diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzgemz" "b/data_all_eng_slimpj/shuffled/split2/finalzzgemz"
new file mode 100644--- /dev/null
+++ "b/data_all_eng_slimpj/shuffled/split2/finalzzgemz"
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\n\nLet  $A_{1},$ $A_{2},$ $A_{3},$ $A_{4}$ be four non-collinear and\nnon-coplanar points and a positive real number (weight) $B_{i}$\ncorrespond to each point $A_{i},$ for $i=1,2,3,4.$\n\nThe weighted Fermat-Torricelli problem for four non-collinear\npoints and non-coplanar points in $\\mathbb{R}^{3}$ states\nthat:\n\n\\begin{problem} Find a unique (fifth) point $A_{0}\\in \\mathbb{R}^{3},$\nwhich minimizes\n\n\\[f(X)=\\sum_{i=1}^{4}B_{i}\\|X-A_{i}\\|,\\]\nwhere $\\|\\cdot\\|$ denotes the Euclidean distance and $X\\in\n\\mathbb{R}^{3}.$\n\n\\end{problem}\n\n\n\nThe existence and uniqueness of the weighted Fermat-Torricelli\npoint and a complete characterization of the solution of the\nweighted Fermat-Torricelli problem for tetrahedra has been\nestablished in \\cite[Theorem~1.1, Reformulation~1.2 page~58,\nTheorem~8.5 page 76, 77]{Kup\/Mar:97}).\n\n\\begin{theorem}{\\cite{BolMa\/So:99},\\cite{Kup\/Mar:97},\\cite{KupitzMartini:94}}\\label{theor1}\nLet there be given four non-collinear points and non-coplanar\npoints $\\{A_{1},A_{2},A_{3},A_{4}\\},$ $A_{1}, A_{2},\nA_{3},A_{4}\\in\\mathbb{R}^{3}$ with corresponding positive\nweights $B_{1}, B_{2}, B_{3}, B_{4}.$ \\\\\n(a) The weighted Fermat-Torricelli point $A_{0}$ exists and is\nunique. \\\\\n(b) If for each point $A_{i}\\in\\{A_{1},A_{2},A_{3},A_{4}\\}$\n\n\\begin{equation}\\label{floatingcase}\n\\|{\\sum_{j=1, i\\ne j}^{4}B_{j}\\vec u(A_i,A_j)}\\|>B_i,\n\\end{equation}\n\n for $i,j=1,2,3$  holds,\n then \\\\\n ($b_{1}$) the weighted Fermat-Torricelli point $A_{0}$ (weighted floating equilibrium point) does not belong to $\\{A_{1},A_{2},A_{3},A_{4}\\}$\n and \\\\\n ($b_{2}$)\n\n\\begin{equation}\\label{floatingequlcond}\n \\sum_{i=1}^{4}B_{i}\\vec u(A_0,A_i)=\\vec 0,\n\\end{equation}\nwhere $\\vec u(A_{k} ,A_{l})$ is the unit vector from $A_{k}$ to\n$A_{l},$ for $k,l\\in\\{0,1,2,3,4\\}$\n (Weighted Floating Case).\\\\\n (c) If there is a point $A_{i}\\in\\{A_{1},A_{2},A_{3},A_{4}\\}$\n satisfying\n \\begin{equation}\n \\|{\\sum_{j=1,i\\ne j}^{4}B_{j}\\vec u(A_i,A_j)}\\|\\le B_i,\n\\end{equation}\nthen the weighted Fermat-Torricelli point $A_{0}$ (weighted\nabsorbed point) coincides with the point $A_{i}$ (Weighted\nAbsorbed Case).\n\\end{theorem}\n\n\n\n\n\n\n\n\n\n\n\nWe consider the following open problem:\n\n\\begin{problem}\nFind an analytic solution with respect to the weighted\nFermat-Torricelli problem for tetrahedra in $\\mathbb{R}^{3},$ such\nthat the corresponding weighted Fermat-Torricelli point is not any\nof the given points.\n\\end{problem}\n\nIn this paper, we present an analytical solution for the weighted\nFermat-Torricelli problem for regular tetrahedra  in\n$\\mathbb{R}^{3}$ for $B_{1}>B_{4},$ $B_{1}=B_{2}$ and\n$B_{3}=B_{4},$ by expressing the objective function as a function\nof the linear segment which is formed by the middle point of the\ncommon perpendicular $A_{1}A_{2}$ and $A_{3}A_{4},$  and the\ncorresponding weighted Fermat-Torricelli point $A_{0}$ (Section~2,\nTheorem~\\ref{theortetr}). It is worth mentioning that this\nanalytical solution of the weighted Fermat-Torricelli problem for\na regular tetrahedron is a generalization of the analytical\nsolution of the weighted Fermat-Torricelli point of a quadrangle\n(tetragon) in $\\mathbb{R}^{2}$ (see in \\cite{Zachos:14}).\n\n\n\nBy expressing the angles $\\angle A_{i}A_{0}A_{j}$ for\n$i,j=1,2,3,4$ for $i\\ne j$  as a function of $B_{1},$ $B_{4}$ and\n$a$ and taking into account the invariance property of the\nweighted Fermat-Torricelli point (geometric plasticity) in\n$\\mathbb{R}^{3},$ we obtain an analytical solution for some\ntetrahedra having the same weights with the regular tetrahedron\n(Section~3, Theorem~\\ref{theorquadnntetrah}).\n\n\n\n\n\n\\section{The weighted Fermat-Torricelli problem for regular tetrahedra: The case  $B_{1}=B_{2}$ and $B_{3}=B_{4}.$ }\n\n\n\n\nWe shall consider the weighted Fermat-Torricelli problem for a\nregular tetrahedron $A_{1}A_{2}A_{3}A_{4},$ for $B_{1}>B_{4},$\n$B_{1}=B_{2}$ and $B_{3}=B_{4}.$\n\nWe denote by $a_{ij}$ the length of the linear segment $A_iA_j,$\nby $A_{12}A_{34}$ the common perpendicular of $A_{1}A_{2}$ and\n$A_{3}A_{4}$ where $A_{12}$ is the middle point of $A_{1}A_{2}$\nand $A_{34}$ is the middle point of $A_{3}A_{4},$ by $A_{0}$ the\nweighted Fermat-Torricelli point of $A_{1}A_{2}A_{3}A_{4}$ by $O$\nthe middle point of $A_{12}A_{34}$ ($A_{12}O=A_{34}O$), by $y$ the\nlength of the linear segment $OA_{0}$ and $\\alpha_{ikj}$ the angle\n$\\angle A_{i}A_{k}A_{j}$ for $i,j,k=0,1,2,3,4, i\\neq j\\neq k.$ We\nset $a_{ij}\\equiv a,$ the edges of $A_{1}A_{2}A_{3}A_{4}$\n(Fig.~\\ref{fig1}).\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.2]{TetrahedronFT1new}\n\\caption{The weighted Fermat-Torricelli problem for a regular\ntetrahedron $B_{1}=B_{2}$ and $B_{3}=B_{4}$ for\n$B_{1}>B_{4}$}\\label{fig1}\n\\end{figure}\n\n\\begin{problem}\\label{sym1}\nGiven a regular tetrahedron $A_{1}A_{2}A_{3}A_{4}$ and a weight\n$B_{i}$ which corresponds to the vertex $A_{i},$ for $i=1,2,3,4,$\nfind a fifth point $A_{0}$ (weighted Fermat-Torricelli point)\nwhich minimizes the objective function\n\n\\begin{equation}\\label{obj1}\nf=B_{1}a_{01}+B_{2} a_{02}+ B_{3} a_{03}+B_{4} a_{04}\n\\end{equation}\nfor $B_{1}>B_{4},$ $B_{1}=B_{2}$ and $B_{3}=B_{4}.$\n\\end{problem}\n\nWe set\n\n\n\n\\begin{eqnarray}\\label{analsolrtetrahedron1}\n&&s\\equiv -a^6 B_1^{12}+2 a^6 B_1^{10} B_4^2+a^6 B_1^8 B_4^4-4 a^6\nB_1^6 B_4^6+a^6 B_1^4 B_4^8+2 a^6 B_1^2 B_4^{10}-a^6\nB_4^{12}+2 \\sqrt{2}\\nonumber{}\\\\\n&&{}\\surd(a^{12} B_1^{22} B_4^2-8 a^{12} B_1^{20} B_4^4+29 a^{12}\nB_1^{18} B_4^6-64 a^{12} B_1^{16} B_4^8+98 a^{12} B_1^{14}\nB_4^{10}-112 a^{12} B_1^{12} B_4^{12}+\\nonumber{}\\\\\n&&{}+98 a^{12} B_1^{10} B_4^{14}-64 a^{12} B_1^8 B_4^{16}+29\na^{12} B_1^6 B_4^{18}-8 a^{12} B_1^4 B_4^{20}+a^{12} B_1^2\nB_4^{22})\n\\end{eqnarray}\n\n\nand\n\n\\begin{eqnarray}\\label{analsolrtetrahedron2}\nt\\equiv -\\frac{a^4 B_1^4}{4 s^{1\/3}}+\\frac{a^4 B_1^2 B_4^2}{2\ns^{1\/3}}-\\frac{a^4 B_4^4}{4 s^{1\/3}}-\\frac{s^{1\/3}}{4\n\\left(B_1^4-2 B_1^2 B_4^2+B_4^4\\right)}.\n\\end{eqnarray}\n\n\n\\begin{theorem}\\label{theortetr}\nThe location of the weighted Fermat-Torricelli point $A_{0}$ of\n$A_{1}A_{2}A_{3}A_{4}$ for $B_{1}=B_{2},$ $B_{3}=B_{4}$ and\n$B_{1}>B_{4}$ is given by:\n\n\\begin{eqnarray}\\label{analsolrtetrahedron}\n&&y=-\\frac{\\sqrt{t}}{2}+\\nonumber{}\\\\\n&&{}\\frac{1}{2} \\sqrt{\\frac{a^4 B_1^4}{4 s^{1\/3}}-\\frac{a^4 B_1^2\nB_4^2}{2 s^{1\/3}}+\\frac{a^4 B_4^4}{4 s^{1\/3}}+\\frac{2 \\left(-8\n\\sqrt{2} a^3 B_1^2-8 \\sqrt{2} a^3 B_4^2\\right)}{\\sqrt{t} \\left(64\nB_1^2-64 B_4^2\\right)}+\\frac{s^{1\/3}}{4 \\left(B_1^4-2 B_1^2\nB_4^2+B_4^4\\right)}}\\nonumber{}\\\\\n\\end{eqnarray}\n\n\n\n\\end{theorem}\n\n\\begin{proof}[Proof of Theorem~\\ref{theortetr}:]\n\nTaking into account the symmetry of the weights $B_{1}=B_{4}$ and\n$B_{2}=B_{3}$ for $B_{1}>B_{4}$ and the symmetries of the regular\ntetrahedron $A_{1}A_{2}A_{3}A_{4}$ the objective function\n(\\ref{obj1}) of the weighted Fermat-Torricelli problem\n(Problem~\\ref{sym1}) could be reduced to an equivalent Problem:\nFind a point $A_{0}$ which belongs to the midperpendicular\n$A_{12}A_{34}$ of $A_{1}A_{2}$ and $A_{3}A_{4}$ and minimizes the\nobjective function\n\n\\begin{equation}\\label{obj12}\n\\frac{f}{2}=B_{1}a_{01}+B_{4} a_{04}.\n\\end{equation}\n\n\n\n\nWe express $a_{01},$ $a_{02},$ $a_{03}$ and $a_{04}$ as a function\nof $y:$\n\n\\begin{equation}\\label{a01}\na_{01}^{2}=\\left(\\frac{a}{2}\\right)^{2}+\\left(\\frac{a\\frac{\\sqrt{2}}{2}}{2}-y\\right)^{2},\n\\end{equation}\n\n\n\\begin{equation}\\label{a02}\na_{02}^{2}=\\left(\\frac{a}{2}\\right)^{2}+\\left(\\frac{a\\frac{\\sqrt{2}}{2}}{2}-y\\right)^{2},\n\\end{equation}\n\n\n\\begin{equation}\\label{a03}\na_{03}^{2}=\\left(\\frac{a}{2}\\right)^{2}+\\left(\\frac{a\\frac{\\sqrt{2}}{2}}{2}+y\\right)^{2},\n\\end{equation}\n\n\n\\begin{equation}\\label{a04}\na_{04}^{2}=\\left(\\frac{a}{2}\\right)^{2}+\\left(\\frac{a\\frac{\\sqrt{2}}{2}}{2}+y\\right)^{2},\n\\end{equation}\n\n\nwhere the length of $A_{12}A_{34}$ is $\\frac{a\\sqrt{2}}{2}.$\n\nBy replacing (\\ref{a01}) and (\\ref{a04}) in (\\ref{obj12}) we get:\n\n\\begin{equation}\\label{obj13}\nB_{1}\\sqrt{\\left(\\frac{a}{2}\\right)^{2}+\\left(\\frac{a\\frac{\\sqrt{2}}{2}}{2}-y\\right)^{2}}+B_{4}\\sqrt{\\left(\\frac{a}{2}\\right)^{2}+\\left(\\frac{a\\frac{\\sqrt{2}}{2}}{2}+y\\right)^{2}}\n\\to min.\n\\end{equation}\n\nBy differentiating (\\ref{obj13}) with respect to $y,$ and by\nsquaring both parts of the derived equation, we get:\n\n\\begin{equation}\\label{fourth1}\n\\frac{B_{1}^2 \\left(\\frac{a\\frac{\\sqrt{2}}{2}}{2}-y\\right)^{2}\n}{\\left(\\frac{a}{2}\\right)^{2}+\\left(\\frac{a\\frac{\\sqrt{2}}{2}}{2}-y\\right)^{2}}=\\frac{B_{4}^2\n\\left(\\frac{a\\frac{\\sqrt{2}}{2}}{2}+y\\right)^{2}}{\\left(\\frac{a}{2}\\right)^{2}+\\left(\\frac{a\\frac{\\sqrt{2}}{2}}{2}+y\\right)^{2}}\n\\end{equation}\n\nwhich yields\n\n\\begin{equation}\\label{fourth2}\n64 y^4 \\left(B_1^2-B_4^2\\right)-8 \\sqrt{2} a^3 y\n\\left(B_1^2+B_4^2\\right)+3 a^4\\left(B_1^2-B_4^2\\right)=0.\n\\end{equation}\n\nBy solving the fourth order equation (\\ref{fourth2}) with respect\nto $y$ , we derive two complex solutions and two real solutions\n(see in \\cite{Shmakov:11} for the general solution of a fourth\norder equation with respect to $y$) which depend on $B_{1}, B_{4}$\nand $a.$ One of the two real solutions with respect to $y$ is\n(\\ref{analsolrtetrahedron}). The real solution\n(\\ref{analsolrtetrahedron}) gives the location of the weighted\nFermat-Torricelli point $A_{0}$ at the interior of\n$A_{1}A_{2}A_{3}A_{4}$ (see fig.~\\ref{fig1}).\n\n\n\\end{proof}\n\n\n\n\n\nWe shall state the Complementary weighted Fermat-Torricelli\nproblem for a regular tetrahedron (\\cite[pp.~358]{Cour\/Rob:51}),\nin order to explain the second real solution which have been\nobtained by (\\ref{fourth2}) with respect to $y$ (see also in\n\\cite{Zachos:14} for the case of a quadrangle).\n\n\\begin{problem}\\label{sym2complementary}\nGiven a regular tetrahedron $A_{1}A_{2}A_{3}A_{4}$ and a weight\n$B_{i}$ (a positive or negative real number) which corresponds to\nthe vertex $A_{i},$ for $i=1,2,3,4,$ find a fifth point $A_{0}$\n(weighted Fermat-Torricelli point) which minimizes the objective\nfunction\n\n\\begin{equation}\\label{obj1}\nf=B_{1}a_{01}+B_{2} a_{02}+ B_{3} a_{03}+B_{4} a_{04}\n\\end{equation}\nfor $\\|B_{1}\\|>\\|B_{4}\\|,$ $B_{1}=B_{2}$ and $B_{3}=B_{4}.$\n\\end{problem}\n\n\\begin{proposition}\\label{theortetrcomp1}\nThe location of the complementary weighted Fermat-Torricelli point\n$A_{0}^{\\prime}$ (solution of Problem~\\ref{sym2complementary}) of\nthe regular tetrahedron $A_{1}A_{2}A_{3}A_{4}$ for\n$B_{1}=B_{2}<0,$ $B_{3}=B_{4}<0$ and $\\|B_{1}\\|>\\|B_{4}\\|$ is the\nexactly same with the location of the corresponding weighted\nFermat-Torricelli point $A_{0}$ of $A_{1}A_{2}A_{3}A_{4}$ for\n$B_{1}=B_{2}>0,$ $B_{3}=B_{4}>0$ and $\\|B_{1}\\|>\\|B_{4}\\|.$\n\n\\end{proposition}\n\n\\begin{proof}[Proof of Proposition~\\ref{theortetrcomp1}:]\nTaking into account theorem~\\ref{theortetr}, for  $B_{1}=B_{2}<0,$\n$B_{3}=B_{4}<0$ we derive:\n\n\n\n\\begin{equation}\\label{compl1}\n\\vec{B_{1}}+\\vec{B_{2}}+\\vec{B_{3}}+\\vec{B_{4}}=\\vec{0}\n\\end{equation}\n\nor\n\n\\begin{equation}\\label{compl2}\n(-\\vec{B_{1}})+(-\\vec{B_{2}})+(-\\vec{B_{3}})+(-\\vec{B_{4}})=\\vec{0}.\n\\end{equation}\n\nFrom (\\ref{compl1}) and (\\ref{compl2}), we derive that the\ncomplementary weighted Fermat-Torricelli point $A_{0}^{\\prime}$\ncoincides with the weighted Fermat-Torricelli point $A_{0}.$ We\nnote that the vectors $\\vec{B}_{i}$ may change direction from\n$A_{i}$ to $A_{0},$ simultaneously, for $i=1,2,3,4.$\n\n\\end{proof}\n\n\\begin{proposition}\\label{theortetrcomp2}\nThe location of the complementary weighted Fermat-Torricelli point\n$A_{0}^{\\prime}$ (solution of Problem~\\ref{sym2complementary}) of\nthe regular tetrahedron $A_{1}A_{2}A_{3}A_{4}$ for\n$B_{1}=B_{2}<0,$ $B_{3}=B_{4}>0$ or $B_{1}=B_{2}>0,$\n$B_{3}=B_{4}<0$ and $\\|B_{1}\\|>\\|B_{4}\\|$ is given by:\n\n\\begin{eqnarray}\\label{analsoltetrahedrcom}\n&&y=\\frac{\\sqrt{t}}{2}+\\nonumber\\\\\n&&{}+\\frac{1}{2} \\sqrt{\\frac{a^4 B_1^4}{4 s^{1\/3}}-\\frac{a^4 B_1^2\nB_4^2}{2 s^{1\/3}}+\\frac{a^4 B_4^4}{4 s^{1\/3}}-\\frac{2 \\left(-8\n\\sqrt{2} a^3 B_1^2-8 \\sqrt{2} a^3 B_4^2\\right)}{\\sqrt{t} \\left(64\nB_1^2-64 B_4^2\\right)}+\\frac{s^{1\/3}}{4 \\left(B_1^4-2 B_1^2\nB_4^2+B_4^4\\right)}}\\nonumber\\\\\n\\end{eqnarray}\n\n\n\n\\end{proposition}\n\n\\begin{proof}[Proof of Proposition~\\ref{theortetrcomp2}:]\n\nConsidering (\\ref{obj13}) for $B_{1}=B_{2}<0,$ $B_{3}=B_{4}>0$ or\n$B_{1}=B_{2}>0,$ $B_{3}=B_{4}<0$ and $\\|B_{1}\\|>\\|B_{4}\\|$ and\ndifferentiating (\\ref{obj13}) with respect to $y\\equiv\nOA_{0}^{\\prime},$ and by squaring both parts of the derived\nequation, we obtain (\\ref{fourth2}) which is a fourth order\nequation with respect to $y.$ The second real solution of $y$\ngives (\\ref{analsoltetrahedrcom}). Taking into account the real\nsolution (\\ref{analsoltetrahedrcom}) and the weighted floating\nequilibrium condition\n$\\vec{B_{1}}+\\vec{B_{2}}+\\vec{B_{3}}+\\vec{B_{4}}=\\vec{0}$ we\nobtain that the complementary weighted Fermat-Torricelli point\n$A_{0}^{\\prime}$ for $B_{1}=B_{2}<0,$ $B_{3}=B_{4}>0$ coincides\nwith the complementary weighted Fermat-Torricelli point\n$A_{0}^{\\prime\\prime}$ for $B_{1}=B_{2}>0,$ $B_{3}=B_{4}<0$ (\nFig.~\\ref{fig4} ). From (\\ref{analsoltetrahedrcom}), we derive\nthat the complementary $A_{0}^{\\prime}$ is located outside the\nregular tetrahedron $A_{1}A_{2}A_{3}A_{4}$ (Fig.~\\ref{fig4}).\n\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.2]{TetrahedronFT2new}\n\\caption{The complementary weighted Fermat-Torricelli point\n$A_{0}^{\\prime}$ of a regular tetrahedron $A_{1}A_{2}A_{3}A_{4}$\nfor $B_{1}=B_{2}>0$ and $B_{3}=B_{4}<0$ or $B_{1}=B_{2}<0$ and\n$B_{3}=B_{4}>0$  for $\\|B_{1}\\|>\\|B_{4}\\|$}\\label{fig4}\n\\end{figure}\n\n\n\n\n\\end{proof}\n\n\\begin{example}\\label{tetr1}\nGiven a regular tetrahedron $A_{1}A_{2}A_{3}A_{4}$ in\n$\\mathbb{R}^{3},$ $a=1, B_{1}=B_{2}=2.5,$ $B_{3}=B_{4}=1$ from\n(\\ref{analsolrtetrahedron}) and (\\ref{analsoltetrahedrcom}) we get\n$y=0.198358$ and $y=0.539791,$ respectively, with six digit\nprecision. The weighted Fermat-Torricelli point $A_{0}$ and the\ncomplementary weighted Fermat-Torricelli point\n$A_{0^{\\prime}}\\equiv A_{0}$ for $B_{1}=B_{2}=-2.5$ and\n$B_{3}=B_{4}=-1$ corresponds to $y=0.198358.$ The complementary\nweighted Fermat-Torricelli point $A_{0}^{\\prime}$ for\n$B_{1}=B_{2}=-2.5$ and $B_{3}=B_{4}=1$ or $B_{1}=B_{2}=1.5$ and\n$B_{3}=B_{4}=-1$ lies outside the regular tetrahedron\n$A_{1}A_{2}A_{3}A_{4}$ and corresponds to\n$y=0.539791>\\frac{A_{12}A_{34}}{2}=\\frac{\\sqrt{2}}{4}.$\n\\end{example}\n\n\n\n\nWe proceed by calculating the angles $\\alpha_{i0j},$ for\n$i,j=0,1,2,3,4.$\n\n\\begin{proposition}\\label{anglestetragon}\nThe angles $\\alpha_{i0j},$ for $i,j=0,1,2,3,4,$ are given by:\n\n\n\\begin{equation}\\label{alpha102}\n\\alpha_{102}=\\arccos{\\left(1-\\frac{a^2}{2\\left(\\left(\\frac{a}{2}\\right)^{2}+\\left(\\frac{\\frac{a\\sqrt{2}}{2}}{2}-y\\right)^{2}\\right)}\\right)},\n\\end{equation}\n\n\\begin{equation}\\label{alpha304}\n\\alpha_{304}=\\arccos{\\left(1-\\frac{a^2}{2\\left(\\left(\\frac{a}{2}\\right)^{2}+\\left(\\frac{\\frac{a\\sqrt{2}}{2}}{2}+y\\right)^{2}\\right)}\\right)},\n\\end{equation}\n\nand\n\n\\begin{equation}\\label{alpha401}\n\\alpha_{104}=\\alpha_{203}=\\alpha_{103}=\\alpha_{204}=\\arccos{\\frac{\\left(\\frac{\\frac{a\\sqrt{2}}{2}}{2}-y\\right)^{2}+\\left(\\frac{\\frac{a\\sqrt{2}}{2}}{2}-y\\right)^{2}-\\frac{a^{2}}{2}}{2\n\\sqrt{\\left(\\frac{a}{2}\\right)^{2}+\\left(\\frac{\\frac{a\\sqrt{2}}{2}}{2}-y\\right)^{2}}\n\\sqrt{\\left(\\frac{a}{2}\\right)^{2}+\\left(\\frac{\\frac{a\\sqrt{2}}{2}}{2}+y\\right)^{2}}}}.\n\\end{equation}\n\n\n\n\n\\end{proposition}\n\n\\begin{proof}[Proof of Proposition~\\ref{anglestetragon}:]\n\nTaking into account the cosine law in $\\triangle A_{1}A_{0}A_{2},$\n$\\triangle A_{3}A_{0}A_{4},$ $\\triangle A_{1}A_{0}A_{4},$\n$\\triangle A_{2}A_{0}A_{4},$ $\\triangle A_{1}A_{0}A_{3},$\n$\\triangle A_{2}A_{0}A_{4},$ and (\\ref{analsolrtetrahedron}), we\nobtain (\\ref{alpha102}), (\\ref{alpha304}) and (\\ref{alpha401}),\nrespectively.\n\n\\end{proof}\n\n\\begin{corollary}{\\cite[Theorem~4.3, p.~102]{NSM:91}}\\label{regular}\nIf $B_{1}=B_{2}=B_{3}=B_{4},$ then\n\\begin{equation}\\label{regultetrsol}\n\\alpha_{i0j}=\\arccos{\\left(-\\frac{1}{3}\\right)},\n\\end{equation}\nfor $i,j=1,2,3,4$ and $i\\ne j.$\n\\end{corollary}\n\n\\begin{proof}\nBy setting $y=0$ in (\\ref{alpha102}), (\\ref{alpha304}) and\n(\\ref{alpha401}), we obtain (\\ref{regultetrsol}).\n\\end{proof}\n\n\n\n\\section{The weighted Fermat-Torricelli problem for tetrahedra in the three dimensional Euclidean Space: The case  $B_{1}=B_{2}$ and $B_{3}=B_{4}.$ }\n\nWe consider the following lemma which gives the invariance\nproperty (geometric plasticity) of the weighted Fermat-Torricelli\npoint for a given tetrahedron\n$A_{1}^{\\prime}A_{2}^{\\prime}A_{3}^{\\prime}A_{4}^{\\prime}$ in\n$\\mathbb{R}^{3}$ (\\cite[Appendix~AII,pp.~851-853]{ZachosZu:11})\n\n\n\n\n\\begin{lemma}{\\cite[Appendix~AII,pp.~851-853]{ZachosZu:11}}\\label{tetragonnntetrah}\nLet $A_1A_2A_{3}A_4$ be a regular tetrahedron in $\\mathbb{R}^{3}$\nand each vertex $A_{i}$ has a non-negative weight $B_{i}$ for\n$i=1,2,3,4.$ Assume that the floating case of the weighted\nFermat-Torricelli point $A_{0}$ occurs:\n\\begin{equation}\\label{floatingcasetetr1}\n\\|{\\sum_{j=1, i\\ne j}^{4}B_{j}\\vec u(A_i,A_j)}\\|>B_i.\n\\end{equation}\nIf $A_0$ is connected with every vertex $A_i$ for $i=1,2,3,4$ and\na point $A_{i}^{\\prime}$ is selected with corresponding\nnon-negative weight $B_{i}$ on the ray that is defined by the line\nsegment $A_0A_i$ and the tetrahedron\n$A_{1}^{\\prime}A_{2}^{\\prime}A_{3}^{\\prime}A_{4}^{\\prime}$ is\nconstructed such that:\n\n\\begin{equation}\\label{floatingcasequad2}\n\\|{\\sum_{j=1, i\\ne j}^{4}B_{j}\\vec\nu(A_{i}^{\\prime},A_{j}^{\\prime})}\\|>B_i,\n\\end{equation}\nthen the weighted Fermat-Torricelli point $A_{0}^{\\prime}$ of\n$A_{1}^{\\prime}A_{2}^{\\prime}A_{3}^{\\prime}A_{4}^{\\prime}$ is\nidentical with $A_{0}.$\n\\end{lemma}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nWe consider a tetrahedron $A_{1}A_{2}A_{3}A_{4}^{\\prime}$ which\nhas as a base the equilateral triangle  $\\triangle\nA_{1}A_{2}A_{3}$ with side $a$ and the vertex $A_{4}^{\\prime}$ is\nlocated on the ray $A_{0}A_{4},$  with corresponding non-negative\nweights $B_{1}=B_{2}$ at the vertices $A_{1}, A_{2}$ and\n$B_{3}=B_{4}$ at the vertices $A_{3}, A_{4}^{\\prime}.$\n\nAssume that we choose  $B_{1}$ and $B_{4}$  non negative weights\nwhich satisfy the inequalities (\\ref{floatingcasetetr1}),\n(\\ref{floatingcasequad2}) and $B_{1}>B_{4},$ which correspond to\nthe weighted floating case of $A_{1}A_{2}A_{3}A_{4}$ and\n$A_{1}A_{2}A_{3}A_{4}^{\\prime}.$\n\n\n\nWe denote by $a_{i4^{\\prime}}$ the length of the linear segment\n$A_{i}A_{{4}^{\\prime}},$ the angle $\\angle\nA_{i}A_{k}A_{{4}^{\\prime}}$ for $i,j,0,1,2,3,4, i\\neq j,$ by\n$h_{0,12}$ the height of $\\triangle A_{0}A_{1}A_{2}$ from $A_{0}$\nto $A_{1}A_{2},$ by $\\alpha$ the dihedral angle between the planes\n$A_{0}A_{1}A_{2}$ and $A_{3}A_{1}A_{2}$ and by\n$\\alpha_{g_{4^{\\prime}}}$ the dihedral angle between the planes\n$A_{3}A_{1}A_{2}$ and $A_{4^{\\prime}}A_{1}A_{2}$ and by $A_{0}$\nthe corresponding weighted Fermat-Torricelli point of the regular\ntetrahedron $A_{1}A_{2}A_{3}A_{4}.$\n\n\n\n\\begin{theorem}\\label{theorquadnntetrah}\nThe location of the weighted Fermat-Torricelli point\n$A_{0^{\\prime}}$ of a tetrahedron $A_{1}A_{2}A_{3}A_{4}^{\\prime}$\nwhich has as a base the equilateral triangle  $\\triangle\nA_{1}A_{2}A_{3}$ with side $a$ and the vertex $A_{4}^{\\prime}$ is\nlocated on the ray $A_{0}A_{4}$ for $B_{1}=B_{2}$ and\n$B_{3}=B_{4},$ under the conditions (\\ref{floatingcasetetr1}),\n(\\ref{floatingcasequad2}) and $B_{1}>B_{4},$ is given by:\n\n\n\\begin{equation}\\label{a04prime}\na_{04^{\\prime}}=\\sqrt{a_{20}^{2}+a_{24^{\\prime}}^2-2\na_{24^{\\prime}}\\left(\\sqrt{a_{02}^2-h_{0,12}^2}\\cos\\alpha_{124^{\\prime}}+h_{0,12}\\sin\\alpha_{124^{\\prime}}\\cos(\\alpha_{g_{4^{\\prime}}}-\\alpha)\\right)}\n\\end{equation}\n\nwhere\n\n\\begin{equation}\\label{alpha}\n\\alpha=\\arccos{\\frac{\\frac{a_{02}^2+a_{23}^2-a_{03}^2}{2\na_{23}}-\\sqrt{a_{2}^2-h_{0,12}^2}\\cos\\alpha_{123}}{h_{0,12}\\sin\\alpha_{123}}}\n\\end{equation}\n\nand\n\n\\begin{equation}\\label{height012}\nh_{0,12}=\\sqrt{\\frac{4a_{01}^{2}a_{02}^2-(a_{01}^2+a_{02}^2-a_{12}^2)^2}{4\na_{12}^2}}.\n\\end{equation}\n\n\n\\end{theorem}\n\n\n\\begin{proof}[Proof of Theorem~\\ref{theorquadnntetrah}:]\n\nFrom lemma~\\ref{tetragonnntetrah}, we get $A_{0^{\\prime}}\\equiv\nA_{0}.$ Therefore, we get the relations (\\ref{a04prime}) and\n(\\ref{alpha}) from a generalization of the cosine law in\n$\\mathbb{R}^{3}$ which has been introduced for tetrahedra in\n\\cite[Solution of Problem~1, Formulas (2.14) and\n(2.20),p.~116]{Zachos\/Zou:09}.\n\n\\end{proof}\n\n\\begin{remark}\nWe may consider a tetrahedron\n$A_{1}A_{2}A_{3^{\\prime}}A_{4^{\\prime}}$ by placing\n$A_{3^{\\prime}}$ on the ray defined by $A_{0}A_{3}$ and\n$A_{0^{\\prime \\prime}}$ is the corresponding weighted\nFermat-Torricelli point. Taking into account\nlemma~\\ref{tetragonnntetrah}, we get $A_{0^{\\prime\\prime}}\\equiv\nA_{0^{\\prime}}\\equiv A_{0}.$\n\\end{remark}\n\n\n\n\n\n\n\n\nThe author is sincerely grateful to Professor Dr. Vassilios G.\nPapageorgiou for his very valuable comments, many fruitful\ndiscussions and for bringing my attention to this particular\nproblem.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe word cosmology stems from the Greek words $\\kappa{}\\acute{o}{}\\sigma{}\\!\\mu{}o\\zeta$ (meaning ``order'', as opposed to chaos, or ``world'') and $\\lambda{}\\acute{o}{}\\gamma{}o{}\\zeta$ (which means ``word'', or ``discourse''), and cosmology is thus the ancient\nstudy of the universe as an ordered whole. The discipline was revolutionised in the twentieth century with the advent of Einstein's general theory of relativity, which had a dramatic impact on our understanding of the evolution of the universe. Where previously it was assumed that matter and energy interact against the static backdrop of space and time, it has now became apparent that space and time are intertwined, and that the expansion of the universe does not take place in ``spacetime'', it actually \\emph{creates} spacetime.\n\nFrom this point of view, it is perhaps unsurprising that fundamentally new processes can take place in the context of general relativity. We will, here, analyze the physics of black holes and wormholes, and their relationship to the proposed generation of ``baby universes'' (also referred to in the literature as ``child universes'') --- regions of spacetime that are initially connected to our own universe, but which causally disconnect from us and inflate, becoming self-contained cosmoses.  In the 1980s, it was proposed that such baby universes could, in theory, be manufactured in a future particle accelerator and that they would appear from the outside to be miniature black holes~\\cite{bib:2017BigBanLitRooMer}. More recently, it has been suggested that baby universes may form spontaneously in our universe and that, if this process occurs, it may have implications for the development of a theory of quantum gravity, unifying general relativity with the physics of the micro-realm, quantum mechanics~(see, e.g.,~\\cite{bib:1993NucPhy__B305208Amb}).\n\nIn what follows, we will first discuss how black holes are tightly connected with the causal structure of spacetime (section~\\ref{sec:bhw}). We will introduce the concepts of the chronological or causal past and future of a subset of events in spacetime, and, then, focus our attention on the basic elements that can be used to describe the asymptotic structure of spacetime, i.e. the spacetime structure at infinity (both, in space and time). These concepts are central to a rigorous definition of a black hole; moreover, they can be naturally related to an intuitive understanding of these objects. We shall also define the related concept of a wormhole -- a tunnel linking two regions of spacetime -- which plays an integral role in baby-universe formation.\n\nIn section~\\ref{sec:inf}, we will outline the motivation for the theory of inflation, which forms the current cosmological paradigm, and also provides the mechanism for baby-universe generation. We will then explain how inflation theory gives rise to the baby-universe proposal, in section~\\ref{sec:buf}. (This process is also known, in the literature, as spacetime tunneling with black hole\/wormhole creation.) We will provide an essential account of the most relevant physical aspects of the process, with an emphasis on the relationship with the physics of the vacuum.\n\nSection~\\ref{sec:con} will, finally, contain a short summary of the main ideas discussed in this contribution, and a qualitative description of some open problems in the field, including: the feasibility of making a baby universe in a real-world particle accelerator; proposed signatures of lab-made universes; and the possibility of baby universes arising spontaneously in the vacuum (and the implications of such a process for quantum gravity).\n\n\n\\section{\\label{sec:bhw}Black Holes and Wormholes}\n\nThe conception of the idea of a black hole -- as a region from which\nnot even light can escape -- perhaps surprisingly, predates general relativity. In 1784, the English clergyman and natural philosopher John Michell, considering Newton's corpuscular theory of light,\npondered whether `light particles' may interact with gravity and would thus have an associated escape velocity. (The escape velocity is the speed that is sufficient for a body to be able to move arbitrarily far away from another body, that, for simplicity, we can consider much more massive. For instance, the escape velocity from the Earth is about $11\\;\\mathrm{km}\/\\mathrm{sec}$, while the escape velocity from the Sun is about $620\\;\\mathrm{km}\/\\mathrm{sec}$.)