diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzaqdu" "b/data_all_eng_slimpj/shuffled/split2/finalzzaqdu" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzaqdu" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nIn this paper we would like to continue our analysis of the relation\nbetween supersymmetry and cosmic censorship, which we started in\n\\cite{US}. We have observed that the parameters of the static dilaton\nblack holes \\cite{GM} considered as bosonic solutions of $N=4$\nsupergravity are constrained due to the existence of supersymmetric\npositivity bounds. The effect of imposing these supersymmetric bounds\non the parameters of black hole solutions is the same as imposing cosmic\ncensorship: they prevent the solutions from exhibiting naked\nsingularities. Based on this example which generalizes the\nReissner-Nordstr\\\"om black hole case considered in the framework of\n$N=2$ supergravity \\cite{GH} we conjectured that, in general,\nsupersymmetry may act as a cosmic censor for static configurations in\nasymptotically flat spaces.\n\nThe generic feature of theories with global supersymmetry (without\ngravity) is the fact that the energy is non-negative, since the\nHamiltonian is a square of supersymmetry charges \\cite{Z}. It looks\nplausible that the cosmic censorship role of local supersymmetry is the\ngeneralization of the role of global supersymmetry as warrant of the\npositivity of energy in supersymmetric non-gravitational theories.\n\nIt may also happen that supersymmetry will help us in justifying the\ncosmic censorship hypothesis for certain nonsupersymmetric theories,\njust as it happened with the proof of the positivity of energy in\nGeneral Relativity. In that case it was enough to know that this theory\ncan be consistently {\\it embedded} into supergravity \\cite{DT}.\n\nIn our previous work we have investigated only static (non-rotating)\nblack holes. The supersymmetric positivity bound of the\nEinstein-Maxwell theory embedded in $N=2$ ungauged supergravity implies\n$m^2 \\geq q^2$ \\cite{GH}, \\cite{US}, which guarantees that the static\ncandidates to end-points of black hole evaporation (i.e. the\nReissner-Nordstr\\\"om solutions) have an event horizon covering the\nsingularity.\n\nIn the stationary case, though, it was not quite clear whether one could\nderive the analogous bound $m^2 \\geq a^2 + q^2$ for rotating Kerr-Newman\n(KN) black holes from supersymmetry alone. (Here $a$ is angular\nmomentum per unit mass.) Moreover, it was proven by Tod \\cite{T} that\nall the KN solutions with $m^2 = q^2$ admit Killing spinors. The KN\nblack hole is a configuration with $m^2 - a^2 - q^2\\geq 0$. The extreme\none has $ m^2 - a^2 - q^2= 0$. Any configuration with $m^2 = q^2$ and\nnon-vanishing angular momentum is far below extremality, which means\nthat the singularity is not covered by any event horizon.\n\nIn fact Tod proved that a whole class of stationary metrics including\nIsrael-Wilson-Perjes metrics \\cite{IW} admit $N=2$ supergravity Killing\nspinors. These solutions have been shown by Hartle and Hawking\n\\cite{HH} to have, in general, naked singularities. Therefore in\n\\cite{US} we restricted our conjecture about supersymmetry as the cosmic\ncensor only to static (and not stationary) asymptotically flat\nsolutions.\n\nNote, however, that the appearance of a naked singularity at $m^2 < a^2+\nq^2$ is a very subtle effect. At $m^2 = a^2+ q^2$ the singularity is\ndeeply hidden under the horizon. An infinitesimally small decrease of\nmass (or increment of angular momentum) immediately destroys the horizon\nand makes the singularity naked. In situations in which small causes\nmay have large effects, quantum corrections may be very important.\n\nIn particular, all supersymmetric KN black holes with a given charge\nhave the same mass, $m= |q|$, independently of their angular momentum.\nIn other words, they correspond to degenerate energy eigenstates. This\ndegeneracy, being a consequence of supersymmetry, can sometimes be\nremoved by quantum effects. And indeed, as we will see, with an account\ntaken of the trace anomaly, only the state with $a=0$ (the nonrotating\nReissner-Nordstr\\\"om black hole) remains supersymmetric.\n\n\\section{Supersymmetry of Israel-Wilson-Perjes metrics}\n\nWe will start by rederiving Tod's result using the standard language of\nfield theory rather than Newman-Penrose spinor language. The first\nanalysis of supersymmetric configurations of $N=2$ supergravity was\nperformed by Gibbons and Hull in \\cite{GH} using the standard field\ntheory spinors. They found that the static Papapetrou-Majumdar (PM)\nmetrics, and in particular, the extremal Reissner-Nordstr\\\"om black hole\nmetrics are supersymmetric\\footnote{We will not consider pp-wave spaces\nin this paper.}. Later Tod \\cite{T} found that, in addition to these\nconfigurations, some other configurations, in particular, some\nstationary metrics, also admit supercovariantly constant spinors. The\nclass of such metrics admitting $N=2$ Killing spinors is known in\nGeneral Relativity \\cite{KRAM} as the class of conformal-stationary\nEinstein-Maxwell fields with conformally flat 3-dimensional space, or\nIsrael-Wilson-Perjes (IWP) metrics. The PM metrics are just the static\nIWP metrics.\n\nThe IWP metrics and the corresponding electromagnetic fields can be\ncompletely described\\footnote{Our notation are given in \\cite{US}, and\n\\cite{TOdual}. In particular, hatted indices are the curved space\nones.} in terms of a time-independent complex function $V$:\n\\begin{eqnarray}\n&&ds^2 = (V\\bar V ) (dt + {\\vec w} d{\\vec x})^2 - (V\\bar V )^{-1}\\,\n(d{\\vec x} )^2 \\ , \\quad\n\\nonumber \\\\\n\\nonumber \\\\\n&&{\\vec \\nabla} \\times {\\vec w} = - i (V \\bar {V})^{-1}\n{\\vec \\nabla} \\log\\; (V \/ \\bar V)\\, ,\n\\nonumber \\\\\n\\nonumber \\\\\n&&F_{0i} = E_i = {\\textstyle\\frac{1}{2}}\\;\n\\partial_{\\hat \\imath} \\; (V+\\bar V)\\, ,\n\\nonumber \\\\\n\\nonumber \\\\\n&&{}^*F_{0i} = iB_i = {\\textstyle\\frac{1}{2}}\\;\n\\partial_{\\hat \\imath} \\; (V - \\bar V)\\, .\n\\label{IW}\n\\end{eqnarray}\n\nThis configuration will be a solution of the Einstein-Maxwell equations\nof motion in absence of matter if the complex function $V$ is chosen to\nbe the inverse of a harmonic function:\n\\begin{equation}\n\\triangle V^{-1} = 0 \\ , \\qquad \\qquad \\triangle \\bar V ^{-1}= 0\n\\ ,\n\\label{triangle}\n\\end{equation}\nwhere $\\triangle$ is the flat-space Laplacian in ${\\vec x}$.\n\nFor real $V$ these configurations are stationary and correspond, as we\nhave said, to the PM solutions \\cite{KRAM}. These are the only regular\nblack hole solutions in the IWP class. All solutions with complex $V$\nhave naked singularities, according to Hartle and Hawking \\cite{HH}. In\nparticular, the solution presented in eqs. (\\ref{IW}) includes the\ncharged KN solution with arbitrary angular momentum and charge equal to\nits mass, $m^2=q^2$. The KN charged rotating black hole solution is\ngiven by\n\\begin{eqnarray}\nds^2 &=& \\left(1- {2mr - q^2\\over r^2 + a^2 \\cos^2 \\theta}\\right)dt^2\n- {2a\\sin^2 \\theta\\, (2mr -q^2)\\over r^2 + a^2 \\cos^2 \\theta}dtd\\phi\n\\nonumber\\\\\n&-& (r^2 + a^2 \\cos^2 \\theta )\\left({dr^2 \\over r^2 + a^2 + q^2 -\n2mr} + d\\theta^2\\right )\n\\nonumber\\\\\n& -& \\sin^2\\theta\\left( r^2 + a^2 + {a\\sin^2 \\theta\\, (2mr -q^2)\\over\nr^2 + a^2 \\cos^2 \\theta} (2mr - q^2) \\right) d\\phi^2 \\ .\n\\label{KN}\n\\end{eqnarray}\nWhen $m^2=q^2$ and $a$ is arbitrary, this metric can be brought to the\nform of eq. (\\ref{IW}) \\cite{IW}. In Cartesian coordinates the complex\nharmonic function is\n\\begin{equation}\nV= 1 + {m\\over \\sqrt {x^2+y^2 +(z-ia)^2} } \\ .\n\\label{V}\n\\end{equation}\nIn terms of more suitable oblate spheroidal coordinates $x+iy = [(r-m)^2\n+ a^2]^{1\/2}\\sin \\theta e^{i\\phi}, \\; z = (r-m) \\cos \\theta$, the\nfunction $V$ takes the form\n\\begin{equation}\nV= 1 + {m\\over r- m - i a \\cos\\theta }\\, ,\n\\label{R}\n\\end{equation}\nand\n\\begin{equation}\nV\\bar V = {(r-m)^2- a^2 \\cos^2\\theta \\over r^2 + a^2\n\\cos^{2}\\theta}\\, ,\n\\label{VV}\n\\end{equation}\nso the Euclidean 3-metric becomes\n\\begin{equation}\n(d{\\vec x})^2 =\\Bigl((r-m)^2 + a^2 \\cos^2 \\theta\\Bigr)\\left({dr^2\n\\over (r-m)^2 + a^2 } + d \\theta^2\\right) + \\Bigl((r-m)^2 + a^2\\Bigr)\n\\sin^2 \\theta d\\phi^2 \\ .\n\\label{dx}\n\\end{equation}\nThe corresponding $\\vec{\\omega}$ is\n\\begin{equation}\n{\\vec w}\\cdot d{\\vec x}= {(2mr -m^2) a \\sin^2 \\theta \\over (r-m)^2 +\na^2 \\cos^2 \\theta } d\\phi \\ .\n\\label{w}\n\\end{equation}\nSubstituting eqs. (\\ref{VV}), (\\ref{dx}) and (\\ref{w}) into eq.\n(\\ref{IW}) and comparing the result with eq. (\\ref{KN}) one can see\nthat this particular IWP metric with function $V$ given by eq.\n(\\ref{V}) coincides with that of a KN charged rotating black hole with\n$m^2=q^2$.\n\nWith this in mind we can analyse the problem of supersymmetry for the\ngeneral class of metrics (\\ref{IW}), since the KN solution with\n$m^2=q^2$ is a particular case of such solutions. We will come back to\nthis specific solution when analysing the contribution of the trace\nanomaly to the equations of motion.\n\nConsider now the supersymmetry transformation of the gravitino field in\n$N=2$ supergravity:\n\\begin{equation}\n{\\textstyle\\frac{1}{2}} \\delta_{\\epsilon} \\Psi_{\\mu I} = \\nabla _\\mu\n\\epsilon_I - {\\textstyle\\frac{1}{2}}\\epsilon_{IJ}\n\\sigma^{ab}F_{ab} \\gamma_\\mu \\epsilon^J \\ , \\qquad I,J = 1,2.\n\\end{equation}\n\nWe want to find time-independent {\\it Killing spinors} $\\epsilon^{I}$,\ni.e. spinors for which the above expression vanishes,\n\\begin{equation}\n\\delta_{\\epsilon} \\Psi_{\\mu I} =0 \\ ,\n\\end{equation}\nand whose partial time derivative vanishes too,\n\\begin{equation}\n\\partial_{\\hat{0}}\\epsilon_{I}=0 \\ .\n\\label{time}\n\\end{equation}\n\nWe already know from Tod's work that these equations will have\nnontrivial solutions for IWP metrics, and so we will substitute eqs.\n(\\ref{IW}) in it, and we will look for the Killing spinors which we know\nto exist. But we will not require the field configurations to satisfy\nany specific equations of motion like the Einstein-Maxwell equations of\nmotion in absence of matter or any other equations. Thus $V$ will not\nbe constrained to be the inverse of a harmonic function as in eq.\n(\\ref{triangle}) and will remain arbitrary for most of our discussion.\n\nIt is convenient to express the supersymmetry transformation of the\ngravitino in terms of Dirac spinors $\\epsilon=\\epsilon^{1}+\\epsilon_{2}$\nand $\\psi_{\\mu}=\\psi_{\\mu}^{1}+\\psi_{\\mu2}$,\n\\begin{equation}\n{\\textstyle\\frac{1}{2}}\\delta_{\\epsilon}\\psi_{\\mu}=\n\\nabla_{\\mu}\\epsilon+{\\textstyle\\frac{1}{2}}\\sigma^{ab}F_{ab}\n\\gamma_{\\mu}\\gamma_{5}\\epsilon\\, .\n\\end{equation}\nIf we express the chiral Majorana spinors $\\epsilon_{I}$ in terms of\ntwo-component Weyl spinors $\\tilde{\\epsilon}_{I}$ according to our\nconventions we have\n\\begin{equation}\n\\epsilon_{I}=\n\\left(\n\\begin{array}{c}\n\\tilde{\\epsilon}_{I}\\\\\n0\n\\end{array}\n\\right)\n\\, ,\n\\hspace{1cm}\n\\epsilon^{I}=\n\\left(\n\\begin{array}{c}\n0 \\\\\n\\tilde{\\epsilon}^{I}\n\\end{array}\n\\right)\n\\, ,\n\\hspace{1cm}\n\\epsilon=\n\\left(\n\\begin{array}{c}\n\\tilde{\\epsilon}_{2} \\\\\n\\tilde{\\epsilon}^{1}\n\\end{array}\n\\right)\n\\, .\n\\end{equation}\nFirst we take the time component of the Killing equation\n$\\delta_{\\epsilon} \\Psi_{\\hat{0} I} =0$. Using the time-independence of\nthe spinors we are looking for, eq. (\\ref{time}), we arrive to\n\\begin{equation}\n\\left(\n\\begin{array}{c}\n\\sigma^{i} [\\omega_{0}^{+0i}\\tilde{\\epsilon}_{2}-\ni F^{+0i}\\tilde{\\epsilon}^{1}] \\\\\n\\\\\n\\sigma^{i} [-\\omega_{0}^{+0i}\\tilde{\\epsilon}^{1}-\ni F^{+0i}\\tilde{\\epsilon}_{2}]\n\\end{array}\n\\right)=0\\, ,\n\\end{equation}\nwhich, upon use of eqs. (\\ref{IW}), implies the following relation\nbetween the Killing spinors:\n\\begin{equation}\n\\tilde{\\epsilon}^{1} = - i\\left({\\overline{V}\/V}\\right)^{\\frac{1}{2}}\n\\tilde{\\epsilon}_{2} \\, ,\n\\label{relation1}\n\\end{equation}\nor, in terms of the chiral Majorana spinors\n\\begin{equation}\n\\epsilon^{1}+(\\overline{V}\/V)^{\\frac{1}{2}}\\gamma^{0}\\epsilon_{2}=0\\, .\n\\label{relation2}\n\\end{equation}\nNow we take the spatial components of the Killing equation\n$\\delta_{\\epsilon} \\Psi_{i}=0$. Using eqs. (\\ref{IW}) and the relation\nbetween the spinors eq. (\\ref{relation1}) we get the following two\nequations:\n\\begin{eqnarray}\n\\partial_{\\hat{\\imath}}(\\overline{V}^{\\frac{1}{2}}\\tilde{\\epsilon}^{1})\n& = & 0\\; ,\n\\nonumber \\\\\n\\partial_{\\hat{\\imath}}(V^{\\frac{1}{2}}\\tilde{\\epsilon}_{2}) & = &0\\; ,\n\\end{eqnarray}\nwhich imply for the chiral Majorana spinors,\n\\begin{eqnarray}\n\\epsilon^{1} & = & \\overline{V}^{\\frac{1}{2}}\\epsilon_{(0)}{}^{1}\\; ,\n\\nonumber \\\\\n\\epsilon_{2} & = & V^{\\frac{1}{2}}\\epsilon_{(0) 2}\\; ,\n\\end{eqnarray}\nwhere $\\epsilon_{(0)}{}^{1}$ and $\\epsilon_{(0) 2}$ are constant chiral\nMajorana spinors. These equations will be consistent with eq.\n(\\ref{relation2}) if the constant spinors themselves satisfy\n\\begin{equation}\n\\epsilon_{(0)}{}^{1} + \\gamma^{0} \\epsilon_{(0) 2} = 0\\ .\n\\end{equation}\n\nLet us stress that the fundamental difference between supersymmetric\nconfigurations with naked singularities and without them among the IWP\nclass is the presence or absence of imaginary part in the function $V$.\nThis is the only function in our Ansatz, which solves Killing spinor\nequations and allowed Tod to find supersymmetric configurations without\nreference to any equation of motion.\n\n\\section{Consistency condition for unbroken supersymmetry}\n\nConsider the classical Einstein-Maxwell action\n\\begin{equation}\nS_{EM}= - {1 \\over 4}\\int d^4 x\\sqrt{-g}\\; (R+F^2)\\, ,\n\\end{equation}\nwhich is the bosonic sector of $N=2$ supergravity. The effective\nequations of motion are\n\\begin{equation}\n\\frac{\\delta S_{EM}}{\\delta g^{\\mu\\nu}} = J_{\\mu\\nu}\\, , \\qquad\n\\frac{\\delta S_{EM}}{\\delta A_{\\mu}} = J^{\\mu} \\, .\n\\label{eqmo}\n\\end{equation}\nThe two tensors $J_{\\mu\\nu}$ and $J^{\\mu}$ are the ``right-hand side\" of\nthe metric and electromagnetic vector potential equations of motion.\nThese two tensors vanish for classical (on-shell) configurations but we\nare going to consider general configurations obeying the equations of\nmotion with $J_{\\mu\\nu}$ and $J^{\\mu}$ nonvanishing in general. The\nnotation emphasizes the fact that $J_{\\mu\\nu}$ is different from the\nclassical electromagnetic energy-momentum tensor that appears in the\nEinstein-Maxwell theory. Later on we will be interested in\nsemiclassical configurations for which these tensors are induced by\nquantum corrections.\n\nIn \\cite{KO} we have derived some consistency conditions (Killing Spinor\nIdentities) that any supersymmetric configuration has to satisfy. We\nare going to show that the only configurations which satisfy these\nidentities are those with $V$ real.\n\nTo find the $N=2$ supergravity Killing Spinor Identities we need the\nfunction\n\\begin{equation}\n\\Omega \\equiv \\sum_b J_b \\delta_{\\epsilon}\\, \\phi^b = J_{\\mu\\nu}\n\\delta_{\\epsilon}\\, g^{\\mu\\nu}+J^{\\mu}\\delta_{\\epsilon}\\, A_{\\mu}\\, ,\n\\end{equation}\nwhere the supersymmetry transformation of the metric is denoted by\n$\\delta_{\\epsilon} g^{\\mu\\nu}$, and that of the vector field by\n$\\delta_{\\epsilon} \\, A_{\\mu}$. Now we have to differentiate this\nfunction over the gravitino field, and the result has to vanish when\n$\\epsilon^{I}$ is a Killing spinor. From now on we will assume this to\nbe so. Then the Killing Spinor Identities take the form\n\\begin{equation}\nJ^{\\mu\\nu}\\overline{\\epsilon}^{I}\\gamma_{\\nu} + \\frac{1}{2}\\;\nJ^{\\mu}\\overline{\\epsilon}_{\\;J}\\; \\epsilon^{JI} = 0 \\ .\n\\label{example}\n\\end{equation}\nThis equation was derived from supersymmetry and therefore the spinor in\nthis equation is anticommuting. However, the identity must hold for\ncommuting spinors as well. Using commuting spinors it is simple to\nderive the consequences of the Killing Spinor Identities for IWP\nconfigurations. Using the algebraic relation (\\ref{relation2}), which\nis valid also for commuting Killing spinors, one can derive the\nfollowing relation between the function $V$ and the bilinear\ncombinations of commuting Killing spinors:\n\\begin{equation}\n\\bar \\epsilon^{I} \\gamma_a \\epsilon_{I} = (2i\\;|V| \\, ,\n\\quad \\vec 0) \\, ,\n\\qquad \\bar \\epsilon_{I} \\epsilon_{J} \\epsilon^{IJ} = -2i\\; V \\, .\n\\end{equation}\nNow we may consider eq. (\\ref{example}), where the spinor is commuting.\nWe multiply this equation by the commuting spinor $\\epsilon_{I}$, sum\nover the index $I$, and we get for the IWP metrics \\begin{equation}\nJ^{\\mu 0}|V|- \\frac{1}{2}\\; J^{\\mu} V = 0\\, , \\end{equation} which\nimplies, for complex $V$, $J^{\\mu 0}=J^{\\mu}=0$. We are left with\n\\begin{equation}\nJ^{\\mu\\nu}\\overline{\\epsilon}^{I}\\gamma_{\\nu}=0\\, .\n\\label{reduced}\n\\end{equation}\nNow we can multiply this equation by a spinor $\\eta_{I}$ such that\n$\\overline{\\epsilon}^{I}\\gamma_{\\nu}\\eta_{I}\\equiv p_{\\nu}\\neq 0$.\nThis gives\n\\begin{equation}\nJ^{ij}p_{i}=0\\, ,\n\\end{equation}\nwhich means that\n\\begin{equation}\nJ^{ij}=(\\eta^{ij}-\\frac{p_{i}p_{j}}{p^{2}})f\\, .\n\\label{Jp}\n\\end{equation}\nFinally, if we multiply eq. (\\ref{reduced}) by $\\gamma_{\\nu} \\eta^{J}\n\\epsilon_{IJ}$ and take into account that $J^{\\mu\\nu}$ is a symmetric\ntensor, we get\n\\begin{equation}\nJ^{\\mu\\nu}\\overline{\\epsilon}^{I}\\gamma_{\\nu}\\gamma_{\\mu}\\eta^{J}\n\\epsilon_{IJ}=\nJ^{\\mu\\nu}g_{\\mu\\nu}\\overline{\\epsilon}^{I}\\eta^{J}\n\\epsilon_{IJ}=0\\, .