n$$\\alpha_s(u,\\vartheta):=\\lim_{\\ub\\to\\ub_s^+} \\Omega^{-1}\\chih (u,\\ub,\\vartheta)-\\lim_{\\ub\\to\\ub_s^-} \\Omega^{-1}\\chih (u,\\ub,\\vartheta),$$\nand \n$$\\alpha_r:=\\alpha_{\\infty}-\\delta(\\ub_s)\\alpha_s.$$\nWe now show that $\\alpha_s$ and $\\alpha_r$ have the desired property. By Theorem \\ref{convergencethm2}, $\\alpha_s$ belongs to $L^\\infty_uL^\\infty(S)$. That $\\alpha_s\\neq 0$ follows from the fact that $\\chih$ has a jump discontinuity across $\\ub=\\ub_s$, which is proved in Proposition \\ref{alphalimit}.\n\nIt remains to show that $\\alpha_r$ belongs to $L^\\infty_u L^\\infty_{\\ub} L^\\infty(S)$. To show this, we consider the measure of the half open interval $[0,\\ub)$ using the measure $\\alpha_r(u,\\vartheta)$:\n\\begin{equation*}\n\\begin{split}\n&(\\alpha_r(u,\\vartheta))([0,\\ub))\\\\\n=&(\\Omega^{-1}\\chih)(u,\\ub,\\vartheta)-\\lim_{\\tilde{\\ub}\\to\\ub_s^+}(\\Omega^{-1}\\chih)(u,\\tilde{\\ub},\\vartheta)+\\lim_{\\tilde{\\ub}\\to\\ub_s^-}(\\Omega^{-1}\\chih)(u,\\tilde{\\ub},\\vartheta)-(\\Omega^{-1}\\chih)(u,\\ub=0,\\vartheta)\\\\\n&+\\int_0^{\\ub} (\\trch\\chih)(u,\\tilde{\\ub},\\vartheta) d\\tilde{\\ub}\\\\\n=&\\lim_{\\tilde{\\ub}\\to\\ub_s^-}\\int_0^{\\tilde{\\ub}} \\frac{\\partial}{\\partial\\ub}(\\Omega^{-1}\\chih)(u,\\tilde{\\ub}',\\vartheta)d\\tilde{\\ub}'+\\lim_{\\tilde{\\ub}\\to\\ub_s^+}\\int_{\\tilde{\\ub}}^{\\ub} \\frac{\\partial}{\\partial\\ub}(\\Omega^{-1}\\chih)(u,\\tilde{\\ub}',\\vartheta)d\\tilde{\\ub}'\\\\\n&+\\int_0^{\\ub} (\\trch\\chih)(u,\\tilde{\\ub},\\vartheta) d\\tilde{\\ub}.\n\\end{split}\n\\end{equation*}\nBy Proposition \\ref{giwsmoothness}, $\\frac{\\partial}{\\partial\\ub}(\\Omega^{-1}\\chih)(u,\\ub,\\vartheta)$ is a bounded function away from the the hypersurface $\\Hb_{\\ub_s}$. Thus $(\\alpha_r(u,\\vartheta))([0,\\ub))$ can be expressed as an integral over $[0,\\ub)$ of an $L^\\infty_u L^\\infty_{\\ub}L^\\infty(S)$ function, as desired.\n\\end{proof}\n\n\n\n\n\\bibliographystyle{hplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nLet an evolution equation $u_t=f[u]$ admits a symmetry $u_\\tau=g[t,u]$ (that is, vector fields $\\partial_t$ and $\\partial_\\tau$ commute), then the stationary equation $g[t,u]=0$ defines a constraint which is consistent with the dynamics in $t$. Moreover, if $\\partial_\\tau$ belongs to a commutative Lie subalgebra of the higher symmetries, then the stationary equation inherits this subalgebra. It is well known that many important classes of exact solutions satisfy such stationary equations, including the finite-gap, multisoliton and rational solutions. On the other hand, if $\\partial_\\tau$ contains some members of the additional noncommutative Lie subalgebra of symmetries then the constraint leads to a Painlev\\'e type equation. Such solutions are also considered in the literature for a long time, but, for the understandable reasons, they are studied much worse.\n\nIn this paper we study the solutions of the Volterra lattice $u_{n,t}=u_n(u_{n+1}-u_{n-1})$ which satisfy the stationary equation for the master-symmetry (plus lower order terms). An analogous, more simple constraint was studied in papers \\cite{Its_Kitaev_Fokas_1990,Fokas_Its_Kitaev_1991}, where the evolution in $n$ was governed by the discrete Painlev\\'e equation dP$_1$ and the evolution in $t$ was governed by the Painlev\\'e equation P$_4$. In our case, the respective equations are dP$_{34}$ and P$_5$ (or P$_3$, for degenerating parameters). The corresponding set of solutions is not so small and contains a family of solutions depending on three essential parameters, which are regular for all $n,t$. \n\nSection \\ref{s:constriants} contains the definition of the constraint under study, lowering of its order and reduction to the Painlev\\'e equations. In Section \\ref{s:regular}, we define a subclass of regular solutions. It is characterized by the special choice of initial data at the fixed singular point $t=0$ and by certain restrictions on the values of parameters, ensuring the absence of poles at $t\\ne0$. These solutions describe small-scale oscillations in a region that grows linearly with increase of $t$. Such behavior is typical for generic solutions outside the soliton sector with the initial data in the form of sharp spikes. The constraint which we study is of interest as an example of exact solution (in terms of the Painlev\\'e transcendents) for this mode. However, so far these solutions are investigated only numerically.\n\nSection \\ref{s:halfline} deals with the case when, in addition to the constraint equations, the condition $u_0=0$ is satisfied. Then the lattice equations admit a reduction on the half-line $n>0$, and the equations P$_5$ and P$_3$ are reduced, respectively, to the confluent hypergeometric equation and the Bessel equation. Section \\ref{s:asymp} contains asymptotic formulas for the regular solutions related to the hypergeometric equation.\n\nThe class of solutions in study contains, in particular, the solution with the most simple initial data in the form of the unit step $u_0=0$, $u_n(0)=1$ for $n>0$. It was constructed in our previous paper \\cite{Adler_Shabat_2018} by comparing of the Wronskian representation for the Volterra lattice solution \\cite{Leznov_1980} with the well known result from the combinatorics about the Hankel transform for the Catalan numbers \\cite{Aigner_1999, Stanley_1999, Layman_2001}. Now we are able to reverse this construction and to apply the constraint equations for computing of the Hankel transform. In Section \\ref{s:det}, this is done for the coefficients of the Taylor expansion of the Kummer function.\n\n\\section{Constraints for the Volterra lattice}\\label{s:constriants}\n\nThe Volterra lattice\n\\begin{equation}\\label{ut}\n u_{n,t}=u_n(u_{n+1}-u_{n-1})\n\\end{equation}\npossesses the symmetries\n\\begin{align*}\n u_{n,t_2}&=u_n(h_{n+1}-h_{n-1}),\\quad h_n:=u_n(u_{n+1}+u_n+u_{n-1}),\\\\\n u_{n,\\tau_0} &= tu_n(u_{n+1}-u_{n-1})+u_n,\\\\\n u_{n,\\tau_1} &= tu_n(h_{n+1}-h_{n-1})\n +u_n\\bigl(\\bigl(n+\\tfrac{3}{2}\\bigr)u_{n+1}+u_n-\\bigl(n-\\tfrac{3}{2}\\bigr)u_{n-1}\\bigr).\n\\end{align*}\nThe flow $\\partial_{\\tau_0}$ corresponds to the scaling transformation and $\\partial_{\\tau_1}$ is the master-symmetry which generates the commutative Lie subalgebra of the symmetries by the formula $\\partial_{t_{k+1}}=[\\partial_{\\tau_1},\\partial_{t_k}]$, starting from $\\partial_{t_1}=\\partial_t$ \\cite{Cherdantsev_Yamilov_1995, Adler_Shabat_Yamilov_2000}. Both sequences $\\partial_{t_k}$ and $\\partial_{\\tau_k}$ are infinite (and the flows $\\partial_{\\tau_k}$ are non-local for $k>1$), but in this paper we will need only the above members of the whole hierarchy of the symmetries.\n\nThe stationary equation for any linear combination of the symmetries is a constraint compatible with (\\ref{ut}). The equations which correspond to the commutative symmetries only, bring to the algebro-geometric (in particular, multisoliton) solutions. The simplest example involving a noncommutative symmetry is given by equation\n\\[\n u_{n,t_2}+2u_{n,\\tau_0}=0\n\\] \n(the coefficient at the second term is fixed by scaling and the term $u_{n,t}$ can be neglected, due to the shift $t\\to t-\\mathop{\\rm const}$). After dividing by $u_n$, the 5-point difference equation appears\n\\[\n h_{n+1}-h_{n-1} +2t(u_{n+1}-u_{n-1})+2=0.