\n\nMichell calculated that\n\\begin{quote}\n\t``\\dots if the semi-diameter of a sph\\ae{}re of the same density with the sun were to exceed that of the sun in the proportion of $500$ to $1$, a body falling from an infinite height towards it, would have acquired at its surface a greater velocity than that of light, and consequently, supposing light to be attracted by the same force in proportion to its vis inerti\\ae{}, with other bodies, all light emitted from such a body would be made to return towards it, by its own proper gravity \\dots''~\\cite{bib:1784PhiTraRoySoc_74_35Mic}\n\\end{quote}\n\nPierre-Simon Laplace also independently developed a notion of Newtonian black holes, or ``dark stars''~\\cite{bib:1799AllGeoEph__4__1Lap}. Note that, in this historical context, there was no modern conception of spacetime, only Newtonian space plus time. There was also no bound to the speed of propagation of interactions: action at a distance allows for the physical effects of perturbations to propagate instantaneously across all space. The idea of a massive body that can trap light is thus not tied in any way to a dynamical spacetime.\n\nThis situation, of course, changes drastically with the general theory of relativity; although it took about half a century after the theory was proposed for the discovery of what we, nowadays, consider the first mathematically rigorous realization of a black hole (the Schwarzschild solution)~\\cite{bib:1916SitKonPreAkaWisBer189}, to provide a clear understanding of spacetime structure related to black holes. The reason for this long process (please, see~\\cite{bib:1987ThrHunYeaGra199Isr} for a detailed account of it), can be substantially traced back to the fact that it is standard, in pre-relativistic physics, to associate a metric interpretation to the values of coordinates, and associate, in this way, a direct physical meaning to the values of tensor components. This approach is, however, very dangerous (and conceptually wrong) in general relativity. The gradual recognition of this danger went together with the development of different coordinate systems that progressively clarified distinctive properties of the Schwarzschild solution; this fact is also witnessed by the change of the name of these objects to ``black holes''.\n\n\\begin{figure}\n\t\\begin{center}\n\t\\includegraphics[width=9cm]{fig_ChronologicalCausal}\n\t\\caption{\\label{fig:crocaupasfut}In this figure we exemplify some of the concepts described in the main text. For graphical convenience, we chose a coordinate system, in which the light cones have the same shape as in special relativity, when the speed of light is set to $1$. The points \\texttt{E}$_{1}$, \\texttt{E}$_{2}$, and \\texttt{E}$_{3}$ are events in spacetime. The curve $\\gamma _{3 \\to 2}$ is an everywhere timelike curve (at every point the vector tangent to the curve falls inside the light cone). The arrows show the \\emph{past}$\\;\\to\\;$\\emph{future} direction, i.e., this curve is future directed. Since, it connects the event \\texttt{E}$_{3}$ with the event \\texttt{E}$_{2}$, according to the definition, \\texttt{E}$_{2}$ is in the chronological future of \\texttt{E}$_{3}$. On the curve $\\gamma _{2 \\to 1}$ the arrows point, instead, in the \\emph{future}$\\;\\to\\;$\\emph{past} direction. The curve is a causal curve: in particular it is nowhere timelike, and everywhere lightlike (at every point the tangent vector to the curve falls exactly on the light cone). According to the definitions given in the main text, \\texttt{E}$_{1}$ is in the causal past of \\texttt{E}$_{2}$. Intuitively, we can understand that there can be no past directed timelike curves connecting \\texttt{E}$_{2}$ to \\texttt{E}$_{1}$: thus \\texttt{E}$_{1}$ is not in the chronological past of \\texttt{E}$_{2}$.}\n\t\\end{center}\n\\end{figure}\n\nIn what follows, we will qualitatively discuss an intuitive definition of a black hole, in terms of the causal structure of spacetime, following the approach that can be found in the classical book by Hawking and Ellis~\\cite{bib:1975CamUniPreHawEll}. Several ingredients are central in this understanding, and will be introduced below. One of them is the identification of events in a given spacetime ${\\mathcal{M}}$ that can be affected by another event, or by a subset of them. This is what goes under the name of the \\emph{chronological} (resp., \\emph{causal}) \\emph{future} of a set ${\\mathcal{U}} \\subset {\\mathcal{M}}$ of events: this is the set of all events that can be connected by a \\emph{future directed timelike} (resp., \\emph{non spacelike}) curve starting from a given event in the set. Thus, if an event \\texttt{q} can be reached, starting from a given event \\texttt{p}, by moving along a future directed curve that at every point remains inside (resp., not outside) the light cone, then we say that \\texttt{q} is in the chronological (resp., causal) future of \\texttt{p}. Of course, analogous characterizations can be given for the \\emph{chronological} (resp., \\emph{causal}) \\emph{past}, and we will indicate the causal past of the set ${\\mathcal{U}} \\subset {\\mathcal{M}}$ by $J ^{-} ( {\\mathcal{U}} )$.\n\nA second ingredient is the understanding that, given an event \\texttt{p} in spacetime, it could be possible to find one (or more) subsets ${\\mathcal{A}} \\subset {\\mathcal{M}}$ of events to the past of \\texttt{p}, such that the knowledge of what happens at these events is enough to completely determine what happens at \\texttt{p}. In this case, the event \\texttt{p} is in what is called the \\emph{future domain of dependence of} ${\\mathcal{A}}$. Again, a corresponding definition can be given for the \\emph{past domain of dependence} of \\texttt{p}, or, in general, of a larger set of events. It is thanks to concepts like the ones briefly outlined above, that we are able to rigorously speak of the causal relationships among events in spacetime.\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[height=6.0cm]{fig_MinkowskiNonCompact}\n\t\t\\hskip 1.9cm\n\t\t\\includegraphics[height=6.0cm]{fig_MinkowskiCompact}\n\t\t\\caption{\\label{fig:infstr}On the left we show the various types of infinity in Minkowski spacetime. The light cone structure identifies timelike, spacelike, and null vectors at every event. Starting from every event, we can remain at the same spatial position and move (dotted lines) to infinity in time toward the future (reaching $i ^{+}$, timelike future infinity) or toward the past (timelike past infinity, $i ^{-}$). Or, we can consider, at fixed time, points far away in space, reaching spacelike infinity, $i _{0}$. Another possibility is to move at infinity, both, in space and time, at the speed of light: toward the future we, then, reach future null infinity, ${\\mathscr{I}} ^{+}$, while toward the past we reach past null infinity, ${\\mathscr{I}} ^{+}$. It is common to perform a suitable (conformal) transportation to obtain a compact version (Penrose diagram) of (half of) the diagram shown on the left. This is shown on the right, where the notation is the same used on the left; in addition, the fact that spacetime is empty\/flat at infinity is explicitly indicated by the double diagonal lines.}\n\t\\end{center}\n\\end{figure}\n\nThere is, however, another fundamental aspect in our intuitive understanding of a black hole, and it is the fact that, in some sense, it has the possibility to trap us. To make this concept rigorous is more subtle, because, conceptually, a \\emph{test} body that can ``only'' move several light years away from a given point in space (given enough time, of course) is as trapped to this point as a body that can just move a few millimeters away. This is why the concept of being trapped inside some region is more conveniently expressed as the \\emph{impossibility to move as far away as we want}, or, in other words, to move \\emph{infinitely far away}. We thus realize that, at least at the very first stage, it might be necessary to consider the asymptotic structure of spacetime to be able to properly understand the concept of a trapping region as a global property of spacetime. A first, convenient, assumption, is to imagine spacetimes that have a structure at large distances similar to the one of Minkowski spacetime, which we are familiar with.~\\footnote{It is possible to generalize these ideas, but we do not really need such generalizations here.} We, then, come back to the unified understanding of spacetime given to us by special relativity. Indeed, several ways to go to \\emph{infinity} exist in Minkowski spacetime. We can consider events as far away as possible from a given point, while keeping the time fixed: this boundary of spacetime is called \\emph{spacelike infinity}, $i _{0}$. Or, we could wait an infinite time, while sitting at a fixed point in space: this is also a boundary of spacetime, which is called \\emph{future timelike infinity}, $i ^{+}$.~\\footnote{A corresponding definition can be given, of course, for \\emph{past timelike infinity.}, $i ^{-}$.} And, finally, we could keep moving away from a given event at the speed of light, reaching what is called \\emph{future null infinity}, ${\\mathscr{I}} ^{+}${}.~\\footnote{Again \\emph{past null infinity}, ${\\mathscr{I}} ^{-}$ is defined in a similar way.}\n\nNow, let us assume that we can describe the whole spacetime ${\\mathcal{M}}$ as a continuous sequence of spacelike hypersurfaces ${\\mathcal{S}} (\\tau)$, indexed by the (continuous, real) parameter $\\tau$: ${\\mathcal{M}} = \\cup _{\\tau} {\\mathcal{S}} (\\tau)$. This description does not need to be unique (and, indeed, it is \\emph{not}!), so we can just pick one of the many that are possible. The parameter $\\tau$ can be interpreted as time, and ${\\mathcal{S}} (\\tau)$ as a constant-time (hyper-)surface. Let us also assume that the spacetime ${\\mathcal{M}}$ has one or more regions with the asymptotic structure defined above. Let us now consider one ${\\mathcal{S}} (\\tau)$. We, then, consider the set of all events that are in the causal past of future null infinity, which according to our previous notation is written as $J ^{-} ({\\mathscr{I}} ^{+})$. We call ${\\mathcal{B}} (\\tau)$ what remains after removing this last set of events from ${\\mathcal{S}} (\\tau)$:\n\\[\n\t{\\mathcal{B}} (\\tau) = {\\mathcal{S}} (\\tau) - J ^{-} ({\\mathscr{I}} ^{+})\n\t\\ .\n\\]\nBy definition, an event \\texttt{b} inside ${\\mathcal{B}} (\\tau)$ cannot be connected by any causal curve to ${\\mathscr{I}} ^{+}$: if such a curve would exist, \\texttt{b} would be inside $J ^{-} ({\\mathscr{I}} ^{+})$, but we have obtained ${\\mathcal{B}} (\\tau)$ exactly by removing such events from ${\\mathcal{S}} (\\tau)$. If it is not empty, ${\\mathcal{B}} (\\tau)$ realizes, at time $\\tau$, a subset of events that cannot send any test signal to future null infinity, and (a connected component in) it is called a \\emph{black hole} on ${\\mathcal{S}} (\\tau)$: it certainly realizes our original intuition! Then, the \\emph{event horizon} can be defined as the boundary of $J ^{-} ({\\mathscr{I}} ^{+})$, i.e. a set of points that can just barely send a light signal to ${\\mathscr{I}} ^{+}$.\n\nOf course, the sets ${\\mathcal{B}} (\\tau)$ and the boundary of $J ^{-} ({\\mathscr{I}} ^{+})$ could be empty sets. However, the Schwarzschild solution is a realization of a spacetime, in which they are not, and it represents the earliest example of a black hole spacetime. We also notice that the Schwarzschild solution contains two asymptotic regions that have a structure similar to the structure at the infinities of Minkowski spacetime.~\\footnote{In general, spacetimes with more than two such regions can also exist, and there are also spacetimes with an infinity of such regions.}\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=11cm]{fig_SchwarzschildCompact}\n\t\t\\caption{\\label{fig:Schsol}Representation of the Schwarzschild solution in terms of a Penrose diagram. The notation is the same as in figure~\\ref{fig:infstr}. We first note that the Schwarzschild spacetime has \\emph{two} asymptotic regions (labels I and II in the figure) with properties similar to the asymptotic structure of Minkowski spacetime (cf. the Penrose diagram in figure~\\ref{fig:infstr}). These regions are causally disconnected, as there is no causal curve starting inside in one of them, and ending in the other. We now focus on region I, which is called the \\emph{black hole} region. If we consider the causal past of the future null infinity in region I, $J ^{-} ({\\mathscr{I}} ^{+})$, we see that it consists of regions I and IV. Region II is thus not in $J ^{-} ({\\mathscr{I}} ^{+})$. This is also true if we consider the causal past of the future null infinity in region III. Moreover, for every event in II it is true that even the fastest light signal will reach the \\emph{zig-zagged} line on the top, which is the Schwarzschild singularity. For some events, the corresponding light cones are drawn together with curves representing events with the same time of the event at the center of the light cone but at a different place (dashed lines), events at the same place but at different times (dotted curves), and light rays directed toward the past and the future (dotted lines). Note that in regions I and III (the static regions), curves of constant time are spacelike (i.e., they flow in the left$\\;\\leftrightarrow\\;$right direction), while curves of constant position are timelike (i.e., they flow in the bottom$\\;\\to\\;$top direction). However, inside the black hole region, the opposite occurs, and the singularity is spacelike. The presence of two causally disconnected regions with the same asymptotic structure of Minkowski (I and III in the picture) is at the heart of the idea of wormholes (see the main text for details).}\n\t\\end{center}\n\\end{figure}\nThis general definition of a black hole, in terms of causal structure, has a concrete realization in solutions to the Einstein equations that can be written in closed form. The first of these solutions was found by Karl Schwarzschild as early as 1915, and it is the first, and typical, example of a black hole. If we consider the form of the metric in the coordinates $(t,r,\\theta,\\varphi)$ (Schwarzschild coordinates,~\\footnote{These coordinates are adapted to the symmetries that characterize Schwarzschild spacetime, which is static (outside the event horizon), and spherically symmetric (consistently, the meaning of the $\\theta$ and $\\varphi$ angles is the same as for polar coordinates in flat spacetime). The \\emph{radial coordinate}, $r$, is called the \\emph{circumferential radius}, because circles with constant $r$ have a length equal to $2 \\pi r$; note, however, that the distance between one point on such a circle, and the center, is \\emph{not} $r$. In a curved spacetime, it is not possible, in general, to find a radial coordinate that satisfies both the properties considered above, and this shows how important it is to consider the precise definition of coordinates, without relying on their names only.} $G$ is Newton's gravitational constant and $c$ is the speed of light)\n\\[\n\td s ^{2}\n\t=\n\t-\n\t\\left( 1 - \\frac{2 G M}{c ^{2} r} \\right) d t ^{2}\n\t+\n\t\\left( 1 - \\frac{2 G M}{c ^{2} r} \\right) ^{-1} d r ^{2}\n\t+\n\tr ^{2} \\left( d \\theta ^{2} + \\sin ^{2} \\theta d \\varphi ^{2} \\right)\n\t,\n\\]\nwe recognize that the above expression has two problematic regions, corresponding to the values $r = 0$ and $r = 2 G M \/ c ^{2}$. Closer and closer to $r = 0$ the spacetime curvature takes larger and larger values, and this point is a physical singularity of spacetime. On the other hand $r = 2 G M \/ c ^{2}$, the gravitational radius, is not such a critical point. For instance, if we change coordinates, we can write the metric in other forms, where the problem at the gravitational radius disappears; however, the one at $r = 0$ always remains. All points identified by a radial coordinate equal to the gravitational radius form the ``event horizon'' of the Schwarzschild black hole. This region behaves as a one-way membrane, that can be crossed when falling toward the singularity at $r = 0$, but that cannot be crossed in the opposite direction. After crossing the event horizon moving to decreasing values of the radial coordinate $r$, the observer cannot reach (nor send any signal) to the outside anymore, and he\/she is forced to keep moving toward $r = 0$.\n\nIn our universe, black holes are realized as stellar mass and supermassive black holes.\nStellar-mass black holes are formed by the gravitational collapse of stars whose masses exceed a certain limit. When a star reaches the end of its life and is no longer able to stop the inward pull of gravity using the outward pressure generated by nuclear fusion, different objects can be formed. White dwarfs, for example, are stellar remnants formed from the gravitational collapse of lower mass stars,\nin which the outward pressure of a degenerate gas of electrons counteracts the pull of gravity. For a non-rotating object, the maximum mass is given by the Chandrasekhar limit, which is about $1.44$ solar masses.\nFor bigger masses of the progenitors, however, the pull of gravity is too strong and gravitational collapse continues. It could possibly be stopped when the density reaches nuclear densities; for masses up to a few solar masses, a neutron star is then formed. For even more massive objects, it is expected that gravitational collapse cannot be stopped, and a stellar mass black hole is then formed.\n\nBlack holes have been proposed to exist in a wide range of masses, from micro black holes that could have been formed as primordial black holes during the early stages of evolution of the universe~\\cite{bib:1971MonNotRoyAstSoc152_75Haw,bib:1981ProThePhy_651443Sat} (or, indeed, may be generated in a particle accelerator due to exotic physics effects~\\cite{bib:2017BigBanLitRooMer})\nto supermassive black holes of billions of solar masses, such as the ones expected to be present at the center of galaxies. \n\nAnother exciting possibility for the spacetime structure, which is suggested by the presence of more than one asymptotic region, are \\emph{wormholes}. Technically, wormholes are allowed by the fact that the Einstein equations, locally, fix the geometry of spacetime, but they do not fix spacetime topology. Following Fuller and Wheeler~\\cite{bib:1962PhyRev128919Ful}, without changing the geometry of spacetime, it is then possible to imagine the two, identical, asymptotic regions that we considered above, as joined by a throat, also called an \\emph{Einstein-Rosen bridge}, or a \\emph{Schwarzschild wormhole}: we obtain, in this way, a multiply connected spacetime. After this seminal work, the idea of a wormhole was revived about a quarter of a century later by Morris and Thorne~\\cite{bib:1988AmeJouPhy_56395Mor}, who put forward the idea of traversable wormholes. The physics of wormholes then flourished, encouraging both the consideration of conceptually challenging realizations (such as, time machines) and technical analysis (especially related to their stability).\nThis will turn out to be a key ingredient for the idea of a baby universe, as we will discuss in the following section~\\ref{sec:buf}.\n\nAlthough wormholes are also allowed by general relativity, as opposed to black holes there is no evidence that they exist. It has been noted that for a wormhole to be traversable -- that is, allowing the passage of particles (or observers) from one region of spacetime to another -- they would need to be propped open and stabilized by exotic matter, with negative energy~\\cite{bib:1989PhyRev__D_39318Vis,bib:1999PhyLet__B260175Ida}. It has also been proposed that (some) black holes may serve as gateways to other universes~\\cite{bib:2016JouCosAstPhy160060Gar,bib:1999TheLifCosSmo}, and in section~\\ref{sec:con} we will discuss proposed signatures of such scenarios, in the context of proposed signatures of baby-universe formation in the lab.\n\nIn the following section we shall outline inflation theory, which sets the current cosmological paradigm for the evolution of the early universe. It is the consideration of inflationary theory that led to one of the aforementioned recent proposals that cosmic black holes may house other universes~\\cite{bib:2016JouCosAstPhy160060Gar}; and it was the development of inflation theory that led to the first proposals that a baby universe could be created in the laboratory~\\cite{bib:1990NucPhy__B339417Fah}, sequestered within a micro black hole, in the 1980s~\\cite{bib:1987PhyRev__D_35174Bla} (section~\\ref{sec:buf}).\n\n\\section{\\label{sec:inf}Inflation Theory as Precursor to Baby-Universe Formation}\n\nThe inflationary paradigm posits that the universe underwent a period of exponential expansion between approximately $10 ^{-36}$ and $10 ^{-32}$ seconds  after the big bang singularity. Below we outline the motivation for the theory, proposed in the 1980s, which also underpins the mechanism of baby-universe formation.\n\nThe fundamental idea behind inflation (and any contemporary cosmological model) is the recognition, from the general theory of relativity, that the universe is not a static entity, but has a complex and rich dynamics. This was cemented by the observations that led to the formulation of Hubble's law, according to which there is a clear linear correlation between our distance from other galaxies, and the velocity at which they are receding from us: the larger the distance, the larger, on average, the recession speed. It was then natural to imagine that, if other galaxies are now receding from us, in the past they should have been closer to us. Thus there could then be an event in the past, where the content of all galaxies in the universe could have been squeezed inside a very small region, where the expansion began. This special event in the history of the universe is now called the big bang singularity (or simply the big bang), and, following general relativity, it singles out not only the birth of our universe, but also of space and time themselves.\n\nAnother two crucial observations about our universe are the following:\n\\begin{enumerate}\n\t\\item our universe looks about the same in every direction we look at: this is usually called \\emph{isotropy};\n\t\\item the universe appears to be the same around every point, i.e. no point seems to have special properties: this is usually called \\emph{homogeneity}.\n\\end{enumerate}\nThey are the defining properties of the \\emph{cosmological principle}. (Of course, if we look at the night sky, we can clearly discern differences between different regions of the sky, and the above two statements seem inconsistent with our visual experience. \nWhat they technically mean is that, by choosing a large enough scale~\\footnote{Say, of the order of $100\\,\\mathrm{Mpc}$, where $1\\,\\mathrm{Mpc} \\approx 3.1 \\cdot 10 ^{19} \\mathrm{km}$.}, the properties of the universe do satisfy homogeneity and isotropy when averaged over such a scale.) These properties of the universe are surprising because there is, in principle, no reason for places in the universe that are very far away from each other to share the same physical properties. On the other hand, as shown by the observations of the cosmic microwave background (CMB), anisotropies and inhomogeneities that we cannot account for are a very tiny effect. One possible explanation for the similarities we see across the sky is \\emph{inflation}. According to inflation, the universe underwent a phase of strongly accelerated expansion almost immediately after the big bang. This expansion separated away regions that were in causal contact at the time, and, even after being separated, these regions kept the \\emph{imprinting} of their (causally connected) early evolution. Inflation, then, resulted in the sudden creation of a very large volume of spacetime, as the universe grew from an initial tiny size, to something that then expanded into the vast collection of superstructures that we see today. Since inflation's development, numerous observations have found evidence corroborating predictions made by the paradigm~\\cite{bib:2013PhyLet__B733112Gut}.\n\nInflation can be naturally related to the description of the fundamental interactions that comes from quantum field theory, and, in particular, with the physics of the quantum vacuum.\n\nIn field theory it is possible to consider field configurations that minimize the potential energy of the field. Since a field realizing one of these configurations has vanishing kinetic energy, then it is in a local minimum of its total energy.\n\nClassically such a configuration is called an \\emph{equilibrium configuration}: for small enough perturbations of an equilibrium configuration the field can acquire a small kinetic energy, but this will not be enough to move far away from the equilibrium configuration. In an ideal setup, the field will perform small oscillations around this configuration, but, in practice, some dissipative effects are also present. Then, the field will thus gradually lose the additional energy that it gained when the equilibrium state was perturbed, and settle back in the configuration corresponding to the local minimum of the energy. Thus, a local minimum of the potential energy is, not only, an equilibrium configuration, but a \\emph{stable} one. In general, there can be many such configurations, possibly with different energies, and, classically, the system can realize in a stable way any such configurations.