\n\\end{equation}\nSince $\\overline{\\epsilon}^{I}\\eta^{J}\\epsilon_{IJ}\\neq 0$, this implies\nthat $J_{\\mu}{}^{\\nu}$. This fact, together with eq. (\\ref{Jp}) proves\nthat, if $V$ is complex, $J^{\\mu\\nu}=J^{\\nu}=0$.\n\nThus, for configurations with $V\\neq \\bar V$, which in general have\nnaked singularities, the consistency conditions for supersymmetry lead\nto relations between the energy-momentum tensor and the Maxwell current\nwhich include a complex function $(V\/\\overline{V})^{\\frac{1}{2}}$. This\nis not acceptable, and the consequence is that the right-hand sides of\nthe Einstein and Maxwell equations have to vanish for supersymmetric\nconfigurations with {\\it complex} $V$ \\footnote{Observe that purely {\\it\nclassical} KN configurations have vanishing $J_{\\mu \\nu}$ and $J^{\\mu}$.\nTherefore, from the purely classical point of view they {\\it are}\nsupersymmetric.}:\n\\begin{equation}\nJ_{\\mu \\nu} =J^{\\mu} = 0 \\ .\n\\end{equation}\nIn particular we have to require the absence of quantum corrections to\nthe right-hand side of the trace of Einstein equation for supersymmetric\nconfigurations with complex $V$:\n\\begin{equation}\nR = g^{\\mu\\nu} J_{\\mu\\nu} = 0\\, .\n\\end{equation}\n\n\\section{The trace anomaly}\n\nThe trace anomaly (also called Weyl anomaly) in gravitational\nfour-dimensional theories was discovered by Capper and Duff about twenty\nyears ago \\cite{CD}. The existence of this anomaly means that the\nconformal invariance under Weyl rescaling of classical gravitational\nfield syste ms does not survive in the quantum theory.\n\nThe trace anomaly of the one-loop on-shell supergravity is given by the\nfollowing expression \\cite{CD}:\n\\begin{equation}\nT= g^{\\mu\\nu} < T_{\\mu\\nu}> \\ = \\, {A\\over 32 \\pi^2} {}^*\nR_{\\mu\\nu\\lambda\\delta} {}^{*}R^{\\mu\\nu\\lambda\\delta} \\ .\n\\end{equation}\nThe coefficient $A$ is known for all fields interacting with gravity.\n\nThe integrated form of the anomaly in Euclidean space expresses the\ntrace of the energy-momentum tensor through the Euler number of the\nmanifold,\n\\begin{equation}\n\\int d^4x \\sqrt{-g} \\;T = A \\, \\chi \\ \\ .\n\\end{equation}\n\nThe fields of $N=2$ supergravity include a graviton, 2 types of\ngravitino and a vector field. As we see in the table, the anomaly\ncoefficient $A= {11\\over 12}$ of pure $N=2$ supergravity does not\nvanish.\n\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline \\hline\n Field & 360A & $ N=2$ & $ N=2$ & $N=2$& $ N=4$ & $ N=4$ \\\\\n {}~ & ~ & supergravity & Yang-Mills & hypermultiplet &\nsupergravity &\nYang-Mills\n\\\\\n\\hline\n$ e_{\\mu}^a$&848&1&0& 0&1&0\\\\\n\\hline\n$\\psi_{\\mu}$&-233&2&0&0&4&0\\\\\n\\hline\n$A_{\\mu}$&-52&1&1&0&6&1\\\\\n\\hline\n$\\chi$&7&0&2&2&4&4\\\\\n\\hline\n$\\phi$&4&0&2&4&1&6\\\\\n\\hline\n$\\phi_{\\mu\\nu}$&364&0&0&0&1&0\\\\\n\\hline\n{}~&~&A=${11\/12}$ & A=$-{1\/12}$ & A=${1\/12}$& A=$0$&A=$0$\\\\\n\\hline \\hline\n\\end{tabular}\n\\vskip 0.3cm\nTable 1: {Anomalies in N=2 and N=4 supermultiplets.}\n\\end{center}\n\n\nThe function ${}^*{}R_{\\mu\\nu\\lambda\\delta}\n{}^{*}R^{\\mu\\nu\\lambda\\delta}$ does not vanish in general for IWP\nconfigurations. In particular, one can calculate this function for the\ncharged KN solution and check that for arbitrary angular momentum and\ncharge equal to its mass $m^2=q^2$ this function does not vanish. One\ncan use for this purpose the values of non-vanishing components of the\nWeyl tensor $C_{abcd}$ and Maxwell tensor $F_{ab}$ given for this\nsolution in \\cite{KRAM} in an isotropic tetrad basis. The expression\nfor the anomaly is given by\n\\begin{equation}\n{}^*R_{\\mu\\nu\\lambda\\delta}\n{}^*R^{\\mu\\nu\\lambda\\delta}= 24 (\\Psi_2 \\Psi_2 + h.c.) - 32 \\;(\\Phi_1\n\\bar \\Phi_1)^2 \\ ,\n\\label{anom}\n\\end{equation}\nwhere\n\\begin{eqnarray}\n\\Psi_2 &= &-{m(r+ia\\cos\\theta)-q^2\\over(r-ia\\cos\\theta)^3\n(r+ia\\cos\\theta)}\\, , \\nonumber\\\\ \\nonumber\\\\ \\Phi_1 &=& {q \\over\n\\sqrt\n2 ( r - i a \\cos \\theta )^2}\\ .\n\\end{eqnarray}\nWe have checked that the function (\\ref{anom}) does not vanish for any\nKN solution with arbitrary values of $m, q, a$ and in particular for\n$m=|q|$. As an additional consistency check we have calculated the\nintegrated form of the eq. (\\ref{anom}) for Reissner-Nordstr\\\"om black\nhole with $m\\geq|q|,\\, a=0$. The result is 2, which agrees with the\nwell-known Euler characteristic of the Schwarzschild and Kerr black\nholes:\n\\begin{equation}\n\\chi = {1\\over 32 \\pi^2}\\int R^{ab} \\wedge\nR^{cd} \\; \\epsilon_{abcd}= 2 \\ .\n\\end{equation}\n\nHow does this affect the conclusion of the previous section? Let's\nconsider now {\\it semiclassical} configurations of this theory, that is,\nconfigurations which satisfy the semiclassical equations of motion\nobtained by adding first-order quantum corrections to the right-hand\nside of the classical equations of motion. These semiclassical\nconfigurations, then, satisfy the equations (\\ref{eqmo}) where the trace\nof $J_{\\mu\\nu}$ is identified with the trace anomaly. This is indeed a\nvery small correction which should not produce big changes in the metric\nof classical configurations. In particular, it is reasonable to expect\nthat classical configurations with nonvanishing imaginary part of $V$\n(as the $m=|q|$ KN configurations) will continue to have a nonvanishing\nimaginary part of $V$ after the quantum corrections have been taken into\naccount.\n\nThe presence of $J_{\\mu}{}^{\\mu}\\neq 0$ and complex $V$ is incompatible\nwith the supersymmetry consistency conditions. Thus, when we embed the\nEinstein-Maxwell theory in a supersymmetric theory for which $A\\neq 0$\n(i.e. the trace anomaly does not vanish) the semiclassical KN\nconfigurations (now including those with naked singularities $m=|q|$)\nare not supersymmetric anymore.\n\nThe question arises immediately how to make the anomaly coefficient $A$\nto vanish. Looking on the table we may observe that the anomaly\nvanishes for any theory which is build out of the $N=4$ multiplet of\nsupergravity and arbitrary number of $N=4$ Yang-Mills multiplets. If we\ndo not want to increase the number of supersymmetries, we may add to\n$N=2$ supergravity $11+n$ $N=2$ vector multiplets and $n$\nhypermultiplets. The anomaly of such system of fields vanishes. What\nhappens, however, with our naked singularity solutions?\n\nWe have found that for all above mentioned theories where the anomaly is\ncancelled, supersymmetric configurations with naked singularities are\nnot solutions of the classical equations of motion anymore. The\nsimplest explanation of this mechanism can be given for the $N=4$\ntheory. We would like to add new fields to the theory in such a way\nthat they propagate in the loop diagrams and cancel the anomaly.\nSimultaneously we want to make only minimal changes in classical field\nequations, in order to preserve our previous solutions. This is not\npossible. Indeed, in the $N=4$ case there is one new equation for the\ndilaton field of the form\n\\begin{equation}\n\\nabla^{2}\\phi - {\\textstyle\\frac{1}{2}}\n e^{-2\\phi}F^{2} =0 \\ .\n\\label{dil}\n\\end{equation}\nThe configurations with naked singularities which we have considered\nbefore had a constant (space- and time-independent) value of the dilaton\nfield and a non-vanishing value of $F^{2}$. Thus they do not satisfy\nequation (\\ref{dil}). In other words, by adding new fields which cancel\nthe anomaly, we are adding new equations which are not satisfied by our\nold solutions with naked singularities. This effect is a consequence of\nthe general structure of the supersymmetric coupling of matter\nmultiplets to vector fields in gravitational multiplets \\cite{toine}.\nIn particular, the coupling of $N=2$ matter multiplets (and we need at\nleast 11 vector multiplets to cancel the anomaly) will also result in\nadditional equations of the type (\\ref{dil}) which will invalidate the\nnaked singularity solutions.\n\nThus, we have found that in the theory under consideration there are no\nstationary supersymmetric solutions with naked singularities. In the\ncase of {\\it static} solutions studied in \\cite{US} this was enough to\nshow that for nonsupersymmetric configurations the singularities are\neven deeper hidden by the horizon, which means that supersymmetry works\nas a cosmic censor. It remains to be seen whether an analogous\nstatement is true for general stationary solutions. In any case, the\nresults obtained above confirm that there exists some deep and\npreviously unexplored relation between the absence of naked\nsingularities and supersymmetry.\n\nWe are grateful to B. Carter, G. W. Gibbons, M. Grisaru, S. Hawking, G.\nHorowitz, A. Linde, P. van Nieuwenhuizen and M. J. Perry for extremely\nuseful discussions. We would like to express our gratitude to the\norganizers of the programme ``Geometry and Gravity\" at the Newton\nInstitute for the most stimulating atmosphere for work and for the\nfinancial support. The work of R. K. was also supported by NSF grant\nPHY-8612280 and the work of T. O. was supported by European Communities\nHuman Capital and Mobility programme grant.\n\n\\vskip 1cm\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe Rauzy Gasket $\\mathcal G$ is a compact subset of the standard 2-simplex,\n\t$\n\t\t \\Delta\n\t\t= \n\t\t \\{ \n\t\t (x, y, z) \n\t\t \\;:\\; \n\t\t x, y, z \\geq 0,\\ %\n\t\t x + y + z = 1\n\t\t \\}.\n\t$ \nIt plays the role of an exceptional set in the theory of interval exchange transformations and other settings, and is the limit set of the iterated function scheme for the three weak projectivised linear maps $T_1, T_2, T_3: \\Delta \\to \\Delta$, defined by\n\t\\begin{gather*}\n\t\t \tT_1(x,y,z) \n\t\t= \n\t\t \t\\left(\n\t\t \t\t\\frac{1}{2-x},\n\t\t \t\t\\frac{y}{2-x}, \n\t\t \t\t\\frac{z}{2-x}\n\t\t \t\\right),\n\t\\\\\n\t\t \tT_2(x,y,z) \n\t\t= \n\t\t\t \\left(\n\t\t\t\t \\frac{x}{2-y}, \n\t\t\t\t \\frac{1}{2-y}, \n\t\t\t\t \\frac{z}{2-y}\n\t\t\t \\right),\n\t\\\\\n\t\t\tT_3(x,y,z)\n\t\t= \n\t \t\t\\left(\n\t \t\t\t\\frac{x}{2-z},\n\t \t\t\t\\frac{y}{2-z}, \n\t \t\t\t\\frac{1}{2-z}\n\t \t\t\\right);\n\t\\end{gather*}\ni.e., $\\mathcal G$ is the smallest non-trivial closed set such that $\\mathcal G = \\bigcup_{j=1}^3 T_j (\\mathcal G)$. \n \n %\n \t\\begin{figure}\n \t\t\\centerline{\\includegraphics[height=5.8cm]{rauzy2}}\n \t\t\\caption{The Rauzy gasket}\n\t\\end{figure}\n\\smallskip\nThe gasket has an interesting history, appearing for the first time in 1991 in the work of Arnoux and Rauzy \\cite{ar}, in the context of interval exchange transformations, where it was conjectured that $\\Leb(\\mathcal G)=0$.\nThe gasket was rediscovered by Levitt in 1993, in a paper which also included a proof (due to Yoccoz) that $\\Leb(\\mathcal G)=0$.\nThe gasket $\\mathcal G$ emerged for a third time in the work of De Leo and Dynnikov \\cite{dd}, this time in the context of Novikov's theory of magnetic induction on monocrystals (see \\cite{rauzy dynamics} for the dichotomy between this and \\cite{ar}). They gave an alternative proof that $\\Leb(\\mathcal G)=0$ and proposed the stronger result $\\dim_H(\\mathcal G)<2$. \nNovikov and Maltsev \\cite{mn} also conjectured the stronger bound $\\dim_H(\\mathcal G) < 2$, which was rigorously established by Avila, Hubert and Skripchenko \\cite{ahs}. \nEmpirical estimates in \\cite{dd} suggest $\\dim_H(\\mathcal G) \\approx 1.72$, and a lower bound was shown in \\cite{G-R-Matheus}. Lastly, Fougeron used semiflows and thermodynamic techniques to show $\\dim_H(\\mathcal G) < 1.825$ \\cite{fou}. Using completely elementary methods, we show the following improved upper bound. \n\n\\begin{thm}\\label{short}\n\t$\\dim_H(\\mathcal G) \\leq 1.7407$.\n\\end{thm}\n\nThe Rauzy Gasket has a number of interesting recent applications. Gamburd, Magee and Ronan \\cite{gmr} showed asymptotic estimates for integer solutions of the Markov-Hurwitz equations featuring $\\dim_H(\\mathcal G)$.\nHubert and Paris-Romaskevich in \\cite{hp} considered triangular tiling billiards, modelling refraction in crystals. The gasket $\\mathcal G$ parameterises triangles admitting trajectories which escape non-linearly to infinity and closed orbits which approximate fractal-like sets.\n\nIn section 2 we give the technical result which leads to the bound in Theorem \\ref{short}. This is formulated in terms of certain infinite matrices.\nIn section 3 we give elementary preliminary bounds on the area and diameter of small triangles given as the images of $\\Delta$ under compositions of the maps $T_1$, $T_2$ and $T_3$.\nIn section 4 we use these to obtain a bound for the dimension, provided an associated sequence of real numbers $X_n$ converges to zero.\nIn sections 5 and 6 we present the core of the proof. In section 5, we use the estimates from section 3 to bound $X_n$ in terms of expressions satisfying an iterative relation.\nIn section 6 we use the renewal theorem to deduce that $X_n \\to 0$ under the hypotheses of Theorem \\ref{long}. Finally, in section 7 we apply Theorem \\ref{long} empirically to deduce the bound in Theorem \\ref{short}.\n\nA fuller account appears in \\cite{thesis}.\n\n\\section{A formal statement}\n\nThe bound in Theorem \\ref{short} is a special case of a decreasing sequence of upper bounds, indexed by a parameter $m \\in \\mathbb N_{\\geq 2}$. Each bound can be described using powers of an infinite matrix.\n\\begin{definition}[An index set]\n\tLet $\\mathcal V = \\bigcup_{k=1}^{m-1}\\mathcal V_k$ denote the finite set where, for $k0$. Then, we can consider the square matrix $B$ indexed by $\\mathcal V \\cup \\{m\\}$, defined by\n\n\t\t$$\n\t\t\t\tB_{\\underline v,\\underline w}\n\t\t\t= \n\t\t\t\t\\begin{cases}\n\t\t\t\t\t\\displaystyle \n\t\t\t\t\t\\max_{\n\t\t\t\t\t x \\in \\Delta_{\\underline w}\n\t\t\t\t\t }\n \t\t\t\t\t\t(2-x_j)^{-3\\delta - \\lambda (1-\\delta)},\n\t\t\t\t&\n\t\t\t\t\t\\hbox{ if } \\underline w \\mapsto_j \\underline v;\n\t\t\t\t\\\\\n\t\t\t\t\t\t\t0, &\\hbox{ otherwise}.\n\t\t\t\t\\end{cases}\n\t\t$$\n\t%\n\t(Note that $B_{\\mathbin{\\mathpalette\\make@circled\\star}, \\underline v} = B_{\\underline v,m} = 0$ for all $\\underline v\\in \\mathcal V\\cup \\{m\\}$.)\n\\end{definition}\n\n\\begin{definition}\nTo define the infinite matrix, we introduce the following values, for $k \\in \\mathbb N$.\n\t$$ \n\t\t\ta_k \n\t\t= \n\t\t\t\\left(\n\t\t\t\t\\frac{k+1}{2k+1}\n\t\t\t\\right)^{3\\delta + \\lambda (1-\\delta)} \n \t\t+ \n \t\t\t\\frac{1}{8^\\delta}\n \t\t\t\\left(\n \t\t\t\t\\frac{k+1}{2k+1}\n \t\t\t\\right)^{\\lambda (1-\\delta)}\n \t\t\\qquad\n \t\t\\hbox{and}\n \t\t\\qquad\n\t\t\tb_k \n \t\t= \n \t\t\t\\left(\n \t\t\t\t\\frac{k+2}{k+3}\n \t\t\t\\right)^{3\\delta + \\lambda (1-\\delta)},\n \t$$\n %\n\\end{definition}\n\nWe now extend the finite matrix $B$ to give an infinite matrix.\n\n\\begin{definition}[An infinite matrix]\n\tFixing $\\delta > 0$, we define the infinite matrix $D$ indexed by $\\mathcal V \\cup\\{m, m+1, \\ldots\\}$ as follows.%\n\t\\footnote{Here, $N-1$ is the cardinality of $\\mathcal V$, and in the ordering of $\\mathcal V\\cup\\{m\\}$, we take $\\mathbin{\\mathpalette\\make@circled\\star}$ to be first and $m$ last.}\n\t%\n\t\t\\begin{align*}\n\t\t\t\tD\n\t\t&= \n\t\t\t\\begin{pmatrix}\n B_{1,1}&\\cdots&B_{1,{N-1}}&B_{1, N} + a_m&a_{m+1}&a_{m+2}& \\cdots\n\t\t\t\t\\\\\n\t\t\t\t\tB_{2,1}&\\cdots&B_{2, {N-1}}&B_{2,N}&0&0& \\cdots\n\t\t\t\t\\\\\n\t\t\t\t\t\\vdots&\\ddots&\\vdots&\\vdots&\\vdots &\\vdots & \\cdots\n\t\t\t\t\\\\\n\t\t\t\t\tB_{{N},1}&\\cdots&B_{{N},{N-1}}&B_{N,N}&0&0 & \\cdots \n\t\t\t\t\\\\\n\t\t\t\t\t0&\\cdots&0&b_m&0&0& \\cdots\n\t\t\t\t\\\\\n\t\t\t\t\t0&\\cdots&0&0&b_{m+1}&0& \\cdots\n\t\t\t\t\\\\\n\t\t\t\t\t0&\\cdots&0&0&0&b_{m+2}& \\cdots\n\t\t\t\t\\\\\n\t\t\t\t\t\\vdots&\\vdots&\\vdots&\\vdots&\\vdots&\\vdots& \\ddots\n\t\t\t\\end{pmatrix} \n\t\t\\\\\n\t\t&=\n\t\t\t\\begin{pmatrix}\n\t\t\t\t\t0&\\cdots&0&a_m&a_{m+1}&a_{m+2}& \\cdots\n\t\t\t\t\\\\\n\t\t\t\t\tB_{2,1}&\\cdots&B_{2, {N-1}} &0&0&0& \\cdots\n\t\t\t\t\\\\\n\t\t\t\t\t\\vdots&\\ddots&\\vdots&\\vdots&\\vdots &\\vdots & \\cdots\n\t\t\t\t\\\\\n\t\t\t\t\tB_{{N},1}&\\cdots&B_{{N},{N-1}}&0&0&0 & \\cdots \n\t\t\t\t\\\\\n\t\t\t\t\t0&\\cdots&0&b_m&0&0& \\cdots\n\t\t\t\t\\\\\n\t\t\t\t\t0&\\cdots&0&0&b_{m+1}&0& \\cdots\n\t\t\t\t\\\\\n\t\t\t\t\t0&\\cdots&0&0&0&b_{m+2}& \\cdots\n\t\t\t\t\\\\\n\t\t\t\t\t\\vdots&\\vdots&\\vdots&\\vdots&\\vdots&\\vdots& \\ddots\n\t\t\t\\end{pmatrix}.\n\t \\end{align*}\n\t%\n\\end{definition}\n \nLet $(D^k)_{i,j}$ be the entry corresponding to the $(i,j)$th entry of the $k$th power of $D$. Our main technical result is the following, which we will rephrase later in a more explicit way.\n\n\\begin{thm}\\label{long}\n If $\\delta > \\frac {1-\\lambda}{3-\\lambda} = 0.037\\ldots$ satisfies $\\sum_{k=1}^\\infty (D^k)_{1,1}\\leq 1$, then $\\dim_H(\\mathcal G) \\leq 1 + \\delta$.\n\\end{thm}\n\nIn particular, letting $m=9$ and estimating numerically the value of $\\delta \\approx 0.7407$ (rounding up to four decimal places) giving equality in the hypothesis of Theorem \\ref{long} gives the bound in Theorem \\ref{short} in the introduction as a corollary (see \\S 7).\n\n\n\\section{Triangle estimates}\n\nIn this section we collect together elementary but useful estimates for the triangles $\\Delta_{\\underline i}$.\n\n\\begin{lemma}[Area estimate]\\label{area}\n If $\\underline i = (i_1, \\ldots, i_n) \\in \\{1,2,3\\}^n$ and $j \\in \\{1,2,3\\}$ \n then, writing $j \\underline i = (j, i_1, \\ldots, i_n)$, we have that\n %\n $$\n \\frac\n {\\area(\\Delta_{j \\underline i})}\n {\\area(\\Delta_{\\underline i})}\n \\leq\n \\max_{x \\in \\Delta_{\\underline i}}\n (2-x_j)^{-3}.