\n\\]\nIt can be easily reduced to the 3-point constraint\n\\begin{equation}\\label{IKF}\n u_n(u_{n+1}+u_n+u_{n-1})+2tu_n+n+(-1)^nb+c=0,\n\\end{equation}\nmoreover, a straightforward computation proves that it is consistent with (\\ref{ut}) if and only if the integration constants $b$ and $c$ do not depend on $t$. This equation, known as the discrete Painlev\\'e equation dP$_1$, turns the lattice equations (\\ref{ut}) into a coupled system for the variables $u_{n-1},u_n$ which is equivalent to the continuous Painlev\\'e equation P$_4$ for the function $y=u_n$ \\cite{Its_Kitaev_Fokas_1990, Fokas_Its_Kitaev_1991, Grammaticos_Ramani_1998, Grammaticos_Ramani_2014}:\n\\begin{equation}\\label{P4}\n y''=\\frac{(y')^2}{2y}+\\frac{3}{2}y^3+4ty^2+2(t^2-\\alpha)y+\\frac{\\beta}{2y},\n\\end{equation}\n\\[\n \\alpha=\\frac{1}{2}(n-3(-1)^nb+c),\\quad \\beta=-(n+(-1)^nb+c)^2.\n\\]\nThe mapping $(u_{n-1},u_n)\\mapsto(u_n,u_{n+1})$ defines one of the B\\\"acklund transformations for (\\ref{P4}).\n\nIn this paper, our main goal will be to investigate another, more complicated case\n\\[\n u_{n,\\tau_1} -4au_{n,\\tau_0} -du_{n,t}=0.\n\\]\nHere, the coefficient $a$ can be scaled either to 0 or to 1, and the shift of $t$ makes possible to remove the term $u_{n,t_2}$. Like in the previous example, this 5-point constraint can be reduced to a 3-point one, although it is less obvious in this case. First, we notice that the equation takes the following form, after dividing by $u_n$:\n\\[\n \\widetilde G_{n+1}+\\widetilde G_n=0 \\quad\\Rightarrow\\quad G_n=\\widetilde G_n+4(-1)^nb=0, \n\\]\nwhere we denote\n\\begin{equation}\\label{Gn}\n G_n=(q_{n+2}+q_{n+1})u_{n+1}-(q_n+q_{n-1})u_n-4a(q_{n+1}-q_n)+4(-1)^nb=0\n\\end{equation}\nand\n\\begin{equation}\\label{qn}\n q_n=2tu_n+n-d.\n\\end{equation}\nNext, we lower the order by use of the integrating factor:\n\\[\n (q_{n+1}+q_n)G_n=\\widetilde F_{n+1}-\\widetilde F_n=0 \\quad\\Rightarrow\\quad \n F_n=\\widetilde F_n-4c=0,\n\\]\nwhere $F_n$ is the left hand side of equation (\\ref{unn}) below. \n\n\\begin{statement}\nThe lattice (\\ref{ut}) is consistent with the equation\n\\begin{equation}\\label{unn}\n F_n=(q_{n+1}+q_n)(q_n+q_{n-1})u_n-4(aq^2_n+(-1)^nbq_n+c)=0,\n\\end{equation}\nwhere $q_n=2tu_n+n-d$, for any constants $a,b,c,d$ and under assumption that $q_{n+1}+q_n\\ne0$ at some inital moment $t=t^*$, for all $n$.\n\\end{statement}\n\\begin{proof}\nA straightforward computation proves the identities\n\\begin{align*}\n G_{n,t}&=u_{n+1}(G_{n+1}+G_n)-u_n(G_n+G_{n-1}),\\\\ \n F_{n,t}&= u_n(q_n+q_{n-1})G_n+u_n(q_{n+1}+q_n)G_{n-1},\n\\end{align*}\nassuming, like for the constraint (\\ref{IKF}), that the integration constants $b$ and $c$ do not depend on $t$. Let equation $F_n=0$ is satisfied for $t=t^*$, then also $G_n=(F_{n+1}-F_n)\/(q_{n+1}+q_n)=0$ for $t=t^*$. Then it follows from the above identities that $G_n=F_n=0$ for all $t$ such that the solution $u_n(t)$ exists.\n\\end{proof}\n\n\\begin{remark}\\label{rem:confinement} \nThe stipulation that $q_{n+1}+q_n$ do not vanish at $t=t^*$ is not very essential. We need it only in order to provide $G_n(t^*)=0$. However, in the situation when $q_{n+1}+q_n=0$ for some $n$, we can directly require that the initial conditions satisfy the equality $G_n=0$, then the constraint (\\ref{unn}) will be preserved as before. In other words, if we consider equation (\\ref{unn}) as a mapping $(u_n,u_{n+1})\\mapsto(u_{n+1},u_{n+2})$ then the equality $q_{n+1}+q_n=0$ determines a singularity which is eliminated by use of the equation $G_n=0$.\n\\end{remark}\n\nSimilar to the case (\\ref{IKF}), the constraint (\\ref{unn}) turns the lattice equations (\\ref{ut}) into a coupled system for the variables $u_{n-1},u_n$, and the shift $n\\mapsto n+1$ defines a B\\\"acklund transformation for the latter. This system is equivalent, after some additional changes, to the Painlev\\'e equations P$_5$ or P$_3$, depending on the value of the parameter $a$. The following statement is verified by straightforward substitution, for which it is convenient to completely rewrite equations (\\ref{ut}) and (\\ref{unn}) it terms of $q_n$:\n\\begin{gather}\n\\label{qt}\n q_{n,t}=\\frac{1}{2t}(q_n-n+d)(q_{n+1}-q_{n-1}),\\\\\n\\label{qnn}\n (q_{n+1}+q_n)(q_n+q_{n-1})= \\frac{8t(aq^2_n+(-1)^nbq_n+c)}{q_n-n+d}.\n\\end{gather}\nNotice, that (\\ref{qnn}) coincides, up to a scaling of parameters, with dP$_{34}$ equation \\cite{Grammaticos_Ramani_2014}. \n\n\\begin{statement}\nLet functions $q_n(t)$ satisfy equations (\\ref{qt}), (\\ref{qnn}). If $a\\ne0$ then functions \n\\[\n y_n(t)=1-\\frac{8at}{q_{n+1}(t)+q_n(t)}\n\\]\nsatisfy the P$_5$ equation \n\\begin{equation}\\label{P5}\n y''=\\Bigl(\\frac{1}{2y}+\\frac{1}{y-1}\\Bigr)(y')^2-\\frac{y'}{t}\n +\\frac{(y-1)^2}{t^2}\\Bigl(\\alpha y+\\frac{\\beta}{y}\\Bigr)\n +\\gamma\\frac{y}{t}+\\delta\\frac{y(y+1)}{y-1},\n\\end{equation}\nwith parameters\n\\[\n \\alpha=\\frac{b^2-4ac}{8a^2},\\quad \\beta=-\\frac{(a+(-1)^nb)^2}{8a^2},\\quad\n \\gamma=-2a(2n-2d+1),\\quad \\delta=-8a^2.\n\\]\nIf $a=0$ then functions\n\\[\n y_n(z)=\\frac{1}{2z}(q_{n+1}(t)+q_n(t)),\\quad t=z^2,\n\\]\nsatisfy the P$_3$ equation: \n\\begin{equation}\\label{P3}\n y''=\\frac{(y')^2}{y}-\\frac{y'}{z} +\\frac{1}{z}(\\alpha y^2+\\beta) +\\gamma y^3+\\frac{\\delta}{y},\n\\end{equation}\n\\[\n \\alpha=-4n+4d-2,\\quad \\beta=-4(-1)^nb-8c,\\quad \\gamma=4,\\quad \\delta=-16b^2.\n\\]\n\\end{statement}\n\n\\section{Regular solutions}\\label{s:regular}\n\nIn general, solutions of equations (\\ref{ut}), (\\ref{unn}) may have singularities in $t$. Such solutions are of interest as well, but in this paper we restrict ourselves by consideration of a special family of solutions, such that functions $u_n(t)$ are continuously differentiable on the whole axis $t\\in\\mathbb{R}$, for all $n\\in\\mathbb{Z}$. This regularuty condition strictly fixes the initial data of the lattice at $t=0$, that is, at the singular point of the system (\\ref{ut}), (\\ref{unn}). Indeed, since the values $u_n(0)$ are finite for a regular solution, hence $q_{n+1}(0)+q_n(0)=2n-2d+1$ and we obtain from (\\ref{unn}) that, if $d$ is not a half-integer then\n\\begin{equation}\\label{und0}\n u_n(0)= a + \\frac{4(-1)^nb(n-d)+4c+a}{4(n-d)^2-1},\\quad d\\not\\in\\frac{1}{2}+\\mathbb{Z}.\n\\end{equation}\nTherefore, for the fixed values of $a,b,c$ and $d$, we are talking about one special solution of equations (\\ref{ut}) and (\\ref{unn}) (while the general solution is 2-parametric). In terms of the P$_5$ equation (\\ref{P5}), this solution corresponds to the functions $y_n(t)$ without singularity at $t=0$, with the initial data\n\\[\n y_n(0)=1,\\quad y'_n(0)=-\\frac{2\\delta}{\\gamma}=-\\frac{8a}{2n-2d+1}.\n\\]\nIn the case when $d=\\frac{1}{2}+k$ is half-integer, the integrating factor $q_{n+1}+q_n$ which we used for derivation of (\\ref{unn}) turns into 0 for $t=0$ and $n=k$, and we have to use equation (\\ref{Gn}) instead of (\\ref{unn}), as it was explained in Remark \\ref{rem:confinement}. For $t=0$, it takes the form\n\\[ \n (n-k+1)u_{n+1}(0)-(n-k-1)u_n(0)=2a-2(-1)^nb.\n\\]\nFrom here, all $u_n(0)$ are uniqueley defined, except for $u_k(0)$ and $u_{k+1}(0)$ which are chosen independently on the rest values in such a way that their sum is constant:\n\\begin{equation}\\label{unk0}\n\\begin{gathered} \n u_n(0)=a+b\\frac{(-1)^n(2n-2k-1)+(-1)^k}{2(n-k)(n-k-1)},~~ n\\ne k,k+1,\\\\\n u_{k+1}(0)+u_k(0)=2a-2(-1)^kb,\\quad d=\\frac{1}{2}+k,~ k\\in\\mathbb{Z}.\n\\end{gathered} \n\\end{equation}\nThese inital data can be viewed as a limiting case of (\\ref{und0}) with $c=(-1)^kb\/2-a\/4$, for $d\\to\\frac{1}{2}+k$.\n\nUnder certain relations between parameters, it is possible that the equality $u_m(0)=0$ is fulfilled which splits the lattice (\\ref{ut}) into two independent systems for $nm$. We consider this case in the rest sections in more details, and now we will assume that $u_n(0)\\ne0$ for all integer $n$.