\n\n\\emph{Quantum mechanically}, however, the system has a non-zero probability to move to a lower energy minimum, even if a small perturbation does not provide it with enough energy to realize the same transition classically. This process is called \\emph{quantum tunneling} (and it is responsible for many relevant physical phenomena, including the ones at the heart of the operation of modern electronic devices, like transistors). We are here interested in the effect of quantum tunneling on the vacuum structure~\\cite{bib:1977PhyRev__D_15292Col,bib:1984NucPhy__B245481Rub,bib:1994PhyRev__D_40103Tan}, and, in particular, in the case when the vacuum structure interacts with gravity~\\cite{bib:1980PhyRev__D_21330Col,bib:1994PhyRev__D_50644Tan}. Indeed, in curved spacetime, vacuum energy can act as a source of the gravitational field, and interesting effects may appear.\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=9cm]{fig_vacuumPotentialTunnel}\n\t\t\\caption{\\label{fig:scafiepot}We show an example of a theory for two scalar fields, $\\phi ^{(1)}$ and $\\phi ^{(2)}$, with a potential possessing two local minima, $V _{\\mathrm{f}}$ and $V _{\\mathrm{t}}$ (with reference to the main text, here, $\\vec{\\phi} = (\\phi ^{(1)} , \\phi ^{(2)})$). The absolute minimum is $V _{\\mathrm{t}}$, and corresponds to the \\emph{true vacuum}, while $V _{\\mathrm{f}}$ is the \\emph{false vacuum}. If the field configuration corresponding to $V _{\\mathrm{f}}$ is realized, the system is deep inside the potential well, and classically stable. However, quantum mechanically there is a non-zero probability for the system to tunnel from the false vacuum configuration to the true vacuum. We remember that, in general, the fields depend on time and position $\\phi ^{(1,2)} = \\phi ^{(1,2)} ( \\vec{x} , t)$. Then, generically, there is no reason for this transition to affect, instantly, all space. It is more natural to expect a dynamical picture, in which the transition may take place at different times in different places: then, this would create different domains of the new vacuum state, evolving inside the old one.}\n\t\\end{center}\n\\end{figure}\nFollowing the work of Coleman and De Luccia, let us consider a model, in which the potential $V (\\vec{\\phi})$ for a collection of scalar fields $\\vec{\\phi}$ has the properties shown in figure~\\ref{fig:scafiepot}. The contribution of the potential to the action, according to general relativity, is of the form\n\\[\n\t\\int d ^{4} x \\sqrt{-g} V (\\vec{\\phi})\n\t\\ ,\n\\]\nwhere, $\\sqrt{-g}$ is the determinant of the spacetime metric, and, it is, in general, a non-trivial function of the coordinates. We then see that even a constant $V (\\vec{\\phi})$ contributes non-trivially, exactly because of the presence of $\\sqrt{-g}$. More than this, if we write the action of general relativity in the presence of a cosmological constant\n\\[\n\t\\int d ^{4} x \\sqrt{-g} \\{ R - 2 \\Lambda \\}\n\t,\n\\]\nwhere $R$ is a function describing the curvature of spacetime and $\\Lambda$ is a constant, we see that adding just a constant to $V (\\vec{\\phi})$ above, contributes exactly as a cosmological constant does in the last equation. This means that effects that could be considered as non-essential in the absence of gravity, can result in absolutely non-trivial physical effects in general relativity. One such consequence would have been to cause the early universe to inflate, expanding at an exponential rate. Another potential effect, the creation of a baby universe by triggering inflation\nwithin a patch of our current universe, is discussed below.\n\n\\section{\\label{sec:buf}Baby-Universe Formation}\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=11cm]{fig_babyUniverseFull}\n\t\t\\caption{\\label{fig:babunifor}The structure of wormhole spacetimes (visualized in the \\emph{background} by the throat connecting two asymptotically flat regions) is a key element in the baby universe formation process. In the superimposed panels, from the bottom-left to the top-right, we show the initial vacuum region (\\emph{first panel}), followed by the early time expansion before quantum tunneling (\\emph{second panel}); then, we visualize the configuration immediately after the quantum tunneling process (\\emph{third panel}, with the newly formed universe on the other side of a wormhole throat); finally, (\\emph{fourth panel}) expansion continues with the baby universe creating its own space, and causally disconnecting from the parent spacetime.}\n\t\\end{center}\n\\end{figure}\n\nBaby-universe formation is a process that combines all the features discussed above; i.e., it is the result of the quantum mechanical transition between inequivalent vacuum states, when we take into account the coupling with gravity, and the characteristic features of black hole\/wormhole spacetimes that we briefly discussed in section~\\ref{sec:bhw}. All the above ingredients are essential for a consistent description of the process, which, qualitatively, can proceed as follows:\n\\begin{enumerate}\n\t\\item[i.] In the current universe we consider a region, which does not realize the minimum energy (\\emph{true}) vacuum.\n\t\\item[ii.] Classically the region can be stable despite its raised energy, and is thus called a \\emph{false} vacuum.\n\tBut quantum mechanically the probability that the false vacuum decays to the true vacuum by quantum tunneling is non-zero, and this process will occur, sooner or later. The interesting point is that quantum tunneling can be realized in two different ways:\n\t\\begin{enumerate}\n\t\t\\item[iii.a] the region tunnels in the same asymptotic region that the starting false vacuum region was; then, the region expands at the expense of the false vacuum region;\n\t\t\\item[iii.b] the region tunnels into a different asymptotic region from the one inhabited by the starting false vacuum region; i.e., after the tunneling expansion takes place, the region that underwent tunneling evolves by creating its own spacetime, outside of the region from which it originated.\n\t\\end{enumerate}\n\\end{enumerate}\nIn the first scenario (steps i., ii., and iii. a), the parent universe is cannibalized by the new, more stable vacuum; the susceptibility of our own universe's vacuum to such a catastrophic fate has been investigated in light of Higgs data from the Large Hadron Collider (LHC), see for instance \\cite{bib:2017BigBanLitRooMer002}.\nBaby-universe formation, however, is the distinct process described by i., ii., and iii.b: as a result, the region that underwent tunneling expands into a completely new universe (the baby universe). Initially, this child region is connected to its parent spacetime via a wormhole tunnel. From the perspective of the parent universe, this wormhole gateway appears only as a minute black hole. As the baby universe inflates, however, this wormhole tunnel is pinched closed, causally disconnecting the baby from the original, parent, spacetime~\\cite{bib:1987PhyRev__D_35174Bla}.\nThis process can be intuitively related to our discussion of the causal structure of black hole spacetimes. Indeed, a standard tunneling process would correspond to a tunneling of the vacuum region that just makes it larger within the same parent spacetime. On the other hand, one can recognize that the tunneling process leading to the formation of a baby universe has the peculiar property to result in the production of an additional asymptotic region, i.e., tunneling also creates a new region of spacetime for the subsequent expansion of the baby universe outside of the parent spacetime~\\cite{bib:2015JouExpThePhy120460}.\n\nNote that the possibility for this baby-universe process to be realized strictly relies on the very peculiar properties of gravity described in terms of general relativity, i.e.\n\\begin{enumerate}\n\t\\item[A.] there are spacetime structures that can contain several (causally disconnected) asymptotic regions;\n\t\\item[B.] differences in vacuum energy can result in differences in the spacetime structure, because energy\/momentum density and their flows act as sources of the gravitational field, according to the Einstein equations;\n\t\\item[C.] spacetime can be (and is) created during cosmological evolution as the universe expands.\n\\end{enumerate}\nQuantum tunneling plays two significant roles in the process: Firstly, it allows the transition between different vacuum states, described above, which would otherwise be impossible in a non-quantum scenario. Secondly, it also makes it possible for a baby universe to be created starting from 'reasonable' initial conditions~\\cite{bib:1990NucPhy__B339417Fah}, as early recognized,\n\\begin{quote}\n\t``For inflation at a typical grand unified theory scale of $10 ^{14}\\:\\mathrm{GeV}$, for example, the universe emerges at $t \\approx 10 ^{-35}\\:\\mathrm{s}$ from a region of false vacuum with a radius of only $\\sim 10 ^{-24}\\:\\mathrm{cm}$ and a mass of only $\\sim 10\\:\\mathrm{kg}$.''\n\\end{quote}\nThis result clearly raises the questions of how feasible it is to make a baby universe in a real-world laboratory -- potentially in a future particle accelerator -- and whether it may be possible to detect its production. Some proposals will be discussed in the next section, along with the notion that baby universes could arise spontaneously within our universe.\n\n\\section{\\label{sec:con}Conclusion and Open Questions Regarding the Feasibility of Baby-Universe Production}\n\nIn this contribution, we have first considered black holes and wormholes in terms of the causal structure of spacetime. In particular, we have emphasized the richness and complexity of the nature of spacetime in general relativity. The consequences of this for the description of our universe can sometimes be fascinating and counterintuitive. In this context, the dynamics of spacetime naturally plays a central role, and we, qualitatively, reviewed such a role in the context of the physics of inflation,\nand its connection with vacuum decay. A unique realization of all these ideas is baby-universe formation, a process in which a new domain of spacetime formed by quantum tunneling in a pre-existing universe can evolve by creating its own space, and eventually (causally) disconnecting from the parent spacetime. We remarked how compelling it is to consider that the initial conditions for the creation of such a universe are such that it is not unreasonable to expect that they could be realized somewhere within our universe: in the literature, this possibility is known as creating a universe in the laboratory.\n\nWhen considering the possibility of making a baby universe in a laboratory within our universe, many issues need to be addressed. These include (but are not limited to):\n\\begin{enumerate}\n\t\\item Whether it is possible to find, or generate, a seed (a patch of starting false vacuum), which can quantum tunnel to create a baby cosmos, as outlined in step iii.b, in the section above.\n\t\\item Whether future (or even current) particle accelerators can reach the energies required to kick-start the inflationary process in that seed.\n\t\\item Whether it would be possible to detect a baby universe -- should one be created -- by distinguishing it from an ``ordinary'' mini black hole.\n\t\\item Whether baby universes are being spontaneously created in the vacuum -- and what physical implications this may have for theories of quantum gravity.\n\\end{enumerate}\nRegarding the question of potential seeds for baby-universe formation, it has been proposed that inflation could be triggered in a monopole (a particle hypothesized to exist by some grand unified theories)~\\cite{bib:1994PhyRev__D_50245Lin,bib:2006PhyRev__D_74024Sak}, if the monopole is subjected to high enough energies in a particle accelerator~\\cite{bib:2017BigBanLitRooMer}, or in certain exotic matter that has negative mass~\\cite{bib:1989PhyRevLet_63341Bar,bib:1990PhyRev__D_42262Har,bib:2008PhyRev__D_77125Gue} (which is hypothesized to exist in some versions of string theory). Such objects have yet to be found (although there are current searches underway to find monopoles at the LHC and elsewhere\\footnote{Indeed, the Monopole and Exotics Detector at the LHC (MoEDAL) collaboration is searching for a particle carrying magnetic charge by using an array of plastic nuclear-track detectors~\\cite{bib:2017PhyRevLet118061Ach}.}), which provides a significant obstacle to any attempts to carry out such a process in practice. It is also debatable whether next-generation particle\naccelerators could reach energies required to generate a baby universe; this may require certain predictions of string theory to hold true, enhancing the effect of gravity at the scale of elementary particles~\\cite{bib:2001PhyRevLet_87161Dim,bib:2002PhyRev__D_65056Gid,bib:2010Nat466426Mer}. At best, if these conditions hold, it has been claimed that planned accelerators, which may be built in coming decades, might reach the required energies~\\cite{bib:2017BigBanLitRooMer001}.\n\nEven assuming these hindrances are overcome, and a baby universe is generated in some future particle collider, it is not clear that physicists running the machine would be able to tell the difference between the production of a child cosmos and a standard micro black hole (predicted to exist by some string theory models~\\cite{bib:2017BigBanLitRooMer001,bib:2001PhyRevLet_87161Dim,bib:2002PhyRev__D_65056Gid}). Current exotic physics searches at the LHC focus on the Hawking radiation signature (in the form of a shower of particles predicted to emanate in all directions from decaying micro black holes). It has been suggested that the rate of emission of this radiation would differ between a mini black hole that contains a baby universe and one that does not~\\cite{bib:1991PhyRev__D_44333Lar}. Intriguingly, as mentioned in section~\\ref{sec:bhw}, it has also recently been proposed that some inflationary models predict the existence of parallel universes hidden within supermassive black holes -- and that indirect observational evidence for their presence may be acquired from the mass distribution of black holes in the universe~\\cite{bib:2017JouCosAstPhy170050Den}. In particular, it has recently been claimed that large mass, primordial black holes, might always contain a baby universe in their interior, because they would be formed by an expanding domain that cannot displace the exterior spacetime: in this case, a \\emph{no-go} theorem that would put an upper-limit on the black hole mass, would be avoided as the process does not take place in an asymptotically flat spacetime, but in a cosmological background \\cite{bib:2017JouCosAstPhy170050Den}.\n\nFinally, the possibility that baby universes could be spontaneously generated in the vacuum of today's universe has been investigated. It has been claimed that the high energy density excitations,\nresponsible for ultraviolet divergences in quantum field theories, including quantum gravity, are likely to be the source of baby universes which carry them out of the original spacetime. Baby-universe formation could therefore be responsible for UV regularization in quantum field theories, which takes into account gravitational effects~\\cite{bib:2008IntJouModPhy__D_17501Gue,bib:2008ProThePhy120985Ans}.\n\n\\section*{Acknowledgements}\n\nE.~G. and Z.~M. gratefully acknowledge financial support from the Foundational Questions Institute (FQXi) through a mini grant, and would also like to thank the University of Bahamas for hosting the event ``A Big Bang in a Little Room''. Z.~M. would like to acknowledge partial support from the John Templeton Foundation for research on the topic. S.~A. would like to gratefully acknowledge hospitality by the Department of Physics of Kyoto University, while working on ideas related to the discussion in this paper.\nMoreover, the authors would like to thank A.~Vilenkin, for insightful discussions about some of the ideas present in this paper.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:introduction}Introduction:\\protect\\\\ }\nSuspensions of fibers are found in a wide variety of applications, including mixing of elongated particles with concrete to enhance strength, tailoring the rheological properties of drilling fluids,  and the production of paper from wood \\citep{bivins2005new, lundell2011fluid, hassanpour2012lightweight, elgaddafi2012settling}. During processing and transport, these materials are subjected to shear deformations, which cause translation, rotation, bending, and breaking of the fibers. These phenomena modify the microstructure in the suspension, which in turn affects the final product's physical and mechanical properties. Therefore, it is desirable to have an accurate prediction of the rheological characteristics of suspensions in order to optimize these processes in industrial facilities. However, rigid fiber suspensions exhibit a complex rheological behavior due to the gamut of determining variables involved, including fiber-fiber, fiber-wall, and fiber-matrix interactions, as well as events such as fiber breakage and migration. The complex rheological behavior includes non-Newtonian flow properties such as shear thinning, finite normal stress differences, yielding, and jamming \\citep{goto1986flow, kitano1981rheology, bounoua2016shear,snook2014normal,keshtkar2009rheological,tapia2017rheology}, making their flow challenging to predict and control.\n\n\nOne of the crucial variables governing suspension rheology is fiber orientation. Hence, most of the constitutive modeling efforts to date have focused on explaining the suspension viscosity by linking fiber orientations to the suspension viscosity. Experimental measurements and theoretical analyses have shown that a higher fiber alignment with the flow results in a lower viscosity when compared to random orientation states. \nA close approximation of the motion of the fiber is frequently achieved by applying Jeffery's exact solution to the motion of an isolated ellipsoid in an infinite Newtonian fluid \\citep{jeffery1922motion}. \\cite{folgar1984orientation} developed the widely used variation of Jeffery's equations based on rotational diffusion and allowed for the influence of interactions on orientation. These tests from \\cite{folgar1984orientation}, Stover \\citep{stover1992observations}, and Petrich \\citep{petrich2000experimental} conducted on the distribution of fiber orientation demonstrated that an increase in the fiber concentration caused a modest change in the orientation of the fibers during steady shear towards the flow-velocity-gradient plane, which led to an increase in the suspension's viscosity as a whole. Originated by  \\cite{hinch1975constitutive,hinch1976constitutive}, the constitutive equation of fiber orientation-induced extra stress in a Newtonian viscous matrix was developed by Dinh and Armstrong \\citep{dinh1981rheology,dinh1984rheological}. Furthermore, Dinh-Armstrong's constitutive model has been used to predict the transient shear viscosity \\citep{sepehr2004rheological,eberle2009using}; the change of fiber orientation was determined with a strain reduction factor of slowing down the orientation response \\citep{folgar1984orientation, advani1990numerical,wang2008objective}. Moreover, Dinh-Armstrong's model has been extended in nonlinear Newtonian viscous liquids, including the Carreau model \\citep{ferec2009modeling}, the power law model \\citep{ferec2016effect}, and the Bingham model \\citep{ferec2017rheological}. The majority of the previous studies have focused on the dilute and semi-dilute regime; however, concentrated fiber suspensions have not been studied as much \\citep{batchelor1971stress,goddard1976stress,goddard1976tensile,shaqfeh1990hydrodynamic}. In concentrated suspension, direct interactions between fibers are typical. So, the constitutive model constructed for dilute\/semi-dilute suspension must be modified, or new models must be reconstructed \\citep{babkin1989constitutive} for the concentrated suspensions. Pipes and coworkers constructed the non-Newtonian constitutive relationships for hyper-concentrated fiber suspensions with an oriented fiber assembly. However, their model is only limited to concentrated suspensions with fibers of very large aspect ratio (>100), where the fibers are required to be arranged with a very high degree of collimation \\citep{pipes1991constitutive,pipes1992anisotropic,pipes1994non}.\nMoreover, the studies that only consider hydrodynamic interactions or pure mechanical contacts \\citep{sundararajakumar1997structure} cannot predict shear-thinning rheology, hinting at the need for a complete physical understanding of micro-mechanics of fiber suspensions.\n\n \n\n\n\n\n\n\nInter-fiber interactions between nearby fibers play a significant role in determining the stress in dense fiber suspensions \\citep{sundararajakumar1997structure,lindstrom2007simulation,khan2021rheology}. This is due to the fact that contacts produce friction and resistance to rolling motion in addition to generating normal forces between the fibers. As the contribution of \nfrictional contacts to the suspension stress becomes dominant in concentrated suspensions, it is crucial to utilize an accurate contact model that can capture their rate-dependent rheological behavior. However, the simplified model of a constant friction coefficient employed in earlier studies may not be applicable to real fibers and hence, cannot accurately predict the rate-dependence of suspension rheology \\citep{salahuddin2013study,lindstrom2008simulation}. Experimental measurements\nreveal that the coefficient of friction is not a constant and depends on the normal force \\citep{brizmer2007elastic,lobry2019shear,khan2021rheology}. Using the load-dependent friction coefficient model, it has been shown that we can quantitatively predict the shear thinning rheology \\citep{khan2021rheology}. \n\n\n\n\n\n\nIn many industrial applications, carrying suspensions at high solid volume fractions is essential to maximize transportation and reduce energy consumption. Prior rheological research mainly dealt with suspensions at relatively small volume fractions \\citep{folgar1984orientation, advani1990numerical,petrie1999rheology,bibbo1987rheology,chaouche2001rheology,salahuddin2013study}. The prediction of Phan \\textit{et al.} \\citep{phan1991flow}  agrees reasonably well with the Miliken \\textit{et al.} \\citep{forth1989milliken} data of relative viscosity at a volume fraction less than 10\\%. At moderately higher volume fractions, it was observed that the specific viscosity increased with the cube of the volume fraction. The constitutive model for dilute suspensions \\citep{giesekus1962elasto, leal1971effect, hinch1973time} can only predict that the specific viscosity is proportional to the volume fraction; the transition from linear to cubic behavior in the relative viscosity vs. the volume fraction is beyond the scope of the dilute-suspension theory. Consequently, this constitutive model for dilute suspensions was extended to the semi-dilute regime to capture the suspension's transition from linear to cubic behavior \\citep{phan1991new}. However, to the best of the author's knowledge, there has been no constitutive  model to capture the relative viscosity of the  concentrated suspensions. Furthermore, for fibers with large aspect ratios ($AR=L\/d$, where $L$ and $d$ are the fiber length and diameter, respectively), it\nis challenging to identify rheology measurements for volume fractions above 0.10. Measurements are only available for volume fractions up to $\\phi = 0.15$ or $\\phi = 0.17$ \\citep{bibbo1987rheology,bounoua2016apparent}, even for aspect ratios as high as 17 or 18, while Bibb\u00f3 \\textit{et al.} \\citep{bibbo1987rheology} made measurements as high as $\\phi = 0.23$ for smaller aspect ratios of $AR = 9$. Additionally, the volume fraction for non-colloidal fibers at which the shear stresses diverge, and the flow of the suspension ceases (i.e.,\njams) was only determined for a lower aspect ratio, $AR = 14$ \\citep{tapia2017rheology}.\nTherefore, it is challenging to describe and predict the rheological characteristics of suspensions close to jamming at different aspect ratios due to the lack of a model accurately capturing the underlying physics.\n\nThis paper aims to resolve these limitations by quantifying the effect of increasing volume fraction, coefficient of friction, and  fiber aspect ratio on the rheology of relatively rigid fiber suspensions using constitutive equations that can accurately capture these effects. We perform extensive numerical simulations by varying\nthe volume fraction, aspect ratio, coefficient of friction, and shear rate. Informed by the numerical results, we propose a viscosity model that expresses the suspension rheology in terms of the shear stress and the fiber aspect\nratio. The proposed model is based on two diverging stress-\nindependent rheological behaviors, where a function of stress can interpolate the properties between two extremes. To this end, we briefly discuss the governing equations, inter-fiber interactions, and simulation conditions in Sec. \\ref{sec:methodology}. We then illustrate the constitutive model in Sec.  \\ref{sec:model}. Finally, in Sec. \\ref{sec:result}, we demonstrate the efficacy of the model by applying it to our simulation data in predicting the suspension rheology. The model\nrequires the knowledge of rheological data at low and high shear limits\nalong with an interpolating function of applied shear stress to\npredict the relative viscosity at intermediate stress values.\n\n\n\n\\section{\\label{sec:methodology}Methodology:\\protect\\\\ }\nThis section describes the models and algorithms used to simulate the shear flow of the suspension of fibers. We consider a suspension of N fibers of aspect ratio $AR$ in a shear flow with top and bottom walls moving in the opposite direction with imposing a shear rate $\\dot{\\gamma}$. We have used the same numerical method to simulate the dense fiber suspensions as in our previous study \\citep{khan2021rheology}.