\n $$\n %\n\\end{lemma}\n\n\\begin{proof}\n By a change of variables,\n %\n $$\n \\area (\\Delta_{j\\underline i}) \n = \n \\area (T_j\\Delta_{\\underline i})\n =\n \\int_{\\Delta_{\\underline i}}\n \\Jac T_j(x)\n \\;\\mathrm dx\n \\leq\n \\max_{x \\in \\Delta_{\\underline i}}\n \\big( \\Jac {T}_j(x) \\big)\n \\area (\\Delta_{\\underline i}).\n $$\n %\n To complete the proof, we now show that $\\Jac T_j (x) = (2-x_j)^{-3}$. If $j=1$, with respect to the orthogonal basis\n $( \\frac1{\\sqrt2}(\\frac\\partial{\\partial x_2}-\\frac\\partial{\\partial x_3}),\\frac1{\\sqrt6}(2\\frac\\partial{\\partial x_1}-\\frac\\partial{\\partial x_2} -\\frac\\partial{\\partial x_3}))$,\n the derivative map $DT_j(x)$ takes the following form:\n %\n \\begin{equation}\n DT_1(x)\n =\n \\frac 1 {(2-x_1)^2}\n \\left(\n \\begin{array}{cc}\n 1 & 0 \\\\\n \\frac1{\\sqrt{3}}(x_3-x_2) & 2-x_1 \\\\\n \\end{array}\n \\right).\n \\label{eq-derivative as matrix}\n \\end{equation}\n %\n Thus, $\\Jac T_1 (x) = (2-x_1)^{-3}$. The other cases ($j\\neq 1$) follow by symmetry.\n\\end{proof}\n\n\\begin{rem}\n In fact, one can deduce a simple formula for $\\area(\\Delta_{\\underline i})$ from the precise form of the above jacobean, using matrix products (see \\cite{thesis}).\n\\end{rem}\n\nWe can apply a similar reasoning to the associated diameters.\n\n\\begin{lemma}[Diameter estimate]\\label{diameter}\nIf $\\underline i = (i_1, \\ldots, i_n) \\in \\{1,2,3\\}^n$ and $j \\in \\{1,2,3\\}$ then\n $$\n \\frac\n {\n \\diam(\\Delta_{j \\underline i})\n }\n {\n \\diam(\\Delta_{\\underline i})\n }\n \\leq \n \\max_{ x \\in \\Delta_{\\underline i}}\n (2-x_j)^{-\\lambda}.\n $$\n\\end{lemma}\n\n\\begin{proof}\n We now consider the operator norm, $\\|DT_j(x)\\|$, of the derivatives $T_j(x)$ (i.e., with respect to the ambient metric in $\\mathbb R^3$). Considering $j=1$, the matrix in \\eqref{eq-derivative as matrix} satisfies\n %\n $$\n \\big\\|DT_1(x)\\big\\|^2\n = \n \\frac \n \n \n {2b^2}\n \\left(\n \n 1 + a + b\n +\n \\sqrt{\n (1+ a + b)^2\n -\n 4b\n }\n }\n \\right\n ,\n $$\n %\n where $a = \\frac13(x_2-x_3)^2$ and $b = (2-x_1)^2$. Using the simple bound $a \\leq \\frac13 (1-x_1)^2$, one has that, for all $x \\in \\Delta$,\n %\n $$\n \\big\\|DT_1(x)\\big\\|^2\n \\leq \n f(x_1)\n :=\n \\frac{\n \n 2x_1^2 - 7x_1 + 8 +2(1-x_1) \n \\sqrt{\n \n x_1^2-5x_1+7\n \n }\n \n }\n {\n \n (2-x_1\n }.\n $$\n %\n We observe (with the help of a computer) that the function $t\\mapsto \\log(f(t))\/\\log(2-t)$ is increasing on $[0,1)$, and converges to $-2\\lambda$ as $t \\to 1^{-}$. Therefore $\\|DT_1(x)\\| \\leq \\sqrt{f(x_1)} \\leq (2-x_1)^{-\\lambda}$. The result for $j=1$ then follows simply from the mean value theorem. The cases of $j\\neq1$ subsequently follow by symmetry.\n\\end{proof}\n\nIn view of the previous lemmas, it is useful to have more explicit bounds involving $(2-x_j)^{-1}$ depending on where the point $x \\in \\Delta$ approximately lies. We henceforth denote\n $$\n A_{n,k}\n = \n \\{\n \\underline i \\in \\{1,2,3\\}^n \n \\;:\\; \n i_1 = \\cdots = i_k \\neq i_{k+1}\n \\}\n \\qquad(1\\leq k < n)\n $$\nand work with the following simple bounds.\n\n\\begin{lemma}\n If $\\underline i \\in A_{n,k}$ and $i_1=i,j,k \\in \\{1,2,3\\}$ are distinct\n then, for any $\\delta>0$, \n %\n \\begin{gather*}\n \\max_{x \\in \\Delta_{\\underline i}}\n (2-x_i)^{-1}\n \\leq\n \n \\frac\n {k+2}\n {k+3}\n \n ,\n \\\\\n \\phantom{\n \\qquad\n \\hbox{and}} \n \n \\max_{x \\in \\Delta_{\\underline i}}\n (2-x_j)^{-1}\n \\leq\n \n \\frac\n {k+1}\n {2k+1}\n \n , \n \\qquad\n \\hbox{and}\n \\\\\n \\max_{x \\in \\Delta_{\\underline i}}\n \\left\\{\n (2-x_j)^{-3\\delta}\n + \n (2-x_k)^{-3\\delta}\n \\right\\}\n\\leq \\left( \\frac{k+1}{2k+1}\\right)^{3\\delta} + \\frac 1 {8^\\delta}.\n \\end{gather*}\n\\end{lemma}\n\n\\begin{proof}\n By symmetry, it suffices to consider $\\underline i = (i_1, \\ldots, i_n)$ with $i_1 = \\cdots = i_{k} = 1$ and $i_{k+1} \\neq 1$. Then $\\Delta_{\\underline i}$ is contained in \n %\n \\begin{equation}\n \\label{R_k}\n \\text{cl}\\big( \n T^k(\\Delta) \\setminus T^{k+1}(\\Delta)\n \\big)\n =\n \\bigg\\{\n x \\in \\Delta\n \\;:\\;\n \\frac\n {k} \n {k+1}\n \\leq\n x_1\n \\leq\n \\frac\n {k+1} \n {k+2} \n \\bigg\\}\n \\end{equation}\n %\n (cl denoting the topological closure), where we have used that $T^k(\\Delta) = \\{x \\in \\Delta\\,:\\,x_1\\geq \\frac{k}{k+1}\\}$. The first two bounds follow directly:\n $$\n (2-x_1)^{-1}\n \\leq \n \\left(\n 2-\\frac{k+1}{k+2}\n \\right)^{-1}\n =\n \\frac{k+2}{k+3}\n \\qquad\\text{and}\n \\qquad\n (2-x_j)^{-1}\n \\leq \n \\left(\n 2-\\frac{1}{k+1}\n \\right)^{-1}\n =\n \\frac{k+1}{2k+1}. \n $$\n %\n \n For the third bound, we observe that the function $f(x) := (2-x_2)^{-3\\delta} + (2-x_3)^{-3\\delta}$ is convex on $\\Delta$ and thus takes its maximum on $T_1^k(\\Delta)$ at one of its vertices:\n %\n $$\n \\max_{\\Delta_{\\underline i}} \n f \n \\leq \n \\max_{T_1^k(\\Delta)}\n f \n = \n \\max \n \\left\\{ \n f\\circ T_1^k(1,0,0), \n f\\circ T_1^k(0,1,0), \n f\\circ T_1^k(0,0,1)\n \\right\\}.\n $$\n %\n More explicitly, noting that $f\\circ T_1^k(1,0,0) = f(1,0,0) \\leq f\\circ T_1^k(0,1,0) = f\\left( \\frac k {k+1}, \\frac 1 {k+1},0 \\right) = f\\left(\\frac k {k+1},0, \\frac 1 {k+1} \\right) = f\\circ T_1^k(0,0,1)$, one ha\n %\n $$\n \\max_{\\Delta_{\\underline i}}f \n \\leq\n \\left(\n 2-\\frac{1}{k+1} \n \\right)^{-3\\delta}\n +\n 2^{-3\\delta}\n = \n \\left(\\frac{k+1}{2k+1}\\right)^{3\\delta} + \\frac 1 {8^\\delta},\n $$\n \n as required.\n\\end{proof}\n\n\\section{Cover estimates}\n\nThe upper bound on the dimension in Theorem \\ref{long} is based on finding a value $\\delta \\in (0,1)$ so that Lemma 4.1 below applies.\n\nIts proof is simple and based on a simple sequence of open covers of $\\mathcal G$, $\\mathcal U_n$, each obtained by covering the set of $n$th level triangles, $\\{\\Delta_{\\underline i} \\hbox{ : } |\\underline i| = n\\}$.\n\n\\begin{lemma}\\label{dimbound}\n Assume $\\delta>0$ and that the sequence \n %\n $$\n X_n \n := \n \\sum_{|\\underline i| = n}\n \\area(\\Delta_{\\underline i})^{\\delta}\n \\diam(\\Delta_{\\underline i})^{1-\\delta} \n \\to \n 0 \n \\qquad\\hbox{as}\n \\qquad\n n \\to \\infty.\n $$\n %\n Then $\\dim_H(\\mathcal G) \\leq 1 + \\delta.$\n\\end{lemma}\n\n\t\t\\begin{figure}[hbt]\n\t\t\t\\begin{tikzpicture}\n\t\t\n\t\t\t\\foreach \\x\/\\y in {5\/2}\n\t\t\t{\n\t\t\t\t\\draw[fill = yellow] (0,0) -- (0,\\y) \n\t\t\t\t\t\tnode[midway,anchor = east]{$\\displaystyle \\frac{2\\area(\\Delta_{\\underline i})}{{\\diam(\\Delta_{\\underline i})}}$} \n\t\t\t\t\t-- (\\x,\\y) -- (\\x,0);\n\t\t\t\t\n\t\t\t\t\\draw[thick, fill = white] (0,0) -- (\\x\/5,\\y) -- (\\x,0) -- (0,0) node[midway,anchor=north]{$\\diam(\\Delta_{\\underline i})$};\n\t\t\t\n\t\t\t\t\n\t\t\t\t\\draw[dotted] (\\x+2,0.5*\\y) -- (\\x+2, 0.5*\\y);\n\t\n\t\t\t\t\\draw[fill = blue, opacity = 0.3] (\\x+1.5,0) -- (\\x+1.5,\\y) -- (2*\\x+1.5,\\y) -- (2*\\x+1.5,0) -- (\\x+1.5,0);\n\t\t\t\n\t\t\t\t\\draw[fill = yellow] (\\x+1.5,0) -- (\\x+1.5,\\y) -- (2*\\x+1.5,\\y) -- (2*\\x+1.5,0) -- (\\x+1.5,0);\n\t\t\t\n\t\t\t\n\t\t\t\t\\foreach \\n in {0,1,2,...,4}{\n\t\t\t\t\n\t\t\t\t\t\\draw[fill=blue, opacity = 0.1] (\\x+2+\\n,0.5*\\y) circle (1.41423);\t\t\t\n\t\t\t\t}\n\t\t\t\t\n\t\t\t\t\\draw(\\x+1.5,0) -- (\\x+1.5,\\y) -- (2*\\x+1.5,\\y) -- (2*\\x+1.5,0) node[midway,anchor = west]{$a$} -- (\\x+1.5,0) node[midway, anchor = north]{$b$};\t\t\n\t\t\t\t\n\t\t\t\t\\foreach \\n in {0,1,2,...,4}{\n\t\t\t\t\t\\draw[fill=black] (\\x+2+\\n,0.5*\\y) circle (0.05);\t\t\n\t\t\t\t}\t\t\n\t\t\t}\n\t\t\t\n\t\t\t\\draw (4\/3,1) node {$\\Delta_{\\underline i}$};\n\t\t\t\n\t\t\t\\end{tikzpicture}\n\t\t\\caption[Covering a triangle by disks]{Left: covering $\\Delta_{\\underline i}$ by a rectangle. Right: covering a rectangle by open disks.}\n\t\t\\label{figr-cover rectangles}\n\t\t\\end{figure}\n\t%\n\n\\begin{proof}\n As illustrated in Figure 2, each triangle $\\Delta_{\\underline i}$ is contained in a rectangle with side lengths $b = \\diam(\\Delta_{\\underline i})$ and $a = 2 \\area(\\Delta_{\\underline i})\/ \\diam(\\Delta_{\\underline i})$, which can in turn be covered by $[2b\/a]$ disks of diameter $2a$. That is, $\\Delta_{\\underline i}$ can be covered by $\\diam(\\Delta_{\\underline i})^2\/ \\area(\\Delta_{\\underline i})$ disks of diameter $4\\area(\\Delta_{\\underline i})\/ \\diam(\\Delta_{\\underline i})$.\n Denoting by $\\mathcal U_n$ the union of the disks covering all of the triangles $\\Delta_{\\underline i}$, this gives\n \n $$\n \\sum_{U \\in \\mathcal U_n}\n \\diam(U)^{1+\\delta}\n \\leq \n 4^{1+\\delta} \n \\sum_{|\\underline i|=n}\n \\left( \n \\frac\n { \\diam(\\Delta_{\\underline i})^2}\n { \\area(\\Delta_{\\underline i})}\n \\right)\n \\left(\n \\frac{\n \\area(\\Delta_{\\underline i})}\n {\\diam(\\Delta_{\\underline i})}\n \\right)^{1+\\delta}\n = \n 4^{1+\\delta} X_n.\n $$\n %\n The result follows from the standard definition of the Hausdorff dimension (see, e.g., \\cite{falconer}).\n\\end{proof}\n\n\\begin{rem}\n Perhaps surprisingly, this lemma appears to give a significant improvement on working exclusively with either $\\diam(\\Delta_{\\underline i})$ or $\\area(\\Delta_{\\underline i})$ (as in \\cite{fou} and \\cite{ar}).\n\\end{rem}\n\n\\smallskip\nTo apply Lemma \\ref{dimbound} we want to bound $X_n$ from above using a partition of\n$\\{1,2,3\\}^n$. \nWe then apply different bounds to the triangles $T_j(\\Delta_{\\underline i})$ according to which element of the partition the index $\\underline i$ lies in.\n\n\\begin{definition}[Partitioning up the sequences]\n For each $n > m$, we can partition\n %\n $$\n \\{1,2,3\\}^n \n :=\n \\bigcup_{\\underline v \\in \\mathcal V}\n A_{n,\\underline v}\n \\cup\n \\bigcup_{k=m}^{n-1}\n A_{n,k}\n \\cup \n \\bigcup_{j=1}^3 \n \\{(j^n)\\},\n $$\n %\n where $A_{n,k}$ is as above and $A_{n,\\underline v} = \\{\\underline i \\in \\{1,2,3\\}^n\\;:\\; (i_1,\\ldots, i_{m+1}) \\sim \\underline v\\}$, recalling the equivalence relation $\\sim$ on page 3.\n %\n\n\\end{definition}\n\nThis partition naturally gives the following components of $X_n$, for $n>m$.\n\n\\begin{definition}\n For $n > m$ and any $\\alpha \\in \\mathcal V \\cup \\{m,\\dots,n-1\\}$, we write\n %\n $$\n X_{n,\\alpha}\n :=\n \\sum_{\\underline i \\in A_{n,\\alpha}}\n \\area(\\Delta_{\\underline i})^\\delta\n \\diam(\\Delta_{\\underline i})^{1-\\delta}.\n $$\n %\n\\end{definition}\n\nWe now derive bounds on $X_{n+1}$ using estimates on $X_{n,\\underline v}$ and $ X_{n,k}$.\n\n\\section[Inductive bounds]{Inductive bounds on $X_{n,\\underline v}$ and $X_{n,k}$}\n\nWe now turn to the problem of showing that $X_n \\to 0$ as $n \\to \\infty$, which will enable us to apply Lemma \\ref{dimbound}.\nOur method, broadly speaking, is based on obtaining inductive bounds bounding\n\n\t\\begin{enumerate}\n \t\t\\item\n\t\t\t$X_{n+1,k+1}$ in terms of $X_{n,k}$, and\t\n\t\t\\item\n \t\t\t$X_{n+1,\\underline v}$ in terms of $X_{n,\\underline w}$ and $X_{n,k}$.\n\t\\end{enumerate}\nThese will prove useful in applying the renewal theorem in the next section.\n\n\\subsection{Bounds of terms indexed by numbers}\n\n\nThe next simple lemma is independent of $m$, and features the constants $b_k$.\n\n\\begin{lemma}[Number Lemma]\\label{number}\n Suppose $\\delta \\in (0,1)$. Then, for all $n > k$,\n %\n \\begin{gather*}\n X_{n+1,k+1} \n \\leq \n b_k X_{n,k}\n \n \n \n \n \n \n \n \n .\n \\end{gather*}\n %\n\\end{lemma}\n \n\\begin{proof}\n \n Since $j\\underline i \\in A_{n+1,k+1}$ if and only if $\\underline i \\in A_{n,k}$ and $j = i_1$, we may write \n %\n \\begin{align*}\n X_{n+1,k+1} \n &=\n \\sum_{\\underline i \\in A_{n,k}} \n \\area(\\Delta_{i_1\\underline i})^\\delta\n \\diam(\\Delta_{i_1\\underline i})^{1-\\delta}\n \\\\\n &\\leq\n \\sum_{\\underline i \\in A_{n,k}}\n \\left(\n \\frac\n {k+2}\n {k+3}\n \\right)^{3\\delta}\n \\left(\n \\frac\n {k+2}\n {k+3}\n \\right)^{\\lambda(1-\\delta)}\n \\area(\\Delta_{\\underline i})^\\delta\n \\diam(\\Delta_{\\underline i})^{1-\\delta}\n \\\\\n &= \n b_k X_{n,k},\n \\end{align*}\n %\n where we have applied Lemmas \\ref{area} and \\ref{diameter}.\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\\end{proof}\n\n\\subsection{Bounds of terms indexed by words}\n\nWe can bound $X_{n+1,\\underline v}$, $n > m,$ with $\\underline v \\in \\mathcal V\\cup\\{m\\}$, using the matrix $B$ and the constants $a_k$, as formulated in the next lemma. \n\n\\begin{lemma}[Word Lemma]\\label{word}\n Let $\\delta\\in (0,1)$, $n > m$ and $\\underline v \\in \\mathcal V \\cup \\{m\\}$. Then, recalling $\\mathbin{\\mathpalette\\make@circled\\star} = (1,2^n)$, there exists a positive sequence $c_n$ such that $\\sum_n c_n < \\infty$ and\n %\n \\begin{align*}\n X_{n+1,\\mathbin{\\mathpalette\\make@circled\\star}} \n &\\leq \n c_n \n + \n \\sum_{k=m}^{n-1}\n a_k \n X_{n,k}\n \\qquad\n \\hbox{and}\n \\\\\n X_{n+1,\\underline v} \n &\\leq\n \\sum_{\\underline w \\in \\mathcal V}\n B_{\\underline v, \\underline w}\n X_{n,\\underline w}\n \\qquad\n \\hbox{if}\n \\ %\n \\underline v \\neq \\mathbin{\\mathpalette\\make@circled\\star}.\n \\end{align*}\n %\n\\end{lemma}\n \n\\begin{proof}\n For the first inequality, note that $j\\underline i \\in A_{n+1,\\mathbin{\\mathpalette\\make@circled\\star}}$ if and only if $j\\neq i_1$ and $\\underline i \\in \\bigcup_{k=m}^{n-1} A_{n,k} \\cup \\{(1^n),(2^n),(3^n)\\}$. Therefore, by Lemmas \\ref{area} and \\ref{diameter}, using symmetry, one has that\n %\n \\begin{align*}\n X_{n+1,1}\n &= \n 6 \n \\area(\\Delta_{(2,1^n)})^\\delta\n \\diam(\\Delta_{(2,1^n)})^{1-\\delta}\n + \n \\sum_{k=m}^{n-1}\n \\sum_{\\underline i \\in A_{n,k}}\n \\sum_{j \\neq i_1}\n \\area(\\Delta_{j\\underline i})^\\delta\n \\diam(\\Delta_{j\\underline i})^{1-\\delta}\n \\\\\n &\\leq\n c_n \n + \n \\sum_{k=m}^{n-1}\n \\sum_{\\underline i \\in A_{n,k}}\n \\sum_{j \\neq i_1}\n \\area(\\Delta_{\\underline i})^\\delta\n \\diam(\\Delta_{j\\underline i})^{1-\\delta}\n \\left(\n \\frac \n {k+1}\n {k+2}\n \\right)^{\\lambda(1-\\delta)} \n \\\\\n &\\leq\n c_n \n + \n \\sum_{k=m}^{n-1}\n \\sum_{\\underline i \\in A_{n,k}}\n \\area(\\Delta_{\\underline i})^\\delta\n \\diam(\\Delta_{\\underline i})^{1-\\delta}\n \\left(\n \\frac \n {k+1}\n {k+2}\n \\right)^{\\lambda(1-\\delta)} \n \\left(\n \\left(\n \\frac \n {k+1}\n {k+2}\n \\right)^{3\\delta}\n +\n \\frac 1 {8^\\delta}\n \\right)\n \\\\\n &=\n c_n \n + \n \\sum_{k=m}^{n-1}\n a_k X_{n,k},\n %\n \\end{align*}\n %\n where $c_n := 6 \\area(\\Delta_{(2,1^n)})^\\delta \\diam(\\Delta_{(2,1^n)})^{1-\\delta}$, as required.\n \n Regarding the second inequality: The combinatorics above and in the previous proof show that $\\underline i \\in \\bigcup_{k=m}^{n-1} A_{n,k} \\cup \\{(1^n),(2^n),(3^n)\\}$ implies $j\\underline i \\not \\in A_{n+1,\\underline v}$ for every $\\underline v \\in \\mathcal V \\cup\\{m\\} \\setminus \\{\\mathbin{\\mathpalette\\make@circled\\star}\\}$. Fixing such a $\\underline v$ and considering the contrapositive, one sees that $j\\underline i \\in A_{n+1,\\underline v}$ implies that $\\underline i \\in A_{n,\\underline w}$ and $\\underline w \\mapsto_{j'}\\underline v$ for some $\\underline w, j'$. Assuming $(i_1,\\ldots, i_{m+1}) = \\underline w$ for simplicity, we have that $j' = j$ and $\\Delta_{\\underline i}\\subset \\Delta_{\\underline w}$. Hence, by Lemmas \\ref{area} and \\ref{diameter},\n %\n $$\n \\frac\n {\n \\area(\\Delta_{j\\underline i})^\\delta\n \\diam(\\Delta_{j\\underline i})^{1-\\delta}\n }\n {\n \\area(\\Delta_{\\underline i})^\\delta \n \\diam(\\Delta_{\\underline i})^{1-\\delta}\n }\n \\leq \n \\max_{\\underline x \\in \\Delta_{\\underline w}}\n (2-x_j)^{-3\\delta - \\lambda(1-\\delta)}\n =: \n B_{\\underline v,\\underline w}.\n $$\n %\n Thus, we may write the following.\n %\n \\begin{align*}\n X_{n+1,\\underline v}\n \n \\sum_{j=1}^3\n \\sum_{\\substack{\\underline w \\in \\mathcal V:\\\\[2pt]\\underline w \\mapsto_j \\underline v}}\n \\sum_{\\underline i \\in A_{n,\\underline w}}\n \\area(\\Delta_{j\\underline i})^\\delta\n \\diam(\\Delta_{j\\underline i})^{1-\\delta}\n \n &\\leq\n \\sum_{\\substack{\\underline w \\in \\mathcal V:\\\\B_{\\underline v,\\underline w}\\neq 0}}\n \\sum_{\\underline i \\in A_{n,\\underline w}}\n B_{\\underline v,\\underline w}\n \\area(\\Delta_{\\underline i})^\\delta\n \\diam(\\Delta_{\\underline i})^{1-\\delta} \n \\\\\n &=\n \n \\sum_{\\underline w \\in \\mathcal V}\n B_{\\underline v,\\underline w}\n X_{n, \\underline w},\n \\end{align*}\n %\n as required.\n \n Finally, a direct computation of the vertices of $\\Delta_{(2,1^n)} = T_2T_1^n(\\Delta)$ shows that its diameter and area are proportional to $n^{-1}$ and $n^{-2}$ respectively, as $n \\to \\infty$. Hence $c_n$ is proportional to $n^{-1-\\delta}$ as $n\\to \\infty$, and is thus summable for every $\\delta>0$, completing the proof.\n\\end{proof}\n \n \\section{Renewal Theorem}\n \nWe now use the iterative bounds of the last section to show that $X_n \\to 0$ as $n \\to \\infty$, under the hypotheses of Theorem \\ref{long}. This uses the following mild adaptation of the classical renewal theorem of Feller \\cite[p.330]{feller}.\n\n\\begin{thm}\n\tSuppose that the sequences $(Y_n)_{n=0}^\\infty$,\n\t$(\\mu_n)_{n=1}^\\infty$ and $(\\nu_n)_{n=1}^\\infty$ are non-negative and satisfy $\\sum_{n=1}^\\infty \\mu_n < 1$, $\\sum_{n=1}^\\infty \\nu_n < \\infty$ and\n\t%\n\t\t$$\n\t\t\tY_n \\leq \\nu_n + \\sum_{k=1}^n \\mu_k Y_{n-k}\n\t\t$$\n\t%\n\tfor each $n\\in \\mathbb N$. Then $Y_n \\to 0$ as $n \\to \\infty$.\n\\end{thm}\n \nWe apply this theorem with $Y_n = X_{n+m+1,\\mathbin{\\mathpalette\\make@circled\\star}}$ to give Theorem \\ref{long}, which can be rewritten as follows.