\n\nThe absence of singularity at $t=0$ does not guarantee that the solution is regular for all $t$, this requires an additional study. The numeric experiments show that, for the solution family under scrutiny, the crucial property is related with the signs of $u_n(0)$: is all $u_n(0)$ are of one and the same sign (positive, without loss of generality) then the solution is regular for all $n,t$; in contrast, if there are $u_n(0)$ with different signs then the solution acquires a singularity at a finte $t$.\n\n\\begin{remark}\nThe question about the regularity criterium fot the solutions of lattice equations (\\ref{ut}) with generic inital data is open. Regular solutions with different signs do exist: a simple explicit example is the solution\n\\begin{equation}\\label{expsol}\n u_{2n} = -\\frac{\\beta(n+\\delta)e^{2\\beta t}}{\\alpha+e^{2\\beta t}},\\quad \n u_{2n+1}=\\frac{\\beta(\\alpha n+\\gamma)}{\\alpha+e^{2\\beta t}},\n\\end{equation}\nwhich, apparently, does not have singularities for $\\alpha\\ge0$, and also the stationary solution $u_{2n}=\\alpha$, $u_{2n+1}=\\beta$ with constants of different signs. However, the nonalternating solutions are of primary interest. In many papers, this requirement is simply postulated; sometimes, the Volterra lattice is wrtten down in the variables $p_n=\\sqrt{u_n}$.\n\\end{remark}\n\n\\begin{figure}[t]\n\\centerline{\\includegraphics[width=70mm]{fig1a.pdf}\\quad\\includegraphics[width=70mm]{fig1b.pdf}}\n\\caption{Solution of the Volterra lattice with the initial data $u_n(0)=1-\\frac{2}{4n^2-1}$}\n\\label{fig:arrow}\n\\end{figure}\n\nFor $a=0$ and nonzero $b,c$, the initial data (\\ref{und0}), (\\ref{unk0}) always change the sign and there are no regular solutions. If $a\\ne0$ then one can set $a=1$ without loss of generality. In addition, taking into account the shift of $n$, one can assume that $d\\in(-\\frac{1}{2},\\frac{1}{2}]$. Then the condition of the positivity of the initial data reduces to inequalities\n\\[\n bd-c-d^2>0,\\quad b(d-1)+c+(d-1)^2>0,\\quad b(d+1)+c+(d+1)^2>0\n\\]\nwhich cut off a bounded region in the parameter space (for a fixed $d$ it is a triangle in the $b,c$ plane) and the corresponding solutions are regular. Fig.\\,\\ref{fig:arrow} demonstrates a typical solution from this family, corresponding to the values $a=1$, $b=0$, $c=-3\/4$ and $d=0$. The initial profile is shown on the left plot. It quickly collapses and generates a zone of small-scale oscillations (with the period comparable to the lattice spacing), which has an arrow-shaped profile and expands at a constant speed in both directions with increasing $t$ (for the negative $t$, the direction of the arrowhead changes). For nonzero $b$ and $d$, the initial data look a bit more complicated, but the general behavior of the solution remains the same. Moreover, the picture does not change much if we take the initial data that do not satisfy the constraint (\\ref{unn}), but are close to (\\ref{und0}). Apparently, this behavior is typical for solutions with generic initial data in the form of sharp spikes (as opposed to solutions of the soliton type, which are formed when the initial data are relatively gently sloping). Fig.\\,\\ref{fig:2arrow} shows a solution for initial data which differ from 1 at two points. Each spike generates oscillations of the type described, which form an interference pattern after fusion.\n\n\\begin{figure}[t]\n\\centerline{\\includegraphics[width=70mm]{fig2a.pdf}\\quad\\includegraphics[width=70mm]{fig2b.pdf}}\\bigskip\n\n\\centerline{\\includegraphics[width=70mm]{fig2c.pdf}\\quad\\includegraphics[width=70mm]{fig2d.pdf}}\n\\caption{Solution of the Volterra lattice with the initial data\\\\ $u_{-40}(0)=0.25$, $u_{40}(0)=2$, $u_n(0)=1$, $n\\ne\\pm40$.}\n\\label{fig:2arrow}\n\\end{figure}\n\nThus, this is a fairly common mode in the Volterra lattice that deserves to be studied. It would be interesting to obtain its description from the point of view of the inverse scattering method. The constraint (\\ref{unn}), with positive initial data, is of interest as an exact solution example for this mode, in terms of the Painlev\\'e transcendents. More precisely, here we use not all solutions of P$_5$, but only one, which is distinguished by the regularity condition at $t=0$. However, this solution does not seem to be expressed in terms of classical special functions, at least for the general values of parameters.\n\n\\section{Solutions on the half-line}\\label{s:halfline}\n\nAssume that, in addition to the constraint (\\ref{unn}), the condition $u_0=0$ is fulfilled (which is also a constraint, consistent with the lattice equations (\\ref{ut})). In this case, the lattice splits into two unrelated subsystems for $n<0$ and $n>0$. It is enough to consider solutions on the half-line $n>0$. First of all, we notice that if $u_0=0$ then the system of ordinary differential equation for the variables $u_0,u_1$ (which is equivalent, in general, to a Painlev\\'e equation) reduces to the Riccati equation for $u_1$.\n\n\\begin{statement}\\label{st:Riccati}\nLet $u_n$ be a solution of the lattice (\\ref{ut}) governed by the constraints (\\ref{unn}) and $u_0=0$. Then the function $u_1(t)$ satisfies the equation\n\\begin{equation}\\label{u1'}\n u'_1+u^2_1-\\Bigl(4a+\\frac{2d-3}{2t}\\Bigr)u_1-\\frac{2(a-b)}{t}=0\n\\end{equation}\nwhich is linearizable via the substitution $u_1=f'\/f$:\n\\begin{equation}\\label{f''}\n tf''+\\Bigl(\\frac{3}{2}-d-4at\\Bigr)f'-2(a-b)f=0.\n\\end{equation}\n\\end{statement}\n\\begin{proof}\nSubstituting of $u_0=0$ into equations (\\ref{unn}) for $n=0$ and $n=1$ gives\n\\begin{gather*}\n -ad^2+bd-c=0,\\\\\n \\frac{1}{4}(q_2+q_1)(q_1-d)u_1 -aq^2_1 +bq_1 -c =0,\n\\end{gather*}\nwhere $q_n=2tu_n+n-d$. Subtracting one equation from another and dividing by $q_1-d$, we obtain\n\\[\n \\frac{1}{4}(q_2+q_1)u_1-a(q_1+d)+b=0.\n\\]\nThis is equivalent to (\\ref{u1'}), taking into account the relation $u'_1=u_1u_2$ which follows from equation (\\ref{ut}). The passage to equation (\\ref{f''}) is standard.\n\\end{proof}\n\nThus, $u_0=0$ and $u_1$ is constructed by solving equation (\\ref{f''}), then all functions $u_{n+1}$ for $n=1,2,\\dots$ are constructed by recurrent relations, either by use of the lattice itself:\n\\begin{equation}\\label{recD}\n u_{n+1}=\\frac{u'_n}{u_n}+u_{n-1},\n\\end{equation}\nor, without using differentation, by relation\n\\begin{multline}\\label{rec0}\n\\quad u_{n+1}=-u_n-\\frac{1}{t}(n-d+1\/2)\\\\\n +\\frac{a(2tu_n+n-d)^2+(-1)^nb(2tu_n+n-d)+bd-ad^2}{tu_n(tu_n+tu_{n-1}+n-d-1\/2)}\\qquad\n\\end{multline}\n(this is the constraint equation (\\ref{unn}) with $c=bd-ad^2$ resolved with respect to $u_{n+1}$). \n\nAs in the previous section, we will consider only regular solutions, and such that $u_n\\ne0$ for $n>0$ (since otherwise the solution is constrained to a finite interval). In this case, equations (\\ref{und0}), (\\ref{unk0}) are slightly refined and we arrive to the following statement. As before, it guarantees only regularity at $t=0$; the regularity for all $t$ is related with the constant sign property of the sequence (\\ref{und00}).\n\n\\begin{statement}\\label{st:un00}\nLet $u_0=0$ and let $u_n\\ne0$ be a solution of the lattice (\\ref{ut}) for $n>0$, satisfying the constraint (\\ref{unn}) and regular at $t=0$. Then\n\\begin{equation}\\label{und00}\n u_1(0)=\\frac{4(b-a)}{2d-3},~~ u_n(0)=\\frac{a(n-d)^2+(-1)^nb(n-d)+d(b-ad)}{(n-d)^2-1\/4},~~ n>1,\n\\end{equation}\nwhere\n\\begin{equation}\\label{bdnek}\n d\\ne\\frac{1}{2}+k,\\quad b\\ne a(2k-1),\\quad b\\ne 2a(d-k),\\quad k=1,2,3,\\dots\n\\end{equation}\n\\end{statement}\n\\begin{proof}\nThe expression for $u_n(0)$, $n>1$, is obtained from (\\ref{und0}) by substituting $c=bd-ad^2$. Here, the values $d=\\frac{1}{2},-\\frac{1}{2},-\\frac{3}{2},\\dots$ are admissible, since the corresponding denominators do not vanish. If $d\\ne\\frac{1}{2}$ then the expression for $u_1(0)$ is also found from (\\ref{und0}) and if $d=\\frac{1}{2}$ then it is obtained from (\\ref{unk0}) for $k=0$ (or directly from equation (\\ref{u1'})). Solving the inequalities $u_{2k-1}(0)\\ne0$ for the obtained initial data brings to restrictions $b\\ne a(2k-1)$; the inequalities $u_{2k}(0)\\ne0$ yield $b\\ne 2a(d-k)$.