\n\n\n\n\n\\subsection{Governing equations}\nAs the fibers are suspended in a fluid flow, the hydrodynamic forces acting on the fibers need to be related to their deformation to get their configuration in the flow. Thus, the elasticity equation of slender bodies is solved. Next, the governing equations for fluid flow and motion of flexible fibers are introduced.\n\\subsubsection{Fluid flow}\n\nThe suspending fluid is considered an incompressible viscous fluid  with a constant density that is governed by the Navier-stokes equations and the continuity equation.  \n\\begin{equation}\n\\frac{{\\partial \\mathbf{u}}}{{\\partial t}} + \\mathbf{\\nabla} \\cdot (\\mathbf{u} \\otimes \\mathbf{u}) =  - \\nabla p + \\frac{1}{{{\\mathop{\\rm \\textit{Re}}\\nolimits} }}{\\mathbf{\\nabla} ^2}\\mathbf{u} + \\mathbf{f},\n\\label{NS}\n\\end{equation}\n\\begin{equation}\n\\mathbf{\\nabla} \\cdot \\mathbf{u} = 0,\n\\end{equation}\nwhere $\\mathbf u$ is the velocity field, $p$ is the pressure, $\\mathbf f$ is the volume force arises from the interactions of the suspending fibers, and $Re = \\rho\\dot{\\gamma}L^2\/\\eta$ is the Reynolds number, where $\\rho$ is the density of the fluid, $\\eta$ is the dynamic viscosity of the suspending fluid, and $L$ is the characteristic length scale which is also the fiber length.\n\n\\subsubsection{Dynamics of flexible slender bodies}\n\nAs the fibers are considered continuous one-dimensional objects, Euler-Bernoulli equations can be derived for the motion of flexible fibers as \\citep{segel2007mathematics} : \n\\begin{equation}\n\\Delta \\rho \\frac{{{\\partial ^2}{\\mathbf{X}}}}{{\\partial {t^2}}} = \\frac{\\partial }{{\\partial s}}(T\\frac{{\\partial {\\mathbf{X}}}}{{\\partial s}}) - \\frac{{{\\partial ^2}}}{{\\partial {s^2}}}({B^*}\\frac{{{\\partial ^2}{\\mathbf{X}}}}{{\\partial {s^2}}}) + \\Delta \\rho {\\mathbf{g}} - {\\mathbf{F}} + {{\\mathbf{F}}^{f}},\n\\label{non-dimensional_fil}\n\\end{equation}\nwhere $s$ is the curvilinear coordinate along the fiber, $\\mathbf X$ is the position of the Lagrangian points on the fiber axis, $T$ the tension, $B^* = EI$ the bending rigidity with $E$ the modulus of elasticity of the fiber and $I$ the second moment of inertia around the filament axis, $\\mathbf{g}$ is the gravitational acceleration, $\\mathbf{F}$ is the fluid-solid interaction force per unit length on the fiber by the surrounding fluid, and  $\\mathbf{F}^f$ is the net interaction force on the fiber due to neighboring fibers. Finally, $\\Delta \\rho$ is the linear density difference between the fiber and the surrounding fluid defined as: \n\\begin{equation}\n\\Delta \\rho  = {\\rho _f} - \\rho {A_f},\n\\label{density}\n\\end{equation}\nwhere $\\rho_f$ is the actual fiber linear density, and $A_f$ is the sectional area of the fiber. The fibers are considered as inextensible but can bend \\citep{huang2007simulation,pinelli2017pelskin}. The inextensible constraint is expressed as : \n\\begin{equation}\n \\frac{{\\partial \\mathbf{X}}}{{\\partial s}}.\\frac{{\\partial \\mathbf X}}{{\\partial s}} = 1.\n \\label{inextensible}\n\\end{equation}\nFor the case of neutrally buoyant fiber, $\\Delta \\rho = 0$. Therefore, the left-hand side and the gravitational term on the right-hand side go to zero. For a neutrally buoyant case, the non-dimensional form  of Eq.~\\ref{inextensible} remains unchanged and Eq.~\\ref{non-dimensional_fil}\ncan be expressed as : \n\\begin{equation}\n    0 = \\frac{\\partial }{{\\partial s}}(T\\frac{{\\partial {\\mathbf{X}}}}{{\\partial s}}) - \\frac{{{\\partial ^2}}}{{\\partial {s^2}}}({B}\\frac{{{\\partial ^2}{\\mathbf{X}}}}{{\\partial {s^2}}}) - {\\mathbf{F}} + \\mathbf {F}^f\n\\label{intermediate}    \n\\end{equation}\nwith the following characteristics scales: $L$ for length, $U_{\\infty} = \\dot{\\gamma}L$ for velocity, $\\dot{\\gamma}^{-1}$ for time, $\\rho_f$ for reference density, $\\rho_f L^2\\dot{\\gamma}^2$ for tension, and $\\rho_f \\dot{\\gamma}^2L$ for force. Therefore, the dimensionless bending stiffness $B=\\frac{B^*}{\\rho_f\\dot{\\gamma}^2L^4}$ measures the ratio of the convective time scale to the elastic time scale. So, as  $B$ decreases, fibers become more flexible. In this study, we fix $B$ at a higher value which ensures a negligible bending of the fiber. Since the left-hand side of Eq.~\\ref{intermediate} is zero, in order to avoid singularity in the coefficient matrix, the equation is modified as:  \n\n\\begin{equation}\n\\frac{{{\\partial ^2}\\mathbf X}}{{\\partial {t^2}}} = \\frac{{{\\partial ^2}{\\mathbf X_{fluid}}}}{{\\partial {t^2}}} + \\frac{\\partial }{{\\partial s}}(T\\frac{{\\partial \\mathbf X}}{{\\partial s}}) - B\\frac{{{\\partial ^4}\\mathbf X}}{{\\partial {s^4}}} - \\mathbf F + \\mathbf {F}^f,\n\\label{pinal}\n\\end{equation}\nwhere the first term on the right-hand side is the fluid particle acceleration which is identical to the left-hand side for the neutrally buoyant fibers. As the fibers are suspended in the fluid medium, we impose zero force, moment, and tension at the free ends.\n\\begin{equation}\n    \\frac{{{\\partial ^2}\\mathbf X}}{{\\partial {s^2}}} = 0, \\frac{{{\\partial ^3}\\mathbf X}}{{\\partial {s^3}}} = 0, T = 0\n\\label{bc_fiber}    \n\\end{equation}\n\nFinally, We use the immersed boundary method (IBM) \\citep{peskin1972flow} to couple the fluid and solid fibers' motions. For the details of the numerical method, the readers are referred to our previous publication \\citep{khan2021rheology}.\n\n\\subsection{\\label{sec:short}Short range interactions:\\protect\\\\ }\nEven though the hydrodynamic interactions are well resolved using the IBM, the short-range interactions need a fine Eulerian mesh that increases computational cost. So, we use the proposed model to calculate the short-range interactions. The short-range interaction, $\\mathbf F^f=\\mathbf F^{lc} +\\mathbf F^c$, is split into lubrication correction $\\mathbf{F}^{lc}$ and contact force $\\mathbf{F}^c$, respectively. The implementation of lubrication correction $\\mathbf{F}^{lc}$ can be found in our previous study \\citep{khan2021rheology}.\n\n\\subsubsection{Contact force}\n\nWith increasing volume fraction, the surrounding fibers hinder the free rotation of fibers, giving rise to fiber-fiber contacts that influence the microstructure. Microstructure influences the macroscopic rheological properties of the suspension, such as relative viscosity. We model the contact interaction between the fibers as it is done in the discrete element method (DEM). We assume that the contact between the fibers takes place through hemispherical asperity. The asperity deformation is defined by surface overlap $\\delta=h-h_r$, and contact happens when $\\delta \\leq 0$. Here, $h$ is the inter-fiber surface separation, and $ h_r$ is the surface roughness, as shown in figure~\\ref{fig:fiber}. \n\\begin{figure}\n\\centering\n\\includegraphics[width=1.0\\linewidth]{figure\/fiber2.eps}\n\\caption{A sketch of the roughness model, $L$ and $d$ are the length and diameter of the fiber, respectively, $h_r$ is the roughness height, and $\\delta=h-h_r$ is the surface overlap. Contact occurs when $\\delta \\le 0$. Dots along the axes of the fibers indicate Lagrangian points.}\\label{fig:fiber}\n\\end{figure}\nThe deformation of asperities\nresults in normal, $\\mathbf F_n$, and tangential, $\\mathbf F_t$ force on the fiber surface. The normal contact force $\\mathbf F_n$ is given by the Hertz law:\n\\begin{equation}\\label{eq:hertz}\n\\mathbf F_n=- F_0\\left(\\frac{|\\delta|}{L}\\right)^{3\/2} \\mathbf{n},\n\\end{equation}\nwhere $F_0\/L^{3\/2}$ is the normal stiffness that can be evaluated as a function of fiber's mechanical properties such as elastic modulus, Poisson's ratio, Young's modulus, and roughness size \\citep{lobry2019shear, more2020effect}. In this study, we use $F_0$ as the characteristic contact force scale. Coulomb's friction law gives the tangential force:\n\\begin{equation}\n\\mathbf F_t=\\mu|\\mathbf F_n|\\frac{\\mathbf F_t}{|\\mathbf F_t|},\n\\end{equation}\nwhere $\\mu$ is the friction coefficient. \n\\subsubsection{Friction model}\nResearchers have utilized a constant friction coefficient when examining the dynamics of fiber suspensions numerically \\citep{stickel2009rheology,banaei2020numerical}. However, a constant coefficient fails to predict the shear rate-dependent suspension viscosity.  In practice, the coefficient of friction depends on many factors, such as the fiber material and the roughness size, which are implicitly included in the normal force via the normal stiffness $F_0\/L^{3\/2}$ \\citep{lobry2019shear, more2020effect}. Hence, a normal load-dependent coefficient of friction is a more accurate physical description of $\\mu$ than a simple constant. We use the Brizmer model \\citep{brizmer2007elastic} for $\\mu$, derived from the measurements between a hemisphere and a flat surface and validated using the finite element analysis \\citep{brizmer2007elastic}, which makes it applicable to a wide range of materials and conditions \\citep{brizmer2007elastic,lobry2019shear,more2020effect}. It is given by:\n\\begin{equation}\\label{eq:frictionlaw}\n\\mu  = 0.27\\coth \\left[ {{0.27{\\left({\\frac{{|\\mathbf F_n^{(i,j)}|}}{{{ F_0}}}} \\right)}^{0.35}}} \\right],\n\\end{equation}\nwhere $F_0$ is the characteristic contact force scale introduced in Eq.~\\ref{eq:hertz}. Eq.~\\ref{eq:frictionlaw} is a decreasing function of $|\\mathbf F_n^{(i,j)}|\/F_0$ as illustrated in figure~\\ref{fig:mu_normal_force}.\n\\begin{figure}\n\\centering\n\\includegraphics[width=1.0\\linewidth]{figure\/mu_force.eps}\n\\caption{Friction coefficient $\\mu$ as a function of the dimensionless contact normal force (Eq. \\ref{eq:frictionlaw}).}\\label{fig:mu_normal_force}\n\\end{figure}\nThus, the coefficient of friction decreases with increasing the normal force (which is equivalent to increasing asperity deformation from Eq.~\\ref{eq:hertz}) between the contacting fibers and attains a plateau at high normal load values. Before we address the fibers with load-dependent friction coefficient, we will present results with constant $\\mu$. This will help to quantitatively understand the original case of load-dependent friction.\n\n\n \n\n\n\\subsection{Stress and bulk rheology calculation}\nWe compute the bulk stress in the suspension by volume averaging the viscous fluid stress and the stress generated by the presence of fibers and inter-fiber interactions. The calculation of bulk stress, including different contributions, is described elsewhere \\citep{banaei2020numerical,khan2021rheology}. In this work, we present the relative viscosity to define the rheological behavior of\nthe suspension. The relative viscosity $\\eta_r$ is defined as: \n\\begin{equation}\n     \\eta_r = \\frac{\\eta_{eff}}{\\eta},\n\\end{equation}\nwhere $\\eta_{eff}$ is the effective viscosity of the suspension. The relative viscosity in terms of bulk stress is: \n\\begin{equation}\n     \\eta_r  = 1 + {\\Sigma}{_{xy}^f},\n\\end{equation}\nwhere ${\\Sigma}{_{xy}^f}$ is time and space averaged shear stress arising from the presence of the fibers. ${\\Sigma}{_{xy}^f}$ is non-dimensionalized by the product of suspending fluid viscosity $\\eta$ and shear rate $\\dot\\gamma$. As $F_0$ is considered as the characteristic contact force scale, the shear rate scale is given by $\\dot{\\gamma}_0 = F_0\/\\pi\\eta d^2$.  The dimensionless shear rate provides an estimate of the relative importance of contact to the hydrodynamic forces defined as:\n\\begin{equation}\n    \\dot {\\Gamma}  = \\frac{\\dot{\\gamma} }{\\dot{\\gamma}_0} =\\frac{\\dot{\\gamma}}{F_0\/\\pi \\eta d^2} ,\n\\label{gamma}\n\\end{equation}\nBeing the characteristic stress scale  $\\sigma_0 = F_0\/\\pi d^2$, the dimensionless stress can be defined as:\n\\begin{equation}\n    \\tilde{\\sigma}= \\frac{\\sigma}{\\sigma_0}=\\frac{\\dot{\\gamma} }{\\dot{\\gamma}_0}{\\Sigma}{_{xy}} =\\dot {\\Gamma}{\\eta_r} ,\n\\label{sigma}\n\\end{equation}\n\n\\subsection{Simulation conditions}\n\\subsubsection{Boundary conditions and domain size}\nThe fibers are suspended in a channel with upper and lower walls moving in the opposite direction with a magnitude of $U_{\\infty}=\\dot{\\gamma}L$ in the x-direction. The wall is subjected to no-slip and no-penetration boundary conditions, and periodicity is assumed in the stream-wise ($x$) and span-wise ($z$) directions. Initially, we place fibers randomly in the simulation domain of size  $5L\\times5L\\times8L$ and $80\\times80\\times128$ grid points in the stream-wise ($x$), wall normal ($y$), and span-wise direction ($z$), respectively. A schematic diagram of the computational configuration and coordinate system is shown in  figure~\\ref{fig:geometry}. The averaged steady state suspension viscosity changes minimal (less than 2 \\%) in simulations with bigger domain,  higher grid, and time resolutions such as 1.5, 2, 2.5, and 3 times the current domain, grid, and time resolution, Moreover, 36 Lagrangian points over the fiber length are enough to resolve the case with the highest fiber aspect ratio. The required time step to capture the fiber dynamics is $\\Delta t = 10^{-5}$. The suspension is simulated until a statistically steady viscosity is observed and mean values after discarding the initial transients are presented. \n\n\\begin{figure}\n  \\centerline{\\includegraphics[width=0.8\\linewidth]{figure\/3-D_final.eps}\n  \\caption{Simulation setup of the shear flow of a fiber suspension. The top and bottom walls move with velocities $U_{\\infty} = \\dot{\\gamma}L$ in the directions shown by the arrows.}\n\\label{fig:geometry}\n\\end{figure}\n\n\\subsubsection{Range of parameters investigated} \nThe aim of this study is to quantify the effect of varying fiber aspect ratios, volume fractions, and shear rates on the rheology  of the suspension. Hence, we simulate suspensions of almost rigid fibers in a shear flow by varying the aspect ratio in the range of $10 \\leq AR \\leq 36$. The simulations were carried out for dimensionless shear rate in the range of $0.1\\leq \\dot\\gamma\/\\dot\\gamma_0\\leq100$ for a range of volume fractions $0.03 \\leq \\phi \\leq 0.45$. The dimensionless bending rigidity, $B$, was set to 5.0 which ensures a negligible bending of the fiber.\nThe range of parameters explored in the present work is summarized in table \\ref{tab:my-table}.\n\n\\begin{table}\n\\caption{Range of parameters explored in this study}\n \n  \\begin{ruledtabular}\n  \\begin{tabular}{cccc}\n      $\\phi$  & $AR$   &   $\\dot\\gamma\/\\dot\\gamma_0$ & $\\mu$\\\\\\hline\\\\\n      \n       $0.03-0.47$   &~ $10-36$~ & ~$0.1-100$~ &~~ $0-11$~ \\\\\n  \\end{tabular}\n \n  \\label{tab:my-table}\n\\end{ruledtabular}\n\\end{table}\n\n \n  \n\n    \n  \n     \n    \n   \n       \n \n  \n\n\n\n\\section{\\label{sec:model}The constitutive model:\\protect\\\\ }\n\n\nWe first demonstrate simulations where the friction coefficient is maintained constant before addressing the fibers with a load-dependent friction coefficient. Thus, we reflect on the significant impact of this microscopic parameter on the suspension viscosity. Recent numerical investigations on frictional non-Brownian particle suspensions have shown that the friction coefficient, considered constant, has a significant impact on the effective viscosity \\citep{mari2014shear,gallier2014rheology,gallier2018simulations,singh2018constitutive}. From these studies, it is clear that the jamming volume fraction depends significantly on the friction coefficient. More recently, \\cite{singh2018constitutive} explained how the viscosity of non-colloidal suspensions changes with the microscopic particle friction coefficient by showing that the jamming volume fraction decreases when the microscopic particle friction coefficient goes from 0 to 1. We point out that simulations of granular flows \\citep{silbert2010jamming} and experiments with discontinuous shear-thickening suspensions \\citep{fernandez2013microscopic} have shown that there is a link between the amount of jamming and the microscopic friction coefficient. Therefore, we start our analysis by deriving correlation laws for the dependence of suspension jamming fraction with constant friction coefficients. After that, the correlation laws will be utilized to describe the rheology of suspensions for cases with load-dependent friction.\n\nIn the beginning, we report the simulations for suspensions of fibers in which the friction coefficient is held at a fixed value in the range of 0 to 15. The following correlation law is fitted for each value of $\\mu$ to describe the variation of the relative viscosity against the volume fraction.\n\\begin{equation}\n    \\eta_r (\\mu,\\phi,AR) = \\alpha(\\mu,AR)\\bigg(1-\\frac{\\phi}{\\phi_m(\\mu, AR)}\\bigg)^{-0.90},\n    \\label{eq:viscosity_vol}  \n\\end{equation}\nwhere $\\alpha$ and $\\phi_m$ are friction and aspect ratio dependent constant coefficients. Here, we model the dependence of viscosity on volume fraction as $\\phi_m^{-0.90}(\\phi_m-\\phi)^{-0.90}$. The form is consistent with the proposed relationships in the recent work concerning the suspension of fibers \\citep{khan2021rheology,tapia2017rheology}. Figure~\\ref{fig:viscosity_vol_mu_const_AR} presents the results of the simulation along with the relevant correlation laws for two different values of the friction coefficient ($\n\\mu =2$ and $\\mu = 5$) for aspect ratios $AR = 10$ and $AR = 36$.\n\\begin{figure*}\n\\centering\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n \n  \\includegraphics[width=1.0\\linewidth]{figure\/viscosity_vol_mu_const_AR18.eps}\n  \\caption{}\n  \\label{fig:viscosity_vol_mu_const_AR18}\n\\end{subfigure\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\includegraphics[width=1.0\\linewidth]{figure\/viscosity_vol_mu_const_AR33.eps}\n  \\caption{}\n  \\label{fig:viscosity_vol_mu_const_AR33}\n\\end{subfigure}\n\\caption{ Relative viscosity as a function of volume fraction for friction coefficient $\\mu =2, 5$ for (a) $AR = 10$ and (b) $AR = 36$. Solid lines: best fit with Eq.~\\ref{eq:viscosity_vol}}.\n\\label{fig:viscosity_vol_mu_const_AR}\n\\end{figure*}\nThe friction and aspect ratio dependent parameters ($\\alpha$ and $\\phi_m$) are found empirically from our simulations as: \n\n\n\\begin{eqnarray}\n{\\alpha(\\mu,AR)} =&& \\alpha^{\\mu_\\infty}(AR)+\\bigg(\\alpha^{\\mu_0}(AR)-\\alpha^{\\mu_\\infty}(AR)\\bigg)\\nonumber\\\\\n&&\\frac{\\textrm{exp}\\big(-X^\\alpha \\textrm{atan}(\\mu)\\big)-\\textrm{exp}(-\\pi X^\\alpha \/2)}{1-\\textrm{exp}(-\\pi X^\\alpha \/2)},\n\\\\\n{\\phi_m(\\mu,AR)} = && \\phi_m^{\\mu_\\infty}(AR)+\\bigg({\\phi_m}^{\\mu_0}(AR)-\\phi_m^{\\mu_\\infty}(AR)\\bigg)\\nonumber\\\\\n&&\\frac{\\textrm{exp}\\big(-X^q \\textrm{atan}(\\mu)\\big)-\\textrm{exp}(-\\pi X^q \/2)}{1-\\textrm{exp}(-\\pi X^q \/2)}\n\\label{eq:constant_fric_mu_alpha}\n\\end{eqnarray}\nas shown by fits in figure~\\ref{fig:phi_m_a_mu}. Here, the superscript $\\mu_0$ and $\\mu_\\infty$ denote the corresponding parameter when the coefficient of friction is zero and infinity \\citep{lobry2019shear}. Note that all the parameters are independent of the coefficient of friction and are reported in table~\\ref{tab:fitting_phi_a}.\n\\begin{figure*}\n\\centering\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n \n  \\includegraphics[width=1.0\\linewidth]{figure\/phi_m_fixed_AR.eps}\n  \\caption{}\n  \\label{fig:phi_mu}\n\\end{subfigure\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\includegraphics[width=1.0\\linewidth]{figure\/a_fixed_AR.eps}\n  \\caption{}\n  \\label{fig:alpha_mu}\n\\end{subfigure}\n\\caption{ (a) Jamming volume fraction $\\phi_m$ (b) pre-factor $\\alpha$ as a function of $\\mu$. Solid lines: best fit with Eq.~\\ref{eq:constant_fric_mu_alpha}.}.\n\\label{fig:phi_m_a_mu}\n\\end{figure*}\n\\begin{table}\n\\caption{Calibrated model parameters for Eq.~\\ref{eq:constant_fric_mu_alpha} for  aspect ratios $AR = 10$ and  $AR = 36$}\n  \\begin{ruledtabular}\n  \\begin{tabular}{cccccccc}\n  \n      $AR$  & $\\phi_m^0$   &   $\\phi_m^{\\infty}$  & $X^q$ &   $\\alpha^0$  &$\\alpha^{\\infty}$  &$X^{\\alpha}$\\\\[1pt]\n      \\hline\n      $10$  &~ $0.40$~ & ~$0.47$~ &~$0.11$~ &~$0.85$~ &~$20.9$~ &~$-7.532$~\\\\\n     \n      $36$  &~ $0.12$~ & ~$0.17$~ &~$0.11$~ &~$0.8015$~ &~$0.82$~&~$-7.532$~\\\\\n   \n       \n  \\end{tabular}\n  \n  \\label{tab:fitting_phi_a}\n  \\end{ruledtabular}\n\\end{table}\nIn particular, for aspect ratio 18, the jamming volume fraction seems to approach the limit $\\phi_m^{\\infty} = 0.47$ as the friction coefficient goes to $\\infty$ and is equal to $\\phi_m^{0} = 0.40$ for $\\mu =0$. These values are consistent with the data found in the literature {\\citep{williams2003random, khan2021rheology}}.\n\n\\definecolor{red}{rgb}{1,0,0}\n\\definecolor{blue}{rgb}{0,0,1} \n\n\n\\newcommand{\\bt}{\\textcolor{blue}{$\\blacktriangle$}} \n\\newcommand{\\textcolor{blue}{$\\blacksquare$}}{\\textcolor{blue}{$\\blacksquare$}}\n\n\\newcommand{\\textcolor{blue}{$\\blackc$}}[2][blue,fill=blue]{\\tikz[baseline=-0.5ex]\\draw[#1,radius=#2] (0,0) circle ;}%\n\n\\newcommand{{\\color{red}$\\circ$}}{{\\color{red}$\\circ$}}\n\\newcommand{{\\color{red}$\\triangle$}}{{\\color{red}$\\triangle$}}\n\\newcommand{{\\color{red}$\\square$}}{{\\color{red}$\\square$}}\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1.0\\linewidth]{figure\/mu_stress.eps}\n\\caption{Average coefficient of friction over all contacting fiber pairs as a function of shear stress. Red symbol corresponds to data for $AR = 10$ having volume fractions ({\\color{red}$\\square$}) $\\phi =0.20$, ({\\color{red}$\\triangle$}) $ \\phi = 0.30$, and ({\\color{red}$\\circ$}) $ \\phi = 0.35$. Blue symbol corresponds to data for $AR = 36$ having volume fractions (\\textcolor{blue}{$\\blacksquare$}) $\\phi =0.03$, (\\bt) $\\phi = 0.05$, and (\\textcolor{blue}{$\\blackc$}{3.0pt}) $\\phi = 0.08$. The solid line shows the effective coefficient of friction from Eq.~\\ref{eq:mu_stress}. }\\label{fig:mu_stress}\n\\end{figure}\n\n\nThe above analysis shows that the simulations with a constant friction coefficient do not provide rate-dependent viscosity in the suspension. Hence, We now turn to the analysis with load-dependent friction between the fibers in the suspension. However, the correlation derived earlier with a constant coefficient of friction will aid quantitative analysis of the case with a load-dependent friction coefficient. \n\n\nFrom the shear rate-dependent viscosity, we observe that  as the  reduced  shear rate  increases,  the  viscosity  decreases due to the reduction of friction coefficient \\citep{khan2021rheology}. It is possible to gain a better understanding of this transition from high to low viscosity by examining the average friction coefficient, $\\mu_{avg}$, as a function of the reduced shear stress, $\\tilde{\\sigma} $ ($\\eta_r\\dot{\\gamma}\/\\dot{\\gamma_0}$),  as shown in figure~\\ref{fig:mu_stress}. Reduced shear stress $\\tilde{\\sigma} $ quantifies the typical force experienced by  two  contacting  fibers, and $\\mu_{avg}$ is the ensemble average of the coefficient of friction $\\mu$ between all the contacting fiber pairs in the suspension. Intriguingly, regardless of the fiber volume fraction and aspect ratio, the data collapse onto a single curve, denoted by $\\mu^{eff}$, fitted with:\n\\begin{equation}\n    \\mu^{eff} = 0.34 \\textrm{coth}\\Bigg[0.35\\bigg(\\frac{\\tilde{\\sigma}}{20}\\bigg)^{0.45}\\Bigg].\n    \\label{eq:mu_stress}\n\\end{equation}\nThe functional form is similar to the one for the coefficient of friction $\\mu$ dependence on the normal load between the fibers $|\\mathbf{F}^{(i,k)}_n|$ in Eq.~\\ref{eq:frictionlaw}, however, it is not exactly the same. This feature is not immediately apparent; thus, it is worth commenting on. The normal contact force does regulate the microscopic coefficient of friction between contacting fibers. However, the bulk shear stress and mean force are only qualitatively related. Even the relationship between the mean friction coefficient and mean force is difficult to predict due to the highly non-linear relationship between the microscopic friction coefficient and force. This implies that the reduced shear stress may be interpreted as the reduced effective normal force between contacting fibers, which regulates the mean friction coefficient. Therefore, we obtain the following expression for suspension viscosity at a finite stress value in terms of the volume fraction $\\phi$, jamming volume fraction $ \\phi_m$, and pre-factor $\\alpha$.\n\n\n\\begin{eqnarray}\n    {\\eta_r (\\tilde{\\sigma},\\phi,AR)} =&& \\alpha\\Big(\\tilde{\\sigma}(\\mu_eff),AR\\Big)\\nonumber\\\\\n    &&\\bigg(1-\\frac{\\phi}{\\phi_m\\Big(\\tilde{\\sigma}(\\mu_eff)), AR\\Big)}\\bigg)^{-0.90}\n    \\label{eq:viscosity}  \n\\end{eqnarray}\nThe rheological properties in the two extreme stress conditions can be expressed in terms of volume fractions $\\phi$, jamming volume fractions  $\\phi_m^{0,\\infty}$ and model parameters $\\alpha^{0,\\infty}$. Here the superscript 0 and $\\infty$ denote the low and high shear limits, respectively.\n\\begin{eqnarray}\n\\eta_r^0 (\\phi,AR) = \\alpha^0(AR)\\bigg(1-\\frac{\\phi}{\\phi_m^0(AR)}\\bigg)^{-0.90},\n\\label{eq:low_shear}\n\\\\\n\\eta_r^{\\infty}(\\phi,AR) = \\alpha^{\\infty}(AR)\\bigg(1-\\frac{\\phi}{\\phi_m^{\\infty}(AR)}\\bigg)^{-0.90}\n\\label{eq:high_shear}\n\\end{eqnarray}\nThe jamming fraction $\\phi_m$ and the pre-factor $\\alpha$ at intermediate stress $\\tilde{\\sigma}$ can be calculated by interpolating their corresponding values in the low and high-stress limits \\citep{more2020constitutive} as follows: \n\n\\begin{eqnarray}\n \\phi_m(\\sigma,AR) = \\phi_m^0(AR)[1-f\n (\\tilde{\\sigma})]+\\phi_m^{\\infty}\n [f(\\tilde{\\sigma})] \n \\label{eq:jamming_phi}\n\\\\\n \\alpha(\\sigma,AR) = \\alpha^{\\infty}(AR)  [f_1(\\tilde{\\sigma})] +\\alpha^0[1-f_1\n (\\tilde{\\sigma})],\n \\end{eqnarray}\nwhere \n\\begin{eqnarray}\nf(\\tilde{\\sigma})&=& f\\Big(\\tilde{\\sigma}(\\mu_{eff})\\Big)\\nonumber\\\\ =&&\\frac{\\textrm{exp}\\big(-X^q \\textrm{atan}(\\tilde{\\sigma}(\\mu_{eff}))\\big)-\\textrm{exp}(-\\pi X^q \/2)}{1-\\textrm{exp}(-\\pi X^q \/2)}, \n\\\\\nf_1(\\tilde{\\sigma}) &=& f\\Big(\\tilde{\\sigma}(\\mu_{eff})\\Big) \\nonumber\\\\ =&& \\frac{\\textrm{exp}\\big(-X^\\alpha \\textrm{atan}(\\tilde{\\sigma}(\\mu_{eff}))\\big)-\\textrm{exp}(-\\pi X^{\\alpha} \/2)}{1-\\textrm{exp}(-\\pi X^{\\alpha} \/2)} .