\n\\begin{thm}\n If $\\delta > \\frac {1-\\lambda}{3-\\lambda} = 0.037\\ldots$ satisfies\n %\n \\begin{equation}\n \\sum_{k=1}^\\infty \n (B^k)_{m,\\mathbin{\\mathpalette\\make@circled\\star}}\n \\sum_{k=m}^\\infty\n a_k\n \\prod_{i=m}^{k-1}\n b_i\n \\leq \n 1,\n \\label{cond} \n \\end{equation}\n then $\\dim_H(\\mathcal G) \\leq 1 + \\delta$. \n\\end{thm}\n \n\\begin{proof} Assume that \\eqref{cond} holds with a strict inequality (the case of equality follows in the limit, since the LHS is decreasing in $\\delta$), and that $\\delta<1$ (the conclusion being otherwise trivial). Applying Lemma \\ref{number} repeatedly in the first estimate of Lemma \\ref{word} gives\n $$\n X_{n+1,\\mathbin{\\mathpalette\\make@circled\\star}}\n \\leq\n c_n\n +\n \\sum_{k=m}^{n-1}\n a_k\n \\prod_{i=m}^{k-1}\n b_i\n X_{n+m-k,m}.\n $$\nMoreover, the second estimate of Lemma \\ref{word} extends inductively to give, for all $\\hat n > m,$\n %\n $$\n X_{\\hat n,m}\n \\leq\n \\sum^{\\hat n-m-1}_{k=1}\n (B^k)_{m,\\mathbin{\\mathpalette\\make@circled\\star}}\n X_{\\hat n - k,\\mathbin{\\mathpalette\\make@circled\\star}}\n +\n \\sum_{\\substack{\\underline v \\in \\mathcal V:\\\\\\underline v \\neq \\mathbin{\\mathpalette\\make@circled\\star}}}\n (B^{\\hat n-m-1})_{m,\\underline v}\n X_{m+1,\\underline v}.\n $$\n %\n Putting these two together (with $\\hat n = n+m+k$) gives the renewal-style inequality\n %\n $$\n X_{n+1, \\mathbin{\\mathpalette\\make@circled\\star}} \n \\leq\n \\nu_n\n +\n \\sum_{k=1}^{n-m} \n \\mu_k \n X_{n+1-k,\\mathbin{\\mathpalette\\make@circled\\star}} \n ,\n $$\n \n with coefficients\n %\n $$\n \\mu_k\n =\n \\sum_{i+j=k} \n (B^i)_{m,\\mathbin{\\mathpalette\\make@circled\\star} } \n a_{m+j} \n \\prod_{l=m}^{m+j-1} \n b_l\n $$\n %\n and remainder term\n %\n $$\n \\nu_n \n = \n c_n \n + \n \\sum_{k=m}^{n-1} \n a_k \n \\prod_{i=m}^{k-1} \n b_i \n \\sum_{y \\in \\mathcal V-\\{\\mathbin{\\mathpalette\\make@circled\\star} \\}}\n (B^{n-k-1})_{m,y}\n X_{m+1,y}.\n $$\n %\n \n Consider the other hypotheses of the renewal theorem. The hypothesis on the coefficients is precisely our assumption that strengthens \\eqref{cond}: \n %\n $$\n \\sum_{k=1}^\\infty \n \\mu_n\n =\n \\sum_{k=0}^\\infty\n (B^k)_{m, \\mathbin{\\mathpalette\\make@circled\\star}} \n \\sum_{k=m}^\\infty\n a_k\n \\prod_{i=m}^{k-1} \n b_i \n < \n 1.\n $$\n %\n Regarding the summability of the remainder terms $\\nu$: By elementary combinatorics, $B^{m+1}$ has only one zero row and zero column, so this inequality also ensures that the spectral radius $\\rho = \\rho(B) < 1$. In any case, by the Perron-Frobenius theorem there exists $C>0$ such that\n %\n $$\n \\sum_{n=1}^\\infty\n \\nu_n-c_n\n \\leq \n C \n \\sum_{n=1}^{\\infty}\n \\sum_{k=m}^{n-1} \n a_k \n \\prod_{i=m}^{k-1} \n b_i\n \\rho^{n-k}\n =\n C\n \\sum_{k=m}^{\\infty}\n a_k \n \\prod_{i=m}^{k-1} \n b_i \n \\sum_{n=1}^{\\infty}\n \\rho^n\n \n \n $$\n %\n Thus, since $\\rho \\in(0,1)$ and\n %\n $$\n a_k \n \\prod_{i=m}^{k-1}\n b_i \n =\n \\mathcal O\\big(k^{-3\\delta-\\lambda(1-\\delta)}\\big)\n \\qquad (k\\to\\infty),\n $$\n \n we see that the right hand side is summable, since $\\delta > \\frac{1-\\lambda}{3-\\lambda}$. That is, $\\sum_n \\nu_n < \\infty$, and the renewal theorem gives that $X_{n, \\mathbin{\\mathpalette\\make@circled\\star}} \\to 0$ as $n \\to \\infty$.\n \n Finally, we use this to bound $X_n$ via the following lemma.\n\\begin{lemma}\n Given $m \\in \\mathbb N$ and $\\delta\\in(0,1)$, there exists $C > 0$ such that $X_n \\leq C X_{n+m+1, \\mathbin{\\mathpalette\\make@circled\\star}}$.\n\\end{lemma}\n\n\\begin{proof}\n For each $\\underline i \\in\\{1,2,3\\}^n$, the word $\\underline i' = (1,2^m,i_1,\\ldots,i_n)\\in A_{n+m+1,\\mathbin{\\mathpalette\\make@circled\\star}}$. This word satisfies\n %\n $$\n \\frac\n {\\area(\\Delta_{\\underline i'})^\\delta\n \\diam(\\Delta_{\\underline i'})^{1-\\delta}}\n {\\area(\\Delta_{\\underline i})^\\delta\n \\diam(\\Delta_{\\underline i})^{1-\\delta}}\n =\n \\frac\n {\\area(T_1T_2^m\\Delta_{\\underline i})^\\delta\n \\diam(T_1T_2^m\\Delta_{\\underline i})^{1-\\delta}}\n {\\area(\\Delta_{\\underline i})^\\delta\n \\diam(\\Delta_{\\underline i})^{1-\\delta}}\n \\geq\n K^{(1+\\delta)(m+1)},\n $$\n %\n where $K$ is obtained by bounding the \\textit{minimum} singular value of $DT_j(x)$ uniformly over $x \\in \\Delta$: namely, applying the estimates of section 3,\n %\n $$\n \\big\\|\n D(T_j^{-1})\n \\big(T_j(x)\\big)\n \\big\\|\n = \n \\frac\n {\\Jac T_j(x)}\n {\\|DT_j(x)\\|}\n \\geq\n (2-x_j)^{\\lambda-3}\n \\geq\n 2^{\\lambda-3}\n =:\n K.\n $$\n %\n In particular, with $C = K^{(1+\\delta)(m+1)}$\n %\n $$\n X_{n+m+1,\\mathbin{\\mathpalette\\make@circled\\star}}\n \\geq\n \\sum_{|\\underline i| = n}\n \\area(\\Delta_{\\underline i'})^\\delta\n \\diam(\\Delta_{\\underline i'})^{1-\\delta} \n \\geq\n C\n X_n,\n $$\n %\n as required.\n\\end{proof}\n\nThus, $X_{n, \\mathbin{\\mathpalette\\make@circled\\star}} \\to 0$ as $n \\to \\infty$ implies that $X_{n} \\to 0$ as $n \\to \\infty$, and applying Lemma \\ref{dimbound} completes the proof of Theorem \\ref{long}.\n\\end{proof}\n\n\\section{Numerical estimates}\n\nThe usefulness of Theorem \\ref{long} depends on our ability to check the hypothesis for a given candidate value of $\\delta$.\nHowever, since the size of $\\mathcal V$ (and hence of $B$) is exponentially increasing in $m$, such a check requires an amount of computer time and memory exponential in $m$, and with diminishing returns. Fortunately, for relatively small values of $m$, we have bounds on the dimension which improve upon the known bounds.\n \nIn the table below we present the values $\\delta_m$ which, for each value of $m$, give equality in \\eqref{cond}, in the table below. These were calculated using Wolfram Mathematica on a Lenovo ThinkPad X220; for $m=9$, each check of \\eqref{cond} took less than 90 seconds.\n\n \\begin{table}[htb]\n\t\\begin{center}\n \t\t\\begin{tabular}{c|cc@{\\qquad\\qquad}rc|c}\n\t\t\t$m$ & $\\delta_m +1$ &&&$m$ & $\\delta_m +1$ \\\\\\cline{1-2} \\cline{5-6}\n\t\t\t2&1.8285&&&6&1.7534\\\\\n\t\t\t3&1.7978&&&7&1.7475\\\\\n\t\t\t4&1.7764&&&8&1.7435\\\\\n\t\t\t5&1.7624&&&9&1.7407\\\\\n\t\t\\end{tabular}\n\t\\end{center}\n\t\t\\caption{Upper bounds $1+\\delta_m$ on $\\dim_H(\\mathcal G)$ for different choices of $m$, rounded upwards to four decimal places.}\n\t\\end{table}\n\n \n\\begin{rem}\n The elementary nature of this method suggests that a similar approach might be used in other related examples of sets to give bounds on their dimension.\n\\end{rem}\n \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Introduction}\n\n\nIn the absence of disorder, an interacting many-body electron system can be described within the hydrodynamic framework \\cite{andreev, principi, lucas}.\nTypical three-dimensional metals rarely enter into the hydrodynamic regime because the electron-impurity (phonon) scattering\nis stronger than the corresponding electron-electron interactions \\cite{lucas1}. However, it is expected that in a clean two-dimensional\n(2D) electron system, such as modulation doped GaAs systems and high-quality graphene layers, the requirements for hydrodynamics\ncan easily be satisfied.\n\nHydrodynamic characteristics are enhanced in a Poiseuille geometry, where a parabolic flow profile can be realized in a narrow pipe. The fluid in this regime has zero velocity at the boundaries.\nThe electronic analog of the viscous flow in the pipe is a transport in a narrow channel of width $W$ with diffusive scattering at the boundary, driven by the electric field.\nViscous electron flows are expected to occur when\nthe mean free path for electron-electron collision, $l_{ee}$, is much shorter than the sample width,\nwhile the mean free path due to impurity and phonon scattering, $l$, is larger than $W$. It has been predicted that the electrical resistivity of a 2D system is proportional\nto electron shear viscosity, $\\eta=\\frac{1}{4}v_{F}^{2}\\tau_{ee}$, where $v_{F}$ is the Fermi velocity and $\\tau_{ee}$ is the\nelectron-electron scattering time $\\tau_{ee}=l_{ee}\/v_{F}$ \\cite{gurzhi, dyakonov, dyakonov1, dyakonov2, govorov}.\n For example, resistance decreases with the square of temperature,\n$\\rho \\sim \\eta \\sim \\tau_{ee} \\sim T^{-2}$, referred to as the Gurzhi effect, and with the square of sample width\n$\\rho \\sim W^{-2}$. The boundary conditions can be characterized by a diffusive scattering or by a slip length $l_{s}$ with extreme cases being\nno-slip ($l_{s} \\rightarrow 0$) and no-stress ($l_{s} \\rightarrow \\infty$) conditions. It is expected that for $l_{s} \\rightarrow \\infty$ no\nGurzhi effect should be detected.\n\nRecently interest in electronic hydrodynamics has arisen from measurements of the transport in graphene, where electron-phonon scattering is relatively weak\n\\cite{levitov, torre, pellegrino, bandurin}. Moreover, a series of updated theoretical approaches has been published \\cite{alekseev, scaffidi, luca, pellegrino2} considering a viscous system in the presence of a magnetic field, which provides additional possibilities to study magnetohydrodynamics.\n\nExperiments on $PdCoO_{2}$ \\cite{moll}, $WP_{2}$ \\cite{gooth},\nand GaAs \\cite{molenkamp, gusev, gusev2, levin} have many features demonstrating the viscous flow of electrons.\nMoreover, the previous study of the giant negative magnetoresistance in high mobility GaAs structure \\cite{haug, hatke, mani, haug2}\ncould be interpreted as a manifestation of the viscosity effects, or interplay\nbetween ballistic and hydrodynamic effects \\cite{alekseev2}.\n\n\\begin{figure}[ht]\n\\includegraphics[width=\\linewidth]{Stokes1.eps}\n\\caption{(Color online) (a) Sketch of the velocity flow profile in the presence of a circular obstacle. (b) Image of the Hall bar device with\nantidot (micro-hole) in the center of the Hall bridge between probes 2-3 (10-9) and 4-5 (8-7).}\\label{fig:1}\n\\end{figure}\n\nThe diffusive scattering condition is the relevant one for most liquid-solid\ninterfaces. The absence of Poiseuille flow and the Gurzhi effect in\ngraphene has been taken as evidence for a specular limit with\na very large slip length \\cite{bandurin}.\n\nIf the slip length is larger than sample size, viscous shear forces can arise, if the fluid flows around an obstacle.\n Flow around a circular disc was considered\nby Stokes a long time ago \\cite{stokes, landau}. In classical two-dimensional\nfluid mechanics, this may lead to a phenomenon referred as the \"Stokes paradox\": no solution of the Stokes equations can be found for which fluid velocity\nsatisfies both the boundary conditions on the body and at infinity \\cite{lamb}.\n\nRecently an electronic analog of the Stokes paradox has been proposed\nfor two-dimensional Fermi liquids \\cite{hruska, guo, lucas1}. Schematically this proposal is illustrated in Figure \\ref{fig:1}a: the resistance of the sample with length $L\\sim W$\nis studied, when a circle obstacle of radius $a_{0} << L$ is located in the middle of the sample \\cite{lucas2, kisilev}. In an electronic liquid, the Stokes paradox\nhas been resolved within the framework of the semiclassical description of quasiparticle dynamics, and a linear response has been obtained due to the momentum relaxation process \\cite{hruska, guo, lucas2}.\nIndeed Ohmic theory predicts that the obstacle will enhance total resistance \\cite{lucas2} :\n\\begin{equation}\nR_{total}=R_{0}+R_{obst},\n\\end{equation}\nwhere $R_{0}$ is obstacle free resistance, and\n$R_{obst}=cR_{0}\\frac{a_{0}^{2}}{L^{2}}$, c is a geometric factor.\nIt is interesting that the Stokes flow around a disc leads to a dramatic consequence beyond Ohmic behaviour: the effective radius of the obstacle $a_{eff}$\nis always larger than the geometric radius $a_{eff} >> a_{0}$ \\cite{lucas2}. More importantly the obstacle resistance decreases with temperature, suggesting that the viscous liquid is essentially always in the regime of specular scattering boundary\nconditions.\n\nIn the present work, we have experimentally examined the transport properties of a mesoscopic 2D electron system with a circular obstacle (antidot or micro-hole).\nAs a reference we also studied a device without an antidot in order to extract the obstacle resistance and determine all relevant viscous parameters,\nwhich provides the comparative analysis between theory and experiment. By tuning the temperature in a wide interval $ 1.5 < T < 70 K$, we show that\nobstacle resistance $R_{obst}$ exhibits a drop as temperature increases (even as $dR_{0}\/dT > O$), in consistence with predictions for the ballistic and hydrodynamic regimes.\n\n\n\\section{Methods}\n\nThe samples were grown by molecular\nbeam epitaxy method. Our samples are high-quality, GaAs quantum wells with a width of 14~nm\nwith electron density $n_{s}=6\\times10^{11} cm ^{-2}$ and a mobility of $\\mu=2.5\\times10^{6} cm^{2}\/Vs$ at T=1.4K.\nOther parameters, such as fermi velocity, mean free path and others are given in Table \\ref{tab1}.\nWe present experimental results on Hall-bar devices.\nThey consist of three, $6 \\mu m$ wide segments of\ndifferent length ($6, 20 , 6 \\mu m$), and 10 contacts. Figure \\ref{fig:1}b shows the image of a typical multiprobe Hall device I.\nThe antidots are located in the middle of the right side and left side segment of the Hall bar by chemical wet etching through the quantum well.\nThe measurements were carried out in a\nVTI cryostat, using a\nlock-in technique to measure the longitudinal $\\rho_{xx}$ resistivity with an\nac current of $0.1 - 1 \\mu A$ through the sample.\n3 Hall bars from the same wafers were studied and showed consistent behaviour. As reference we also measured a Hall bar\nwithout an antidot. Additionally we also studied macroscopic samples, where, it is expected, that the viscous effects are small. These samples have Hall-bar geometry\n(length $l\\times$ width $W = 500 \\mu m \\times 200 \\mu m$) with six contacts.\n\n\\begin{table}[ht]\n\\centering\n\\begin{tabular}{|l|l|l|l||l||l||l|}\n\\hline\n&W&$n_{s}$ & $v_{F}$ & $l$ & $l_{2}$ & $\\eta$\\\\\n&($\\mu m$) & $(10^{11} cm^{2}$) & $(10^{7} cm\/s)$ & $(\\mu m$) & $(\\mu m$) & $(m^{2}\/s)$\\\\\n\\hline\n&$6$& $6.0$ & $3.3$ & $35$ & $3$ & $0.25$\\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{tab1} Parameters of the electron system at $T=1.4 K$. Parameters $l$, $l_{2}$ and $\\eta$ are determined in the text.}\n\\end{table}\n\n\\section{Experiment in reference device and discussion}\n\n\nThe electronic analog of the hydrodynamic regime in the pipe is a electric current in a narrow channel of width $W\\sim 1-10\\mu m$.\nFigure \\ref{fig:1}b shows the image of the Hall bar device with a micro-hole in the center of the Hall bridge. The resistance between different probes\nhas been measured.\n Figure \\ref{fig:2}a shows the longitudinal magnetoresistance for a sample with an antidot and a reference sample without an antidot.\nLongitudinal magnetoresistance of a viscous 2D high mobility system in GaAs has been studied in previous research for different configurations of current and voltage probes \\cite{gusev, gusev2, levin}.\nRemarkably, we find that probe configuration and sample geometry strongly affect the temperature evolution of\nlocal resistance and its value at zero magnetic field. For example, when the current is applied between probes 1 and 6, and voltage is measured between probes 4 and 5 (referred further as C1 configuration),\nthe corresponding resistance $R_{I=1-6;V=4-5}$ increases with temperature T, while the resistance $R_{I=8-7;V=4-5}$, when the current is applied between probes 8 and 7 and voltage is measured between probes 4 and 5 (referred further as C2 configuration), decreases with T and always appears bigger\nthan $R_{I=1-6;V=4-5}$. We attribute such behaviour to enhanced viscosity due to diffusive scattering on the rough edge\nand inhomogeneity of the velocity field, predicted in paper \\cite{alekseev}. Indeed we reproduced these results in the samples studied in this work, and Figure \\ref{fig:2}a shows that\nthe resistance at B=0 in configuration C2 is bigger than the resistance in configuration C1. Moreover, the resistance with an antidot is enhanced\nin comparison with the reference sample in both configurations. One more striking feature is the anomalously large negative magnetoresistance, which is strongly enhanced for configuration C2.\nSatellite peaks are clearly observed in samples with antidots resulting in additional broadening of the total magnetoresistance.\nTherefore, we may conclude here that the effect of the obstacle is adding a series resistor, as has been predicted in paper \\cite{lucas2}.\n\\begin{figure}[ht]\n\\includegraphics[width=\\linewidth]{Stokes2.eps}\n\\caption{(Color online)\n(a) The magnetoresistance of a GaAs quantum well in a Hall bar sample with obstacle and in a reference sample for different configurations, T=1.4K. The schematics\nshow how the current source and the voltmeter are connected for the measurements:\nconfiguration with antidot is shown on the right side, configuration for reference without antidot is shown on the left side.\n(b) Temperature dependent magnetoresistance of a reference Hall bar sample. Dashes are examples illustrating\nmagnetoresistance calculated from Eq. 2 for different temperatures: 1.5 K (blue), 27.7 K(light blue), 44 K (green).\n(c) Relaxation rates $1\/\\tau$ (squares) and $1\/\\tau_{2,ee}$ (circles) as a function of the temperature obtained by fitting\nthe theory with experimental results for the reference sample (black scatters) and a sample with an obstacle (red scatters).\nThick black and red lines is Equation 3, dashes is Equation 4.\n}\\label{fig:2}\n\\end{figure}\nBefore analyzing the obstacle effect, and in order to make this analysis more complete, we present the results of measurements of longitudinal magnetoresistivity $\\rho_{xx}(B)$ in samples without a micro-hole.\nIn order to increase the viscosity effect, we study resistance in C2 configuration.\nFigure \\ref{fig:2}b shows $\\rho_{xx}(B)$ as a function of magnetic field and temperature.