\n\nFor $d=\\frac{1}{2}+k$, $k>0$, we use equations (\\ref{unk0}) instead of (\\ref{und0}). In this case, if follows from the equality $u_0=0$ that\n\\[\n a=b\\frac{2k+1-(-1)^k}{2k(k+1)}\n\\]\nand it is easy to check that then also $u_{2k+1}(0)=0$, so these values of $d$ are rejected.\n\\end{proof}\n\nIf $a=0$ and $b\\ne0$ then the change $z=2\\sqrt{2bt}$, $f(t)=t^{d\/2-1\/4}y(z)$ brings (\\ref{f''}) to the Bessel equation\n\\begin{equation}\\label{Bessel}\n z^2y''+zy'+\\bigl(z^2-(d-\\tfrac{1}{2})^2\\bigr)y=0.\n\\end{equation}\nThe corresponding initial data (\\ref{und00}) are alternating. Numeric experiments show that if $b<0$ (without loss of generality) then the solution acquires the poles at $t<0$, but it tends to 0 for $t>0$ (the corresponding function $y$ is the Bessel function of the imaginary argument). This gives an example of alternating solution which is bounded and regular in the quadrant $n,t>0$. However, this solution is very unstable with respect to the calculation errors and the perturbations of the initial data.\n\n\\section{Asymptotics in the case $a=1$, $u_0=0$}\\label{s:asymp}\n\nIf $a\\ne0$ then the scaling of the independent variable $z=4at$, $f(t)=y(z)$ brings (\\ref{f''}) to the confluent hypergeometric equation\n\\begin{equation}\\label{confl}\n zy''+(\\beta-z)y'-\\alpha y=0,\\quad \\alpha=\\frac{a-b}{2a},\\quad \\beta=\\frac{3}{2}-d.\n\\end{equation}\nSince the function $u_1=f'\/f$ must be regular at $t=0$, we should to choose as $y$ the Kummer function $M(\\alpha,\\beta,z)$ (or ${\\!}_1F_1(\\alpha;\\beta;z)$ in other notation), that is, $f(t)= M(\\alpha,\\beta,4at)$. \n\n\\begin{figure}[t!]\n\\centerline{\\includegraphics[width=70mm]{fig3a.pdf}\\qquad\\includegraphics[width=70mm]{fig3b.pdf}}\n\\caption{Solution with the initial data $u_n(0)=1$, $n>0$. The dashed lines correspond to one or two terms of the asymptotic expansions.}\n\\label{fig:t25}\n\\end{figure}\n\nIn this section we set $a=1$ without loss of generality. If $b=0$ and $d=-\\frac{1}{2}$ then the initial data take especially simple form of the unit step: $u_n(0)=1$ for $n>0$. The corresponding solution, shown of fig.\\,\\ref{fig:t25}, was studied in \\cite{Adler_Shabat_2018} (notice, that we used there an equivalent representation of $f$ in terms of the modified Bessel function: $f(t)=e^{2t}(I_0(2t)-I'_0(2t))=M(\\frac{1}{2},2,4t)$; however, $f$ is not expressed through $I_n$ for the generic initial data (\\ref{und00})). \n\nThe initial data are changed for other admissible values of $b$ and $d$, but the general behaviour of the solution remains the same, exactly as for the case of solutions described in Section \\ref{s:regular}. Moreover, the numeric experiments demonstrate that this mode is stable with respect to small enough perturbations of the step-like initial data (of course, under the condition that the boundary value $u_0=0$ is not changed). Fig.\\,\\ref{fig:rand} shows the evolution of the initial data\n\\[\n u_n(0)=1+0.9\\exp(-0.001(n-30)^2)r_n,\\quad n>0,\n\\] \nwhere $r_n$ is a random value uniformly distributed in $[-1,1]$. This perturbation leads to appearance of soliton-like structures on the pure decay solution background, but the overall asymptotics does not change. \n\n\\begin{figure}[t!]\n\\centerline{\\includegraphics[width=70mm]{fig4a.pdf}\\qquad\\includegraphics[width=70mm]{fig4b.pdf}}\n\\centerline{\\includegraphics[width=70mm]{fig4c.pdf}}\n\\caption{Solution with random perturbation of the initial unit step.}\n\\label{fig:rand}\n\\end{figure}\n\nTo determine the asymptotics, we use the formal expansion of the solution of the Riccati equation (\\ref{u1'}) \n\\[\n u_1=q_0+q_1t^{-1}+q_2t^{-2}+\\dots \n\\]\nSubstitution into equation proves that the leading term may take the values $q_0=4$ or $q_0=0$, while the subsequent coefficients are computed uniquely. This gives two series which correspond to the different asymptotics at $t\\to\\pm\\infty$, which is easy to see by comparing with known asymptotic formulas \\cite{AS}: \n\\[\n M(\\alpha,\\beta,z)=\\left\\{\n \\begin{aligned}\n &\\frac{\\Gamma(\\beta)}{\\Gamma(\\alpha)}e^zz^{\\alpha-\\beta}\\bigl(1+O(|z|^{-1})\\bigr), & \\mathop{\\rm Re} z>0,\\\\ \n &\\frac{\\Gamma(\\beta)}{\\Gamma(\\beta-\\alpha)}(-z)^{-\\alpha}\\bigl(1+O(|z|^{-1})\\bigr), & \\mathop{\\rm Re} z<0. \n \\end{aligned}\\right.\n\\]\nFrom here it follows, for the function $u_1=f'\/f$ and for real $z=4t$, that $u_1\\sim 4-(\\alpha-\\beta)t^{-1}+\\dots$ at $t\\to+\\infty$ and $u_1\\sim 0+\\alpha t^{-1}+\\dots$ at $t\\to-\\infty$. Next, the expansions for all $u_n$ are obtained by intermediate use of equation (\\ref{u1'}) and the lattice equation (\\ref{ut}). We wil restrict ourselves by two first terms of the asymptotics.\n\n\\begin{statement}\\label{st:uinf}\nConsider the solution of the lattice (\\ref{ut}) with $u_0=0$ and the initial data $u_n(0)>0$ for $n>0$ defined by equations (\\ref{und00}), (\\ref{bdnek}) with $a=1$. Then the following asymptotic formulas are valid: for $t\\to+\\infty$\n\\[\n u_n=\n \\left\\{\n \\begin{aligned}\n & \\frac{n(n-2d+b)}{16t^2}+\\frac{n(n-2d+b)(2n-2d+3b)}{128t^3}+O(t^{-4}),\\quad n=0,2,\\dots,\\\\\n & 4-\\frac{2n-2d+b}{2t}\\\\\n & \\qquad -\\frac{n^2-(2d-3b)n+b^2-2bd+1}{16t^2}+O(t^{-3}),\\quad n=1,3,\\dots\n \\end{aligned}\n \\right.\n\\]\nand for $t\\to-\\infty$\n\\[\n u_n=\n \\left\\{\n \\begin{aligned}\n & -\\frac{n}{2t}+\\frac{(2d-b)n}{16t^2}+O(t^{-3}),\\quad n=0,2,\\dots,\\\\\n & -\\frac{n-b}{2t}+\\frac{(2d-b)(n-b)}{16t^2}+O(t^{-3}),\\quad n=1,3,\\dots\n \\end{aligned}\n \\right.\n\\]\n\\end{statement}\n\\begin{proof}\nWe find from equation (\\ref{u1'}) that\n\\[\n u_1(t)=\\left\\{\n \\begin{aligned} \n & 4+\\frac{2d-b-2}{2t}+\\frac{(b+1)(2d-b-2)}{16t^2}+\\dots,\\quad t\\to+\\infty,\\\\\n & \\frac{b-1}{2t}+\\frac{(b-1)(b-2d)}{16t^2}+\\dots,\\quad t\\to-\\infty.\n \\end{aligned}\\right.\n\\]\nIt is easy to prove by induction that expansions for $u_n$ at $t\\to+\\infty$ are of the following form, depending on the parity of $n$:\n\\[\n u_{2j}=\\frac{p_{j,2}}{t^2}+\\frac{p_{j,3}}{t^3}+\\dots,\\quad\n u_{2j+1}=4+\\frac{q_{j,1}}{t}+\\frac{q_{j,2}}{t^2}+\\dots\n\\]\nand that the substitution into (\\ref{ut}) gives the difference equations for the coefficients\n\\begin{gather*}\n q_{j,1}-q_{j-1,1}=-2,\\quad 4p_{j+1,2}-4p_{j,2}=-q_{j,1},\\\\\n p_{j,2}(q_{j,2}-q_{j-1,2})=-p_{j,3},\\quad 16p_{j+1,3}-16p_{j,3}+8q_{j,2}=q^2_{j,1}.\n\\end{gather*}\nSimilarly, the expansions at $t\\to-\\infty$ start from $t^{-1}$ for all $n$: \n\\[\n u_{2j}=p_{j,1}t^{-1}+p_{j,2}t^{-2}+\\dots,\\quad u_{2j+1}=q_{j,1}t^{-1}+q_{j,2}t^{-2}+\\dots\n\\] \nand the coefficients are governed by equations\n\\begin{gather*}\n q_{j,1}-q_{j-1,1}=-1,\\quad p_{j+1,1}-p_{j,1}=-1,\\\\ \n p_{j,1}(q_{j,2}-q_{j-1,2})=-p_{j,2},\\quad q_{j,1}(p_{j+1,2}-p_{j,2})=-q_{j,2}.\n\\end{gather*}\nThe initial data for these equations are given by the coefficients $q_{0,1},q_{0,2}$ of the above series for $u_1$ and the values $p_{0,k}=0$ corresponding to $u_0=0$. In both cases, the solution is easily obtained as polynomials in $j$ \nand we obtain the required formulas by returning to the variable $n$.\n\\end{proof}\n\nFor $t>0$, one can obtain a rough estimate of the decay zone by constructing a triangular region bounded by the plots of one or two terms of the asymptotic expansions, as shown on figs.