\n \\end{eqnarray}\n\nA similar interpolation function was used to interpolate $\\alpha$ and $\\phi_m$ when we considered results with a constant coefficient of friction. Later we find an expression to describe the reduced stress as a function of $u^{eff}$ (Eq.~\\ref{eq:mu_stress}). Thus, we use a similar interpolation function as expressed by $f(\\tilde{\\sigma})$. In addition, the pre-factor $\\alpha$ and jamming volume fraction $\\phi_m$ in the low and high shear stress limits can be expressed in terms of aspect ratio, $AR$ and model parameters \\{$\\bar{\\phi}_m, \\bar{\\alpha}\\}^{0,\\infty}$, \\{$\\hat{\\phi}_m, \\hat{\\alpha\\}}^{0,\\infty}$,  \\{${Q}_{\\alpha,\\phi_m}\\}^{0,\\infty}$, \\{${R}_{\\alpha,\\phi_m}\\}^{0,\\infty}$, and \\{${S}_{\\alpha,\\phi_m}\\}^{0,\\infty}$. Here, the superscript 0 and $\\infty$ denote the model parameters at low and high shear limits, respectively. We use $\\bar{}$ and $\\hat{}$ over model parameters for the fibers with aspect ratios 10 and 36 cases, respectively. \n\n\\begin{eqnarray}\n{\\alpha^{0,\\infty}}=&&\\bar{\\alpha}^{0,\\infty}+\\Big[\\hat{\\alpha}^{0,\\infty} - \\bar{\\alpha}^{0,\\infty}\\Big]\\nonumber\\\\\n&&\\textrm{log}\\Big(\\frac{Q_{\\alpha}^{0, \\infty}}{{\\left( AR\\right)}^{R_{\\alpha}^{0,\\infty}}}\\Big)^{S_{\\alpha}^{0,\\infty}}\n\\label{eq:alpha_aspect}\n\\\\\n{\\phi_m^{0,\\infty}}=&&\\bar{\\phi_m}^{0,\\infty}+\\Big[\\hat{\\phi_m}^{0,\\infty} - \\bar{\\phi_m}^{0,\\infty}\\Big]\\nonumber\\\\\n&&\\textrm{log}\\Big(\\frac{Q_{\\phi_m}^{0, \\infty}}{{\\left( AR\\right)}^{R_{\\phi_m}^{0,\\infty}}}\\Big)^{S_{\\phi_m}^{0,\\infty}}\n\\label{eq:phi_aspect}\n\\end{eqnarray}\nThe value of the calibrated model parameters is reported in table~\\ref{tab:alpha_phi_aspect}.\n\\section{\\label{sec:result}result}\n\n\\subsection{Aspect ratio dependent rheology}\nExperiments \\citep{keshtkar2009rheological, bounoua2016shear} and numerical simulations \\citep{khan2021rheology,tapia2017rheology} have shown that increasing aspect ratio leads to increased suspension viscosity due to a reduction of the jamming fraction. Figure~\\ref{fig:jamming} shows the relative viscosity as a function of fiber volume fraction for the long ($AR = 36$) and short ($AR = 10$) fibers in the low and high shear rate limits with the modified Maron-Pierce fitting curves [Eqs.~\\ref{eq:low_shear},\\ref{eq:high_shear}]. As we present for $AR = 10$ and $AR = 36$ in the main text, we refer to the cases as \"short\" and \"long\" henceforth in the article. Here, the reduction in the jamming volume fraction, $\\phi_m$, with increasing aspect ratio and stress is consistent with experiments \\citep{keshtkar2009rheological, bounoua2016apparent, bounoua2016shear,  tapia2017rheology}. We observe that the calibrated model parameters and jamming volume fraction depend only on the fiber aspect ratio in the low and high shear rate limits. So, they can be expressed in terms of $AR$, as shown in Eqs.~(\\ref{eq:alpha_aspect})-(\\ref{eq:phi_aspect}). Figures~\\ref{fig:phi_AR} and  \\ref{fig:alpha_AR} show that Eqs.~(\\ref{eq:alpha_aspect})-(\\ref{eq:phi_aspect}) are a good fit and accurately capture the effect of increasing fiber aspect ratio on the rheology of the dense fiber suspensions in the low and high shear limits. \n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1.0\\linewidth]{figure\/jamming.eps}\n\\caption{ Relative viscosity of the long ($AR = 36$) and short ($AR = 10$) fiber suspensions for different volume fractions. Dashed and solid lines represent fitting Eqs. in the low $(^0$, Eq.~\\ref{eq:low_shear}$)$ and high $(^{\\infty}$, Eq.~\\ref{eq:high_shear}$)$ stress limits. }\\label{fig:jamming}\n\\end{figure}\n\n\\begin{figure*}\n\\centering\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n \n  \\includegraphics[width=1.0\\linewidth]{figure\/phi_AR.eps}\n  \\caption{}\n  \\label{fig:phi_AR}\n\\end{subfigure\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\includegraphics[width=1.0\\linewidth]{figure\/alpha_AR.eps}\n  \\caption{}\n  \\label{fig:alpha_AR}\n\\end{subfigure}\n\\caption{ Calibrated model parameters as a function of the aspect ratio of the fibers. Dashed and solid lines represent fitting Eqs. (\\ref{eq:alpha_aspect}-\\ref{eq:phi_aspect}) in the low $(^0)$ and high $(^{\\infty})$ shear limits. (a) jamming volume fraction, $\\phi_m$ and (b) calibrated model parameter, $\\alpha$. An increase in the aspect ratio leads to an increase in $\\alpha$ in both the low and high shear rate limits. However, increasing the aspect ratio reduces the jamming volume fraction, $\\phi_m$ due to the increase in relative viscosity \\citep{keshtkar2009rheological, tapia2017rheology, khan2021rheology} .}\n\\label{fig:phi_alpha_AR}\n\\end{figure*}\n\n\n\\begin{table*}\n\\caption{Aspect ratio dependent calibrated model parameters for relative viscosity, $\\eta_r$. }\n  \\begin{ruledtabular}\n\\def~{\\hphantom{1}}\n  \\begin{tabular}{ccccccccccc}\n \n      ${}$  & ${\\{ \\bar{} \\}}^0$   &   ${\\{\\hat{}\\}}^0$  & $Q_{\\{\\}}^0$ &   $R_{\\{\\}}^0$  &$S_{\\{\\}}^0$  &${\\{ \\bar{} \\}}^{\\infty}$   &   ${\\{\\hat{}\\}}^{\\infty}$  & $Q_{\\{\\}}^{\\infty}$ &   $R_{\\{\\}}^{\\infty}$  &$S_{\\{\\}}^{\\infty}$\\\\[1pt]\n      \\hline\\\\\n      $\\alpha$  &~ $145$~ & ~$15$~ &~$2.04$~ &~$1.3$~ &~$0.85$~ &~$4$~&~$0.75$~&~$82.35$~&~$1.25$~&~$0.70$~\\\\\n    \n      $\\phi_m$  &~ $0.23$~ & ~$0.70$~ &~$57$~ &~$1.4$~ &~$0.20$~&~$0.2$~&~$0.57$~&~$82$~&~$1.27$~&~$0.35$~\\\\\n      \n       \n  \\end{tabular}\n  \n  \\label{tab:alpha_phi_aspect}\n   \\end{ruledtabular}\n\\end{table*}\n\nBefore going into detail about stress-dependent rheology, it is important to note that aspect ratio dependence and stress dependence are unrelated. Stress-dependent rheological behavior is recovered using a load-dependent friction coefficient model, whereas aspect ratio dependence is obtained by altering the $AR$ of the fiber. In the absence of load-dependent friction, we only get the aspect ratio dependence rheology as presented in figure~\\ref{fig:jamming} and modeled in Eqs.~(\\ref{eq:alpha_aspect})-(\\ref{eq:phi_aspect}). We will get stress between 0 and $\\infty$ depending on the value of $\\mu$. But the increase in viscosity with aspect ratio will still be observed, which is consistent with earlier simulations \\citep{wu2010numerical, khan2021rheology} and experiments \\citep{keshtkar2009rheological,bounoua2016shear, tapia2017rheology}.\n\n\\subsection{Stress dependent viscosity}\n\nIn this section, we present the shear stress-dependent viscosity for suspensions with varying fiber aspect ratios along with the constitutive equations fitting curves. It has been demonstrated that the rheological properties in the intermediate stress values can be interpolated once the rheological characteristics and the jamming fraction in the low and high shear stress limits are known \\citep{singh2018constitutive,more2020constitutive}. Section \\ref{sec:model} proposes aspect ratio and stress-dependent constitutive equations based on this premise.\n\nFigure~\\ref{fig:viscosity_stress_AR} shows the shear rate dependent relative viscosity for the short ($AR = 10$) and long ($AR=36$) fiber suspensions for the volume fractions investigated in the study.\nThe proposed model accurately predicts relative viscosity in the intermediate stress ($\\tilde{\\sigma}$) regime for both aspect ratios. \n\\begin{figure*}\n\\centering\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n \n  \\includegraphics[width=1.0\\linewidth]{figure\/viscosity_shear_rate_AR_10.eps}\n  \\caption{}\n  \\label{fig:viscosity_stress_AR_10}\n\\end{subfigure\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\includegraphics[width=1.0\\linewidth]{figure\/viscosity_shear_rate_AR_36.eps}\n  \\caption{}\n  \\label{fig:viscosity_stress_AR_36}\n\\end{subfigure}\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\includegraphics[width=1.0\\linewidth]{figure\/viscosity_shear_rate_AR_17_comparison.eps}\n  \\caption{}\n  \\label{fig:stress_com}\n\\end{subfigure\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\includegraphics[width=1.0\\linewidth]{figure\/Jammig_comparison.eps}\n  \\caption{}\n  \\label{fig:jamming_com}\n\\end{subfigure}\n\\caption{Relative viscosity as a function of dimensionless shear stress, $\\tilde{\\sigma}$: (a) for fiber aspect ratio, $AR=10$ and (b) for fiber aspect ratio, $AR=36$. (c) \\& (d) Comparison of the constitutive model with the experimental data from Bibbo \\textit{et al.} \\citep{bibbo1987rheology}. The shear stress was fixed to  $\\tilde{\\sigma} = 0.02$ when comparing the volume fraction-dependent relative viscosity with the experimental data.\nThe solid lines represent Eq.~\\ref{eq:viscosity} with the values of the calibrated model parameters obtained using the simulation data.}.\n\\label{fig:viscosity_stress_AR}\n\\end{figure*}\nIt follows from this model that the reduced force that controls friction, and consequently the suspension relative viscosity, is proportional to the reduced shear stress: \n\\begin{equation}\n    \\frac{F_n}{F_0}=\\frac{\\pi\\eta\\eta_rd^2\\dot\\gamma}{20F_0}=\\frac{\\eta_r(\\dot\\gamma\/\\dot\\gamma_0)}{20}=\\frac{\\tilde\\sigma}{20}\n\\end{equation}\n\nWe can use the values of calibrated parameters obtained from the simulation data to predict the stress-dependent relative viscosity of fiber suspensions with aspect ratio 17.  The prediction is compared to the experimental data from \\cite{bibbo1987rheology}, as shown in figure~\\ref{fig:stress_com}. The model does a satisfactory job of capturing the relative viscosity of the experimental system. For this comparison, we assume $\\sigma_0 = 0.45$ to non-dimensionalize the experimental stress. Finally, we predict the volume fraction-dependent viscosity at a high shear rate for aspect ratios 17 and 51 and compare it with \\cite{bibbo1987rheology}, as shown in figure~\\ref{fig:jamming_com}. The data was only available in the semi-concentrated regime, and our simulation did a good job of capturing the experimental data. \n\nFinally, due to the agreement between the results from the load-dependent friction simulations and the viscosity from Eq.~\\ref{eq:viscosity}, it is possible to estimate the effective coefficient of friction, $\\mu_{eff}$, in the experiment without running new simulations. Once we know the jamming volume fraction from the experiments, $\\mu_{eff}$ can be computed by reversing Eq.~\\ref{eq:constant_fric_mu_alpha}. In this case, $\\mu_{eff}$ is actually the microscopic friction coefficient for the applied load, $F_n=\\pi*d^2\\sigma\/20$, where $\\sigma$ is the experimental stress. For the reported experiment of Bibbo \\textit{et al.} \\citep{bibbo1987rheology}, we find the coefficient friction to be 1.12. We note that it is possible to deduce $\\mu_{eff}$ from the viscosity measurements only if the values of the  experimental  jamming  fraction  belong  to  the variation range  of $\\phi_m$, deduced  from the  numerical simulations  at  constant  friction  coefficient. It is important to remember that the friction coefficient values derived from the viscosity measurements are estimates.\n\n\n\n\\subsection{Flow state diagram}\nThe shear rheology described above is controlled by three dimensionless parameters, namely, dimensionless shear stress $\\tilde{\\sigma}$, volume faction $\\phi$, and aspect ratio $AR$. The results discussed in the paper are presented in a flow state diagram in figure~\\ref{fig:flow_state}. Here in the $\\tilde{\\sigma}-\\phi$ phase space, we identify  $\\phi_m^\\infty$, $\\phi_m^0$, and $\\phi_m (\\tilde{\\sigma})$. In the lower part of the diagram, when the stress is too low, and in the upper part of the diagram, when the stress is large, rheology diverges at $\\phi_m^0$, and $\\phi_m^\\infty$, respectively. So in the two extremes, the relative viscosity is rate-independent. In between, we observe rate-dependent viscosity.\n\nMoreover, the volume fraction at which the suspension jams increases with increasing stress. Previous studies on the suspension of fibers have reported that at low stress values, the suspension does not flow but can flow at higher stress values \\citep{bibbo1987rheology, chaouche2001rheology, switzer2003rheology, keshtkar2009rheological,bounoua2016apparent,bounoua2016normal} meaning that the jamming volume fraction, $\\phi_m$ depends on the shear stress. However, the exact dependence of the jamming fraction on the stress was largely unexplored. Our numerical simulation is superior in the sense that it quantifies the exact dependence of the jamming volume fraction on the applied stress denoted by the solid line in figure~\\ref{fig:flow_state}. In addition, the applicability of the model is further strengthened  in capturing the dependence of jamming volume fraction on the stress for different aspect ratios. \n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1.0\\linewidth]{figure\/flow_state.eps}\n\\caption{$\\tilde{\\sigma}-\\phi$ phase space diagram for fibers $AR = 10, 16$, and 36. Increasing aspect ratio decreases the jamming volume fraction.  The suspension is jammed for the region on the right of the solid curve.}. \\label{fig:flow_state}\n\\end{figure}\n\n\\section{Conclusion}\n\nThis paper presents a constitutive model for frictional fiber suspensions in a steady shear flow. The proposed model quantifies the effects of three independent parameters - fiber volume fraction $\\phi$, dimensionless stress $\\tilde{\\sigma}$, and fiber aspect ratio $AR$. The model quantitatively predicts the viscosity of a  range of aspect ratios and volume fractions once it is calibrated using the zero and high shear viscosities of just a few aspect ratio cases.\n\n\n\n\n\nWe start our analysis by deriving relations describing the dependence of jamming volume fraction on the friction coefficient to reflect the impact of this microscopic parameter on the suspension rheology. However, constant friction cannot explain the rate-dependent rheology in suspensions. Hence, we use the relations derived from the constant coefficient of friction for quantitatively comprehending the case with load-dependent friction coefficient.  This approach assists us in defining the stress-dependent jamming fraction $\\phi_m(\\tilde{\\sigma})$ by interpolating between two jamming fractions at the extreme limit of stresses, in the manner proposed by Wyart and Cates \\citep{wyart2014discontinuous} for shear thickening suspensions. The divergence of viscosity approaching  the interpolated jamming volume fraction follows the form of the two limits, with the viscosity growing as $\\phi_m^{-0.90}(\\phi_m-\\phi)^{-0.90}$. Furthermore, we extend the constitutive model to capture the aspect ratio-dependent rheological behavior of the suspension. Once the aspect ratio-dependent rheological properties in the low and high shear limits are known, we can use the constitutive model to quantify the aspect ratio's effect on the rheology between the two stress limits. The model predictions for the relative viscosity, $\\eta_r(\\phi,\\tilde{\\sigma},AR)$ agree well not only over the full range of parameters it is calibrated on but beyond this range as well as evident from a good agreement between experimental measurements and model predictions. In the end, we display the dependence of the jamming volume on the applied stress and aspect ratio of the fibers through a flow state diagram in the $\\phi - \\tilde{\\sigma}$ plane.\n\nFinally, these results can be used to quantitatively predict the shear-thinning suspension behavior at different fiber aspect ratios and volume fractions. This model can be used to estimate the effective coefficient of friction in the experiment from the jamming volume fraction. The idea thus gained can be utilized in manipulating suspension behavior by changing the fiber size along with mechanisms based on hydrodynamic\ninteractions, fiber surface roughness \\citep{khan2021rheology}, and friction \\citep{salahuddin2013study, switzer2003rheology}. We found that to have higher solid concentrations desired in industrial applications, we should break down the fibers into smaller sizes or reduce the coefficient of friction through surface treatment.\n\n\n\n\n\n\\begin{acknowledgments}\nAMA would like to acknowledge financial support from the Department of Energy via grant EE0008910. \n\\end{acknowledgments}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{#1}}\n\\renewcommand{\\Im}{{\\rm Im }}\n\\renewcommand{\\Re}{{\\rm Re }}\n\n\\def\\one{{\\hbox{ 1\\kern-.8mm l}}}\n\\def\\ch{{\\rm ch \\, }}\n\\def\\tr{{\\rm tr\\,}}\n\\def\\sgn{{\\rm sgn\\,}}\n\\def\\vol{{\\rm vol\\,}}\n\\def\\p{\\partial}\n\\def\\ba{\\bar{a}}\n\\def\\bb{\\bar{b}}\n\\def\\bc{\\bar{c}}\n\\def\\bd{\\bar{d}}\n\\def\\be{\\bar{e}}\n\\def\\bz{\\bar{z}}\n\\def\\bZ{\\bar{Z}}\n\\def\\bW{\\bar{W}}\n\\def\\bD{\\bar{D}}\n\\def\\bA{\\bar{A}}\n\\def\\bB{\\bar{B}}\n\\def\\bR{\\bar{R}}\n\\def\\bS{\\bar{S}}\n\\def\\bT{\\bar{T}}\n\\def\\bU{\\bar{U}}\n\\def\\bPi{\\bar{\\Pi}}\n\\def\\bOmega{\\bar{\\Omega}}\n\\def\\bpartial{\\bar{\\partial}}\n\\def\\bj{{\\bar{j}}}\n\\def\\bi{{\\bar{i}}}\n\\def\\bk{{\\bar{k}}}\n\\def\\bl{{\\bar{l}}}\n\\def\\bm{{\\bar{m}}}\n\\def\\btheta{\\bar{\\theta}}\n\\def\\bpsi{\\bar{\\psi}}\n\\def\\bF{\\bar{F}}\n\\def\\bs{\\bar{s}}\n\\def\\bt{\\bar{t}}\n\\def\\bv{\\bar{v}}\n\\def\\bx{\\bar{x}}\n\\def\\by{\\bar{y}}\n\\def\\bz{\\bar{z}}\n\\def\\btau{\\bar{\\tau}}\n\\def\\bA{\\bar{A}}\n\\def\\bB{\\bar{B}}\n\\def\\bC{\\bar{C}}\n\\def\\bX{\\bar{X}}\n\\def\\bY{\\bar{Y}}\n\\def\\bZ{\\bar{Z}}\n\\def\\bnabla{{\\bar \\nabla}}\n\\def\\GeV{{\\rm GeV}}\n\\def\\TeV{{\\rm TeV}}\n\\def\\un{\\underline{0}}\n\n\\def\\sx{{\\bf x}}\n\\def\\sy{{\\bf y}}\n\\def\\vx{{\\vec{x}}}\n\\def\\vy{{\\vec{y}}}\n\n\n\\def\\cl{C\\ell}\n\\def\\ch{{\\rm ch}}\n\\def\\cod{{\\rm cod}}\n\\def\\tOmega{\\tilde{\\Omega}}\n\\def\\half{\\frac{1}{2}}\n\\def\\ts{\\tilde{\\sigma}}\n\\def\\Dsl{\\gamma\\cdot D}\n\\def\\c{\\check }\n\\def\\cln{C\\ell_n}\n\\def\\clnp{C\\ell_n^+}\n\\def\\clnm{C\\ell_n^-}\n\\def\\rtar{\\rightarrow}\n\\def\\Re{{\\rm Re}}\n\n\n\n\\def\\a{\\alpha}\n\\def\\B{b}\n\\def\\c{\\check }\n\\def\\Cl{C\\ell}\n\\def\\cl{C\\ell}\n\\def\\ch{{\\rm ch}}\n\\def\\cod{{\\rm cod}}\n\\def\\Dsl{\\gamma\\cdot D}\n\\newcommand{\\gpd}{\/\\!\/\\,}\n\\def\\half{\\frac{1}{2}}\n\\def\\KRX{{}^w K(\\CX) }\n\\def\\M{M}\n\\def\\ofd{orientifold}\n\\def\\Re{{\\rm Re}}\n\\def\\tko{{\\rm Twist}_{KO_{\\IZ_2}}(\\CX_w)}\n\\def\\tkr{{\\rm Twist}_{KR}(\\CX_w)}\n\\def\\y{\\yellow}\n\\def\\ZZ{\\mathbb{Z}}\n\\def\\zt{\\IZ_2}\n\n\n\n\n\\title{A Shifted View of Fundamental Physics }\n\n\\author{Michael    Atiyah and  Gregory W. Moore  }\n\n\\abstract{We speculate on the role of relativistic versions of delayed differential equations\nin fundamental physics. Relativistic invariance implies that we must consider both\nadvanced and retarded terms in the equations, so we refer to them as shifted equations.\nThe shifted Dirac   equation has some novel properties. A tentative formulation\nof shifted Einstein-Maxwell equations naturally incorporates a small but nonzero\ncosmological constant. }\n\n\n\\begin{document}\n\n\\section{Prologue by Michael   Atiyah }\n\n The past three or four decades have seen a remarkable fusion of\n theoretical physics with mathematics.  Much of the impetus for this is due to\n Is Singer whose 85th birthday is being celebrated at this meeting.\n He introduced me \\footnote{This joint article is based on the lecture by the first author at Singer 85} to many of the physical ideas and instructed me in the areas of mathematics that had interacted strongly with physics in the first part of the 20th century.  These were differential geometry, flourishing in the wake of Einstein's theory of general relativity and functional analysis, which provided the rigorous background for quantum mechanics as laid down by von Neumann.\n\n\n By contrast my own mathematical background has\n centered round algebraic geometry and topology, the areas\n  of mathematics which have played an increasingly important\n   part in the developments in physics of the last quarter of the 20th century and beyond.\n\n\n I think Is and I were fortunate to be at\n the right place at the right time, working on the kind of\n mathematics which became the new focus of interest in theoretical\n physics of gauge theories and string theory.\n\n\n Looking to the past the great classical era began with\nNewton, took a giant stride with Maxwell and culminated in Einstein's spectacular theory of general relativity.\n\n\n But then came quantum mechanics and quantum\n field theory with a totally new point of view, rather far removed from the geometric ideas of the past.\n\n\n The major problem of our time for physicists is\n how to combine the two great themes of GR, that governs the\n large scale universe, and QM that deals with the very small scale.\n\n At the present time we have string theory, or\nperhaps ``M-theory\", which is a beautiful rich mathematical\nstory and will certainly play an important role in the future of both\nmathematics and physics.  Already the applications of these ideas to mathematics\nhave been spectacular.  To name just a few, we have\n\n\\begin{enumerate}\n\\item Results on the moduli spaces of Reimann surfaces.\n\\item The Jones polynomials of knots and their extension by Witten to ``quantum invariants\" of 3-manifolds.\n\\item Donaldson theory of 4-manifolds and the subsequent emergence of Seiberg-Witten theory.\n\\item Mirror symmetry between holomorphic and symplectic geometry.\n\\end{enumerate}\n\n\nAlthough string theory or M-theory are thought by many to be the ultimate theory combining QM and GR no-one   knows what M-theory really is.  String theory is recognized only as a perturbative theory, but the full theory is still a mystery (one of the roles of the letter M).\n\n\n Some claim that the final\ntheory is close at hand - we are almost there.  But perhaps this is misplaced\noptimism and we await a new resolution based on radical new ideas.  There are,\nafter all, some major challenges posed by astronomical observations.\n\n\\begin{itemize}\n  \\item Dark Matter\n  \\item Dark Energy (with a very small cosmological constant)\n\\end{itemize}\n\n\n Moreover, the direct linkage between the rarified mathematics\nof string theory and the world of experimental physics is, as yet,\n very slender.  A friend of mine, a retired Professor of Physics,\n commentating on a string theory lecture, said it was \"pure poetry\"!\n This can be taken both as criticism and as a tribute.  In the same way\n science fiction cannot compete with the modern mysteries of the quantum vacuum.\n\n\n But, if we need new ideas, where will they come from?  Youth is the traditional source of radical thoughts, but only a genius or a fool would risk their whole future career on the gamble of some revolutionary new point of view.  The weight of orthodoxy is too heavy to be challenged by a PhD student.\n\n\n So it is left to the older generation like me to speculate.  The same friend who likened string theory to poetry encouraged me to have wild ideas, saying \"you have nothing to lose!\" That is true, I have my PhD.  I do not need employment and all I can lose is a bit of my reputation.  But then allowances are made for old-age, as in the case of Einstein when he persistently refused to concede defeat in his battle with Niels Bohr.\n\n\n So my birthday present to Is is to tell him that we senior citizens can indulge in wild speculations!\n\n\\vfill\\eject\n\n\\section{An Exploration}\n\n\n\nThe idea we\\footnote{The work of G.M. is supported by DOE grant DE-FG02-96ER40959. He would\nlike to thank T. Banks and S. Thomas for discussions.}\n want to explore is the use of retarded (or advanced) differential equations in fundamental physics.\nThese equations, also known as\n ``functional differential equations,'' or ``delayed differential equations'' have been much studied by\nengineers and mathematicians, but the applications in fundamental physics have been limited.\nThis idea has a number of different origins:\n\n\\begin{description}\n\\item[(i)] Such retarded differential equations occur in Feynman's thesis \\cite{FeynmanThesis}.\n\\item[(ii)] In the introduction to Bjorken and Drell \\cite{BD}  it is suggested that, if space-time at very small scales is ``granular,\" then one would have to use such equations.\n\\item[(iii)] Their use in the context of quantum mechanics has been advocated by C.K. Raju \\cite{Raju}.\n\\end{description}\n\n\\noindent We will discuss the general idea briefly before going on to explain how to develop a more\nscientific version.  This one of us worked on at an earlier stage as reported in the Solvay Conference\n\\cite{SolvayTalk}.\n\n  \n  Let us begin by looking at the simplified example of a linear retarded differential equation for a  function $x(t)$:\n\\begin{equation}\\label{eq:ret}\n\\dot{x}(t)+kx(t-r)=0\n\\end{equation}\n where the positive number $r$ is the retardation parameter and $k$ is a constant. A \n rescaling of the time variable shows that the equation really only depends on a single\n dimensionless parameter $\\mu = kr$.  Moreover,\n the initial data for such an equation is an arbitrary function $g(t)$ over the interval $[0,r]$.\n  Successive integration then allows us to extend the function for all $t \\geq 0$, while successive differentiation (for smooth initial data $g(t)$) enables us to extend to negative $t$.  A second way to discuss the solutions is to note that the functions\n $x(t) = x_0 e^{- z t\/r}$ solve \\eqref{eq:ret} provided $z = \\mu e^z$. The latter transcendental\n equation has an infinite set of roots tending to $z=\\infty$. (In general, all the roots $z$ have\n nonzero real part, and hence the solutions have an unphysical divergence in the far past or future.)