\n\nIn the hydrodynamic approach, the semiclassical treatment of the transport describes the motion of carriers when the higher order moments of the distribution function are taken into account.\nThe momentum relaxation rate $1\/\\tau$ is determined by electron interaction with phonons and static defects (boundary).\nThe second moment relaxation rate $1\/\\tau_{2,ee}$ leads to the viscosity and contains the contribution\nfrom the electron-electron scattering and temperature independent scattering by disorder \\cite{alekseev, scaffidi}.\nIt has been shown that conductivity is determined by two independent $\\textit{parallel}$ channels of electron momentum relaxation:\nthe first is due to momentum relaxation time and the second due to viscosity \\cite{alekseev, scaffidi}.\nThis approach allows the introduction of the magnetic field dependent viscosity tensor and the derivation of the magnetoresisivity tensor \\cite{alekseev, scaffidi, luca}:\n\\begin{equation}\n\\rho_{xx}= \\rho_{0}^{bulk}\\left(1+\\frac{\\tau}{\\tau^{*}}\\frac{1}{1+(2\\omega_{c}\\tau_{2,ee})^{2}}\\right),\\,\\,\\,\n\\end{equation}\n\nwhere $\\rho_{0}^{bulk}=m\/ne^2\\tau$, $\\tau^{*}=\\frac{W(W+6l_{s})}{12\\eta}$, viscosity $\\eta=\\frac{1}{4}v_{F}^{2}\\tau_{2,ee}$.\n\nAll relaxation rates are given by:\n\\begin{equation}\n\\frac{1}{\\tau_{2,ee}(T)}=A_{ee}^{FL}\\frac{T^{2}}{[ln(E_{F}\/T)]^{2}}+\\frac{1}{\\tau_{2,0}},\n\\end{equation}\n\nwhere $E_{F}$ is the Fermi energy, and the coefficient $A_{ee}^{FL}$ be can expressed via the Landau interaction parameter.\nThe relaxation rate $\\frac{1}{\\tau_{2,0}}$ is not related to the electron-electron collisions, since any process responsible for relaxation of the second moment\nof the distribution function, even scattering by static defect, gives rise to viscosity \\cite{alekseev}.\nThe momentum relaxation rate is expressed as:\n\\begin{equation}\n\\frac{1}{\\tau}=A_{ph}T+\\frac{1}{\\tau_{0}},\n\\end{equation}\nwhere $A_{ph}$ is the term responsible for the phonon scattering, and $\\frac{1}{\\tau_{0}}$ is the scattering rate due to static disorder\n(not related to the second moment relaxation rate $\\frac{1}{\\tau_{2,0}}$). It is worth noting that above 40 K the scattering from polar LO\nphonons becomes important and the scattering time deviates from simple linear dependence on temperature \\cite{haris, kawamura}).\n\n\nWe fit the magnetoresistance curves in Figure \\ref{fig:2}b and the resistance in zero magnetic field with the 3 fitting parameters :\n$\\tau(T)$, $\\tau^{*}(T)$ and $\\tau_{2,ee}(T)$.\nWe compare the temperature dependence of $\\frac{1}{\\tau_{2,ee}(T)}$ and $\\frac{1}{\\tau(T)}$ with theoretical predictions given by Equations 3 and 4,\nwhich is shown in Figure \\ref{fig:2}c.\nThe following parameters\nare extracted:\n$1\/\\tau_{2,0}=0.95\\times10^{11}$ s, $A_{ee}^{FL}=0.35\\times10^{9} s^{-1}K^{-2}$,\n$A_{ph}=0.5\\times10^{9}sK^{-1}$ and $1\/\\tau_{0}=0.65\\times10^{10} s$,\nwhich are correlated with previous studies \\cite{gusev,levin}. Note, however, that a discrepancy with Equations 3 and 4 is found at high temperatures,\nwhich we attributed to the inelastic process due to scattering by LO phonons .\nRelaxation time $\\tau^{*}(T)$ depends on $\\tau_{2,ee}(T)$ and the boundary slip length $l_{s}$.\nComparing these values, we find that $l_{s}=3.2 \\mu m < L$. Our data are in good agreement with the theoretical prediction for the case\nwhen the slip length is temperature independent.\nTable 1 shows the mean free paths : $l=v_{F}\\tau$, $l_{2}=v_{F}\\tau_{2,ee}$ and viscosity, calculated with parameters extracted from the fit of experimental data.\n\nIn the last part of this section, we wish to discuss the influence of the ballistic effect on negative magnetoresistance in our reference samples.\nAs we already mentioned in the introduction, a previous study of the magnetoresistance in high mobility two dimensional GaAs system\ndemonstrated giant two-scale negative magnetoresistance consisting of a narrow temperature independent peak near zero magnetic field and\n shoulder-like magnetoresistance, which strongly depends on the temperature \\cite{haug2}. The model \\cite{alekseev2} proposes, that the temperature independent\n peak is attributed to the ballistic effects, while shoulder is attributed to the hydrodynamic effects due to flowing between randomly located\n macroscopic \"oval\" defects. It is worth noting that, because we observe small size peaks in magnetoresistance in C1 configuration (Figure \\ref{fig:2}a),\n ballistic contribution, predicted in the model \\cite{alekseev2} can have a non-negligible effect at least at low temperature. We present two arguments justifying, that ballistic effect\n is smaller than hydrodynamic contribution. First, we have demonstrated that magnetoresistance and $R(T)$ strongly depend on the configuration ( C1 or C2),\n which is unlikely to be attributed to the ballistic effect \\cite{gusev, gusev2, levin}. For example, ballistic contribution can not describe\n the resistance drop with temperature ( Gurzhi effect), observed in our samples \\cite{gusev}. Second,\n our giant negative magnetoresitance strongly depends on temperature and can be successfully described within a hydrodynamic framework \\cite{alekseev}\n in wide temperature range, in contrast to the T-independent peak observed in paper \\cite{haug2}. However, even though both ballistic and hydrodynamic\n contribution are equally important at low temperature, at high temperature, the viscosity effect becomes dominant, and all our conclusion can be applied\n equally well to the samples with and without obstacle.\n\n\n\n\n\\section{Experiment: obstacle resistance}\n\nIn this section, we focus on the study of resistance in samples with an obstacle. Figure \\ref{fig:3}a shows the magnetoresistance for samples with an\nobstacle for both configurations C1 and C2. One can see small satellite peaks making the central peak wider in comparison with the reference sample.\nWe attribute these oscillations to geometrical resonance effects, which are pronounced in 2D charged liquids \\cite{beenakker, roukes}.\nWe perform numerical simulations of the electron trajectories in ballistic structures for different obstacle sizes (for details see Supplementary material).\nThe results of theses simulations (dots) for $a_{0}=1 \\mu m$ are compared to the experimental data. One can see, that the width of the magnetoresistance\ncurve roughly corresponds to the experimental data, while the position of the peak is slightly shifted to a higher\nmagnetic field in comparison with the experiment.\n\n\\begin{figure}[ht]\n\\includegraphics[width=\\linewidth]{Stokes3.eps}\n\\caption{(Color online)\n(a) Magnetoresistance for a sample with an obstacle in C1 and C2 configurations, T=4.2K. The dots represent results for the billiard model.\nThe magnetoresistance of samples with different obstacle radii and in the reference sample (without obstacle) for configurations C1 (b) and C2 (c), T=4.2K.}\\label{fig:3}\n\\end{figure}\n\nMagnetoresistance as a function of the magnetic field for different radii $a_{0}$ is shown in Figures \\ref{fig:3}b,c for two configurations C1 and C2.\nThe diameter of the antidot has been measured directly from an optical microscope image (Figure \\ref{fig:1}b) with precision $0.1\\mu m$. The effective\nantidot diameter is larger than the lithographic one due to the depletion region, which, however, in our high density sample does not exceed $0.05\\mu m$.\nWe estimate this value from the assumption that the width of the region where the potential increases from the bottom to the Fermi energy is of the same order as the Fermi\nwavelength for typical electron concentrations \\cite{ando}.\nTraces for the reference sample without an obstacle are shown for comparison. One can see that the resistance\nwith an obstacle is always larger than the reference resistance. Resistance of a sample with an antidot radius of $a_{0}=1.3 \\mu m$ is higher than the resistance with $a_{0}=1.4 \\mu m$, probably due to radius uncertainty ($\\pm 0.05 \\mu m$). Viscosity effects are enhanced in C2 configurations and below\nwe focus on the results obtained from this probe configuration.\n\n\nFigure \\ref{fig:4}a shows the evolution of magnetoresistance with temperature for samples with an obstacle in C2 configuration.\nWe fit a central peak with the Lorentzian curve (Eq.2). Note that this peak is absent in magnetoresistance for C1 configuration (Figure \\ref{fig:2}a and Figure \\ref{fig:3}a)\nbecause it is overlapped by satellite peaks. As for the reference sample, we used the 3 fitting parameters :\n$\\tau(T)$, $\\tau^{*}(T)$ and $\\tau_{2,ee}(T)$.\nFigure \\ref{fig:2}c shows the relaxation rates $1\/\\tau(T)$, and $1\/\\tau_{2,ee}(T)$ for an obstacle sample in comparison with the reference sample as a function of temperature.\nOne can see that the rate $1\/\\tau_{2,ee}(T)$ is following the dependencies of Eqs. 3-4 with parameters\n$1\/\\tau_{2,0}=1.15\\times10^{11}$ s, $A_{ee}^{FL}=0.9\\times10^{9} s^{-1}K^{-2}$, while the rate $1\/\\tau(T)$ is\nsaturated at low temperatures, and it is unlikely that it can be described by the acoustic phonon scattering mechanism. The difference between\nrates $1\/\\tau_{2,ee}(T)$ for obstacle and reference samples can be attributed to uncertainty in the determination of the Lorentz curve\nwidth due to the satellite ballistic peak. The momentum relaxation rate is extracted from resistivity at zero magnetic field, which is enhanced\nin the obstacle samples.\n\n\n\\begin{figure}[ht]\n\\includegraphics[width=\\linewidth]{Stokes4.eps}\n\\caption{(Color online)\n(a) Temperature dependent magnetoresistance of a sample with obstacle ($a_{0}=1.3 \\mu m$). Dashes is\nmagnetoresistance calculated from Eq. 1 for 4.2 K with parameters taken from fit with the reference sample's magnetoresistance.\n(b) Temperature dependent resistivity of a sample with an obstacle, reference sample and macroscopic sample. Triangulares are\nresistivity calculated from Eq. 2. Solid line represents resistivity due to acoustic phonon scattering. (c) Relative obstacle resitivities\nfor samples with different obstacle radii. Colors solid line represents calculations from Equation 8 with numerical parameters taken from magnetoresistance\nmeasurements in the reference sample. Black solid line represents calculations without the Stokes paradox effect. Obstacle resistance exhibits\na drop with decreasing temperature ($d\\rho_{obst}\/dT < 0$).} \\label{fig:4}\n\\end{figure}\nThe temperature dependence of resistivity at zero magnetic field for different obstacle radii and the reference sample in configuration C2 is shown in Figure \\ref{fig:4}b.\nNote, that for our approximately square-shaped devices (Figure \\ref{fig:1}b), 2D resistivity practically equals the resistance: $R=1.6\\rho$, and below we discuss the resistivity behaviour.\nOne can see that resistance (resistivity) decreases in the temperature interval $1.5 K < T< 45 K $ and increases at higher temperatures.\nWe argue here that the ballistic (quasiballistic) contribution is described by the first term Equation 2, and\ncomparison with theory proves that it is much smaller than the viscosity contribution described by the second term. Below we repeat several keyword arguments which\njustify this conclusion and which have been discussed in previous publications \\cite{gusev, levin}. First, the resistivity for C2 configuration decreases with temperature and follows the Gurzhi law $\\rho\\sim T^{-2}$\nat least at low T (see Figure \\ref{fig:2}c) \\cite{gusev}. In contrast, resistivity in macroscopic samples increases with T and follow the linear law $\\rho\\sim T$ (below 40 K), due to acoustic phonon scattering (see\nFigure \\ref{fig:4}b) \\cite{haris, kawamura}.\nTherefore, we would expect that resistivity due to moment relaxation is temperature independent (scattering with static defects or boundary) or increases with T (due to the phonon scattering mechanism).\nSecond, the electron-electron scattering obeys the power law $\\frac{1}{\\tau_{2,ee}(T)}\\sim T^{2}$ (the logarithmic term is weakly T-dependent) \\cite{gusev, levin}, instead of the linear T law\nexpected for phonon scattering. We compared the experimental dependence of $\\rho(T)$ in zero magnetic field with theoretical models and\nobtained a good agreement (see Figure \\ref{fig:4}b -triangulares). Finally, resistivity strongly depends on the probe configuration (Figure \\ref{fig:2}a), which is unlikely to be attributable to the ballistic effect.\nIndeed, we calculated the ballistic contribution\nin our sample geometry and found only weak dependence on the configuration, which disagrees with our observations.\n\nIn the Figure \\ref{fig:4}b, we can see that resistivity of the samples with obstacles is always larger than the resistivity of the reference sample\nwithin the investigated temperature range. The enhanced obstacle resistivity $\\rho_{obst}(T)=\\rho_{total}(T)-\\rho_{0}(T)$ as a function of temperature\nis shown in Fig. \\ref{fig:4}c for two obstacle radii. For comparison we demonstrate the resistivity measured in a macroscopic sample $\\rho_{macr}$. Conventional Ohmic behaviour\nis expected in this device:\nbelow 40 K,\nmacroscopic resistivity displays simple linear temperature dependence due to acoustic phonon scattering (shown by solid line), while at higher temperatures\nscattering from polar LO phonons starts to become important. Indeed $d\\rho_{macr}\/dT >0$ in the entire interval of temperatures. In contrast\nobstacle resistance shows $d\\rho_{obst}\/dT < 0$ in the same temperature region.\n\n\n\\section{Theory and discussion}\n\nSimplified Ohmic theory predicts that obstacle resistivity should be proportional to obstacle free resistance and the square of the\nobstacle radius \\cite{lucas} $\\rho_{obst}(T) \\sim \\rho_{0}(T)(\\frac{a_{0}}{L})^{2}$. Therefore, one might expect that obstacle resistivity just\nreproduces the temperature dependence of the Ohmic resistivity. The solid line in Figure \\ref{fig:4}c represents the resulting\nobstacle resistivity without viscosity effects, when only phonon scattering (acoustic and LO phonons) is\ntaken into account. It predicts a very strong ($\\sim 10$ times) increase of $R_{obst}(T)$ with temperature, which indeed\ndisagrees with our experiments. One may conclude here that the T- coefficient of obstacle resistance is attributed to the combination\nof two effects: viscous flow of electrons in a narrow sample and the hydrodynamic flow around the obstacle.\n\n\nAs we already mentioned in the introduction, a lot of theoretical effort has gone into the resolution of the Stokes paradox\nin two-dimensional charged liquids. The main result is that the effective radius of the obstacle is larger\nthan the geometric radius $a_{0}$ and depends on temperature. The inverse scattering length drastically affects electron flow\nbehaviour in the presence of an obstacle: $\\frac{1}{l_{eff}}=\\frac{1}{l}+\\frac{1}{l_{2}}$\n\nThree different regimes of the transport have been considered \\cite{lucas2}:\n\n(i) Diffusive: in this limit $a_{0} >> \\sqrt{l_{eff}l_{2}}$, and effective radius is give by\n\\begin{equation}\na_{eff}=a_{0}.\n\\end{equation}\n\n(ii) Ballistic: in this limit $l_{eff} >> a_{0}$, and effective radius is give by\n\\begin{equation}\n a_{eff}^{2}=\\frac{a_{0}l_{2}}{2}.\n\\end{equation}\n\n(iii) Hydrodynamic : in this limit\n\n$l_{eff} << a_{0}<< \\sqrt{l_{eff}l_{2}}$, and effective radius is give by\n\\begin{equation}\n a_{eff}^{2}=\\frac{l_{eff}l_{2}}{ln(\\frac{l_{eff}l_{2}}{a_{0}^{2}})}.\n\\end{equation}\n\n\\begin{figure}[ht]\n\\includegraphics[width=\\linewidth]{Stokes5.eps}\n\\caption{(Color online)\nThe characteristic lengths $l$, $l_{2}$ and $l_{eff}$ (dashes) as a function of temperature.\nDots-parameters obtained from magnetoresistance measurements in the two reference samples. Fit of characteristic length\nwith parameters indicated in the main text. Horizontal lines- the width of the sample $W$ and\ndiameter of the obstacle $a_{0}$. Ballistic and hydrodynamic regimes on the length scale of the disk are shown.}\\label{fig:5}\n\\end{figure}\n\nThis difference in the parameter regimes leads to markedly different physical behavior in the transport.\nIt is remarkable that, in the hydrodynamic regime, the effective radius only weakly depends on the actual radius $a_{0}$.\nIn order to compare our results with theoretical predictions for corresponding transport limits, we\ncalculate relevant electron parameters as a function of temperature. Figure \\ref{fig:5} represents temperature\ndependence of the characteristic lengths $l$, $l_{2,ee}$ and $l_{eff}$ extracted from experiments on the two reference samples.\nOne can see that the viscous regime conditions $l_{2,ee} < W < l$ are satisfied in all temperature intervals, which is justified\nby observation of the Gurzhi effect below $T < 40 K$. Since obstacle radius is much smaller than the width of the sample,\nthe hydrodynamic limit for the Stokes effect requires higher temperatures $T > 40 K$. Model \\cite{lucas2} predicts\na general behavior for the effective obstacle radius, which covers all transport regimes:\n\n\\begin{eqnarray}\na_{eff}^{2}\\approx l_{eff} l_{2}\\Bigg\\{\\left(1-\\frac{2l_{eff}}{l_{2}}\\right)\\times \\nonumber \\\\\n \\times\\log \\left[\\frac{l_{2}}{l_{eff}}\\left(\\sqrt{1+\\left(\\frac{2l_{eff}}{a_{0}}\\right)^{2}}-1\\right)+1\\right] \\nonumber \\\\\n + \\sqrt{1+\\left(\\frac{2l_{eff}}{a_{0}}\\right)^{2}}-1 \\Bigg\\}^{-1}.