\\;\\ref{fig:t25} and \\ref{fig:rand}. In particular, an upper bound for the wedge point $n_0(t)$ of the decay zone can be obtained by solving the inequality $4-\\frac{2n_0-2d+b}{2t}>1$, which gives, apparently, $n_0<3t+\\mathop{\\rm const}$. More accurate estimates can be obtained by taking the next asymptotic terms. For the negative $t$, the solution is well approximated by the first term of the asymptotics.\n\nWe conclude this section with a note on the conservation laws of the Volterra lattice, that is, relations of the form\n\\begin{equation}\\label{conserv}\n \\frac{d}{dt}\\rho^{(k)}_n=\\sigma^{(k)}_{n+1}-\\sigma^{(k)}_n,\n\\end{equation}\nwhere $\\rho^{(k)}_n$ and $\\sigma^{(k)}_n$ depend on a finite number of variables $u_n$. Three simplest conservation laws are given by\n\\begin{gather*}\n \\rho^{(0)}_n=\\log u_n,~~ \\sigma^{(0)}_n=u_{n-1}+u_n,\\qquad\n \\rho^{(1)}_n=u_n,~~ \\sigma^{(1)}_n=u_{n-1}u_n,\\\\\n \\rho^{(2)}_n=\\frac{1}{2}u^2_n+u_nu_{n+1},~~ \\sigma^{(2)}_n=u_{n-1}u_n(u_n+u_{n+1}).\n\\end{gather*}\nIn the case of the problem on the whole line and with the initial data which have the same constant asymptotics for $n\\to\\pm\\infty$, it follows from (\\ref{conserv}) that the quantities $H_k=\\sum_n(\\rho^{(k)}_n-r^{(k)})$ are preserved, where the summation is taken over all integer $n$ and the constant $r^{(k)}$ is chosen so that the sum is well defined. For the lattice truncated by $u_0=0$, the analogous sums over $n>0$ do not preserve, since\n\\[\n \\frac{d}{dt}H_k=\\lim_{n\\to\\infty}\\sigma^{(k)}_n-\\sigma^{(k)}_1\\ne0.\n\\]\nFor solutions with the asymptotics $u_n\\to1$ for $n\\to\\infty$, the sums are regularized as follows\n\\[\n H_0=\\sum^\\infty_{n=1}\\log u_n,\\quad\n H_1=\\sum^\\infty_{n=1}(u_n-1),\\quad\n H_2=\\sum^\\infty_{n=1}\\Bigl(\\frac{1}{2}u^2_n+u_nu_{n+1}-\\frac{3}{2}\\Bigr),\n\\]\nand we have $\\sigma^{(0)}_\\infty=2$, $\\sigma^{(0)}_1=u_1$;\n$\\sigma^{(1)}_\\infty=1$, $\\sigma^{(1)}_1=0$ and \n$\\sigma^{(2)}_\\infty=2$, $\\sigma^{(2)}_1=0$. Then\n\\[\n \\frac{d}{dt}H_0=2-u_1,\\quad \\frac{d}{dt}H_1=1,\\quad \\frac{d}{dt}H_2=2,\n\\]\nand since all three sums are equal to 0 at $t=0$, hence\n\\[\n H_0= \\int^t_0(2-u_1(\\tau))d\\tau,\\quad H_1=t,\\quad H_2=2t.\n\\]\n\n\\section{Determinant identities}\\label{s:det}\n\nIn addition to the recurrent relations (\\ref{recD}), (\\ref{rec0}), there exists the Wronskian representation of the Volterra lattice solution on the half-line, which goes back to the Leznov paper \\cite{Leznov_1980}. It is not very convenient for a practical computing of solutions, but we will show that using it together with explicit expressions for $u_1(t)$ and $u_n(0)$ makes possible to get nontrivial identities for some number sequences.\n\n\\begin{statement}\\label{st:w}\nThe solution of the lattice (\\ref{ut}) on the half-line $n\\ge0$, such that $u_0=0$ and $u_1=f'\/f$, with an arbitrary infinitely differentiable function $f(t)$, is of the form\n\\begin{equation}\\label{uw}\n u_n=\\frac{w_{n-3}w_n}{w_{n-2}w_{n-1}},\\quad n=0,1,2,\\dots,\n\\end{equation}\nwhere $w_{-3}=0$, $w_{-2}=w_{-1}=1$ and, for $k\\ge0$, \n\\begin{equation}\\label{ww}\n w_{2k}=\n \\left|\\begin{matrix}\n f & f' & \\dots & f^{(k)} \\\\\n f' & f'' & \\dots & f^{(k+1)}\\\\\n \\vdots & \\vdots & \\ddots &\\vdots \\\\\n f^{(k)} & f^{(k+1)} & \\dots & f^{(2k)}\n \\end{matrix}\\right|,\\quad\n w_{2k+1}=\n \\left|\\begin{matrix}\n f' & f'' & \\dots & f^{(k+1)} \\\\\n f'' & f''' & \\dots & f^{(k+2)}\\\\\n \\vdots & \\vdots & \\ddots &\\vdots \\\\\n f^{(k+1)}& f^{(k+2)} & \\dots & f^{(2k+1)}\n \\end{matrix}\\right|.\n\\end{equation}\n\\end{statement}\n\\begin{proof}\nWe will prove that $w_n$ satisfy equations\n\\begin{equation}\\label{wt}\n w_nw'_{n+1}-w'_nw_{n+1}=w_{n-1}w_{n+2},\\quad n=-2,-1,0,1,\\dots,\n\\end{equation}\nthen it is easy to check that substitution (\\ref{uw}) gives a (unique) solution of the lattice equations (\\ref{ut}) with $u_0=0$ and $u_1=f'\/f$. \n\nFor $n=-2,-1$, the relations (\\ref{wt}) are verified directly. For $n\\ge0$, let $W(A)$ denote the Wronskian of an arbitrary finite sequence $A$ of smooth functions (possibly empty). For $n=2k$ we set $A=f^{(1)},\\dots,f^{(k)}$, then\n\\begin{alignat*}{2}\n &w_{n-1}=W(A), &\\quad& w_n=W(f^{(0)},A) =(-1)^kW(A,f^{(0)}),\\\\\n &w_{n+1}=W(A,f^{(k+1)}),&& w_{n+2}=W(f^{(0)},A,f^{(k+1)}) =(-1)^kW(A,f^{(0)},f^{(k+1)}).\n\\end{alignat*}\nSimilarly, for $n=2k+1$ we set $A=f^{(0)},\\dots,f^{(k)}$, then\n\\begin{alignat*}{2}\n &w_{n-1}=W(A), &\\quad& w_n=W(1,A)=(-1)^{k+1}W(A,1),\\\\\n &w_{n+1}=W(A,f^{(k+1)}),&& w_{n+2}=W(1,A,f^{(k+1)})=(-1)^{k+1}W(A,1,f^{(k+1)}).\n\\end{alignat*}\nIn both cases, equation (\\ref{wt}) is satisfied due to the identity\n\\[\n W(A,b)\\frac{d}{dt}W(A,c)-\\frac{d}{dt}W(A,b)W(A,c)=W(A)W(A,b,c),\n\\]\nwhere $b,c$ are arbitrary smooth functions. In order to prove it, it is sufficient to consider both left and right hand sides as the differential operators with respect to $c$ and to compare their kernels and the coefficients at the highest derivative.\n\\end{proof}\n\nThe determinants of the form (\\ref{ww}) for the number sequences $f_0,f_1,f_2,\\dots$ are actively studied in combinatorics. Recall that the Hankel transformation for such a sequence is the sequence of determinants of size $n\\times n$ with $(i,j)$-th element equal to $f_{i+j-2}$. This mapping is not one-to-one, since the determinant of size $n$ involves $2n-1$ members of the sequence. To get a one-to-one mapping, one can use simultaneously Hankel transforms for the sequence itself and the sequence without the zero member, provided that all determinants do not vanish.\n\nFor instance, the Hankel transform of the Catalan numbers $1$, $1$, $2$, $5$, $14$, $42$, $132$, $429,\\dots$ is the sequence $1,1,1,\\dots$ and the same is true for the sequence of the Catalan numbers with the first member dropped \\cite{Aigner_1999, Stanley_1999, Layman_2001}. In \\cite{Adler_Shabat_2018}, we associated these identities of the Catalan numbers with the Wronskian representation of the solution with the initial data $u_0=0$, $u_n(0)=1$, $n>0$ and derived the Riccati equation for $u_1(t)$ from this.\n\nNow we are able to reverse and to generalize this result. Indeed, the Wronskians $w_{2k}$ and $w_{2k+1}$ define the Hankel transformations for the sequences $f,f',\\dots$ and $f',f'',\\dots$, respectively. By setting $t=0$ in (\\ref{ww}), we obtain the Hankel transformations for the coefficients of the Taylor expansion of the function $f(t)$ (here, the regularity for other values of $t$ is not necessary) and it remains to compare the result with the known initial data by substituting into equation (\\ref{uw}) at $t=0$.\n\n\\begin{statement}\\label{st:Hankel}\nLet a solution of the lattice equation (\\ref{ut}) on the half-line $n\\ge0$ be given by equations (\\ref{uw}), (\\ref{ww}) with the function $f$ given by the Taylor expansion\n\\[\n f(t)=f_0+f_1t+\\cdots+f_n\\frac{t^n}{n!}+\\cdots,\n\\]\nthen, for $k=0,1,2,\\dots$, \n\\begin{equation}\\label{wwu0}\n\\begin{gathered}\n h_{2k}=\\left|\\begin{matrix}\n f_0 & \\dots & f_k \\\\\n \\vdots & \\ddots &\\vdots \\\\\n f_k & \\dots & f_{2k}\n \\end{matrix}\\right| = \\prod^k_{j=1}(u_{2j-1}(0)u_{2j}(0))^{k+1-j},\\qquad\\\\\n h_{2k+1}=\\left|\\begin{matrix}\n f_1 & \\dots & f_{k+1} \\\\\n \\vdots & \\ddots &\\vdots \\\\\n f_{k+1} & \\dots & f_{2k+1}\n \\end{matrix}\\right| = u^{k+1}_1(0)\\prod^k_{j=1}(u_{2j}(0)u_{2j+1}(0))^{k+1-j}.\n\\end{gathered}\n\\end{equation}\n\\end{statement}\n\\begin{proof}\nBy setting $h_n=w_n(0)$, we obtain $h_{-2}=h_{-1}=h_0=1$ and the recurrent relation\n\\[\n h_nh_{n-3}=u_n(0)h_{n-1}h_{n-2},\\quad n=1,2,\\dots, \n\\]\nwhich proves the statement by induction.