\n\n\nFrom either approach,  we note that  equation    \\eqref{eq:ret},\n  like the equations of quantum mechanics, has an infinite-dimensional space of initial data:\n   It can be taken to be the Hilbert space $L^2[0,r]$. We will take that as an encouraging sign and,\n without pursuing further the  parallel with quantum theory at present we can ask whether retarded differential equations make any sense in a relativistic framework where there is no distinguished time direction in which to retard. In fact this can be done and there is a natural and essentially unique way to carry this out.  The first observation is that the translation $t\\rightarrow t-r$ has the infinitesimal generator $-r\\frac{d}{dt}$, so that the translation is formally just $\\exp(-r\\frac{d}{dt})$.\n In Minkowski space we need a relativistically invariant version of $\\frac{d}{dt}$, i.e. a relativistically invariant first order differential operator.  But this is just what Dirac was looking for when he invented the Dirac Operator $D$.  The important point is that $D$ acts not on scalar functions but on \\textbf{spinor} fields.  Thus, we can write down a relativistic analogue of \\eqref{eq:ret}  which is a retarded version of the usual Dirac equation\n\\begin{equation}\\label{eq:ret-Dir}\n\\{i\\hbar D-mc+ik \\exp(-rD)\\}\\psi =0\n\\end{equation}\n where $D$ is the Dirac operator and $\\psi$ is a spinor field.  Note that $r$ has dimensions of length\n $L$  whilst  $k$ (slightly different from that\nappearing in \\eqref{eq:ret}) now has the physical dimension $MLT^{-1}$.\n\n  Both $k$ and $r$  are required to be real for physical reasons which will be clarified shortly (see\n\\eqref{eq:quant} and \\eqref{eq:equiv-dev} below).  This appears to be rather a formal equation and one can question whether it makes any sense.  In fact \\eqref{eq:ret-Dir} makes sense for all physical fields $\\psi$, i.e. those which propagate at velocities less than the velocity of light.  Any such $\\psi$ is a linear combination of plane-waves and the operator $\\exp(-rD)$ applied to such a plane-wave component just retards it by $r$ in its own time-direction.\nFor waves which travel with velocity $c$ mathematical arguments based on continuity, or physical arguments using  clocks, require that there be no retardation.\n \n Although we have said that \\eqref{eq:ret-Dir} is a retarded equation the fact that spinors have both positive and negative frequencies implies that it is also an advanced equation.  Perhaps we should use a neutral word such as ``shifted\" instead of advanced or retarded. Having said that we may consider several variants of the shifted\nDirac equation where we replace\n\n\\begin{subequations}\\label{eq:D-mod}\n\\begin{align}\nD & \\to    \\ID_+ := D + \\frac{k}{\\hbar} e^{-r D} \\label{eq:D-plus}  \\\\\nD & \\to  \\ID_- := D - \\frac{k}{\\hbar} e^{r D} \\label{eq:D-minus} \\\\\nD & \\to   \\ID_s := D + \\frac{k}{\\hbar} \\sinh(r D) \\label{eq:D-s} \\\\\nD & \\to   \\ID_c := D + i \\frac{k}{\\hbar} \\cosh(rD) \\label{eq:D-c}\n\\end{align}\n\\end{subequations}\n\nFor brevity we will focus on the modification \\eqref{eq:D-plus} in what follows.\n\n\n\n\\medskip\n\n\nIt is instructive to examine the plane-wave solutions of the modified Dirac equations\nin Minkowski space.\nWe take $\\psi = s(p) e^{-i p\\cdot x\/\\hbar}$ where $s(p)$ is a constant spinor and\nthere is a dispersion relation  $p^2 = p_\\mu p^\\mu = E_0^2\/c^2$, with $E_0> 0$\nrepresenting the inertial rest energy of a particle. Acting on such a wavefunction\n$D \\psi = \\frac{E_0}{c} \\gamma\\cdot \\hat p \\psi $ where $\\gamma\\cdot \\hat p$ squares to $1$.\nLet $s_\\pm(p)$ denote the eigenspinors.\nThe planewave solutions of the shifted Dirac equation\n\\begin{equation}\\label{eq:mod-Dir}\n(i \\hbar \\ID_+ - mc )\\psi =0\n\\end{equation}\nare then $ s_+(p)e^{-i p\\cdot x\/\\hbar}$ and $s_-(p)e^{i p\\cdot x\/\\hbar}$ provided\n\\begin{equation}\\label{eq:ener-q}\n\\frac{E_0}{c}-mc+ik \\exp(\\dfrac{irE_0}{\\hbar c})=0.\n\\end{equation}\nSince the first two terms are real we derive a   quantization condition\n\\begin{equation}\\label{eq:quant}\n\\dfrac{rE_0}{\\hbar c}=(n+1\/2)\\pi \\hspace{15pt} \\textrm{with} \\hspace{15pt}  n \\hspace{15pt} \\textrm{integral}\n\\end{equation}\n Note that, thanks to the half-integer shift,\n  the value $r=0$ is excluded in \\eqref{eq:quant}.\\footnote{Given the speculative connections to\n   quantum mechanics mentioned above it is natural to wonder if this is related to the zero point energy\n   of the harmonic oscillator.} This is related to and explains the factor $i$ in \\eqref{eq:ret-Dir} ($k$ being real).  Replacing $n$ by $n-1$ in \\eqref{eq:quant} is equivalent to changing the sign of $k$ in \\eqref{eq:ret-Dir}.  The two signs of $k$ are on an equal footing and so we should consider both.\n A similar discussion with a massless dispersion relation, i.e. $E_0=0$, shows that there\n are no solutions: It is not possible to retard a massless stable fermion.\n\n\n There are two ways to interpret the quantization condition \\eqref{eq:quant}. First, it is useful to\n rewrite it by recalling\n that the Compton wavelength of a particle of rest energy $E_0$ is $\\lambda_c = \\frac{2\\pi \\hbar c}{E_0}$.\n Thus we have\n\n \\begin{equation}\\label{eq:quant-ii}\n r = \\frac{2n+1}{4} \\lambda_c.\n \\end{equation}\n\nThe first interpretation of this  equation is that it demands\n that different fermions have different retardation\n parameters, given by \\eqref{eq:quant-ii}. We might expect this to become problematic\n when, say, electrons interact with protons, neutrons, or neutrinos.\n A second interpretation declares that there is a universal\n retardation time in Nature, denoted $r$. In this note we will adopt the second\n point of view.\n\nThe question arises as to the magnitude of $r$ and $n$. The hypothesis we\nare entertaining is that the modified Dirac equation \\eqref{eq:mod-Dir}   should\napply to stable fermions whose propagation in vacuum would ordinarily be\ndescribed (to good approximation) by a standard Dirac equation.\nThus we are led to consider protons, neutrons, electrons,\nand neutrinos.   The Compton wavelength of the\nelectron is $\\lambda_c^e \\sim 10^{-12}$m,  while for the proton and neutron we have\n$\\lambda^{p,n}_c\/\\lambda_c^e \\sim 10^{-3}$, whilst the lightest neutrino probably has\n$10^5 < \\frac{\\lambda^{\\nu}_c}{\\lambda_c^e} < 10^9$ \\cite{PDG,Vogel:2008zzb}. Equation \\eqref{eq:quant-ii} shows that $r$ is\nbounded below by $\\lambda_c\/4$. Optimistically taking the smallest nonzero neutrino mass\n $\\sim 1 eV$ we have\n\\begin{equation}\\label{eq:r-bound}\nr \\succsim 10^{-5} {\\rm cm}.\n\\end{equation}\nThis scale is uncomfortably large and we hence take the corresponding integer\nfor the lightest neutrino to be\nof order one.\n\n\n\n Returning to \\eqref{eq:ener-q} we have:\n\\begin{equation}\\label{eq:mc2-dev}\nE_0=mc^2+(-1)^n kc,\n\\end{equation}\nThis is not a deviation from Einstein's formula relating rest energy to mass,\n but simply the relation of the\ninertial rest energy to the parameters of the modified Dirac equation.\n\n\nOf course, the modified Dirac equations will lead to deviations from standard\nphysical results and hence we expect $k$ and $r$ to be small. One\ninteresting deviation is in the equivalence principle. The modified\n Dirac equation can be derived from an action principle\n\n \\begin{equation}\\label{eq:action}\n \\int \\vol \\bar \\psi \\left( i \\hbar \\ID_+ - mc \\right) \\psi\n \\end{equation}\n\n from which one can derive the energy-momentum tensor $T_{\\mu\\nu}$. For our purposes it will\n suffice to consider the action in a weak\n gravitational field\n\n \\begin{equation}\\label{eq:weak-grav}\n ds^2 = - (1-2 \\Phi) dt^2 + dx^i dx^i,\n \\end{equation}\n\n  where $\\Phi$ is the Newtonian\n gravitational potential, and extract the coefficient of $\\Phi$ to obtain the gravitational\n rest energy. In the curved metric \\eqref{eq:weak-grav} the Dirac\n operator $D = \\gamma\\cdot \\p + S + \\cdots $ with $S = \\Phi \\gamma^0 \\p_0 + \\half \\gamma_i \\p_i \\Phi$\n Making this substitution, and using on-shell spinor wavefunctions for a particle\n at rest we find\n\\begin{equation}\\label{eq:equiv-dev}\nT_{00}=E_0\\left(1- \\mu (-1)^n\\right)\n\\end{equation}\nwhere\n\\begin{equation}\\label{eq:def-mu}\n\\mu:= \\frac{kr}{\\hbar}\n\\end{equation}\n is a dimensionless number. We thus find a\n deviation from the equivalence principle\n\n \\begin{equation}\n \\frac{ T_{00} - E_0}{E_0} =(-1)^{n+1} \\mu.\n \\end{equation}\n\n  Consequently, the parameter $\\mu$ will have to be very small (less than about $10^{-13}$) not to contradict observational evidence \\cite{Will:2010uh}.   Since $r$ is uncomfortably large  according to\n  \\eqref{eq:r-bound} it must be that $k\/\\hbar$ is small, something which will prove to be\n  interesting in Section \\ref{sec:Cosmo-Const} below. Indeed\n \n  \\begin{equation}\\label{eq:k-upper}\n  \\frac{k}{\\hbar} < 10^5 \\mu {\\rm cm}^{-1} < 10^{-8 } {\\rm cm}^{-1} .\n  \\end{equation}\n \n  The parameter $\\mu$\n   is a fundamental   constant of our ``theory\" and links together the two key dimensionful parameters $r$, the shift, and $k$, the coefficient that measures the magnitude of the term that shifts the Dirac operator.  Moreover \\eqref{eq:def-mu} involves Planck's constant, reflecting the quantum character of the parameter $kr$.  All these parameters are embodied in our basic choice of the shifted Dirac operator.\n\n\nWe close this section with three sets of remarks.\n\n\\subsection{Remarks regarding the formal modification of the Dirac operator}\n\n\n\\begin{enumerate}\n\n\\item  Let us comment on the modified Dirac equation for the other choices $\\ID_-, \\ID_s, \\ID_c$.\nUsing $\\ID_-$ instead of $\\ID_+$ we obtain the same results by replacing particles with antiparticles.\nFor the modified Dirac equations using $\\ID_c$ and $\\ID_s$\n there is no  quantization condition such as \\eqref{eq:quant}, whilst \\eqref{eq:mc2-dev} is\n replaced by a transcendental relation between $E_0, m, k, r$. The equation with $\\ID_s$ and\n $m=0$ is compatible with a massless dispersion relation but that with $\\ID_c$ is not.  The violation of the\n equivalence principle \\eqref{eq:equiv-dev} is similar in all four cases.\n\n\\item One may ask how to define the shifted version of the Klein-Gordon equation for\nscalar fields. The\nusual relation $D^2 = - \\nabla^2$ is more complicated for modified Dirac operators\n\\eqref{eq:mod-Dir}. One simple possibility is to replace $D^2 \\to \\ID_+ \\ID_-$. The\nlatter can be interpreted as a power series in $D^2$:\n\\begin{equation}\n\\ID_+ \\ID_- = D^2 - 2\\mu D^2 \\left( \\frac{\\sinh r D}{rD}\\right) - \\frac{k^2}{\\hbar^2}\n\\end{equation}\nReplacing $D^2 \\to - \\nabla^2$ in this expression produces a candidate modified Klein-Gordon\noperator.\n\n\n \\item   We can extend, or couple, the Dirac operator to other fields.  Thus coupling to spinors again we get the\n operator $d+\\delta$ acting on all differential forms (where $\\delta$ is the Hodge adjoint of $d$).  We can exponentiate this operator but since it does not preserve the degrees of forms, we cannot just restrict it say to 2-forms.  Treating Maxwell's equations requires more care and we will come back to this later in Section\n \\ref{ref:E-M}.\n\n\n\n\\item  Following Dirac, our search for relativistically\n shifted equations inevitably led us to spinors. An interesting question\n is how such shifted equations interact with supersymmetry. Perhaps one\n should shift field equations in superspace.\n\n\n\\end{enumerate}\n\n\\subsection{Further remarks on advanced and delayed equations in general }\n\n\n\n\\begin{enumerate}\n\n\\item Returning to  \\eqref{eq:ret}, we note that it is an unusual equation since it involves\n a sum of a skew adjoint\n operator with a unitary operator. Thus it involves an element of an\n ``affine operator group.''  This group\n    is in turn a degenerate form  of a semi-simple group, reminiscent of the\nWigner contraction of   semi-simple groups to inhomogeneous orthogonal groups\nsuch as the  Poinar\\'e group.\n\n\\item\nShifting a differential equation in a way that involves both retarded and advanced terms drastically alters its nature, even in the simple 1-dimensional case when  \\eqref{eq:ret} is replaced by\n\\begin{equation}\\label{eq:adv-ret-x}\n\\dot{x}(t)+k\\{x(t-r)+x(t+r)\\}=0.\n\\end{equation}\n This is no longer a simple evolution equation\n and the theory of such equations has hardly been developed.\n However, once again, there is an infinite-dimensional space of\n  exponential solutions $x(t)= e^{-zt\/r}$ where $z$ now satisfies the equation\n\\begin{equation}\\label{eq:tran-2}\nz = \\mu (e^z + e^{-z}).\n\\end{equation}\nAs before, there is an infinite set of roots $z_n$ which tend to infinity for $n \\to \\pm \\infty$.\nAn infinite-dimensional space of solutions is given by the linear combinations\n\\begin{equation*}\n\\sum_n x_n \\exp(-z_n t\/r).\n\\end{equation*}\n\n\n\\item\nThe roots of \\eqref{eq:tran-2} always have nonzero real parts so once again there is\nunacceptable behavior in the past or future. However, it should be noted that in the\nlimit $\\mu \\to \\infty$ these real parts tend to zero. This is particularly\nclear if one takes $k\\to \\infty$ in \\eqref{eq:adv-ret-x}, since the solutions\nplainly become anti-periodic with period $2r$. Thus, for large   $\\mu$ there is a\nfinite-dimensional space of solutions with acceptable oscillatory behavior, at least in a\nrestricted time domain. Of course our interest is in the opposite case of very small $k$, but this might conceivably be related by some kind of duality. The parallel interchange of small and large values of $r$ arises in Fourier theory. In fact Feynman in \\cite{FeynmanThesis} studies an equation similar to\n \\eqref{eq:adv-ret-x} where $r$ represents the distance between a particle and a virtual image.\n  He shows that,   surprisingly, there is a conserved energy. (In fact his system is equivalent to a standard physical one without retardation.) For Feynman $r$ is large while for us $r$ is small.\n\n\n\n\\item For our shifted Dirac operator we expect that  mathematical\ndifficulties of the sort encountered in the previous two remarks arise from the mixing of the positive\nand negative frequencies in the presence of external forces.\nIn standard quantum field theory such mixing of states   forces the introduction\nof Fock space in quantum field theory.  In other words our shifted Dirac\noperator may be easy to define but its mathematical and physical implications\ncan be profound.  A link to quantum theory would not therefore be too surprising.\n\n\n\n\\end{enumerate}\n\n\n\n\\subsection{Some more phenomenological remarks}\n\n\n\\begin{enumerate}\n\n\n\\item We have not attempted to investigate the modified Dirac equation in nontrivial\nelectromagnetic backgrounds. Moreover, we have not attempted to include the effects\nof quantum interactions and thus explore the effects on the standard successes of\nQED such as Bhaba scattering, Compton scattering, the Lamb shift, the anomalous magnetic\nmoment etc.  This would\nbe a logical next step if our present speculations are to be taken further. The\nexistence of nontrivially interacting string field theories gives some hope that such\ninteractions can be sensibly included.\n\n\\item Since the modified Dirac equation \\eqref{eq:mod-Dir} is not compatible with massless fermions it is not\nat all obvious how to include important properties of the standard model such as chiral\nrepresentations of the gauge group and the Higgs mechanism into our framework.\n\n\n\\item It follows from \\eqref{eq:quant-ii} that\n all particles have a Compton wavelength (and hence\n mass) related by a ratio of two odd integers. Of course any\n real number admits such an approximation, but  the numerator\n and or denominator in such an approximation cannot become\n too large without making $r$ too large. For the electron\n and proton these integers must be large, but this is not \n obviously so for neutrinos. Perhaps this idea\n can be tested experimentally as our knowledge of neutrino\n masses improves.\n\n\n\n\n\\end{enumerate}\n\n\n\n\\section{The Shifted Einstein Equations}\\label{sec:Einstein}\n\n\\subsection{Definition}\\label{sec:Shift-Ricci}\n\nEven if we do not know how shifted equations may relate to quantum mechanics we can still ask if there is any way of producing a shifted version of the Einstein equation of General Relativity.\n\n\\medskip\n\n We begin by recalling some standard differential geometry, initially in the\npositive-definite Riemannian version. Recall there  \n are two natural second order differential operators \n active on the space of 1-forms, both generalizing the Laplacian of flat space,\n\n\\begin{description}\n\\item[(i)] the Hodge Laplacian $= (d+\\delta)^2=\\Delta$\n\\item[(ii)] the Bochner Laplacian $= \\nabla^*\\nabla$ where\n\\end{description}\n\n\\begin{equation*}\n\\nabla : \\Omega^1\\rightarrow \\Omega^1 \\otimes \\Omega^1\n\\end{equation*}\n\n\\noindent is the covariant derivative.\n\n\\noindent The Weitzenbock formula asserts that\n\n\\begin{equation}\\label{eq:H-B}\n\\textrm{Hodge - Bochner = Ricci}\n\\end{equation}\n\n\\noindent This, on a compact manifold, was originally used by Bochner to prove that, if the Ricci tensor was positive definite, then there were no harmonic 1-forms and so the first Betti number was zero.\n\n\n\n  Formula \\eqref{eq:H-B} continues to hold for a Lorentzian manifold, though sign conventions have to be carefully checked. We replace $\\nabla^*\\nabla$ with $-\\nabla^2$ and define $\\delta$ using the Hodge star.\n   More fundamentally, we view $D= d+\\delta$ as a Dirac operator coupled to the  spin bundle. Then, we have\n  \n   \\begin{equation}\\label{eq:Mink-H-B}\n   D^2 + \\nabla^2 = \\textrm{Ricci }\n   \\end{equation}\n  \n   as an operator equation on $1$-forms. We are indebted to Ben Mares for his careful calculations to confirm signs.\n\n  Equation \\eqref{eq:Mink-H-B} says that  the Einstein vacuum equations, Ricci $=0$, asserts the equality of the two Laplacians on 1-forms.  This indicates how we might define a shifted Hodge Laplacian   on 1-forms: We replace $D$ by one of the modified Dirac operators \\eqref{eq:D-mod}. There are 10 ways to do this, but we will focus on only three:\n \\begin{subequations}\\label{eq:Ricc-mod}\n \\begin{align}\n \\textbf{Ricci}_{++} := & c(\\ID_+)^2 + \\nabla^2 \\label{eq:Ric-pp} \\\\\n  \\textbf{Ricci}_{+-} := & \\ID_+ \\ID_- + \\nabla^2 \\label{eq:Ricc-pm}\\\\\n \\textbf{Ricci}_{ss} : = &  \\ID_s^2  + \\nabla^2 \\label{eq:Ricc-ss}\n \\end{align}\n \\end{subequations}\n\n\nIn equation \\eqref{eq:Ric-pp} we cannot  simply square the operator $\\ID_+$\n because the resulting operator does not preserve the\n  degree of forms. However, we can then compress it   on 1-forms by taking the composite operator\n\\begin{equation}\\label{eq:Comp}\nc(\\ID_+^2) := P \\left(\\ID_+ \\right)^2 I\n\\end{equation}\n where \\textbf{$I$} is the inclusion $\\Omega^1 \\rightarrow \\Omega^*$ and $P$ is the projection $\\Omega^* \\rightarrow \\Omega^1$. The other two operators do not need compression.\n\n\nThe shifted Einstein equations should then be given by equating \\textbf{Ricci} to zero. Written out,\nthese are the equations\n \\begin{subequations}\\label{eq:Ein-mod}\n \\begin{align}\n  \\textrm{Ricci}-\\frac{2k}{\\hbar} D \\sinh(rD) + \\frac{k^2}{\\hbar^2}\\cosh(2rD)  & = 0  \\label{eq:Ein-pp} \\\\\n   \\textrm{Ricci}-\\frac{2k}{\\hbar} D \\sinh(rD) - \\frac{k^2}{\\hbar^2}  & = 0  \\label{eq:Ein-pm} \\\\\n    \\textrm{Ricci}+\\frac{2k}{\\hbar} D \\sinh(rD) + \\frac{k^2}{\\hbar^2}\\sinh^2(rD)  & = 0  \\label{eq:Ein-ss}\n \\end{align}\n \\end{subequations}\n\n\nIt is  not at all clear how to interpret these operator equations. We will comment further\non this point in Section  \\ref{subsec:Einst-interp} below.\nOne interpretation is to regard\nthe operator as an expansion in $D^2$, then, using \\eqref{eq:Mink-H-B}, convert this to\n an expansion in $\\nabla^2$. Then\nwe can view higher order terms in $\\nabla^2$ as an expansion in low energy and small momenta.\n\n\n\n\n\n\\subsection{The cosmological constant}\\label{sec:Cosmo-Const}\n\nIf we adopt the viewpoint that the shifted Einstein equations \\eqref{eq:Ricc-mod}\ncan be understood as an expansion in low energies then\n the leading    term is a well-defined equation on the metric given by\n replacing $D^2 \\to {\\rm Ricci}$ in \\eqref{eq:Ein-mod}. Using the fact that\n $\\mu$ must be small the modified Einstein equations become approximately\n \\begin{subequations}\\label{eq:Cosmo-Const}\n \\begin{align}\n  \\textrm{Ricci}   & = -\\frac{k^2}{\\hbar^2}   \\label{eq:Cosmo-const-pp} \\\\\n  \\textrm{Ricci}   & = \\frac{k^2}{\\hbar^2}   \\label{eq:Cosmo-const-pm} \\\\\n  \\textrm{Ricci}   & = 0   \\label{eq:Cosmo-const-ss}\n \\end{align}\n \\end{subequations}\n\n    In other words the first correction to the Einstein equation produced by our shifted operators is to introduce a cosmological constant $\\Lambda = \\pm \\frac{k^2}{\\hbar^2}$. This relates our\n     parameter $k$ to cosmology. Evidently, the theory can accommodate all three cases of positive, negative,\n    and zero cosmological constant, and hence   the correct sign of the cosmological constant is hardly a triumph for our approach. The observational data support the case \\eqref{eq:Cosmo-const-pm}.\n    Any logical consideration which distinguishes or rules out some  of the possible shifted Ricci\n    tensors would  be most interesting.\n\n\n\n    The magnitude of $\\Lambda$ is also interesting. As we have seen,\n    observed bounds on violations of the equivalence principle\n    imply that $k\/\\hbar$ is small and hence $\\Lambda$ is small. In fact, the observational\n      evidence\n  \\footnote{See, for example, \\cite{Tegmark:2003ud}.}   gives an order of magnitude for $\\Lambda$\n  corresponding to an energy density $\\rho_{vac} \\sim (1 eV)^4$ and\n  hence $\\Lambda = \\frac{8 \\pi G}{c^4} \\rho_{vac}$ of order\n\\begin{equation}\n\\Lambda \\sim 10^{-56}\\textrm{cm}^{-2}\n\\end{equation}\n and hence\n\\begin{equation}\\label{eq:mu-mag}\n\\frac{k}{\\hbar} \\sim 10^{-28}\\textrm{cm}^{-1},\n\\end{equation}\nmuch smaller than the upper bound \\eqref{eq:k-upper}.  If we now take the order of magnitude of $r$ derived from our quantization condition\nthen it follows from \\eqref{eq:r-bound} that\n\\begin{equation}\\label{eq:mu-fin}\n\\mu \\succsim  10^{-33}.\n\\end{equation}\n\n\n\\subsection{Further comments on interpretation}\\label{subsec:Einst-interp}\n\nIn Section \\ref{sec:Shift-Ricci} we showed formally how to shift the Ricci tensor, giving formula\n\\eqref{eq:Ricc-mod}, and we focused on the lowest order correction which gave the cosmological constant.  We then noted the order of magnitudes that emerged.  Notably the estimate \\eqref{eq:mu-fin} for our dimensionless parameter $\\mu=kr\/\\hbar$.  But, as we noted, it is not at all clear how to interpret the operator equation (acting on 1-forms)\n\\eqref{eq:Ein-mod}.  We will now discuss this question.\n\n\n  In the first place, although $D$ does not preserve the degrees of forms,   a power series in the dimensionless quantity $r^2 D^2=r^2 \\Delta$  should make sense (even non-perturbatively) on the space of 1-forms.\n\n\n  Second it seems reasonable to restrict \\eqref{eq:Ein-mod} to 1-forms which are locally solutions of the wave equation\n\\begin{equation}\\label{eq:wave}\n- \\nabla^2 \\phi=0\n\\end{equation}\nand hence determined by initial conditions along a space-like 3-space.  In view of  \\eqref{eq:H-B}, on such 1-forms $\\phi$, we have\n\n\\begin{equation*}\n\\Delta \\phi = \\textrm{Ricci}\\ (\\phi)\n\\end{equation*}\n\n\\noindent so that $\\Delta =D^2$ acts tensorially on such $\\phi$.  Unfortunately it does not preserve the solutions of \\eqref{eq:wave}, since $\\Delta$ need not commute with the Ricci operator.  However if we are looking for a perturbation expansion in powers of $\\mu$ this procedure might be made to work.\n\n\n An alternative idea is to regard the equations \\eqref{eq:Ein-mod} as equations on an\ninfinite-dimensional Hilbert space, namely $\\Omega^1(M_4)$. That is we consider the equation\n\\begin{equation}\\label{eq:alt}\n( \\ID_+ \\ID_- + \\nabla^2)\\phi=0\n\\end{equation}\nas an equation for the pair $(g,\\phi)$ where $g$ is the metric tensor and $\\phi$ is a 1-form.\n \\footnote{This is reminiscent of Einstein \\ae ther theory. See, for example,\n \\cite{ArkaniHamed:2002sp}. However, a minimal requirement would appear to be that\nthe  restriction to any $x\\in M_4$ of the $\\phi$'s solving  \\eqref{eq:alt} should span $T_x^*M_4$. } \n This needs detailed investigation and again it could first \n be looked at perturbatively in powers of $\\mu$.  \n However it should be emphasized that \\eqref{eq:alt}  makes sense non-perturbatively.\n\nFinally, we remark that in string field theory one very naturally\nruns into exponentials $\\exp[ \\alpha' \\nabla^2] $ acting on fields. This is usually not\nconsidered to be too disturbing since at energy-momenta small compared to the string\nscale such terms are close to $1$, and at energy-momenta on the order of the string\nscale the usual notion of  commutative space and time might be breaking down in any case.\nCertainly, topology changing effects in string theory take place at that scale.\nWitten has in the past speculated that string theory would necessarily lead to a revision\nof quantum mechanics. (His main reason being that there is no dilaton-field\nin 11-dimensional supergravity.)  There is thus some remote connection here to Witten's\nspeculation.\n\n\\section{The Shifted Einstein-Maxwell Equations}\\label{ref:E-M}\n\n  In addition to shifting the Dirac operator and the Ricci operator we should also shift the Maxwell operator\n\n\\begin{equation}\\label{eq:max-op}\nd^*F_A=d^*dA.\n\\end{equation}\nOne way to do this is to  use Kaluza-Klein reduction and the shifted Einstein equations\n in five-dimensional spacetime.  That is, we take the 5-dimensional space $M_5$ to be a\nprincipal circle bundle over the Lorentzian spacetime $M_4$ equipped with  a connection\n$\\Theta$ and a metric:\n\\begin{equation}\\label{eq:KK-ansatz}\nds^2 = e^{2\\sigma} \\Theta^2 + d\\bar s^2\n\\end{equation}\nwhere $d\\bar s^2$ is the pullback of a metric on the four-manifold $M_4$\n and $\\sigma$ is the dilaton field, again pulled back from $M_4$.  