\n\\end{eqnarray}\n\n\nWe compared the prediction of this model with our results, which are shown in Figure \\ref{fig:4}c. The theory predicts\nslightly nonmonotonic behaviour of $\\rho_{obs}(T)$ due to the interplay between $\\rho_{0}(T)$ and $a_{eff}(T)$ dependencies:\nat low temperatures, contribution from obstacle free resistivity is dominant, while at higher temperatures, the effective radius exhibits\na sharp drop due to viscosity. We can see that the predicted results roughly agree with experimental observations due to the approximate character of the\nanalytical calculations. It is because the theory \\cite{lucas2} does not consider collisions with the sample boundary, which lead to a quadratic velocity\nprofile in the sample and a viscous character of the flow even without an obstacle.\n\n\nIt is important to note that $d\\rho_{obst}\/dT < 0$ in the whole temperature interval, which disagrees\nwith macroscopic resistivity behavior ($d\\rho_{macr}\/dT >0$) and mesoscopic total resistivity behaviour (with and without antidots), displaying nonmonotonic behaviour :\n$d\\rho_{total}\/dT <0$ for $1.4 K < T < 40 K $ and $d\\rho_{total}\/dT > 0$ for $ 40 K < T < 80 K$ .\n\n\n\\section{Summary and conclusion}\n\nWe have studied an electronic analog of the Stokes flow around the obstacle in a two-dimensional system in high quality GaAs quantum wells.\nThe resistance of 2D electrons with a micro-hole fabricated in the center of the sample is always enhanced in comparison\nwith obstacle-free devices. Obstacle resistance decreases with temperature even as $d\\rho_{0}\/dT >0$. Experimental results confirm the theoretically predicted significance\nof momentum relaxation in the ballistic and hydrodynamic regimes, which is significantly distinct from conventional Ohmic behaviour.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nContradiction Detection and Natural Language Inference are particularly hard problems from the Natural Language processing (NLP) domain \\cite{cidm2019, icpr2021, pucknat2021detecting}. A variety of Machine Learning (ML) approaches have been introduced to tackle this task, the state-of-the-art being transformer-based methods such as XLM-RoBERTa \\cite{liu_roberta, conneau2019unsupervised}. While achieving overall good classification performance, those methods still lack understanding of linguistic features, and are relying heavily on extensive amounts of data for language model pre-training. To this end, we investigate the well-known SNLI data set \\cite{Bowman2015-EMNLP} with the aim to discover distinct linguistic properties that are important in recognizing contradictions. In addition, we also collect a data set in German language from various online sources, which is being labeled by human annotators for the contradiction detection task. The objectives of this work are two-fold:\n\\begin{itemize}\n \\item We want to find out, whether the types of contradictions differ between the synthetically created SNLI data, and the data we collected from online sources. This will also help us building a model that works well in a real-world application scenario.\n \\item We want to identify relevant linguistic features, that could help an ML model learn to recognize contradictions, without relying on extensive amounts of pre-training data. To this end, we analyze the predictions of an XLM-RoBERTa model with respect to those syntactic and semantic features that lead to wrong classifications.\n \n\\end{itemize}\nBased on those findings, we develop some first ideas for informed ML approaches that would help improve those results in the future, by injecting linguistic knowledge into the learning process. Our analysis is limited to the German language, but many of the results could most likely be applied to other languages as well.\n\n\\section{Related Work}\nThere is little prior work on the subject of linguistically aware modelling for Natural Language Inference. Marneffe et al. \\cite{Marneffe2008-ACL} are the first to comprehensively define Contradiction Detection as a distinct problem, and present some initial methods based on handcrafted semantic features such as antonymity and polarity. They point out, that detecting contradictions is a particularly hard task, because a deep level of language understanding is needed, as well as some background knowledge that cannot necessarily be inferred from the analyzed text alone. \n\nLi et al. \\cite{Li2017} attempt to learn contradiction-specific word embeddings by enforcing words with opposite meaning to be mapped into different regions of feature space. This addresses the issue, that opposite words (antonyms) tend to appear in similar contexts, such that a conventional word embedding model would learn similar embeddings for those. Using this method, the authors report state-of-the-art results on the SemEval task.\n\nA more recent approach is presented by \\cite{limit_bert2019}. They introduce a novel, linguistically aware combination of pre-training objectives for BERT \\cite{devlin2018bert}, including POS-tagging, semantic role labeling and syntactic parsing. The idea is for the model to capture a high level of semantic knowledge which will be helpful in fine-tuning on a downstream task. They achieve competitive results on the GLUE benchmark.\n\nPucknat et al. \\cite{pucknat2021detecting} evaluated different neural network based approaches on the Contradiction Detection task in German language. While XLM-RoBERTa \\cite{liu_roberta, conneau2019unsupervised} achieves the best results of all models under investigation, it still has problems with complicated syntactic structures and real-world language use. This gave rise to the idea of investigating the role of linguistic features more closely, in order to come up with an informed learning approach.\n\n\\section{Data}\nFor our analysis, we use two different data sources: A translated version of the Stanford Natural Language Inference (SNLI) data set, and a collection of real-world examples from various online sources in German language.\n\n\\subsection{SNLI}\nThe Stanford Natural Language Inference (SNLI) data set was first introduced by Bowman et al. \\cite{Bowman2015-EMNLP}. It is the largest collection of human-generated premise and hypothesis pairs for the NLI task to date, with over 570,000 examples. The data was collected in a crowd-source campaign, where both samples and labels were created by human annotators. The final labels were decided upon by a majority vote, thus minimizing noise due to human error and ambiguity. In the original data set, there are three possible labels: \\textit{entailment}, \\textit{neutral} and \\textit{contradiction}. For the purpose of the Contradiction Detection objective, we binarize those labels by consolidating the \\textit{neutral} and \\textit{entailment} labels to \\textit{no contradiction}.\n\nA large portion of the SNLI training set (100 000 examples), as well as the whole validation and test set were machine-translated to German \\cite{cidm2019}, using the DeepL API\\footnote{ https:\/\/github.com\/fraunhofer-iais\/snli\\_translated}. The data set was found to be of overall sufficient quality, but there are some artifacts and inconsistencies, due to the machine translation and the annotation setup. Because of those issues, it is not completely representative of a real-world setting. \n\n\n\\subsection{Online Data Set}\nTo address those shortcomings, we collected a data set from various online sources in German language\\footnote{The data set will be made publicly available upon the acceptance of this work.}. Those include news \\footnote{https:\/\/correctiv.org, https:\/\/nachrichtenleicht.de\/}, tweets \\footnote{https:\/\/twitter.com\/}, company and employer ratings \\footnote{https:\/\/de.trustpilot.com\/, https:\/\/www.kununu.com\/}, game reviews \\footnote{https:\/\/store.steampowered.com\/} and product reviews \\footnote{https:\/\/www.amazon.de\/}.\n\nThe data is being manually annotated by six workers using two different modes. For the first one, the annotators are presented with random examples from all five sources, and shall come up with contradicting or non-contradicting hypotheses for each of those examples. There is also the option to exclude sentences, if no meaningful hypothesis can be found. Since this procedure is quite costly, we additionally use another annotation mode, presenting the annotators with pairs of sentences from the online sources, where the premise and hypothesis have already been matched. To achieve this, the samples are first being grouped into different categories, according to the meta-data from the website (e.g. similar keywords on Twitter). Additionally, a text similarity measure is applied to identify samples that are likely to refer to the same topic. Those text pieces that belong to the same category, and show a high similarity are then being matched and presented to the reviewers as premise-hypothesis pairs. Given this setup, the annotators only have to add the respective label: \"contradiction\", \"no contradiction\" or \"exclude\" (for cases where the sentences do not relate to each other, or one of them makes no sense).\n\nWe create 10 000 data points using the first annotation procedure, and another 31 000 using the second approach. The 10 000 manually created examples are being reviewed by a second annotator, to minimize noise due to subjectivity and human error. After those steps, 531 samples had to be excluded, so that we end up with a total of 40 589 examples. A random 60-20-20 training-validation-test split is being applied to the remaining data set. \n\n\\section{Linguistic Analysis}\nWe perform a linguistic analysis of the two data sets, focussing on the qualitative differences between SNLI and internet data, and those instances that impose problems for the classifier. For this evaluation, an XLM-RoBERTa model \\cite{liu_roberta, conneau2019unsupervised} is used, which has been pre-trained for the Masked Language Modeling task on 100 languages, and fine-tuned on the respective training set for the Contradiction Detection task (translated SNLI \/ online data). For details on the architecture and training procedure, we refer to \\cite{pucknat2021detecting}, as the focus of this paper is an analytical one.\n\n\\subsection{Differences between the two Data Sets}\nThe two analyzed data sets differ primarily in syntactic structure. The first (SNLI) has basic syntactic structures and semantics as well as grammatical simplicity, whereas the second data set (internet data) virtually lives from syntactic versatility. Here, the data sets differ not only in sentence-related verbosity and sentence length but also in the juxtaposition of such sentences. The sentences from the internet data are also formulated more homogeneously and thus come much closer to real language use. Some example sentence pairs from the two data sets can be found in tables \\ref{tab:snli_examples} and \\ref{tab:internet_examples}.\n\n\\begin{table*}[h]\n\\centering\n\\small\n\n\\begin{tabular}{ccc} \\toprule\n \\textbf{Premise} & \\textbf{Hypothesis} & \\textbf{Label} \\\\ \\midrule\n \\begin{minipage}{5cm} \\vspace{1mm} \"Die Qualit\u00e4t der Kette ist sehr gut. Die Kette sieht hochwertig aus und die Lieferung war wirklich schnell :)\" - \\textit{``The quality of the necklace is very good. The necklace looks high-quality and delivery was really fast :)\"} \\vspace{1mm} \\end{minipage} & \\begin{minipage}{5cm} \\vspace{1mm} \"Keine Benachrichtigung \u00fcber Sendung und keine sendungsverfolgung m\u00f6glich. Zu lange Lieferzeiten.\"- \\textit{``No notification on the shipment and shipment tracking not possible. Delivery times too long.\"} \\vspace{2mm} \\end{minipage} & \"contradiction\" \\\\ \\midrule\n \\begin{minipage}{5cm} \\vspace{1mm} \"Das Unternehmen schreibt sich das Thema hoch auf die Fahne. Leider steht es nur da.Angefangen von der M\u00fclltrennung bis zum Versand von E-Teilen, die in \u00fcberdimensionierten Kartons versendet werden.\" - \\textit{``The company claims to highly prioritize the topic. Unfortunately, that is all it does. Starting with waste separation, as well as mailing spare parts in oversized boxes.\"} \\vspace{1mm} \\end{minipage} & \\begin{minipage}{5cm} \\vspace{1mm} \"Es werden gerne mal zum CSD etc. Marketingaktionen gestartet oder das T-Logo in Regenbogenfarben angemalt. Nachhaltig ist das aus meiner Sicht nicht\" - \\textit{``It is common for the company to start marketing campaigns on the occasion of CSD etc., or paint the T-logo in rainbow colors. From my point of view, none of this is sustainable.\"} \\vspace{1mm} \\end{minipage} & \"no contradiction\" \\\\ \n \\bottomrule\n \\end{tabular}\n\\caption{Examples from the internet data set, original German version and English translation (in italic), with binarized labels}\n\\label{tab:internet_examples}\n\\end{table*}\n\n\n\\subsection{Challenges for the NLI model}\nThe model put out faulty analyses whenever it was confronted with grammatically incomplete and incorrect sentences. As soon as one of the sentences showed grammatical deficiencies in the form of sentence breaks (anacoluth) or word cuts, problems arose with the recognition of the reference word or the sentence's meaning. Another area of concern is the record length. The model often failed to recognize the syntactic and semantic keywords\/signifiers when confronted with longer and more complicated sentences.\n\n\\begin{tcolorbox}[notitle,boxrule=0pt,boxsep=0pt,left=1em,right=1em,\ntop=0.5em,bottom=0.5em,colback=gray!10,colframe=gray!10, fontupper=\\color{darkgray}]\n\\textbf{Premise}: ``Ich finde den Ansatz mit den Bio\u00f6len sehr gut. Deshalb habe ich mich auch f\u00fcr eine Bestellung entschieden.\" $-$ \\textit{``I think the approach with the organic oils is very good. That is why i decided to place an order\"}\n\\textbf{Hypothesis:} ``Ich habe selten so viel Kulanz und Entgegenkommen von einem H\u00e4ndler erlebt. Ich habe am dritten Februar bestellt, leider kam das Paket nicht zum angezeigten Liefertermin. Ein kurzer E-Mail Kontakt mit dem H\u00e4ndler zeigte, dass das Paket beim Zusteller verloren gegangen war. Ohne Umst\u00e4nde wurde sofort ein neues Paket losgeschickt, was auch 3 Tage sp\u00e4ter ankam doch leider war dort eine Flasche kaputt. Ein erneuter E-Mail Kontakt und schon wurde die Flasche ersetzt, aber nicht nur das mir wurde auch noch eine Flasche, wegen den ganzen Umst\u00e4nden geschenkt.\" \\textit{``I have rarely experienced so much goodwill and responsiveness from a retailer. I ordered on 3 February, but unfortunately the package did not arrive on the date indicated. A brief email contact with the retailer showed that the parcel had been lost by the delivery company. A new parcel was immediately sent, which arrived three days later, but unfortunately one of the bottles was broken. Another email contact and the bottle was replaced, but not only that, I was also given a bottle as a gift because of all the circumstances.\"}\\vspace{2mm} \\\\ \n\\textbf{Gold label:} No contradiction \\\\\n\\textbf{Prediction}: Contradiction\n\\end{tcolorbox}\n\n\nIn this example, a grammatical construction consisting of two main clauses is juxtaposed with a multi-membered construction, in this case consisting of main and subordinate clauses. The syntactically and semantically important signifier of the premise \"with the bio-oils\" is suppressed in the hypothesis and not explicitly emphasised again. Even if there is a clear connection in terms of content, it is lost in the stringing together of individual semantic hierarchies. A clear semantic analysis is hardly possible for the model in this form due to this structure and the lack of reference words or similarities.\n\nThe following example also shows problems with the accumulation of sentences in juxtaposition to short or even elliptical sentences. The meaning and structure of the conditional construction is radically changed in the hypothesis. Furthermore, it is questionable whether the meaning of such short phrases as \"fits exactly\" can really be determined and related. The same problem naturally occurs with grammatically incomplete sentences. Missing reference words and inter-syntactic relations cannot be sufficiently captured, even though these sentence constructions are accepted in both German and English and may fall under the genre of linguistic devices.\n\n\\begin{tcolorbox}[notitle,boxrule=0pt,boxsep=0pt,left=1em,right=1em,\ntop=0.5em,bottom=0.5em,colback=gray!10,colframe=gray!10, fontupper=\\color{darkgray}]\n\\textbf{Premise}: ``Ja okay, Basic hei\u00dft nicht hochwertig. Auf\ndieses Laken m\u00f6chte man sich nun wirklich nicht legen.\nDas Laken greift sich sehr unangenehm ist eigentlich kaum zu beschreiben, \u00e4hnlich Plastik. Ich hatte dieses vorgesehen f\u00fcr meinen Mieter einer m\u00f6blierten Wohnung, aber ich denke das m\u00f6chte ich ihm nicht zumuten.\" $-$ \\textit{``Yes okay, basic does not mean high quality. You really don't want to lie on this sheet. The sheet is very unpleasant to the touch, it's hard to describe, similar to plastic I had intended this for my tenant in a furnished flat, but I don't think I want to put him through that.\"}\n\\textbf{Hypothesis:} ``Passt genau!\" \\textit{``Fits perfectly!\"}\\vspace{2mm} \\\\ \n\\textbf{Gold label:} No contradiction \\\\\n\\textbf{Prediction}: Contradiction\n\\end{tcolorbox}\n\nIn addition the model struggled with negations. If a negation was not directly related to the signifier, flawed results were produced. To this effect, there have been cases in which additional words could stand out and shift the meaning. Furthermore, the model did not deal well with technical or rare terms and was unable to compare them adequately. At the same time ambiguity, antonyms, homonyms and homonymous verbs were not recognized and could therefore not be linked correctly. The model seemed to recognize metaphors and allegories only to a limited extent. The previously mentioned uncertainty with individual terms, as well as the semantic capturing of individual parts of sentences, lead\nto most errors in terms of recognizing contradictions. \n\n\\begin{tcolorbox}[notitle,boxrule=0pt,boxsep=0pt,left=1em,right=1em,\ntop=0.5em,bottom=0.5em,colback=gray!10,colframe=gray!10, fontupper=\\color{darkgray}]\n\\textbf{Premise}: ``Auf der Demonstration hatten die Demonstranten mit viel Rauch und Nebel zu k\u00e4mpfen.