\n\\end{proof}\n\nWe note that some nontrivial identities follows from here even for the explicit solution\n\\[\n u_{2k-1}=e^t,\\quad u_{2k}=k\n\\]\nwhich is a particular case of solution (\\ref{expsol}) at $\\alpha=0$, $\\beta=-1\/2$, $\\gamma=-2$ and $\\delta=0$. In this case we find, by solving equation $f'\/f=u_1=e^t$, that\n\\[\n f(t)=e^{e^t-1}= \\sum^\\infty_{n=0}B_n\\frac{t^n}{n!},\n\\]\nwhere $B_n$ are the Bell numbers $1,1,2,5,15,52,203,\\dots$ enumerating the partitions of a set of $n$ elements. Here $u_{2k-1}(0)=1$, $u_{2k}(0)=k$ and one obtains easily that $h_{2k}=h_{2k+1}=1!\\cdots k!$ (the superfactorial of $k$). This result is known since 1978~\\cite{Ehrenborg_2000}.\n\nNow, let us apply Statement \\ref{st:Hankel} to solutions desribed in the previous section (however, now we do not set $a=1$). Due to the known expansion of the Kummer function, we have\n\\begin{equation}\\label{Kummer}\n f(t)= M(\\alpha,\\beta,4at)\n = 1+4a\\frac{\\alpha}{\\beta}t+(4a)^2\\frac{(\\alpha)_2}{(\\beta)_2}\\frac{t^2}{2!}+\\dots\n +(4a)^n\\frac{(\\alpha)_n}{(\\beta)_n}\\frac{t^n}{n!}+\\dots,\n\\end{equation}\nwhere $(\\alpha)_n$ is the Pochhammer symbol \n\\[\n (\\alpha)_n=\\alpha(\\alpha+1)\\cdots(\\alpha+n-1),\\quad (\\alpha)_0=1. \n\\]\nIn addition, let us denote\n\\[\n ((\\alpha))_n= (\\alpha)_1\\cdots(\\alpha)_n= \\alpha^n(\\alpha+1)^{n-1}\\cdots(\\alpha+n-1)^1,\\quad ((\\alpha))_0=1; \n\\]\nin particular, the superfactorial is denoted as $((1))_n=1!\\cdots n!$.\n\n\\begin{statement}\\label{st:Poch-Hankel}\nLet\n\\begin{gather*}\n f_n=(4a)^n\\frac{(\\alpha)_n}{(\\beta)_n},\\quad n=0,1,2,\\dots,\\\\\n \\alpha\\ne -n,\\quad \\beta\\ne -n,\\quad \\alpha-\\beta\\ne n,\\quad n=0,1,2,\\dots,\n\\end{gather*}\nthen, for $k=0,1,2,\\dots$, \n\\begin{equation}\\label{Poch-Hankel}\n\\begin{gathered}\n h_{2k}=\\left|\\begin{matrix}\n f_0 & \\dots & f_k \\\\\n \\vdots & \\ddots &\\vdots \\\\\n f_k & \\dots & f_{2k}\n \\end{matrix}\\right| = \\frac{((1))_k(4a)^{k(k+1)}((\\alpha))_k((\\beta-\\alpha))_k}{(\\beta)^{k+1}_k((\\beta+k))_k},\\\\\n h_{2k+1}=\\left|\\begin{matrix}\n f_1 & \\dots & f_{k+1} \\\\\n \\vdots & \\ddots &\\vdots \\\\\n f_{k+1} & \\dots & f_{2k+1}\n \\end{matrix}\\right| = \\frac{((1))_k(4a)^{(k+1)^2}((\\alpha))_{k+1}((\\beta-\\alpha))_k}{(\\beta)^{k+1}_k((\\beta+k))_{k+1}},\n\\end{gathered}\n\\end{equation}\n\\end{statement}\n\\begin{proof}\nSet \n\\[\n b=a(1-2\\alpha),\\quad d=\\frac{3}{2}-\\beta,\n\\] \nthen the initial data (\\ref{und00}) take the form\n\\[\n \\begin{aligned}\n u_1(0)=4a\\frac{\\alpha}{\\beta},\\quad\n &u_{2k}(0)=4a\\frac{k(\\beta-\\alpha+k-1)}{(\\beta+2k-2)(\\beta+2k-1)},\\\\\n &u_{2k+1}(0)=4a\\frac{(\\alpha+k)(\\beta+k-1)}{(\\beta+2k-1)(\\beta+2k)},\n \\end{aligned}\\quad k=1,2,3,\\dots,\n\\]\nand the inequalities for $\\alpha$ and $\\beta$ coincide exactly with the conditions (\\ref{bdnek}) which guarantee that all numerators and denominators are not 0. According to Statements \\ref{st:Riccati} and \\ref{st:w}, the corresponding regular (at $t=0$) solution of the Volterra lattice is defined by equations (\\ref{uw}), (\\ref{ww}) with function (\\ref{Kummer}) and we only have to substitute the initial data into (\\ref{wwu0}) and to arrange the factors. \n\\end{proof}\n\nIn particular, the example with the Catalan numbers $f_n=\\dfrac{(2n)!}{(n+1)!n!}$ corresponds to the choice $a=1$, $b=0$ and $d=-\\frac{1}{2}$ (or, $\\alpha=\\frac{1}{2}$ and $\\beta=2$). In this case the initial data are $u_n(0)=1$ for $n>0$ \nand instead of the general formula, it is easier to use the recurrence relation $h_nh_{n-3}=u_n(0)h_{n-1}h_{n-2}$ directly, which immediately gives that all $h_n=1$.\n\nSimilarly, the central binomial coefficients $f_n=\\dfrac{(2n)!}{(n!)^2}$ correspond to the choice $a=1$, $b=0$ and $d=\\frac{1}{2}$ (or, $\\alpha=\\frac{1}{2}$ and $\\beta=1$). In this case $u_1(0)=2$ and $u_n(0)=1$ for $n>1$ and the recurrent relation yields $h_{2k}=2^k$, $h_{2k+1}=2^{k+1}$. This example is also known in the combinatorics.\n\n\\subsubsection*{Acknowledgements}\n\nThis work was carried out under the State Assignment 0033-2019-0006 (Integrable systems of mathematical physics) of the Ministry of Science and Higher Education of the Russian Federation.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Model 1: estimating the SNARC effect}\n\nThe model is a hierarchical Bayesian linear regression model, where $dRT$ is predicted by stimulus number. A graphical representation of the model can be seen in Figure \\ref{fig:model}. Formally, we define\n\n\\begin{equation}\\label{eq:model}\n dRT_{ij} = \\alpha_i+\\beta_i(j) + \\varepsilon_{ij}\n\\end{equation}\n\n\\noindent\nwhere $i =$ subject number and $j=$ stimulus number. The residuals $\\varepsilon_{ij}$ are assumed to be normally distributed with mean 0 and precision $1\/\\sigma^2$. Thus, we can express the likelihood for the data as\n\n\\begin{equation}\\label{eq:likelihood}\n dRT_{ij} \\sim \\text{Normal}(\\mu_{ij}, 1\/\\sigma^2)\n\\end{equation}\n\n\\noindent\nwhere $\\mu_{ij}=\\alpha_i+\\beta_i(j)$. The prior for each individual-level intercept $\\alpha_i$ is set to be uniformly distributed between -200 and 200. The hierarchical structure is instantiated on slope. First, I set the prior for each individual-level slope $\\beta_i$ to be normally distributed with two hyperparameters: mean $b$ and precision $1\/\\sigma_b^2$. In turn, this requires priors on the hyperparameters -- the prior for $b$ is uniformly distributed between -20 and 20. Note that this range is based on inspection of the slopes obtained in \\citet{viarouge2014}. The prior for group-level slop precision $1\/\\sigma_b^2$ (as well as group-level residual precision $1\/\\sigma^2$) is set as a Gamma distribution with both shape and scale equal to 0.01. \n\n\\begin{figure}\n \\centering\n\\begin{tikzpicture}\n\n \n \\node[obs] (dRT) {$dRT_{ij}$};%\n \\node[latent, above=of dRT] (mu) {$\\mu_{ij}$}; %\n \\node[latent, above=of mu, xshift=-2cm] (beta) {$\\beta_i$};%\n \\node[latent, above=of mu, xshift=2cm] (alpha) {$\\alpha_i$};%\n\n \\node[latent, above=of beta, xshift=-1cm] (b) {$b$};%\n \\node[latent, above=of beta, xshift=1cm] (sigma_b) {$\\sigma_b^2$};%\n\n \\node[latent, left=of dRT, xshift=-2cm] (sigma) {$\\sigma^2$};%\n \n \n \\edge {sigma,mu} {dRT}\n \\edge {beta,alpha} {mu}\n \\edge {b,sigma_b} {beta}\n\n \n \\plate[inner sep=10pt] {plate1} {(dRT)(mu)} {number $j$}; %\n \\plate[inner sep=10pt] {plate2} {(plate1)(beta)(alpha)} {subject $i$}; %\n\n \n \\node[text width=6cm, anchor=west, right] at (3,2)\n {\n \\begin{align*}\n b & \\sim \\text{Uniform}(-20,20)\\\\\n 1\/\\sigma_b^2 & \\sim \\text{Gamma}(0.01,0.01)\\\\\n \\alpha_i & \\sim \\text{Uniform}(-200,200)\\\\\n \\beta_i & \\sim \\text{Gaussian}(b,\\sigma_b^2)\\\\\n \\mu_{ij} &= \\alpha_i + \\beta_i(j)\\\\\n 1\/\\sigma^2 &\\sim \\text{Gamma}(0.01, 0.01)\\\\\n dRT_{ij} & \\sim \\text{Gaussian}(\\mu_{ij},\\sigma^2)\n \\end{align*}\n};\n \\end{tikzpicture}\n\n \n \\caption{Graphical model of the hierarchical Bayesian linear regression model for the SNARC effect. Following the convention of \\citet{lee2014}, nodes represent variables of interest (observed=shaded, latent=unshaded), with dependencies represented via the graph structure.}\n \\label{fig:model}\n\\end{figure}\n\n\\subsection{Fitting the model}\n\n\\subsubsection{Data}\nI fit the model to data collected from 35 participants in a number parity task. The numbers 1, 2, 8, and 9 were presented in the center of a computer screen, after which participants were asked to quickly indicate via a button press whether the number was even or odd. The procedure mirrored that of Experiment 1 of \\citet{pinhas2014}. Participants completed 112 trials of the task under two counterbalanced response rules (either even=left or even=right). This resulted in a collection of 3,920 trials. We removed 259 error trials and an additional 12 trials for which RT exceeded 3 seconds (a total of 6.9\\% of trials). The remaining 3,649 trials were collapsed into $35 \\times 8 \\times 2$ cells by computing median $RT$ for each of the conditions defined by crossing the factors of subject, stimulus number, and response hand. Then, $dRT$ was computed for each combination of subject and number by subtracting left-hand RT from right-hand RT.