The shifted Einstein equations for\n such a metric produce the shifted Einstein-Maxwell equations in four dimensions.\n\n\n In addition to the shifted Einstein-Maxwell equations the\n  equation of motion for the scalar $\\sigma$ is amusing.\n When $F_A=0$ it is simply\n\\begin{equation}\\label{eq:Radius}\n- \\bar\\nabla^2 e^{\\sigma} = \\epsilon {k^2 \\over \\hbar^2} e^{\\sigma}\n\\end{equation}\nwhere $\\epsilon=\\pm 1$,\nso the radius $e^{\\sigma}$ of the KK circle is an eigenfunction of\nthe Laplacian on the four-dimensional spacetime. If we take\nthe metric to be deSitter space, and consider spatially homogeneous\nsolutions to \\eqref{eq:Radius} then the general solution is\n\\begin{equation}\\label{eq:gensol}\ne^{\\sigma}  = a_+ e^{ \\sqrt{\\epsilon} {kc \\over \\hbar} t} + a_- e^{ -\\sqrt{\\epsilon}{kc\\over \\hbar} t}\n\\end{equation}\nThus, in general the   circle radius grows exponentially in the past or the future.\nThis looks potentially devastating since the circle starts\ngrowing or shrinking exponentially on a time-scale\n$\\sim {\\hbar  \\over k c}$. However,  if we take\n$k\/\\hbar \\sim 10^{-28} cm^{-1}$ then  this timescale\nis approximately $(3\\pi)^{-1} \\times 10^{11}$ years. For comparison\nthe age of the universe is approximately $1.4 \\times 10^{10}$ years!\n  The observation that ${1\\over \\sqrt{\\Lambda} c}$ is the same order of magnitude\nas the age of the universe is a version of the famous ``coincidence problem'' in\nmodern cosmology. It means that we just happen to live at an era in\nthe history of the universe when the effects of the cosmological constant\nchange from being negligible to being dominant. In our discussion the\ncoincidence ``problem'' turns out to be a blessing, not a curse.\n\\footnote{Still, the existence of a massless scalar with gravitational\nstrength couplings is problematic in view of fifth-force experiments \\cite{Will:2010uh}.}  Of course, once\nthe exponential behavior\nstarts to become important one should surely no longer use the\nzero-mode approximation (i.e. neglecting the $\\CO(D^2)$ corrections).\n\n\n\n\nWe close with three remarks\n\n\n\\begin{enumerate}\n\n\\item    Clearly the above procedure could be extended to include nonabelian Yang-Mills equations\nby considering Kaluza-Klein reduction on spaces with nonabelian isometry groups.\n\n\\item  This discussion shows that the idea\nof shifting geometric operators is a natural process, akin to geometric quantization.  It would be fascinating to find any relation between the two processes especially since our shifting process includes the gravitational field, a target not yet achieved in quantization.\n\n\\item  The fact that our treatment rests fundamentally on spinors, the Dirac operator, and Kaluza-Klein\n theory suggests possible connections to super-gravity and string theory.\n\n\\end{enumerate}\n\n\n\\section{Conclusion}\n\n\nIn this paper we have, very tentatively, put forward a speculative new idea that seems worth exploring.  The idea is to introduce in a natural geometric way operators which shift (i.e. retard or advance) the basic operators of mathematical physics.  This includes the Dirac, Maxwell and Ricci operators (occurring in the Einstein equations of GR).  The shifting involves just two key physical parameters\n\n\\enlargethispage{1cm}\n\n\\begin{equation} \\label{eq:r-mag}\nr \\succsim 10^{-5} {\\rm cm}\n\\end{equation}\n\\begin{equation}\\label{eq:k-mag}\n\\frac{k}{\\hbar}  \\sim 10^{-28} {\\rm cm}^{-1}.\n\\end{equation}\nHere $r$ measures the timeshift and $k$ measures the magnitude of the shift. There is a natural quantization condition\n \\begin{equation}\\label{eq:quant-ii-con}\n r = \\frac{2n+1}{4} \\lambda_c\n \\end{equation}\nwhere $\\lambda_c$ is the Compton wavelength of a stable fermion. Of course, this has a quantum-mechanical\naspect and involves Planck's constant $\\hbar$.\n\n  The constant $k$ is at first sight arbitrary (except that it clearly must be very small).  However once we introduce our shifted Ricci operator we find that $k^2\/\\hbar^2$ is related to the cosmological constant.  Using the observed value of this gives the estimate \\eqref{eq:k-mag}.\n\n Thus the two key constants $r,k$ are determined by physical observations at the atomic and cosmological scales respectively.  This is a satisfactory situation. It is also reminiscent of some of the\nideas of T. Banks \\cite{Banks:2010tj}.\n\nNote that our approach has a dimensionless fundamental constant linking $r,k$ and $\\hbar$\n\n\\begin{equation}\n\\mu=kr\/\\hbar \\hspace{20pt}\\mu \\succsim  10^{-33}.\n\\end{equation}\n\n\nWe leave the reader with three key questions\n\n\n\\begin{enumerate}\n\n\\item What is the correct interpretation of equation \\eqref{eq:Ein-mod}?\n\n\\item Can   the above ideas can be made into a coherent model of physics, compatible\nwith the  successes of the Standard Models of particle physics and modern cosmology ?\n\n\\item How is this idea of shifted equations related to quantum mechanics?\nWe leave this to others and to the future.  There are tantalizing hints of possible connections,\nnot least the philosophical and mathematical difficulties on both sides!\n\n\n\\end{enumerate}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:sec1}\nMost of the stellar content in our Galaxy forms in cold ($T$\\,$\\sim$\\,10 - 30\\,K) and dense ($n$\\,$>$\\,10$^3$\\,cm$^{-3}$) cores within the Giant Molecular Clouds (GMC) of the Galactic Disk (e.g., Blitz 1991). The main properties of the stellar population, such as the efficiency of the mass conversion into stars and the shape of the initial mass function (IMF) may very well be closely related to the physical properties and mass distribution of the progenitor structures in the parental cloud (e.g. Mc\\,Kee \\& Ostriker 2007).\n\nBeing one of the nearest GMCs in the Galactic disk, the Vela Molecular Ridge (VMR, Murphy \\& May 1991, $l \\approx$ 260$^{\\circ}$-275$^{\\circ}$, $b \\approx \\pm$3$^{\\circ}$), represents an ideal observational target. It is composed \nof four  molecular clouds at a distance between 700 pc (clouds A,\\,C,\\,D) and 2000 pc (cloud B, Liseau et al. 1992), and harbours protostars \nwith masses up to 10 M$_{\\odot}$, both isolated and clustered (Massi et al. 2000, 2003). \nVela-C is the most massive component and hosts the youngest stellar population (Yamaguchi et al. 1999); this latter has been investigated by combining near-infrared data ($J$,\\,$H$,\\,$K$-band images) with far-infrared surveys ($MSX$,\\,$IRAS$) and around thirty isolated protostars and seven embedded young clusters associated with C$^{18}$O clumps have been found (Liseau et al. 1992; Lorenzetti et al. 1993; Massi et al. 2003, Baba et al. 2006). A bright HII region, RCW\\,36, has been associated by Massi et al. (2003) with an early-type star (spectral type O5-B0). A second HII region, RCW\\,34, was originally associated to Vela-C by several authors (Herbst 1975, Murphy \\& May 1991), but recent observations favour a much longer distance ($d$=2.5 kpc, Bik et al. 2010).\nVela-C has been imaged at 250\\,$\\mu$m, 350\\,$\\mu$m and 500\\,$\\mu$m with the Balloon-borne Large Aperture Submillimeter Telescope (\\emph{BLAST}, Pascale et al. 2008) that provided the first census of the compact dust emission in a range of evolutionary stages and lifetimes (Netterfield et al. 2009). \n\nAs a target of the \\emph{Herschel} guaranteed time key program  HOBYS ('\\emph{Herschel} imaging survey of OB Young Stellar objects', Motte et al. 2010), Vela-C has been observed with the \\emph{PACS} (Poglitsch et al. 2010) and \\emph{SPIRE} (Griffin et al. 2010) cameras between 70\\,$\\mu$m and 500\\,$\\mu$m. The extended emission in the form of filaments and ridges has been presented by Hill et al. (2011) who identify five different sub-regions with different column densities. Here we will focus on the determination of the physical parameters of the compact sources, their evolutionary stage and their mass distribution. The latter has been investigated in several star forming regions through different tracers (e.g. Testi \\& Sargent 1998, Motte et al. 1998, 2001, Kramer et al. 1998, Reid \\& Wilson 2006, Enoch et al. 2008, Rathborne et al. 2009, K\\\"{o}nyves et al. 2010, Ikeda \\& Kitamura 2011). \nIn this paper, we intend to enlarge the statistics providing the mass distribution over a wide range, going from subsolar values up to tens \nof solar masses. \n\nThis paper is organized as follows. In Section\\,\\ref{sec:sec2} we describe the observations, the data reduction procedures and the photometric results. In Section\\,\\ref{sec:sec3} we describe the Spectral Energy Distribution (SED) analysis along with the fitted physical parameters. In Section\\,\\ref{sec:sec4} we discuss the evolutionary stages of the sources along with their mass distribution. Our results are then summarized in \nSection\\,\\ref{sec:sec5}.\n\n\n\\section{Observations, data reduction and results}\\label{sec:sec2}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=14cm]{tricro1.eps}\n\\caption{Composite 3-color image of Vela-C : \\emph{PACS} 70\\,$\\mu$m (blue), \\emph{PACS} 160\\,$\\mu$m (green), and \\emph{SPIRE} 500\\,$\\mu$m (red). The blueish regions near the center and towards the bottom right of the map are the HII regions RCW\\,36 and RCW\\,34, respectively. Red and blue diamonds represent the locations of prestellar and protostellar sources, respectively, belonging to the sample of 268 objects used for the SED analyisis. Bigger diamonds indicate sources with M\\,$\\ge$\\,20\\,M$_{\\odot}$, while yellow diamonds indicate a few sources inside RCW\\,34 (see text). The top left segment corresponds to a length of 5 pc in the sky for a distance to Vela-C of 700 pc.}\n\\label{tricro}\n\\end{figure*} \n\nVela-C was observed on May 18, 2010 with the \\emph{Herschel} parallel mode, i.e., using simultaneously \\emph{PACS} at 70\/160 $\\mu$m and \\emph{SPIRE} at 250\/350\/500 $\\mu $m. A common area of $\\sim$ 3 square degrees was covered by both the instruments around the position $\\alpha_{(J2000.0)}$=08$^{\\mathrm{h}}$59$^{\\mathrm{m}}$55$^{\\mathrm{s}}$, $\\delta_{(J2000.0)}$=-43$^{\\circ}$53$^{\\prime}$00$^{\\prime\\prime}$. The field was observed in two orthogonal directions at the scan speed of $20\\hbox{$^{\\prime\\prime}$}$\/s. \nThe data reduction strategy is described in detail in Traficante et al. (2011): here we summarize only the fundamental steps. From archival data to the Level 1 stage, we used scripts prepared in the \\emph{Herschel} Interactive Processing Environment (HIPE, Ott 2010), partially customized compared to the standard pipeline. The obtained time ordered data (TODs) of each bolometric detector were then processed further by means of dedicated IDL routines and finally maps were created using the FORTRAN code ROMAGAL. Images of Vela-C at all wavelengths have been presented by Hill et al. (2011), here for the readers' convenience, we show in Figure \\ref{tricro} the composite 3-color image at 70\/160\/500 $\\mu$m.                                 \n\n\\subsection{Source Detection and Photometry}\\label{sec:sec2.1}\nThe detection and photometry of compact sources have been carried out using the Curvature Threshold Extractor package (CuTeX, Molinari et al. 2011) applying the same strategy as Molinari et al. (2010). Briefly, in each band, sources were identified as peaks in the second-derivative images of the original HOBYS maps, then an elliptical Gaussian fit was performed to provide: 1) the total flux, integrated down to the zero intensity level, 2) the observed FWHM (defined as the geometrical mean of the major and minor ellipse axes), and 3) the peak intensity. The latter, when divided by the local rms noise, allows us to obtain an {\\it a posteriori} estimate of the $S\/N$ ratio. We filtered out all sources with $S\/N<5$.\nFollowing Elia et al. (2010), entries at different wavelengths in the \\emph{PACS}\/\\emph{SPIRE} catalogue have been attributed to the same source based on simple positional criteria. In practice, we associate two sources detected in two different bands if their mutual angular distance does not exceed the radius of the \\emph{Herschel} half-power beam-width (HPBW\\footnote{The values of the \\emph{Herschel} HPBW are 5.0$^{\\prime\\prime}$ at 70\\,$\\mu$m, 11.4$^{\\prime\\prime}$ at 160\\,$\\mu$m, 17.8$^{\\prime\\prime}$\nat 250\\,$\\mu$m, 25.0$^{\\prime\\prime}$ at 350\\,$\\mu$m, and 35.7$^{\\prime\\prime}$ at 500\\,$\\mu$m.}) at the longer wavelength. Around 15\\% of the entries in the catalogue present multiple associations at decreasing wavelength; as a general rule, the closest counterpart was kept without attempting to divide the flux at the longer wavelength among the sources. We assigned to each source the celestial coordinates of the counterpart at the shortest wavelength, where the spatial resolution is higher. Finally, to exclude artifacts, we discarded sources with an axes ratio $>$ 2 and a position angle randomly changing with wavelength by more than 20$^{\\circ}$.\n\n\n\\begin{table}[h!]\n\\caption{\\label{tab:tab1} Statistics of the 5\\,$\\sigma$ \\emph{PACS}\/\\emph{SPIRE} catalogue.}  \\small\n\\begin{center}\n\\begin{tabular}{cccc}\n\\hline\n   Band                   & \\#     &    Sensitivity limit      &         90\\% Completeness limit$^a$  \\\\\n                          &        &      (Jy)                 &                (Jy)                  \\\\\n\\hline\nTotal entries$^b$         & 1686   &                           &                           -          \\\\\n70 $\\mu$m                 &  658   &\t  0.04                 &                 0.21                 \\\\\n160 $\\mu$m                &  871   &\t  0.09                 &                 0.67                 \\\\\n250 $\\mu$m                &  966   &\t  0.11                 &                 1.05  (17)           \\\\\n350 $\\mu$m                &  697   &\t  0.35                 &                 1.32  (22)           \\\\\n500 $\\mu$m                &  416   &\t  0.46                 &                 1.95                 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\begin{list}{}{}\n\\item[$^{\\mathrm{a}}$] For comparison, in parentheses the \\emph{BLAST} completeness limit are reported, estimated from Netterfield et al. 2009\n(their Figure\\,3). \n\\item[$^{\\mathrm{b}}$] Number of sources detected in at least one band.\n\\end{list}\n\\end{table}\n\n\\subsection{Catalogue statistics}\\label{sec:sec2.2}\nThe complete \\emph{Herschel} catalogue of Vela-C will be published in the next future as part of the HOBYS data products. Here we limit to give in Table \\ref{tab:tab1} a statistical summary of the detected sources. In the second column we list the number of entries in each of the five bands: this number increases with increasing wavelength up to $\\lambda$\\,=\\,250\\,$\\mu$m, than decreases at longer wavelengths. Such behaviour can be explained in the light of the sensitivity limits given in the third column: since the number of entries in the first three bands increases despite the loss of sensitivity, such increasing is a real effect reflecting the intrinsic properties of the cloud. Conversely, in the last two \\emph{SPIRE} bands, the poorer sensitivity limits, together with the loss of angular resolution, are likely responsible for the reduced number of the detected sources.  \n\nThe flux completeness limits (fourth column) are estimated at the 90\\% of confidence level, estimated by recovering artificial Gaussian sources randomly spread over the map and whose fluxes and diameters cover the same ranges found for the observed sources. In Figure\\,\\ref{compl}, we show the percentage of the recovered artificial sources as a function of the flux for each \\emph{Herschel} band.  \nFor comparison, the \\emph{BLAST} completeness limits at 250\\,$\\mu$m and 350\\,$\\mu$m (estimated in Figure\\,3 of Netterfield et al. 2009) are more than a factor of 15 larger than the \\emph{SPIRE} limits at the same wavelengths.   \n\n\n\\begin{figure}\n\\centering\n\n\\includegraphics[width=8cm]{completeness.eps}\n\\caption{Percentage of recovered artificial sources as a function of their fluxes in the five \\emph{Herschel} bands.}\n\\label{compl}\n\\end{figure} \n\n\n\\subsection{Source diameters}\\label{sec:sec2.3}\nIn each of the five bands, the physical FWHM$_{dec}(\\lambda)$ has been derived by deconvolving the observed FWHM with the HPBW  at the same wavelength:\n\\begin{equation}\nFWHM_{dec}(\\lambda) = \\sqrt{FWHM_{\\lambda}^2-HPBW_{\\lambda}^2}. \n\\end{equation}\n\nNoticeably, FWHM$_{dec}(\\lambda)$ increases with $\\lambda$, with mean values of 7.5$^{\\prime\\prime}$, 13.4$^{\\prime\\prime}$, 22.7$^{\\prime\\prime}$, 28.6$^{\\prime\\prime}$, 41.0$^{\\prime\\prime}$ in the five  \\emph{Herschel} bands, respectively. Since we are most interested in tracing the cold dust emission (namely that at temperature $\\la$ 25 K), which is likely poorly related to the 70\\,$\\mu$m flux, we define as angular source diameter ($\\theta$) the FWHM$_{dec}$ measured at 160\\,$\\mu$m, where we detect a considerable number of sources with a good angular resolution (see also Motte et al. 2010). For a  small fraction of sources that remained undetected at 160\\,$\\mu$m, $\\theta$ is the FWHM$_{dec}$ measured at 250\\,$\\mu$m. \n\nWe consider as spatially resolved the detections with $\\theta$ larger than $\\sim$\\,60\\% of the angular resolution at 160\\,$\\mu$m (corresponding to a diameter $D$\\,$\\sim$ 0.025 pc at the Vela-C distance), as also adopted in the Aquila Rift and Polaris by Andr\\'e et al. (2010). \nNoticeably, around $\\sim$ 25\\% of our sources have a diameter between 0.025 pc and 0.05 pc ('cores'), while most sources are larger ('clumps', $0.5\\,\\le D\\,\\le 0.13$ pc); in the following analysis, we will not distinguish any longer between the two categories, and will refer to both of them with the general term 'source'. \n\n\n\\section{Analysis}\\label{sec:sec3}\n\n\\subsection{Flux scaling}\\label{sec:sec3.1}\nThe physical parameters of the detected sources have been derived by fitting their SEDs with a  modified black body curve. Two examples of the observed SEDs are shown in Figure\\,\\ref{fig:seds}. These illustrate a trend seen in many of the SEDs, namely that the {\\it observed} fluxes (depicted with crosses) flatten or even rise longwards $\\lambda \\ge$ 250\\,$\\mu$m.\nThis effect is a direct consequence of the increase of the FWHM$_{dec}(\\lambda)$ with wavelength (Sect.\\,\\ref{sec:sec2.2}), which in turn implies\nincreasing areas over which the emission is integrated. In practice, such flattening indicates that a single-temperature, modified black body fit is not completely adequate to model the observed photometric points. However, since the small amount of available data prevents us from using a multiple-temperature, modified black body model, we restricted the fit only to the innermost portions of the sources, whose spatial scales are defined by the source angular diameter $\\theta$.\nTo estimate the emission coming from these restricted solid angles, we followed the flux scaling method adopted by Motte et al. (2010) and described in detail by Nguy$\\tilde{\\hat{\\rm e}}$n Lu{\\hskip-0.65mm\\small'{}\\hskip-0.5mm}o{\\hskip-0.65mm\\small'{}\\hskip-0.5mm}ng et al. (2011). \nThis method is based on the idea that for quasi-spherical, self-gravitating sources, the radial density law is $\\propto r^{-2}$ (with $r$ $\\sim$ 0.1-1 pc), and thus $M(<r) \\propto r $ (Shu 1977). This relationship implies that fluxes scale linearly with the source radius, provided that the emission is optically thin and the temperature gradient is weak ($T(r) \\propto r^{-q}$, with $q\\,\\sim 0$, see Motte \\& Andr\\'e 2001 for more details). \nAll these conditions have been empirically verified in our case by taking the photometry at different apertures for some isolated sources, and finding a linear increase of fluxes with increasing radii. Hence, fluxes at 250\\,$\\mu$m, 350\\,$\\mu$m, and 500\\,$\\mu$m, were reduced  \nby the ratio FWHM$_{dec}(\\lambda)\/\\theta$. The average values of these factors are 1.9 at 250\\,$\\mu$m, 2.0 at 350\\,$\\mu$m, and 3.2 at\n500\\,$\\mu$m, respectively. In Figure\\,\\ref{fig:seds}, dots represent fluxes after the above correction.\n\n\\begin{figure}\n   \\centering\n \\includegraphics[width=9cm]{seds_paper.eps}\n \\caption{\\label{fig:seds} Examples of modified black body fits to the SEDs of two sources (undetected and detected at 70\\,$\\mu$m, respectively). Crosses and dots mark the data points before and after the correction applied at longer wavelengths to minimize the effects of the poorer spatial resolution (see text). Values of the main fitted parameters are reported.}%\n\\end{figure}\n\n\\subsection{Selection criteria}\\label{sec:sec3.2}\nOnce the \\emph{SPIRE} fluxes were rescaled, we further filtered the catalogue to select a robust sub-sample for SED analysis. The final\nlist was composed of sources fulfilling the following criteria, i.e., sources: \n\\begin{itemize}\n\\item[(1)] with detections in at least three adjacent bands between 160\\,$\\mu$m and 500\\,$\\mu$m, to contain the peak of the emission of the\ncold dust;\n\\item[(2)] without a dip in the SED between three adjacent wavelengths;  \n\\item[(3)] not peaking at 500 $\\mu$m. Note that conditions (2) and (3) exclude most of the artifacts or detections affected by incorrect positional associations or bad photometry; \n\\item[(4)] spatially resolved at 160\\,$\\mu$m;\n\\item[(5)] not presenting multiple associations at $\\lambda$ $\\ge$ 160\\,$\\mu$m, to alleviate confusion problems; \n\\item[(6)] not belonging to the RCW34 region, to exclude sources that could be more distant than 700\\,pc (see Sect.\\,\\ref{sec:sec1}). These sources, identified by eye in the \\emph{Herschel} maps, are depicted in yellow in Figure\\,\\ref{tricro}.\n\\end{itemize}\nThe first three criteria are the most stringent ones, since they lead to a selection of only 388 sources out of 1686 present in the catalogue. Applying criteria (4) to (6), we get a final sample of 268 sources to be fitted. Their locations are depicted in Figure \\ref{tricro} with different colors indicating their evolutionary stage (see below). \n\n\n\\subsection{SED fitting}\\label{sec:sec3.3}\nSince cold dust emission is likely unrelated to the photometry measured at 70\\,$\\mu$m, we fitted the SEDs longward $\\lambda$\\,$\\ge$\\,160\\,$\\mu$m and checked {\\it a posteriori} whether or not the modified black body curve fits also the 70\\,$\\mu$m point.\nThe assumed relation is:\n\n\\begin{equation}\\label{eq:flux}\n{\\rm F}_\\nu=(1-e^{-\\tau_\\mathrm{\\nu}})B_\\nu(T_d)\\Omega\n\\end{equation}\nwhere F$_\\nu$ is the observed flux at the frequency\n$\\nu$, $\\Omega$ is the source area (in sr), B$_\\nu$($T_d$) is the black body function at the dust temperature $T_d$, and $\\tau_\\mathrm{\\nu}$ is the optical depth, \nparametrized as $\\tau_\\mathrm{\\nu}=(\\nu\/\\nu_0)^{ \\beta}$, where $\\nu_0 = c\/\\lambda_0$ is the frequency at \nwhich $\\tau$=1, and $\\beta$ = 2, as predicted by simple dust emission models (Hildebrand 1983). A linear least-squares fit to the scaled fluxes (with their photometric uncertainties) was performed by comparing the observations with a database of models where $T_d$ and $\\lambda_0$ are allowed to vary in the following ranges:\n8\\,K $\\le$ $T_d$ $\\le$\\,40 K; 10\\,$\\mu$m $\\le$ $\\lambda_0$ $\\le$ 500\\,$\\mu$m. \nWe applied the \\emph{PACS} color corrections, amounting to a maximum of 4\\% of the flux, to the model fluxes before comparing them with the data (M\\\"{u}ller et. al 2011).\nAs a check, we also verified that the fitted values of $\\lambda_0$ never exceed 160\\,$\\mu$m (see Sect.\\,\\ref{sec:sec4.1} and Figure\\,\\ref{fig:mass_size}); this insures that the emission is optically thin in the \\emph{SPIRE} wavelength range, namely where flux scaling is performed (see the discussion in Sect.\\,\\ref{sec:sec3.1})\n\nFinally, masses were derived from the dust emission using the relation \n\\begin{equation}\\label{eq:mass}\nM=(d^2 \\Omega\/k_\\mathrm{{300}}) \\tau_\\mathrm{300}\n\\end{equation}\nwhere $k_\\mathrm{300}$\\,=\\,0.1 cm$^2$ g$^{-1}$ is the opacity per unit gas mass computed at $\\lambda$ = 300\\,$\\mu$m assuming a gas-to-dust ratio of 100 (Beckwith et al. 1990), and $d$=700 pc. (The details of the derivation of Eq.\\,\\ref{eq:mass} from Eq.\\,\\ref{eq:flux} will be given in Pezzuto et al., in preparation).\n\n\\subsection{Separating starless from protostellar sources}\\label{sec:sec3.4}\nSince Vela-C was not observed by \\emph{Spitzer} at 24\\,$\\mu$m, which would have revealed embedded sources with adequate sensitivity, we used the \\emph{Herschel} data themselves, and where available, other mid- and far-IR surveys to distinguish starless and protostellar sources. This distinction was primarily done on the basis of the presence of a 70\\,$\\mu$m counterpart (see also Bontemps et al. 2010). Indeed, a tight correlation between 70\\,$\\mu$m fluxes and internal luminosities of protostars\nhas been demonstrated (Dunham et al. 2008), and, furthermore, external heating by the interstellar radiation field cannot \nproduce a 70\\,$\\mu$m emission from a source detectable with \\emph{Herschel}.\nHence, we consider as starless all sources without a 70\\,$\\mu$m counterpart (e.g. Figure\\,\\ref{fig:seds}, left panel). Conversely, \nprotostellar are: {\\it i}) sources whose 70\\,$\\mu$m flux lies above the best modified black body fit by more than 3\\,$\\sigma$ (e.g. Figure\\,\\ref{fig:seds}, right panel), or, {\\it ii}) sources detected in the mid-IR at $\\lambda$ $<$ 70\\,$\\mu$m by other surveys (IRAS\/MSX\/Akari). Out of 268 objects, 218 are starless and 48 are protostellar. Two further sources, not detected at $\\lambda$ $<$ 70\\,$\\mu$m by other surveys, were fitted \nwith a single modified black body curve from 70\\,$\\mu$m to 500\\,$\\mu$m. The temperature of both sources is $\\sim$ 25\\,K, i.e. warmer than starless sources (Sect.