\" $-$ \\textit{``On the demonstration, the demonstrators had to deal with a lot of smoke and fog\"}\\vspace{2mm} \\\\ \n\\textbf{Hypothesis:} ``Die Polizei setzte Tr\u00e4nengas gegen die Demonstranten ein.\" \\textit{``The police used tear gas against the demonstrators.\"}\\vspace{2mm} \\\\ \n\\textbf{Gold label:} No contradiction \\\\\n\\textbf{Prediction}: Contradiction\n\\end{tcolorbox}\n\nThis example shows another interesting problem of the model. The premise receives the state description of an environmental occurrence. \"smoke and fog\" stand together here as a methaporic synonym to the statement of \"tear gas\" contained in the hypothesis. The model cannot recognize the metaphor. The same applies to homonyms and homonymous verbs. Once the meanings are not presented in a direct way, the model cannot analyze possible contradictions due to lack of understanding. Furthermore, such word types can often only be evaluated from the context. Thus, this bivalent problem appears to be a great challenge for the model.\n\nThe greatest error rate, however, was seen in the analysis and assignment of local prepositions. These could only rarely or not at all be distinguished from one another and were treated in the same way by the model although they fulfil a major semantic function.\n\n\\begin{tcolorbox}[notitle,boxrule=0pt,boxsep=0pt,left=1em,right=1em,\ntop=0.5em,bottom=0.5em,colback=gray!10,colframe=gray!10, fontupper=\\color{darkgray}]\n\\textbf{Premise}: ``Tim sitzt neben der Badewanne.\" $-$ \\textit{``Tim sits next to the bathtub.\"}\\vspace{2mm} \\\\\n\\textbf{Hypothesis:} ``Tim w\u00e4scht sich.\" \\textit{``Tim washes himself.\"}\\vspace{2mm} \\\\ \n\\textbf{Gold label:} Contradiction \\\\\n\\textbf{Prediction}: No contradiction\n\\end{tcolorbox}\n\nThis example was not part of one of the data sets, but was created by us to test the ability of the model to recognize semantic differences when the sentence is being slightly altered. We experimented with replacing the respective local preposition by other local prepositions (\"in\/auf\/neben\/unter\"), which, however, give a completely different semantic implication. We wanted to test whether the model's prediction would change, but it yielded the same result as the original sentence. Furthermore, we added a subordinate clause (``... w\u00e4hrend eine Frau sich w\u00e4scht\" \/ ``while a woman is washing herself\") that directs the meaning to another object and yet the model stuck to an incorrect analysis and made no distinctions among the prepositions or the sentence content. It was interesting that these errors did not occur with other preposition types.\n\n\n\n\n\\section{Conclusion and Future Work}\nWe presented a first comprehensive, in-depth linguistic analysis of two data sets for the Contradiction Detection task in German language. In doing so, we discovered a number of syntactic and semantic features that pose a challenge to the transformer model. These valuable insights open a variety of opportunities for future research. One possibility would be to define additional pre-training tasks that capture syntactic and semantic knowledge. For example, part of speech tagging could be helpful, given that the model has problems with prepositional references and missing verbs. It could also be worthwhile exploiting semantic webs like WordNet or GermaNet to gain knowledge about antonyms and synonyms.\n\nAnother option is to use formalized knowledge to pre-train the model, as described by \\cite{laura_informed}. This could be achieved, by constructing characteristic, simplified training examples for types of contradictions that the model struggles with, based on the linguistic knowledge we gathered. Those could also be produced in a semi-automated fashion, by applying rules inferred from the external knowledge. \nGiven that the current model has difficulties capturing the semantics of metaphors and rare terms, it would most likely be worthwhile to look into domain-specific fine-tuning methods. For example, in the context of financial textual data analysis \\cite{sifa2019towards, hillebrand2022kpi} information-extraction could be applied as a pre-training task.\n\nFinally, it would also be interesting to validate our findings by conducting a quantitative analysis, namely to test in how many cases the classifier gives an incorrect prediction for a sample that shows a specific linguistic phenomenon.\n\n\\section*{Acknowledgements}\nWe would like to thank Pascal Binias, Moritz Haidl, Torben Hoffmann, Matilda Heinen, Lars Otte and Henning Otte for manually annotating the internet data set.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:Introduction}\n\nExtremal black holes play a fundamental role in general relativity, high energy physics and astronomy. It has been reported \\cite{rees2005} that $70\\%$ of stellar black holes (such as Cygnus X-1 \\cite{extremalobservation2} and GRS 1915+105 \\cite{extremalobservation1}) are near-extremal, suggesting that near-extremal black holes are ubiquitous in the universe. It has also been argued \\cite{brenneman-spin} that many supermassive black holes (such as the ones in the center of MCG--06-30-15 \\cite{xrayextremal} and NGC 3783 \\cite{brennemanpaper}) are near-extremal. The spins of the astrophysical black holes in all these works are below the widely predicted upper bound $a \\approx 0.998M$, which is called the Thorne limit \\cite{Thorne1974}. Note that more recent works suggest that it may be possible to go beyond the Thorne limit in the astrophysical setting \\cite{Skadowski2011}. Identifying observational \\emph{signatures} that indicate the presence of black holes that are sufficiently close to extremality may be fruitful for investigating whether astrophysical black holes with spins beyond the Thorne limit exist; see for example \\cite{gralla2016}. Extremal black holes also have interesting theoretical properties. For example, they saturate geometric inequalities for the total mass, angular momentum and charge \\cite{dainprl6, reiris2013, alaeeprl}. Moreover, they have zero temperature and hence they play an important role in the study of Hawking radiation \\cite{haw95} and in string theory \\cite{stromingerextremalentropy}. Their near-horizon limits yield new solutions to the Einstein equations with conformally invariant properties classified in \\cite{nearhorizonrn, h07, hol10}. Applications in quantum gravity have been obtained in \\cite{strominger-extremal-holography, conformalnhek, cft-extremal-cosmo} and gravitational and electromagnetic signatures of the near-horizon geometry have been presented in \\cite{gralla2016, grallastrominger}. \n\nAn important aspect of extremal black holes is that they exhibit intriguing dynamical properties. Perturbations of various types suffer from a ``horizon instability'' \\cite{aretakis4,aretakis1,aretakis2,hj2012,hm2012} according to which derivatives transversal to the event horizon of dynamical quantities grow asymptotically in time along the event horizon. The source of this instability is the existence of a charge (i.e.\\ a surface integral) $H$ which is conserved along the horizon. We remark that, under the presence of superradiance, a sequence of zero-damped quasinormal modes has been found \\cite{glampedakisfull,zeni13} leading to an amplified version of the horizon instability \\cite{zimmerman1} on such backgrounds. For another type of gravitational instability, we refer to \\cite{luis}. \n\nIn this letter, we address the no-hair hypothesis in the case of extremal black holes. The no-hair hypothesis postulates that the only externally observable classical parameters of black hole spacetimes are the mass, electric charge and angular momentum; all other dynamical information (known as ``hair'') is ``lost'' behind the event horizon rendering it permanently inaccessible to external observers. The uniqueness theorems (see e.g.~\\cite{alexakisduke}) and stability theorems (see e.g.~\\cite{Dafermos2016}) provide a first confirmation of the no-hair hypothesis for sub-extremal black holes. In the extremal case, however, the aforementioned conserved charge $H$ on the event horizon may be viewed as another classical parameter of the black hole.\nOn the other hand, all natural quantities (e.g.~translation-invariant derivatives of all orders) decay in time away from the horizon. For this reason, $H$ can be thought of as ``horizon hair'' for the extremal black hole \\cite{harvey2013}. \n\nAn open problem discussed in \\cite{hm2012, ind2018} is the measurement of the horizon hair $H$ by \\textbf{far-away} observers who receive radiation from the near-horizon region. Such observers live in the spacetime region where the distance $r$ from the black hole is large and comparable in size to $t$, the standard time coordinate. This region is modelled by null infinity. In this letter, \\textit{we show that the horizon hair $H$ of scalar perturbations on Extremal Reissner--Nordstr\\\"{o}m (ERN) is measurable from null infinity}, providing thus a resolution to the above open problem (see Section \\ref{sec:ObservingTheHorizonInstabilityFromNullInfinity}). This result has not been seen before in the literature and appears here for the first time. Previous works \\cite{hm2012, ori2013} (see Section \\ref{sec:AsymptoticsForERN} for a review and more details) showed that the horizon hair can be read off at constant distances $r$ or distances $r$ that are much smaller than $t$, but they did not address the measurement of $H$ from null infinity.\n\n Our result suggests that 1) \\textit{extremal black holes admit classical externally measurable hair} and 2) \\textit{the horizon instability could potentially serve as an observational signature}. \n Another implication is that scalar perturbations also admit a conserved charge inside the black hole, on the Cauchy (inner) horizon, whose value is equal to that of the event horizon hair $H$. This directly implies that the conserved charge on the Cauchy horizon is also measurable from null infinity. Hence, \\textit{our result provides a new mechanism that can be used to read off information at the event horizon and at the Cauchy horizon from null infinity}. We further note that our mathematically rigorous argument uncovers a new connection with soft hair (see also the discussion in Section \\ref{sec:ReviewOfSubExtremalRN}). \n\n\n\n\n\n\n\\section{The horizon hair $H[\\psi]$ of ERN}\n\\label{sec:TheHorizonHairHPsi}\n\nWe next briefly recall the horizon instability of extremal black holes. We will consider scalar perturbations $\\psi$ solving the wave equation $\\Box_g\\psi=0$ where $g$ is the ERN metric which in ingoing EF coordinates $(v,r,\\theta,\\varphi)$ takes the form\n\\[g=-Ddv^2+2dvdr+r^2(d\\theta^2+\\sin^2\\theta d\\varphi^2),\\]\nwhere $D=\\left(1-\\frac{M}{r}\\right)^2$. The event horizon corresponds to $\\mathcal{H}^{+}=\\left\\{r=M\\right\\}$. The vector field $T=\\partial_v$ is stationary and normal to $\\mathcal{H}^{+}$, whereas $\\partial_r$ is translation-invariant $([\\partial_r ,T]=0)$ and transversal to $\\mathcal{H}^{+}$. Let $\\Sigma_0$ be a spherically symmetric Cauchy hypersurface which crosses the event horizon and terminates at null infinity (for example, we can take $\\Sigma_0$ to be $\\{v=0 \\}$ for $r\\leq 2M$ and $\\{u=0 \\}$ for $r\\geq 2M$, where $u,v$ are the standard double null coordinates) and let $\\Sigma_{\\tau}=F_{\\tau}(\\Sigma_0)$ where $F_{\\tau}$ is the flow of the vector field $T$. We denote by $\\partial_{\\rho}$ the radial vector field that is tangential to $\\Sigma_{\\tau}$ and satisfies $\\partial_{\\rho}r=1$. Let $S_{\\tau}=\\mathcal{H}^{+}\\cap \\Sigma_{\\tau}$. Then, the following surface integrals\n\t\\begin{equation}\nH[\\psi]:=-\\frac{M^2}{4\\pi}\\int_{S_{\\tau}}\\partial_r (r\\psi) \\, d\\Omega\n\\label{introhorizonH}\n\\end{equation} \n\tare independent of $\\tau$ and hence are conserved on $\\mathcal{H}^{+}$ for all solutions $\\psi$ to the wave equation on ERN. Here $d\\Omega=\\sin\\theta d\\theta d\\varphi$. We will refer to $H[\\psi]$ as the horizon hair of $\\psi$. In fact, there exists an infinite number of analogous conserved charges $H_{\\ell}[\\psi]$ for each angular momentum $\\ell$ appearing in the spherical harmonic decomposition of $\\psi$ \\cite{aretakis2}, with $H[\\psi]=H_0[\\psi]$.\n\t\n\tWe next consider \\textit{outgoing} perturbations which arise from compactly supported and horizon penetrating ($H\\neq 0$) initial data. It turns out that the following instability results on $\\mathcal{H}^{+}$ \\cite{aretakis1,aretakis2}: 1) \\underline{\\textbf{Non-decay}}: $\\partial_r \\psi|_{\\mathcal{H}^{+}}\\sim -\\frac{1}{M} H[\\psi] $ as $\\tau\\rightarrow \\infty$, 2) \\underline{\\textbf{Blow-up}}: $\\partial_r \\partial_r \\psi|_{\\mathcal{H}^{+}} \\sim \\frac{1}{M^3}H[\\psi] \\cdot \\tau$ \t as $\\tau\\rightarrow \\infty$. \nMore generally $\\partial_r^{k}\\psi|_{\\mathcal{H}^{+}}\\sim c_k\\cdot H[\\psi]\\cdot \\tau^{k-1}$ where \n$ c_k = (-1)^{k}\\frac{1}{M^3}\\frac{k!}{(2M^2)^{k-1}}$ for $k\\geq 1$. The quantity $H$ can be given a physical interpretation by considering the energy density measured by incoming observers at $\\mathcal{H}^+$: $\\boldsymbol{T}_{rr}[\\psi]\\sim M^{-6}\\cdot H^2[\\psi]$, where $\\boldsymbol{T}$ is the energy-momentum tensor, and hence does not decay along $\\mathcal{H}^{+}$. On the other hand, all physically relevant quantities decay in time \\underline{away} from the horizon. Murata--Reall--Tanahashi's numerical simulations \\cite{harvey2013} of the evolution of the Einstein--Maxwell-scalar field system for perturbations of ERN suggest that the horizon instability persists in the fully non-linear setting. This instability is also relevant for \\emph{near-extremal} black holes where it is expected to be a \\emph{transient phenomenon}, see for example \\cite{harvey2013}. For other extensions of this instability we refer to \\cite{aretakis4,aretakis1,aretakis2,hj2012,hm2012,sela2, aag1,aretakis2013,harveyeffective,khanna17,aretakis2012,bizon2012,zimmerman1,sela,ind2018,ori2013,murata2012, dd2012,cardoso-2017,zimmerman5,zimmerman2,zimmerman3,berti2017,aretakis3,zimmerman4, gajic}. \n\nOne can also define a conserved charge for scalar perturbations on the Cauchy horizon $\\mathcal{C}\\mathcal{H}^{+}$ in the black hole interior of ERN (conserved charges can be defined on any hypersurface with vanishing surface gravity \\cite{hj2012, aretakisglue}):\n\t\\begin{equation}\n\\underline{H}[\\psi]:=-\\frac{M^2}{4\\pi}\\int_{\\underline{S}_{\\tau}}\\partial_r (r\\psi) \\, d\\Omega,\n\\label{introhorizonHintro}\n\\end{equation} \nwhere $\\underline{S}_{\\tau}=\\{u=\\tau\\}\\cap \\mathcal{C}\\mathcal{H}^{+}$ and $\\partial_r$ is taken with respect to the outgoing EF coordinates $(u,r,\\theta,\\varphi)$ in the interior region. In contrast to the sub-extremal case, the spherical mean of outgoing perturbations is continuously differentiable at the Cauchy horizon \\cite{gajic, harvey2013} and hence $\\underline{H}[\\psi]$ is well-defined. An important corollary of the precise late-time asymptotics (see Section \\ref{sec:RigorousAsymptotics}) is the relation\n\\begin{equation}\n\\label{eq:HeqHbar}\nH[\\psi]=\\underline{H}[\\psi]\n\\end{equation}\n for all outgoing perturbations $\\psi$.\n \\begin{figure}[H]\n\t\\begin{center}\n\t\t\t\t\\includegraphics[scale=0.17]{hair-inout.pdf}\n\\end{center}\n\\vspace{-0.6cm}\n\\caption{\\footnotesize A Penrose diagrammatic representation of the spacetime regions of interest. The conserved charge on the Cauchy horizon is equal to the horizon hair $H[\\psi]$ on the event horizon.}\\normalsize\n\t\\label{fig:hairinout}\n\\end{figure}\n\\vspace{-0.4cm}\n\\section{Measurements at null infinity}\n\\label{sec:ObservingTheHorizonInstabilityFromNullInfinity}\n\nWe define the following expression involving the radiation field $r\\psi|_{\\mathcal{I}^{+}}$ of scalar perurbations $\\psi$ on any (sub-extremal or extremal) RN spacetime:\n\\begin{equation}\n\\boxed{s[\\psi]:=\\frac{1}{4M}\\lim_{\\tau\\rightarrow \\infty}\\tau^2\\cdot (r\\psi)\\big|_{\\mathcal{I}^{+}} \\!\\!+\\!\\frac{1}{8\\pi}\\int_{\\mathcal{I}^{+}\\cap\\left\\{\\tau\\geq 0\\right\\}}\\!\\!\\! r\\psi\\big|_{\\mathcal{I}^{+}} \\, d\\Omega d\\tau.}\n\\label{radfieldexpr}\n\\end{equation}\nIn order to compute $s[\\psi]$, it actually suffices to know the radiation field for large times $\\tau\\geq \\tau_{\\text{late}}$ (for arbitrarily large $\\tau_{\\text{late}}$). Indeed, the second term on the right hand side of \\eqref{maineq} is equal to \n\\[-\\frac{1}{2M}\\int_{\\mathcal{I}^+\\cap\\{\\tau=\\tau_{\\text{late}} \\}} r^3\\partial_{\\rho}(r\\psi)d\\Omega\\!+\\frac{1}{8\\pi}\\int_{\\mathcal{I}^{+}\\cap\\{\\tau\\geq \\tau_{\\text{late}}\\}}\\!\\!\\! r\\psi d\\Omega d\\tau.\\]\nWe obtain the following identity on sub-extremal and extremal RN:\n\\begin{numcases}{s[\\psi]=}\n\\label{maineq}\nH[\\psi] & in extremal RN,\\\\\n\\label{maineqsub}\n0 & in sub-extremal RN,\n\\end{numcases}\nwhere in \\eqref {maineq} $\\psi$ is an outgoing scalar perturbation on ERN and in \\eqref{maineqsub} $\\psi$ is an initially compactly supported scalar perturbation on sub-extremal RN. \\textbf{Identity \\eqref{maineq} appears here for the first time and it shows that the horizon hair $H$ (and consequently, the horizon instability) is measurable purely from null infinity.} A sketch of the derivation of \\eqref{maineq} is given in Section \\ref{sec:AProofOfTheAsymptotics}. Furthermore, in view of identity \\eqref{maineqsub} (discussed further in Section \\ref{sec:ReviewOfSubExtremalRN} below) and the fact that $H[\\psi] \\neq 0$, \\textbf{the expression $s[\\psi]$ provides an observational signature of extremal black holes}. One could also expect $s[\\psi]$ to be useful in a transient sense to provide an observational signature for near-extremal black holes.