\n\n\\subsubsection{Results}\nThe regression model parameters were estimated using R \\citep{r} and JAGS \\citep{jags}. Posterior sampling consisted of 3 MCMC chains, each containing 100,000 draws. The first 5000 draws of each chain were discarded as ``burn-in'' samples, leaving 285,000 samples remaining. These remaining samples were thinned by a factor of 10, leaving a final sample of 28,500 posterior draws for each parameter. Visual inspection of trace plots indicated that all chains converged appropriately. Additionally, the Gelman-Rubin statistic $\\hat{R}=1.001$ for each parameter, indicating that the Markov chains for each parameter converged to the appropriate stationary distribution \\citep{gelman1992,gelman2013}.\n\nSince the model is hierarchical, I was able to estimate posterior distributions for each $\\alpha_i$ and $\\beta_i$ (i.e., each participant's intercept and slope, respectively). Further, I estimated the posterior distribution of $b$, which is the group level mean slope. The flexibility of this model allows one to ask many questions about the SNARC effect, both at the group level and the individual level. To illustrate, I will investigate whether there was an overall SNARC effect for the group \\citep[c.f.,][]{fias1996}. This can be answered by looking at the posterior distribution of the group-level $b$.\n\nThe posterior distribution of the group-level slope $b$ is depicted in Figure \\ref{fig:model1-posterior}. As can be seen in the figure, the mass of the distribution is centered over the posterior mode $b=-11.5$. Further, the 95\\% HPDI for the slope $b$ is $[-15.6, -7.3]$. Finally, since the posterior distribution is a \\emph{probability distribution}, we can compute the probability that $b<0$ (that is, the probability that there is a non-zero SNARC effect). This probability is greater than 0.999. Other Bayesian tools can be applied to this model, such as computing a Bayes factor comparing a null-SNARC model to our obtained model via the Savage-Dickey density ratio \\citep{wagenmakers2010}. Briefly, this method amounts to computing the density of the value $b=0$ in the posterior distribution divided by the density of $b=0$ in the prior (i.e., $\\text{Uniform}(-20,20)$). Doing this resulted in a Bayes factor of approximately 280,000 to 1 in favor of the alternative hypothesis, indicating overwhelming support for an overall SNARC effect.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.9\\linewidth]{model1-posterior.jpeg}\n \\caption{Posterior distribution of the group-level slope $b$ for Model 1, indicating a substantial SNARC effect across the group.}\n \\label{fig:model1-posterior}\n\\end{figure}\n\n\n\\section{Model 2: estimating the numerical distance effect}\n\nAs with Model 1, this model is also a hierarchical Bayesian linear regression model, where $RT$ is predicted by numerical distance. For this specific instantiation of the model, I will index numerical distance via the \\emph{ratio} between compared numbers. For example, \\citet{fazio2014} used four ratio bins as predictors of RT. A graphical representation of the model can be seen in Figure \\ref{fig:model2}. Formally, we define\n\n\\begin{equation}\\label{eq:model2}\n RT_{ij} = \\alpha_i+\\beta_i(j) + \\varepsilon_{ij}\n\\end{equation}\n\n\\noindent\nwhere $i =$ participant number and $j=1,\\dots,4$ equals ratio bin number. The residuals $\\varepsilon_{ij}$ are again assumed to be normally distributed with mean 0 and precision $1\/\\sigma^2$. The likelihood for the data is\n\n\\begin{equation}\\label{eq:likelihood}\n RT_{ij} \\sim \\text{Normal}(\\mu_{ij}, 1\/\\sigma^2)\n\\end{equation}\n\n\\noindent\nwhere $\\mu_{ij}=\\alpha_i+\\beta_i(j)$. The prior for each individual-level intercept $\\alpha_i$ is set to be uniformly distributed between 0 and 2000. The prior for each individual-level slope $\\beta_i$ to be normally distributed with two hyperparameters: mean $b$ (with uniform prior between -100 and 100) and precision $1\/\\sigma_b^2$ (with a $\\text{Gamma}(.01,.01)$ prior). Finally, the prior for the group-level residual precision is $\\text{Gamma}(.01,.01)$.\n\n\\begin{figure}\n \\centering\n\n\n\\begin{tikzpicture}\n\n \n \\node[obs] (RT) {$RT_{ij}$};%\n \\node[latent, above=of RT] (mu) {$\\mu_{ij}$}; %\n \\node[latent, above=of mu, xshift=-2cm] (beta) {$\\beta_i$};%\n \\node[latent, above=of mu, xshift=2cm] (alpha) {$\\alpha_i$};%\n\n \\node[latent, above=of beta, xshift=-1cm] (b) {$b$};%\n \\node[latent, above=of beta, xshift=1cm] (sigma_b) {$\\sigma_b^2$};%\n\n \\node[latent, left=of dRT, xshift=-2cm] (sigma) {$\\sigma^2$};%\n \n \n \\edge {sigma,mu} {dRT}\n \\edge {beta,alpha} {mu}\n \\edge {b,sigma_b} {beta}\n\n \n \\plate[inner sep=10pt] {plate1} {(dRT)(mu)} {number $j$}; %\n \\plate[inner sep=10pt] {plate2} {(plate1)(beta)(alpha)} {subject $i$}; %\n\n \n \\node[text width=6cm, anchor=west, right] at (3,2)\n {\n \\begin{align*}\n b & \\sim \\text{Uniform}(-100,100)\\\\\n 1\/\\sigma_b^2 & \\sim \\text{Gamma}(0.01,0.01)\\\\\n \\alpha_i & \\sim \\text{Uniform}(0,2000)\\\\\n \\beta_i & \\sim \\text{Gaussian}(b,\\sigma_b^2)\\\\\n \\mu_{ij} &= \\alpha_i + \\beta_i(j)\\\\\n 1\/\\sigma^2 &\\sim \\text{Gamma}(0.01, 0.01)\\\\\n dRT_{ij} & \\sim \\text{Gaussian}(\\mu_{ij},\\sigma^2)\n \\end{align*}\n};\n \\end{tikzpicture}\n\n \n \\caption{Graphical model of the hierarchical Bayesian linear regression model for the numerical distance effect. Following the convention of \\citet{lee2014}, nodes represent variables of interest (observed=shaded, latent=unshaded), with dependencies represented via the graph structure.}\n \\label{fig:model2}\n\\end{figure}\n\n\\subsection{Fitting the model}\n\n\\subsubsection{Data}\nI fit the model to the RT data from \\citet{fazio2014}. Fifty-five 5th graders completed a symbolic number comparison task, in which they were asked to choose which of two Arabic numerals was larger. Each child completed 40 trials using stimulus numbers ranging from 5 to 21. Of the 40 trials, 10 came from each of 4 ratio bins: 1.15 - 1.28, 1.28 - 1.43, 1.48 - 1.65, and 2.46 - 2.71. The ratio for each stimulus pair was defined as the quotient obtained when dividing the larger number by the smaller number. In all, participants completed 2,200 trials. I removed 95 error trials and 6 additional trials for which RT exceeded 5 seconds (a total of 4.6\\% of trials). The remaining 2,099 trials were collapsed into $55 \\times 4$ cells by computing median $RT$ for each of the conditions defined by crossing the factors of subject and ratio bin.\n\n\\subsubsection{Results}\nThe regression model was fit using the same procedure as in Model 1. As before, visual inspection of trace plots indicated that all chains converged appropriately, with $\\hat{R}=0.001$ for all parameters.\n\nThe posterior distribution of the group-level slope $b$ is depicted in Figure \\ref{fig:model2-posterior}. As can be seen in the figure, the mass of the distribution is centered over the posterior mode $b=-65.9$, with 95\\% HPDI for the slope $b$ equal to $[-83.4, -50.7]$. In addition to estimating the group-level slope, we can test the existence of the numerical distance effect via a Bayes factor, which as before I computed using the Savage-Dickey density ratio. This Bayes factor was approximately 1.7 million to 1 in favor of the alternative hypothesis, indicating overwhelming support for an overall numerical distance effect.