\\,\\ref{sec:sec3.4}). On this basis we surmise that these sources are embedded Class\\,0 protostars (Andr\\'e et al. 1993).   \n\n\n\\subsection{Physical parameters}\\label{sec:sec3.5}\nIn this section, we discuss the statistics of the key physical parameters measured directly from the observations or derived from SEDs fits.\nThese quantities are listed in Table\\,\\ref{tab:tab2} and plotted in Figure\\,\\ref{fig:isto}.\n\nStarless sources exhibit a very narrow temperature distribution (top left panel), strongly peaked at $T_d$ $\\sim$ 10 K. The high cutoff is at $T_d \\sim$ 15\\,K, while the low one coincides with the minimum temperature of 8\\,K considered in the fit. On average, protostellar sources are warmer than starless sources; their temperature distribution spreads almost uniformly between 9\\,K and 15\\,K, though a few sources are found with $T_d$ up to 25\\,K. The overall temperature distribution (not shown in Figure\\,\\ref{fig:isto}) resembles that found from \\emph{BLAST} data of the whole Vela region \n(Netterfield et al. 2009). This latter distribution, however, does not show the sharp $\\sim$\\,10\\,K peak, but rather has a shallower shape peaking at $\\sim$ 12\\,K. This marginal difference may be due to both the better\nangular resolution of \\emph{Herschel}, which allows deeper penetration into colder regions, and to the inclusion of warmer regions (e.g., Vela-D, Olmi et al. 2009) in the Netterfield et al. sample. \nIt is also interesting to compare the compact source distribution with the average temperature of the surrounding cloud. The latter was derived by Hill et al. (2011) by fitting a modified black body to the \\emph{Herschel} SED of each pixel. In the parts of the cloud where no compact sources are detected, the temperature is fairly constant at $T_d$\\,$\\sim$\\,14 K (black line in Figure\\,\\ref{fig:isto}). Practically all the starless sources are colder than the ambient medium, a result expected for dense structures heated only by external radiation.  Interestingly, the same result is found for most of the protostellar sources. This finding reasonably indicates that the external envelope we are tracing is moderately heated by the internal protostar(s), which is therefore likely to be in the very early phases of the evolution.\n\nThe mass distribution (top right panel of  Figure\\,\\ref{fig:isto}) shows a nearly monothonic decrease from lower to higher masses.\nThere are no significant differences between the average masses of starless and protostellar sources ($\\sim$\\,5 M$_{\\odot}$, Table\\,\\ref{tab:tab2}); this suggests\nthat the envelope remains substantially unperturbed by the internal gravitational collapse and, again, that the protostellar sources are relatively unevolved, as little mass accretion is taking place.\nNoticeably, 35 sources (of which 7 protostellar and 28 starless) have 10\\,$\\le$ $M$\/M$_\\odot$\\,$\\le$\\,20 and 8 (of which 1 protostellar and 7 starless) have 20\\,$\\le$ $M$\/M$_\\odot$\\,$\\le$\\,55, whose locations are depicted with big crosses in Figure\\,\\ref{tricro}. Apart from a few exceptions, these massive objects are rather compact (with diameters between 0.04 pc and 0.07 pc), hence they are potentially able to form individually massive stars.\nFor comparison, Hill et al. (2011) found in Vela-C nine objects with masses between 13\\,M$_{\\odot}$ and 70\\,M$_{\\odot}$. We consider the agreement between the two determinations satisfactory, given the different source extraction methods (Hill et al. 2011 takes into account in the photometry the bias created by bright rimmed effects and source multiplicity) and the different prescriptions under which the SED fitting has been performed (for example, Hill et al. did not apply flux scaling).\n\nThe source diameter distributions are presented in the bottom left panel of Figure\\,\\ref{fig:isto}. For starless sources, it spreads from the resolution limit of 0.025 pc up to 0.13 pc, peaking around 0.05 pc. Such dimensions, as anticipated in Sect.\\,\\ref{sec:sec2}, span from those of progenitors of single stars, up to those typical of large clumps, that may give rise to the\nformation of protostellar clusters. Conversely, protostellar sources are significantly more compact; this suggests that no further fragmentation will occur and that the majority of the protostellar sources likely host single stellar objects or stellar systems with a small number of members.  \nThe small sizes of protostellar sources can be alternatively (and additionally) explained by considering that the internal source(s) of heating likey causes a temperature gradient and hence a more peaked intensity distribution that is fitted with a compact spatial extent.\n\nFinally, in the bottom right panel of Figure\\,\\ref{fig:isto}, we plot the distributions of luminosity in the \\emph{Herschel} bands ($L_\\mathrm{FIR}$). This quantity has been computed by interpolating the corrected fluxes between all available wavelengths between 70\\,$\\mu$m and 500\\,$\\mu$m.\nFor starless sources, $L_\\mathrm{FIR}$  is a good approximation of bolometric luminosity; typically, it ranges between 0.05\\,L$_\\odot$ and 0.5\\,L$_\\odot$, with a small percentage of objects at larger luminosities not exceeding 5 L$_\\odot$.\nFor protostellar sources, $L_\\mathrm{FIR}$ has to be considered as a lower limit to the bolometric luminosity because of the lack of near- and mid-infrared data points in the SEDs.\n\n\n\\begin{table*}\n\\caption{\\label{tab:tab2}. Statistics of the main physical parameters of the sources in Vela-C. We\ngive median, average, minimum and maximum values.}  \\small\n\\begin{center}\n\\begin{tabular}{c|ccc|ccc}\n\\hline\nParameter                          & \\multicolumn{3}{c|}{Starless (218)} &  \\multicolumn{3}{c}{Protostellar (48)}\\\\ \n                                   &  median     &  average  & min-max            &  median   &  average    & min-max      \\\\              \n\\hline\n$M$ (M$_{\\odot}$)                  &   3.3       & 5.5       &  0.13-55.8         & 2.7       & 4.8         &  0.15-29.1   \\\\\n$T_d$ (K)                          &   10.0      & 10.3      &  8.0-15.2          & 11.4      & 12.8        &  9.0-24.2    \\\\\n$D$ (pc)                           &   0.064     & 0.067     &  0.025-0.13        & 0.040     & 0.040       &  0.025-0.07  \\\\\n$L_\\mathrm{FIR}$ (L$_{\\odot}$)     &   0.17      & 0.22      &  0.04-4.8          & 0.6       & 8.0         &  0.08-138    \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n\\begin{figure}\n   \\centering\n \\includegraphics[width=9cm]{isto.eps}\n \\caption{\\label{fig:isto} Histograms of the main physical parameters of the sources in Vela-C (top left: temperature; top right: mass; bottom left: diameter; bottom right: \\emph{Herschel} luminosity). Starless and protostellar sources are separately highlighted in red and blue, respectively. The black line in the top left panel represents the temperature of the surrounding cloud, taken from Hill et al. (2011), while that in the bottom left panel highlights the spatial resolution limit.}\n\\end{figure}\n\n\n\n\\section{Discussion}\\label{sec:sec4}\n\\subsection{Virial analysis of starless sources}\\label{sec:sec4.1}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=8cm]{mass_size.eps}\n\\caption{\\label{fig:mass_size} Mass vs. diameter diagram for the sources in Vela-C. Red filled circles and red open circles are prestellar and unbound sources, respectively. Blue filled circles are protostellar sources. Dotted lines are the {\\it loci} of sources with $M$\\,=\\,0.5 M$_{BE}$, for $T$\\,=\\,8 K and $T$\\,=\\,20 K, while the dashed vertical line represents the spatial resolution limit. We also show the curve representing the relation between mass and diameter in the case $\\lambda_0$=160\\,$\\mu$m. This relation shifts towards higher masses with increasing $\\lambda_0$: hence, since all our sources are located below the reported curve, they have optically thin emission at $\\lambda > 160\\,\\mu m$.\n}\n\\end{figure}\n\nStarless sources are defined as {\\it prestellar} if they are gravitationally bound (e.g., Andr\\'e et al. 2000, Di\nFrancesco et al. 2007), hence they can potentially form one or more stars. In principle, spectroscopic observations would be required in order to derive the virial masses and to securely establish the sources' dynamical states. Since such observations are not available for Vela-C, we have assumed thermal pressure support and neglected the internal turbulence. In this simplified view, the virial mass can be surrogated by the critical Bonnor-Ebert mass:\n\n\\begin{equation}\nM_\\mathrm{BE} \\approx 2.4 R_\\mathrm{BE} a^2\/G\n\\end{equation} \nHere $a$ is the sound speed at the source temperature\\footnote{$a$=$\\sqrt{k_B T_d \/ \\mu}$, where k$_B$ is the Boltzmann constant, T$_d$ is the fitted temperature, and $\\mu$ = 2.33 m$(H)$ is the mean molecular weight, with m$(H)$ the atomic hydrogen mass.},  $G$ is the gravitational constant, and R$_{BE}$ the Bonnor-Ebert radius (in pc). We take R$_\\mathrm{BE}$\\,=\\,$D$\/2.\nStarless sources with $M\/M_\\mathrm{BE}$ $\\ge$ 0.5 are selected as being gravitationally bound (Pound \\& Blitz 1993) and\ncandidate prestellar. By this definition, a large number of prestellar sources is found (206 objects), i.e., $\\sim$ 94\\% of the starless ones, remarkably higher than the percentage (69\\%) found in the Aquila Rift cloud (K\\\"{o}nyves et al. 2010, Andr\\'e et al. 2010). Moreover, this percentage mantains high (193 objects, i.e.,\\,88\\%) even if we demand that bound sources have $M$ at least equal to 1\\,$M_\\mathrm{BE}$.\n\nTo check further this result, we plot all sources in a mass\\,vs.\\,diameter diagram (Figure\\,\\ref{fig:mass_size}). First we note that, as a consequence of \ncondition (4) in the selection criteria of Sect.\\,\\ref{sec:sec3.2}, all points lie to the right of the spatial resolution limit of 0.025 pc, shown with a vertical dashed line. Second, we plot the relation between mass and diameter as a function of the second variable in the Equation\\,\\ref{eq:mass}, namely the wavelength $\\lambda_0$ at which the optical depth $\\tau$\\,=\\,1. This relation shifts towards higher masses with increasing $\\lambda_0$, therefore all the sources located below (above) the curve for a given value of $\\lambda_0$ have optically thin (thick) emission at $\\lambda > \\lambda_0$.\nAs example, we show such relation for the value $\\lambda_0$\\,=160\\,$\\mu$m, which is the shortest wavelength that we include in the SEDs fits. Since all our sources are located below this curve, their emission is optically thin at wavelengths longer than 160\\,$\\mu$m. \n\nConsistently with our virial classification, most of the prestellar sources in Figure\\,\\ref{fig:mass_size} lie above or between the lines of 0.5 M$_\\mathrm{BE}$, computed at 8\\,K and 20\\,K. We note, however, that a relevant fraction of these objects have a mass up to a factor of ten larger than the Bonnor-Ebert mass at 20\\,K. Reasonably, such objects are dynamically unstable, if turbulence and magnetic field supports against gravity are neglected.\n\nAll the sources classified as unbound are located below the Bonnor-Ebert curve at $T$\\,=\\,8 K. Noticeably, most of them cluster around the value $M$ $\\sim$ 0.2 M$_{\\odot}$. This latter corresponds to the mass detection limit, estimated from the flux sensitivity limits of Table\\,\\ref{tab:tab1} and by assuming \naverage values of $T$ and $\\lambda_0$ in Eqs.(2) and (3). Since such mass limit decreases with increasing temperature, we deduce that we are able to efficiently probe only the unbound sources particurlarly warm. Indeed, for the sub-sample of this category (12 objects)  we get $\\langle T_d \\rangle$=13.8\\,K, namely higher than the value of $\\langle T_d \\rangle$=10.3\\,K, referring to the whole sample of starless sources (see Table\\,\\ref{tab:tab2}). Noticeably, our inability to detect cold, unbound sources, may explain the very large fraction of prestellar sources with respect to the unbound ones (see above). Moreover, this fraction could further decrease if the most massive prestellar cores were gravitationally unstable, as already noticed above.\n\nFinally, to show more clearly the results of Table\\,\\ref{tab:tab2}, we also plot in the same diagram the positions of protostellar sources. As already noted in  Sect.\\ref{sec:sec3.5}, they cluster in Figure\\,\\ref{fig:mass_size} at diameters smaller than starless sources, and close to our spatial resolution limit. Since no significant differences in mass (as well as in temperature values) are recognizable with respect to starless sources, this result should reflect an increase in average density.  \n\n\n\\subsection{Luminosity vs. Mass diagram}\n\n\\begin{figure}\n   \\centering\n \\includegraphics[width=9cm]{lmdiag.eps}\n \\caption{\\label{LM} $L_{bol}-M$ diagram for prestellar and protostellar sources in Vela-C (red and blue circles, respectively). Grey solid lines represent the evolutionary tracks for low-mass objects adopted by Molinari et al. (2008), for initial values of 0.5, 1, 5, and 10 M$_{\\odot}$, respectively. An arrow indicates the evolution direction, while green lines delimit the region of transition between Class 0 and Class I sources (Andr\\'e et al. 2000). The red dotted line represents the best-fitting power law ($L_{bol}\\propto M^{0.42\\pm0.04}$) for the distribution of the prestellar sources.}\n\\end{figure}\n\nAnother interesting perspective to be examined is the relation between the bolometric luminosities ($L_{bol}$) and the envelope masses of the sources ($M$). To make such comparison, we take as $L_{bol}$ the $L_\\mathrm{FIR}$ values, even if these latter well represent bolometric luminosities  only for starless sources. Previous studies (e.g. Saraceno et al. 1996, Andr\\'e et al. 2000, Molinari et al. 2008, Elia et al. 2010, Bontemps et al. 2010, Henneman et al. 2010) have illustrated that the $L_{bol}$ vs. $M$ plot is a meaningful tool for  the characterization of the evolutionary status of cores and clumps, since tracks representing model predictions can be directly compared with the locations of the observed sources. In such a diagram, a collapsing core is expected to follow initially an almost vertical path (i.e., mass constant with time, and luminosity strongly increasing with accretion), then subsequently protostellar outflow activity increasingly disperses the envelope while luminosity remains almost constant, resulting in a horizontal track.\n\nIn Figure\\,\\ref{LM}, the $L_{bol}$, $M$ pairs of the Vela-C protostellar and prestellar sources are plotted, along with the evolutionary tracks for low-mass objects adopted by Molinari et al. (2008), based on the simplified assumption of a central protostar accreting mass from\nthe envelope at a constant rate  $\\dot{M}=10^{-5}$ M$_{\\odot}$ yr$^{-1}$. More evidently than in previously shown diagrams, a significant distinction between the two populations, although not a complete segregation, is seen. In the same plot, we delimit with green\nlines the transition region between Class 0 and Class I objects (Andr\\'e et al. 2000), which roughly separates the vertical portions of the tracks, where Class 0 objects are expected to be found, from the horizontal ones, where Class I objects are located. Noticeably, the large majority of the protostellar sources populate the region corresponding to Class 0 objects. This is in contrast with our previous results based on SED fitting, according to which 48 sources are young protostars (although not more evolved) and just two sources are identified as candidate Class 0 sources. Such a discrepancy can be however partly reconciled if considering that our underestimate of the bolometric luminosities of protostellar sources reflects underestimates of the actual ages. Moreover, the Andr\\'e et al. (2000) limit separating Class 0 and I could not be completely adequate in Vela, as already found in Taurus by Motte \\& Andr\\'e (2001). \n\nAlso, the distribution of the prestellar sources is quite flat and homogeneously concentrated in a region of the diagram corresponding to a very early stage where no mass accretion has started.\nA tentative power-law fit gives a dependence  $L_{bol}\\propto M^{(0.42\\pm0.04)}$. This slope is shallower than that found by Brand et al. (2001) for their sample of protostellar sources, namely 0.85. A similar scaling relation has been found for the CO luminosity of clumps: from the theoretical point of view, $L_{CO}\\propto M_{vir}^{\\sim 1}$ at the virial equilibrium (Scoville \\& Sanders 1987; see also Wolfire et al. 1993 and references therein); this slope has been found by Rengarajan (1984), while other authours find lower values (e.g. Solomon et al. 1987,\nYonekura et al. 1997, Miyazaki \\& Tsuboi 2000). A direct comparison with our result, however, cannot be performed due to the different classes of sources considered, or different quantities compared.\n\n\n \n\\subsection{Source mass distribution}\n\\begin{figure}\n   \\centering\n \\includegraphics[width=9cm]{newmassspectrum_0.10_0.6.eps}\n \\caption{\\label{fig:massspectrum} Source mass distribution in Vela-C. The error bars correspond to $\\sqrt N$ statistical uncertainties. Continuous red line represents the fit to the linear portion of the mass distribution (namely that consistent with a single straight line within the errorbars) for all the prestellar sources. A similar fit performed on the restricted sample of prestellar sources with diameters less than 0.08\\,pc is represented with a grey dashed line. The derived slopes are reported as well.\nMass distribution of CO clumps (Kramer et al. 1998), the single-star IMF (Kroupa 2001) and the multiple-system IMF (Chabrier 2005) are shown for comparison with an orange, a green, and a blue line, respectively. The dotted vertical line represents the mass completeness limit of prestellar sources.}\n\\end{figure}\n\n \nFigure\\,\\ref{fig:massspectrum} shows the mass distribution of prestellar sources in Vela-C.\nWe estimate a mass completeness\nlimit of $\\sim$ 4 M$_{\\odot}$, derived by assuming the average temperature of prestellar sources \nof Table\\,\\ref{tab:tab2} and the flux 90\\% completeness limit at 160\\,$\\mu$m (see Table\\,\\ref{tab:tab1}).\nNote that this value is significantly better than the \\emph{BLAST} completeness limit ($\\sim$ 14 M$_{\\odot}$ for sources colder than 14\\,K). We fit a power law \n$N$(log$M$) $\\propto$ $M^{\\gamma}$ to the linear portion of the distribution, finding a slope $\\gamma$\\,=-1.1$\\pm$0.2. The uncertainty was determined by considering both the statistical error of the data and the variation of the slope with the histogram binning, which was varied from 0.1 to 0.3 in log($M$\/M$_{\\odot}$).  \nNoticeably, no variations are found in the slope taking into account only the 193 objects for which $M \\ge 1 M_\\mathrm{BE}$, since their mass distribution differs from that of the whole sample only in the mass range below the completeness limit, where the fit is not performed.\n \nThe value of $\\gamma$ we find in Vela-C is shallower than that found by Netterfield et al. ($\\gamma$=-1.9$\\pm$0.2, for sources with $T_d<$ 14\\,K). If we limit the fit to the \\emph{BLAST} mass completeness limit of 14\\,$M_{\\odot}$, however, we obtain a slope $\\gamma\\,=-1.9\\pm0.2$, which reconciles our high-mass end distribution with that obtained with \\emph{BLAST}. More important, and \nthanks to the \\emph{Herschel}'s massive increase in sensitivity, which significantly extends the mass range, a possible change of slope is recognizable at $M \\ga 10$ M$_{\\odot}$. This is close to the value of $\\sim 9\\,$M$_{\\odot}$ where a steepening was probed, for example, in Orion A (Ikeda et al. 2007). Recent theoretical models by Padoan \\& Nordlund (2011) predict such change of slope, even if at lower masses ($M\\,\\sim\\,3-5$\\,M$_{\\odot}$). Unfortunately, no firm conclusion can be drawn on the base of our data because of the poor statistics in the high-mass bins.\n\nCompared with other literature values, our $\\gamma$ value of -1.1 is between the mass distribution slope of CO clumps ($\\gamma$\\,$\\sim$-0.7, Kramer et al. 1998) and that typical of prestellar cores, as for example, that measured in the Aquila Rift ($\\gamma$\\,=\\,-1.45$\\pm$0.2) on a sample of sources with 2\\,$\\la\\,M$\/M$_{\\odot}\\la\\,10$ and diameter typically less than 0.08 pc (K\\\"{o}nyves et al.\\,2010). Hence, our slope reflects the heterogenous nature of our sample, which is composed of objects with a variety of diameters typical of both clumps and cores. Indeed, if we include in the mass distribution fit only sources with diameter less than 0.08 pc (142 objects), we obtain  a slope $\\gamma$=-1.4$\\pm$0.2 (grey dotted line in Figure\\,\\ref{fig:massspectrum}), which well reconciles with the slope of the Aquila Rift. Such a slope is also consistent with the IMF slope for 1.0\\,$\\la\\,M$\/M$_{\\odot}$ (for single-star IMF $\\gamma$=-1.3$\\pm$0.7, Kroupa 2001, while for multiple-systems IMF $\\gamma$=-1.35$\\pm$0.3, Chabrier 2005), likely indicating that fragmentation will proceed less efficiently in these small objects.\n\nFinally, a substantial agremeent within the error bars is provided by the comparison with surveys covering a mass range largerly overlapping that of Vela-C. For example, $\\gamma\\,\\sim$\\,-1.3$\\pm$0.2 for 0.8\\,$\\,< M$\/M$_{\\odot}\\,<6$ in Perseus, Serpens and Ophiucus (Enoch et al. 2008) and $\\gamma$\\,=\\,-1.3$\\pm$0.1 for 3\\,$\\la\\,M$\/M$_{\\odot}\\la\\,60$ in Orion A (Ikeda et al. 2007).\\\\\n\n\\section{Conclusions}\\label{sec:sec5}\nWe have reported on the \\emph{Herschel} observations of the Vela-C star forming region over an area of $\\sim$ 3 squared degrees. From our analysis we obtained the following results:\n\\begin{itemize}\n\\item[-] From a 5\\,$\\sigma$ level catalogue of cold compact sources in five \\emph{Herschel} bands between 70\\,$\\mu$m and 500\\,$\\mu$m, we have selected a robust sub-sample of 268 sources. Their physical diameters indicate that our sample is mainly composed by \ncloud clumps (0.05 pc $ \\la\\,D\\,\\la $ 0.13 pc), together with a $\\sim$\\,25\\% of cores (0.025 pc $ \\la\\,D\\,\\la $ 0.05 pc). \n\\item[-] Based on the detection of a 70\\,$\\mu$m flux, we have identified 218 starless and 48 protostellar sources. For two further sources\nwe do not give a secure classification, but suggest them as candidate Class\\,0 protostars.\n\\item[-] Source physical parameters have been derived from modified black body fits to the SEDs. Both starless and protostellar sources are\non average colder than surrounding medium. This indicates that the radiation from interstellar field and\/or from embedded protostars\nis unefficient in penetrating the cold dust in deep.\n\\item[-] Protostellar sources are on average sligthly warmer and more compact than starless sources. Both these evidences can be ascribed to the presence of an internal source(s) of moderate heating, which also causes a temperature gradient and a more peaked intensity distribution. Moreover, the reduced dimensions of protostellar sources may indicate that they will not fragment further.\n\\item[-] No significant differences are found between the masses of the two groups. These range from sub-solar up to tens of solar masses.\nIn particular, we find 8 objects with $M >$ 20\\,M$_{\\odot}$, which are potential candidate progenitors of high-mass stars.\n\\item[-] More than 90\\% of the starless sources result prestellar (i.e. bound) if a virial analysis is applied. This percentage, however, should be considered as an upper limit, both because our sensitivity does not allow us to efficiently probe the coldest unbound sources and because a number of the massive prestellar sources could be gravitationally unstable.\n\\item[-] A luminosity vs. mass diagram for the two populations of prestellar and protostellar sources has been constructed. Prestellar sources cluster in a well defined region of the diagram corresponding to a very early stage in which no mass accretion\nis expected to have started. A tentative power-law fit to the observed distribution gives a dependence $L_{bol}\\propto M^{(0.42\\pm0.04)}$. Conversely, protostellar sources populate the diagram region corresponding to the early accretion phase.\n\\item[-] The mass distribution of the prestellar sources with $M \\ge$ 4 M$_{\\odot}$ shows a slope of $-1.1\\pm0.2$. This is between that typical of CO clumps and those of cores in closeby star-forming regions, maybe reflecting the heterogeneous nature of our sample, which is a mixture of cores and clumps. We signal a possible a change of slope in the mass distribution for $M \\ga $ 10\\,M$_{\\odot}$, even if the very big errorbars in the higher mass bins prevent us to draw firm conclusions on its reliability.\n\\end{itemize}\n\n \n\n\\begin{acknowledgements}\nSPIRE has been developed by a consortium of institutes led by Cardiff Univ. (UK) and including Univ.Lethbridge (Canada); NAOC\n(China; CEA, LAM (France); IFSI, Univ.Padua (Italy); IAC (Spain); Stockholm Observatory (Sweden); Imperial College London, RAL, UCL-MSSL, UKATC, Univ. Sussex (UK); Caltech, JPL, NHSC, Univ. Colorado (USA). This development has been supported by national \nfunding agencies: CSA (Canada); NAOC (China); CEA, CNES, CNRS (France); ASI (Italy); MCINN (Spain); SNSB (Sweden); STFC and UKSA (UK); and NASA (USA). PACS has been developed by a consortium of institutes led by MPE (Germany) and including UVIE (Austria); KU Leuven, CSL, IMEC (Belgium); CEA, LAM (France); MPIA (Germany); INAF-IFSI\/OAA\/OAP\/OAT, LENS, SISSA (Italy); IAC (Spain). This \ndevelopment has been supported by the funding agencies BMVIT (Austria), ESA-PRODEX (Belgium), CEA\/CNES (France), DLR (Germany),\nASI\/INAF (Italy), and CICYT\/MCYT (Spain). Data processing and maps production has been possible thanks to ASI generous support via contracts I\/005\/07\/0-1, I\/005\/011\/0 and I\/038\/080\/0.\n\\end{acknowledgements}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}