\n\n\nThe remaining conserved charges $H_{\\ell}$ could, in principle, be measured at null infinity in an analogous fashion.\nAnother consequence of \\eqref{maineq}, combined with \\eqref{eq:HeqHbar}, is that the conserved charge $\\underline{H}$ on the Cauchy horizon can be computed from null infinity.\nWe further obtain the following identity on hypersurfaces of constant area radius $r=R>M$ in the strong field region:\n \\begin{equation}\n{H[\\psi]=\\frac{R-M}{4M}\\cdot \\lim_{\\tau\\rightarrow \\infty} \\tau^{2}\\cdot \\psi\\big|_{r=R}},\n \\label{Hfromconstantr}\n \\end{equation}\nconfirming the numerical predictions of \\cite{hm2012} and the heuristic analysis of \\cite{sela,ind2018,ori2013}. \n\n\n\\section{Late-time Asymptotics}\n\\label{sec:RigorousAsymptotics}\n\n\\subsection{Review of sub-extremal RN}\n\\label{sec:ReviewOfSubExtremalRN}\n\nSince higher angular modes $\\psi_{\\geq 1}=\\psi-\\frac{1}{4\\pi}\\int_{\\mathbb{S}^2}\\psi\\,d\\Omega$ decay faster than the spherical mean $\\frac{1}{4\\pi}\\int_{\\mathbb{S}^2}\\psi\\,d\\Omega$, it suffices to project to the spherical mean (and hence, equivalently, it suffices to consider spherically symmetric perturbations). For initial data extending to $\\mathcal{I}^{+}$ on sub-extremal RN, the unique obstruction to inverting $T$ is the non-vanishing of the Newman--Penrose constant $I[\\psi]$, which is a conserved charge along null infinity. This is related to the identity $I[T\\bar{\\psi}]=0$ for all regular solutions $\\bar{\\psi}$ to the wave equation. For compactly supported initial data (satisfying $I[\\psi]=0$), we can construct the time-integral $\\bar{\\psi}$ of $\\psi$ which satisfies $T\\bar{\\psi}=\\psi$ and has finite Newman--Penrose constant $I[\\bar{\\psi}]$. We denote $I^{(1)}[\\psi]=I[\\bar{\\psi}]$. If follows that the unique obstruction to inverting the operator $T^2$ is the non-vanishing of $I^{(1)}[\\psi]$. The relevance of $I^{(1)}[\\psi]$ became apparent in \\cite{paper2} where the precise late-time asymptotics were obtained for compactly supported initial data:\n\\begingroup\n\\squeezetable\n\\begin{table}[H]\n \\begin{tabular}{c |c| c } \n\t\t\n\n\\hline\n $\\psi|_{\\mathcal{H}^{+}}$ &$\\psi|_{r=R}$ & $r\\psi|_{\\mathcal{I}^{+}}$ \\\\ \\hline\n $8I^{(1)}[\\psi]\\cdot \\frac{1}{\\tau^{3}}$ &$8I^{(1)}[\\psi]\\cdot \\frac{1}{\\tau^{3}}$ & $-2I^{(1)}[\\psi]\\cdot {\\tau^{-2}}-8MI^{(1)}[\\psi] \\log\\tau \\cdot \\tau^{-3}$ \n\\\\ \\hline\n \\end{tabular}\n\t\t\\vspace{-0.1cm}\\caption{\\footnotesize Leading order terms in the time asymptotics on sub-extremal RN.}\n\t\\label{subernhorizon}\n\t\\end{table}\n\\endgroup\\normalsize\nThe following expression of $I^{(1)}[\\psi]$ was obtained in terms of compactly supported initial data on $\\Sigma_0$ in \\cite{paper-bifurcate}:\n\\begin{equation}\nI^{(1)}[\\psi]= \\frac{M}{4\\pi}\\int_{\\Sigma_{0}\\cap\\mathcal{H}^{+}}\\!\\!\\psi+\\frac{M}{4\\pi}\\int_{\\Sigma_0} \\nabla \\psi\\cdot n_{\\Sigma_0},\n\\label{i1sigma0}\n\\end{equation}\nwhere the integrals are considered with respect to the induced volume form. It turns out that $I^{(1)}[\\psi]$ can be computed from null infinity:\n\\begin{equation}\nI^{(1)}[\\psi]=\\frac{M}{4\\pi}\\int_{\\mathcal{I}^{+}\\cap\\left\\{\\tau\\geq 0\\right\\}} r\\psi\\big|_{\\mathcal{I}^{+}} \\, d\\Omega d\\tau.\n\\label{i1scri}\n\\end{equation}\nThe integral of the radiation field along $\\mathcal{I}^{+}$ has appeared before in the work of Luk--Oh \\cite{Luk2015} on strong cosmic censorship. \nIt is clear from Table \\ref{subernhorizon} and identity \\eqref{i1scri} that \\eqref{maineqsub} holds for perturbations on sub-extremal RN. Note also that the late-time asymptotics along, say, the event horizon depend solely on the integral of the radiation field along null infinity, confirming previous heuristic work predicting dominance of the weak field dynamics in the late-time evolution. \n\nThe existence of $I^{(1)}[\\psi]$ yields a conservation law which can be recast into an identity between the integral of $r\\psi$ along $\\mathcal{I}^+$ and an analogous integral along $\\mathcal{I}^-$, revealing a tantalizing connection with the presence of a \\emph{soft electric hair} \\cite{Strominger2014, He2014, Hawking2016, Hawking2017}. Indeed, one may formally derive the null infinity conservation law for $r\\psi$ and the conservation of charges associated to soft electric hair for a 2-form $F$ satisfying the Maxwell equations with a source $j$, by integrating the following 4-form equations: $d \\star d\\psi=0$ and $(d\\star F+4\\pi\\star j)\\wedge d\\epsilon=0$, respectively, in suitable spacetime regions and applying Stokes' theorem. Here $\\epsilon$ denotes an arbitrary smooth function that only depends on the angular coordinates. \n\n\\vspace{-0.5cm}\n\n\\subsection{Asymptotics for ERN}\n\\label{sec:AsymptoticsForERN}\n\nWe distinguish three classes of perturbations on ERN: \n\\begingroup\n\\squeezetable\n\\begin{table}[H]\n\\begin{center}\n \\begin{tabular}{ c|c | c } \\hline\n Perturbations & $H$ & $I$ \\\\ \n\\hline\n outgoing & $\\neq 0$ & $=0$ \\\\ \\hline\nstatic moment & $\\neq 0$ & $\\neq 0$ \\\\ \\hline\n ingoing & $=0$ & $=0$ \\\\ \\hline\n \\end{tabular}\n \\end{center}\n\t\\vspace{-0.4cm}\n \\caption{Types of initial data. Here $H$ denotes the conserved charge on $\\mathcal{H}^{+}$ and $I$ denotes the Newman--Penrose constant on $\\mathcal{I}^{+}$.}\n\\end{table}\n\\vspace{-0.4cm}\n\\endgroup\nFor outgoing and ingoing perturbations (with compactly supported initial data) we define the constant $I^{(1)}$ as in \\eqref{i1sigma0} (or, equivalently, in \\eqref{i1scri}). For ingoing perturbations, we also define \n\\begin{equation}\nH^{(1)}[\\psi]:= \\frac{M^2}{4\\pi}\\int_{\\mathcal{H}^{+}}\\!\\! \\psi\\big|_{\\mathcal{H}^{+}}d\\Omega d\\tau.\n\\label{h01}\n\\end{equation} \n We refer to $H^{(1)}[\\psi]$ as the \\textit{time-inverted horizon charge}. \n A physical interpretation of $H^{(1)}[\\psi]$ can be given in terms of the \\textit{dual} scalar field $\\widetilde{\\psi}$ of $\\psi$ defined by $\\widetilde{\\psi}= \\frac{M}{r-M}\\psi\\circ \\Phi\\label{dual}\n$, where $\\Phi$ denotes the Couch--Torrence conformal inversion. It can be easily seen that 1) the duality is self-inverse, 2) $\\psi$ solves the wave equation if and only if $\\widetilde{\\psi}$ solves the wave equation and 3) $H[\\psi] =I[\\widetilde{\\psi}]$. The latter relation was obtained independently in \\cite{bizon2012,hm2012}. It follows that $H^{(1)}[\\psi]:= I^{(1)}[\\widetilde{\\psi}]$. Moreover, in view of \\eqref{i1sigma0} applied to $\\widetilde{\\psi}$, one may obtain an expression for $H^{(1)}$ in terms of the initial data on $\\Sigma_0$. \nWe can now present the precise late-time asympotics along the event horizon:\n\\begingroup\n\\begin{table}[H]\n\\begin{center}\n \\begin{tabular}{ c|c | c } \n\\cline{2-3} & outgoing data & ingoing data \\\\ \n\\hline\n$\\psi|_{\\mathcal{H}^{+}}$ & ${2H \\cdot \\tau^{-1}}$ & $\\boldsymbol{-2H^{(1)} \\cdot \\tau^{-2}}$ \n\\\\ \\hline\n$\\partial_r \\psi|_{\\mathcal{H}^{+}}$ & $ -\\frac{1}{M}\\cdot H$ & $\\boldsymbol{\\frac{2}{M^2}\\cdot H^{(1)}\\cdot {\\tau^{-2}}}$ \\\\ \\hline\n$\\partial_r \\partial_r \\psi|_{\\mathcal{H}^{+}}$\t& $ \\frac{1}{M^3}\\cdot H\\cdot\\tau$ & $ \\boldsymbol{\\frac{1}{M^3}\\cdot H^{(1)}}$ \\\\ \\hline\n$\\partial_r \\partial_r \\partial_r \\psi|_{\\mathcal{H}^{+}}$ & $-\\frac{3}{2M^5} \\cdot H \\cdot\\tau^2$ & $\\boldsymbol{-\\frac{3}{M^5}\\cdot H^{(1)}\\cdot \\tau}$ \\\\ \\hline\n \\end{tabular}\n\t\\vspace{-0.1cm}\\caption{\\footnotesize Asymptotics along the event horizon on ERN for outgoing and ingoing perturbations. \\textbf{The ingoing asymptotics are new and have not appeared before in the literature.} The outgoing asymptotics are consistent with \\cite{hm2012, sela, ori2013}. } \\normalsize\n\t\\label{ernhorizon}\n\\end{center}\n\\end{table}\n\\vspace{-0.7cm}\n\\endgroup\n We present below the precise late-time asymptotics away from the horizon:\n\\begingroup\n\\squeezetable\n\\begin{table}[H]\n\\begin{center}\n \\begin{tabular}{ c|c | c } \n\t \n \\hline\nData & $\\psi|_{r=R}$ & $r\\psi|_{\\mathcal{I}^{+}}$ \\\\ \n\\hline\noutgoing & $ \\frac{4M}{r-M}H \\cdot \\tau^{-2}$ & $\\!\\boldsymbol{\\left(4MH-2I^{(1)}\\right)\\! \\cdot \\!\\tau^{-2}}$\n\\\\ \\hline\n static moment & $\\!\\boldsymbol{4\\left(I+\\frac{M}{r-M}H \\right)\\cdot \\tau^{-2}}$ & $\\boldsymbol{2\\cdot I[\\psi]\\cdot \\tau^{-1}}$ \\\\ \\hline\ningoing & $\\!\\boldsymbol{-8\\left( I^{(1)}+\\frac{M}{r-M}H^{(1)} \\right)\\!\\cdot\\! \\tau^{-3}}$ & $\\boldsymbol{-2I^{(1)}\\cdot \\tau^{-2}}$ \\\\ \\hline\n \\end{tabular}\n\\caption{\\footnotesize Asymptotics away from the event horizon on ERN and specifically on $r=R>M$ and on null infinity $\\mathcal{I}^{+}$. \\textbf{The bold terms are new and appear here for the first time. The late-time asymptotics for $r\\psi|_{\\mathcal{I}^{+}}$, in conjunction with the expression \\eqref{i1scri} for $I^{(1)}$, yield \\eqref{maineq}. } The asymptotic term for $\\psi|_{r=R}$ for outgoing perturbations in the strong field region $\\{r=R\\}$ is consistent with the results presented in \\cite{hm2012, sela, ori2013, Burko2007, zimmerman1,harveyeffective, zimmerman4}. }\n\\label{awayerntablerev}\n\\end{center}\n\\end{table}\n\\endgroup \\normalsize\n \n\n\\section{Sketch of the proof}\n\\label{sec:AProofOfTheAsymptotics}\n\nIn this section we present a summary of the main ideas involved in deriving the late-time asymptotics for \\underline{outgoing} spherically symmetric perturbations on ERN. The full details will be presented in the upcoming paper \\cite{aag7}. \n\n\\textbf{Step 1.} We obtain the asymptotics for $\\psi$ and $T\\psi$ on the event horizon and actually in the spacetime region to the left of the hypersurface $\\gamma_{\\mathcal{H}}=\\{r=M+\\tau^{\\alpha}\\}$ for some $3\/4<\\alpha<1$ (see Figure \\ref{fig:overview}). Indeed, we can estimate $\\partial_u(r\\psi)\\sim 2H u^{-2}$ to the left of $\\gamma_{\\mathcal{H}}$ which after integration from $\\gamma_{\\mathcal{H}}$ yields asymptotics for $r\\psi$ to the left of $\\gamma_{\\mathcal{H}}$. Here $\\partial_u$ is taken with respect to the standard EF double null coordinates $(u,v)$. \n\n\n\\textbf{Step 2.} We derive asymptotics\/estimates for the derivative $\\partial_{\\rho}\\psi$ that is tangential to $\\Sigma_{\\tau}$ as follows: \nintegrating the wave equation along $\\Sigma_{\\tau}$ from the horizon $r=M$ to some $r>M$ we obtain:\\small\n\\begin{equation*}\n\\begin{split}\n&\\ \\ \\ \\ \\ Dr^2\\partial_{\\rho}\\psi(r,\\tau)=\\\\ &\\boldsymbol{2M^2 T\\psi|_{\\mathcal{H}^+}(\\tau)}\n+r^2 T\\psi(r,\\tau) +\\int_{M}^{r}O(r')T\\psi+O(r')T^2\\psi\\,dr'.\n\\end{split}\n\\end{equation*}\\normalsize The bold horizon term is the leading one: $\n\\boldsymbol{2M^2 T\\psi|_{\\mathcal{H}^+}}\\sim \\boldsymbol{-4MH\\cdot \\tau^{-2}}$.\nWe conclude that for any $r>M$:\n\\begin{equation}\n\\begin{split}\n& \\ \\ \\ \\ \\ \\left|\\partial_{\\rho}\\psi(r,\\tau)+\\boldsymbol{4MHD^{-1}r^{-2}\\tau^{-2}}\\right|\\\\&\\leq \\: C\\tau^{-\\frac{5}{2}+\\epsilon}\\cdot D^{-\\frac{3}{2}}r^{-\\frac{1}{2}}+CD^{-1}r^{-2}\\tau^{-2-\\epsilon}.\n\\end{split}\n\\label{partialrhoasymptotics}\n\\end{equation} \n\n\\textbf{Step 3.} We next obtain the late-time asymptotics for $r\\psi$ on $\\gamma_{\\mathcal{I}}=\\{r=\\tau^{\\alpha}\\}$. We use the following splitting identity:\n\\begin{equation}\n\\Big. r \\psi\\Big|_{\\gamma_{\\mathcal{I}}}\\hspace{0.5cm}= \\Big.\\underbrace{r\\partial_\\rho (r\\psi)\\Big|_{\\gamma_{\\mathcal{I}}}}_{\\substack{\\text{contribution from} \\\\ \\text{the right side of $\\gamma_{\\mathcal{I}}$}}}-\\Big.\\underbrace{r^2 \\partial_\\rho\\psi\\Big|_{\\gamma_{\\mathcal{I}}}}_{\\substack{\\text{contribution from} \\\\ \\text{the left side of $\\gamma_{\\mathcal{I}}$}}}.\n\\label{splittingide}\n\\end{equation}\nWe will show that the first (resp.~the second) term on the right hand side of \\eqref{splittingide} can be estimated using properties of the right (resp.~left) side of $\\gamma_{\\mathcal{I}}$. We introduce a new technique, which we call \\textbf{the singular time inversion}. We construct the time integral $\\psi^{(1)}$ of $\\psi$ which solves the wave equation $\\square_g\\psi^{(1)}=0$ and satisfies $T\\psi^{(1)}=\\psi$. Since $H[\\psi]\\neq 0$ we have that $\\psi^{(1)}$ is \\emph{singular} at the horizon; in fact, $\n(r-M)\\cdot \\partial_{\\rho}\\psi^{(1)} =-\\frac{2}{M}\\cdot H[\\psi]$\nclose to the event horizon. On the other hand, $\\psi^{(1)}$ is smooth away from the event horizon and has a well-defined Newman--Penrose constant $I^{(1)}=I[\\psi^{(1)}]<\\infty$. It can be shown that $|r\\psi^{(1)}|\\lesssim \\tau^{-1\/2+\\epsilon}$ as $\\tau\\rightarrow \\infty$ to the right of $\\gamma_{\\mathcal{I}}$. The boundedness of $I^{(1)}$ yields $\\partial_{\\rho}(r\\psi^{(1)})|_{\\gamma_{\\mathcal{I}}}\\sim I^{(1)} v^{-2} \\sim I^{(1)} \\tau^{-2} $ since $v\\sim \\tau$ and $r\\sim \\tau^{\\alpha}$ along $\\gamma_{\\mathcal{I}}$. Hence, we obtain $\\partial_{\\rho}(r\\psi)|_{\\gamma_{\\mathcal{I}}}\\sim I^{(1)} \\tau^{-3}$ and hence \n$r\\partial_{\\rho}(r\\psi)|_{\\gamma_{\\mathcal{I}}}\\sim r\\tau^{-3}\\sim \\tau^{-3+\\alpha}$ along $\\gamma_{\\mathcal{I}}$. We conclude that this term does \\underline{not} contribute to the asymptotics of $r \\psi|_{\\gamma_{\\mathcal{I}}}$. We next derive the precise asymptotics of $r^2 \\partial_\\rho\\psi|_{\\gamma_{\\mathcal{I}}}$. Integrating the wave equation along $\\Sigma_{\\tau}$ for $r=R$ to $r=r_{\\gamma_{\\mathcal{I}}}$ we obtain \n\\begin{equation}\n\\Big|Dr^2\\partial_\\rho\\psi\\big|_{{\\gamma_{\\mathcal{I}}}}-Dr^2\\partial_\\rho\\psi\\big|_{r=R}\\Big|\\lesssim \\int^{r_{\\gamma_{\\mathcal{I}}}}_{R}r|\\partial_\\rho\\big(rT\\psi\\big)|\\,dr.\n\\label{tarho}\n\\end{equation}\nThe right hand side can be shown to be bounded by $\\tau^{-2-\\epsilon}$ for some $\\epsilon>0$ which implies that the asymptotics for $r^2\\partial_{\\rho}\\psi|_{\\gamma_{\\mathcal{I}}}$ can be derived from the asymptotics of $\\partial_{\\rho}\\psi|_{\\{r=R\\}}$. We can now apply \\eqref{partialrhoasymptotics} for $r=R$ to conclude that the asymptotics for $r^2\\partial_{\\rho}\\psi|_{\\gamma_{\\mathcal{I}}}$ and $r\\psi|_{\\gamma_{\\mathcal{I}}}$ depend \\textbf{only} on $H$. Specifically, we obtain as $\\tau\\rightarrow \\infty$:\n\\begin{equation}\nr \\psi\\Big|_{\\gamma_{\\mathcal{I}}} \\sim -{ r^2 \\partial_\\rho\\psi\\Big|_{\\gamma_{\\mathcal{I}}}} \\sim -{Dr^2 \\partial_\\rho\\psi\\Big|_{r=R}}\\sim 4MH\\tau^{-2}.\n\\label{gammaias}\n\\end{equation}\n\n\\textbf{Step 4.} Integrating \\textbf{backwards} the estimate for $\\partial_{\\rho}\\psi$ of the previous steps from $\\gamma_{\\mathcal{I}}$ up to $\\gamma_{\\mathcal{H}}$ and using the asymptotics for $r\\psi|_{\\gamma_{\\mathcal{I}}}$, we obtain the asymptotics for $r\\psi$ in the region between $\\gamma_{\\mathcal{H}}$ and $\\gamma_{\\mathcal{I}}$. \n\n\n\\textbf{Step 5.} In this last step we derive the asymptotics for $r\\psi$ to the right of $\\gamma_{\\mathcal{I}}$ all the way up to null infinity. We use the construction for the singular time integral $\\psi^{(1)}$ once again. Specifically, we derive the asymptotics of the difference $T(r\\psi^{(1)})-T(r\\psi^{(1)})|_{\\gamma_{\\mathcal{I}}}=r\\psi-r\\psi|_{\\gamma_{\\mathcal{I}}}$ in terms of $I^{(1)}=I[\\psi^{(1)}]$:\n\\begin{equation*}\n\\Big|r\\psi|_{\\mathcal{I}^{+}}(\\tau)-r\\psi|_{\\gamma_{\\mathcal{I}}}(\\tau)+ 2I^{(1)}\\cdot \\tau^{-2}\\Big|\\lesssim C \\tau^{-2-\\epsilon}.\n\\end{equation*}\nPlugging in the asymptotics \\eqref{gammaias} of $r\\psi|_{\\gamma_{\\mathcal{I}}}$ yields the asymptotics of the radiation field $r\\psi$ as in Table \\ref{awayerntablerev}.\n\n\\vspace{-0.15cm}\n \\begin{figure}[H]\n\t\\begin{center}\n\t\t\t\t\\includegraphics[scale=0.17]{overview-proof.pdf}\n\\end{center}\n\\vspace{-0.6cm}\n\\caption{\\footnotesize A labeling of the spacetime regions indicating the order in which the late-time asymptotics of $\\psi$ are derived. We see that a delicate global study is needed in order to derive the asymptotics on null infinity. }\n\t\\label{fig:overview}\n\\end{figure}\n\\vspace{-0.4cm}\\normalsize\n\n\n\\section{Concluding remarks}\n\\label{sec:ConcludingRemarks}\n\nThe physical relevance of our results stems from the expectation that the horizon hair of \\textit{axisymmetric} scalar perturbations on Extremal Kerr (EK) can be analogously measured from null infinity. Even though the late-time behavior for fixed non-zero azimuthal modes on EK has been derived by Casals--Gralla--Zimmerman \\cite{zimmerman1}, the precise late-time asymptotics are not known. In fact, a very exciting problem would be to examine potential contributions of the near-horizon geometry to the precise late-time asymptotics for \\textbf{general} (without any symmetry assumptions) scalar, electromagnetic and gravitational perturbations on EK. A closely related problem is to probe the measurability properties of the Lucietti--Reall gravitational instability \\cite{hj2012} of EK from null infinity. The ultimate goal would be of course to study the fully non-linear perturbations of EK in the context of the Einstein-vacuum equations. A simplified but still very interesting problem would be to obtain analogous measurability results for the Murata--Reall--Tanahashi spacetimes \\cite{harvey2013}. \n\n\n\\section{Acknowledgements}\n\\label{sec:Acknowledgements}\n\nWe thank Harvey Reall for his insightful comments. S. Aretakis acknowledges support through NSF grant DMS-1265538, NSERC grant 502581, an Alfred P. Sloan Fellowship and the Connaught Fellowship 503071.\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}