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.9\\linewidth]{model2-posterior.jpeg}\n \\caption{Posterior distribution of the group-level slope $b$ for Model 2, indicating a substantial numerical distance effect across the group.}\n \\label{fig:model2-posterior}\n\\end{figure}\n\n\\section{Comparing the model to the classical approach}\nIn this section, I will demonstrate that the hierarchical Bayesian model developed in this paper can provide better measurement fidelity for the group-level regression slopes that are desired when assessing group-level SNARC effects or NDEs, particularly for small samples. As mentioned earlier, the reason for this increased accuracy is because of a property that is unique to Bayesian inference -- namely the property of \\emph{shrinkage}. When one specifies a group-level prior distribution for the slope (e.g., the $\\text{Uniform}(-20,20)$ distribution that I used in Model 1), this prior is then combined with the data likelihood via Bayes theorem by multiplication. In our case, extreme individual slope estimates (those beyond -20 and 20) vanish by virtue of being multiplied by the prior probability of obtaining those estimates (i.e., probability = 0). The resulting posterior distribution is then \\emph{shrunk} away from these parameter estimates.\n\nTo demonstrate this, I conducted a simulation. I randomly generated $dRT$ values for a small samples of $n=15$ simulated participants as follows. First, I generated 15 random slopes $b_i$, where $b_i\\sim \\text{Gaussian}(-10,1)$. That is, I assumed that individual slopes are drawn from a normal distribution centered at -10, with standard deviation 1. Similarly, I randomly generated 15 random intercepts $a_i$, where $a_i \\sim \\text{Uniform}(-200,200)$. Then, I generated \n\n\\[\n dRT_{ij} = a_i + b_i(j) + \\varepsilon_{ij}\n\\]\n\n\\noindent\nwhere $i=1,\\dots,15$, $j=1,2,8,9$, and $\\varepsilon_{ij} \\sim \\text{Gaussian}(0,100)$. Then, I computed a classical linear regression for each participant $i=1,\\dots,15$, recording each estimated slope $\\hat{b}_i$. Finally, I fit the hierarchical Bayesian model (Model 1) to the overall set of data to compute the posterior distribution of $b$.\n\nThe results can be seen in Figure \\ref{fig:simulation}. Notice that while both the frequentist 95\\% confidence interval and the Bayesian 95\\% highest posterior density interval contain the population mean $b=-10$, the Bayesian estimate is much less variable. This is because compared to the classical linear regression method (e.g., Lorch \\& Myers, 1990), the extreme parameter estimates are shrunk toward the posterior mean. The same story is repeated with hypothesis testing, as well. Indeed, performing a $t$-test on the individual regression slopes \\citep[as in][]{fias1996} results in a non-significant SNARC effect, $t(14)=-1.82$, $p=0.09$, which results in a Type II error. However, a Bayesian hypothesis test (the Savage-Dickey method) yields a Bayes factor that favors the SNARC effect by a factor of 75 to 1. \n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.9\\linewidth]{simulation.jpeg}\n \\caption{Slope estimates from a simulated data set of $n=15$ participants. The solid line represents the posterior density of $b$, the group-level slope estimate for the SNARC effect. The histogram represents the slope estimates obtained from separate classical linear regressions for each individual.}\n \\label{fig:simulation}\n\\end{figure}\n\n\\section{Discussion}\n\nThe purpose of this paper was to introduce a hierarchical Bayesian linear regression framework for measuring group-level and individual-level numerical representations. The models were then applied to two common markers of numerical representations: the SNARC effect, which indexes spatial-numerical associations, and the numerical distance effect (NDE), which indexes representations of numerical magnitude. The models developed here represent a straightforward Bayesian extension of the classical linear regression approach to measuring these effects that was introduced by \\citet{lorch1990} and first applied to the SNARC effect by \\citet{fias1996}.\n\nBoth models were built from a hierarchical Bayesian framework, which is advantageous for several reasons. First, the models assume that the regression slopes (the primary object of interest in these analyses) are drawn from a group-level distribution. This hierarchical definition allows both group-level and individual-level slope estimates to be modeled simultaneously from the data. As such, we can answer questions at either the group-level (i.e., is there a SNARC effect?) or participant-level (i.e., are the slopes associated with ohter measures, such as math achievement?). For example, one might use this framework as a stepping stone to a more complex model, where the group-level slopes $b$ are hypothesized to differ by some independent variable. A concrete instance of this comes from studies in which the authors found group differences in the SNARC effect \\citep[e.g.,][]{fischer2010,cipora2015}. One could perform similar studies by building an additional linear model on these group-level parameters, with priors appropriate to such effects \\citep[e.g., a Cauchy prior, as in][]{rouder2009}. Importantly, since the inference would be done in a Bayesian framework, it is possible to measure evidence for null effects too \\citep{wagenmakers2007}, so the model could be used to test for \\emph{invariances} as well as differences. \n\nAnother advantage to using a hierarchical Bayesian model for numerical representations is that such models tend to have better measurement fidelity. Indeed, one of the advantages of the Bayesian framework in general is the notion of shrinkage, where extreme parameter estimates are ``shrunk'' toward the mean by virtue of the prior \\citep{gelman2013}. This property is particularly salient for small samples, where classical frequentist methods tend to perform poorly. I demonstrated exactly this phenomenon in the simulation above: because of some extreme individual-level slope estimates, the frequentist confidence interval was quite wide, and as a result, we could not detect the SNARC effect. However, the Bayesian highest posterior density interval provided a much more accurate estimate of the true population slope. Critically, in this simulation, only the Bayesian method produced the correct inference.\n\nA final advantage to this modeling framework that I will mention is that all assumptions of the model are made explicit. This may be a new approach to some, especially those who are accustomed to classical inferential software packages for which the statistical assumptions are kept ``under the hood.'' However, I think the approach presented in this paper can be very useful to a wide variety of problems in cognitive psychology, as the researcher can take the models presented here and modify them for any desired context. Indeed, the prior knowledge of a given field can be easily and coherently integrated into the model without too much work.\n\nIn summary, the models developed in this paper will provide a flexible tool that can be used to estimate group-level and individual-level numerical representations quickly and coherently. Further, the methods developed are not specific to numerical cognition, so application to a wide variety of problems should not be too difficult. Regardless of the application, the hierarchical Bayesian linear regression framework presented here provides a powerful, coherent, and accurate measurement model for applied work in cognitive psychology.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}