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+{"text":"\\section{Introduction}\nThe so-called quasi-metric framework (QMF), an alternative geometric framework \nfor formulating relativistic gravitation, was invented a number of years ago \n[1, 2]. Currently, the QMF turns out to have a non-viable status due to its\npredicted properties of the cosmic relic neutrino background (assuming \nstandard neutrino physics) [3]. This status could possibly change if future\nexperiments show evidence of the necessary non-standard neutrino physics needed\nto resolve the apparent conflict with experiment. Now said predicted properties\nof the cosmic neutrino background depend crucially on the neutrino physics, but \nnot on the gravitational sector of the QMF. Therefore, the currently \nnon-viable status of the QMF is valid independently of any particular theory \nof quasi-metric gravity.\n\nHowever, the original theory of quasi-metric gravity (OQG) has a much more \nserious problem. That is, due to the very restricted form postulated for\nquasi-metric space-time geometry, only a partial coupling is possible between \nthe active stress-energy tensor and space-time curvature. Unfortunately, this\nmakes the original theory essentially waveless since said restricted form\nof the quasi-metric space-time geometry makes it fully determinable by the \nmatter sources alone. Unlike general relativity (GR), where the field equations\ndirectly determine the Ricci tensor only, leaving the Weyl curvature free,\nthe OQG leaves free no aspect of quasi-metric space-time geometry. Thus no \nkinds of gravitational waves in vacuum can exist according to the OQG. With \nthe recent direct experimental evidence for GR-like gravitational waves, this \nmeans that the OQG must be abandoned and at best be treated as a waveless \napproximation theory.\n\nNow it turns out that it is not possible to have both a sensible weak-field\nlimit of the contracted Bianchi identities and in addition allowing a full \ncoupling of quasi-metric space-time geometry to the active stress-energy tensor.\nIn fact, if one tries this, the weak field limit of the contracted Bianchi \nidentities will be inconsistent with that of the local conservations laws. \nFortunately, it is possible to avoid this problem by relaxing the postulated \nform of the quasi-metric space-time geometry sufficiently to allow the \nexistence of one extra field equation not being directly coupled to the active \nstress-energy tensor. As we shall see, for weak gravitational fields in vacum, \nthis extra field equation has, by construction, a dynamical structure somewhat \nsimilar to its counterpart in canonical GR. That is, due to some resemblance in\nform to the dynamical structure of canonical GR, the extended field equations \npredict weak GR-like gravitational waves in vacuum. This means that there is \nsome hope that the extended version of quasi-metric gravity may eventually \nturn out to be viable.\n\n\\section{Fully extended quasi-metric gravity}\n\\subsection{General space-time geometry}\nThe QMF has been published in detail in [1] (see also [2]). Here we include \nonly the minimum basics and the required adoptions made to acommodate the \nextended quasi-metric field equations.\n\nIn short, the basic theoretical motivation for introducing the QMF is to \neliminate the in principle enormous number of potential possibilites regarding \ncosmological dynamics and evolution existing for metric theories of gravity. \nSince the Universe is presumely unique, the existence of such multiple \npotential possibilities is a problem and a liability because at most one of \nsaid possibilities is available for testing. This means that the predictive \npower of a cosmology based on metric theories of gravity is in general weak; \neither one must try to solve said problem in some {\\em ad hoc} manner, or one \nis at best limited to fitting a number of cosmological parameters in a \nconsistent way. On the other hand the QMF solves said problem in a geometrical\nway; in quasi-metric space-time there is no room at all for potential \ncosmological dynamics involving the cosmic expansion. This follows from the \nidea that the cosmic expansion should be described as a general phenomenon not \ndepending on the causal structure associated with any pseudo-Riemannian \nmanifold. In other words, within the QMF, the cosmic expansion has nothing to \ndo with causality or dynamics; rather it is an ``absolute'' intrinsic property \nof quasi-metric space-time itself. And as we shall see in this section, the \ngeometrical structure of quasi-metric space-time ensures that this alternative \nway of describing the cosmic expansion is mathematically consistent and \nfundamentally different from its counterpart in GR. In what follows it is \nshown how said motivation is realized geometrically.\n\nThe geometrical basis of the QMF consists of a 5-dimensional \ndifferentiable manifold with topology ${\\cal M}{\\times}{\\bf R}_1$, where \n${\\cal M}={\\cal S}{\\times}{\\bf R}_2$ is a Lorentzian space-time manifold, \n${\\bf R}_1$ and ${\\bf R}_2$ both denote the real line and ${\\cal S}$ is a \ncompact 3-dimensional manifold (without boundaries). That is, in addition to \nthe usual time dimension and 3 space dimensions, there is an extra time \ndimension represented by {\\em the global time function} $t$ introduced as a\nglobal time coordinate on ${\\bf R}_1$. The reason for introducing this extra \ntime dimension is that by definition, $t$ parametrizes any change in the \nspace-time geometry that has to do with the cosmic expansion. By construction, \nthe extra time dimension is degenerate to ensure that such changes will have \nnothing to to with causality. Mathematically, to fulfil this property, the \nmanifold ${\\cal M}{\\times}{\\bf R}_1$ is equipped with two degenerate \n5-dimensional metrics ${\\bf {\\bar g}}_t$ and ${\\bf g}_t$. The metric \n${\\bf {\\bar g}}_t$ is found from field equations as a solution, whereas the \n``physical'' metric ${\\bf g}_t$ can be constructed (locally) from \n${\\bf {\\bar g}}_t$ (for details, see refs. [1, 2]).\n\nThe global time function is unique in the sense that it splits quasi-metric \nspace-time into a unique set of ``distinguished'' 3-dimensional spatial \nhypersurfaces called the {\\em fundamental hypersurfaces (FHSs).} Observers \nalways moving orthogonally to the FHSs are called {\\em fundamental observers \n(FOs)}. The topology of ${\\cal M}$ indicates that there also exists a unique \n``preferred'' ordinary global time coordinate $x^0$. We use this fact to \nconstruct the 4-dimensional quasi-metric space-time manifold $\\cal N$ by \nslicing the submanifold determined by the equation $x^0=ct$ out of the \n5-dimensional differentiable manifold. (It is essential that this slicing is \nunique since the two global time coordinates should be physically equivalent; \nthe only reason to separate between them is that they are designed to \nparametrize fundamentally different physical phenomena.) The general \ndomain of applicability of the 5-dimensional degenerate metric fields \n${\\bf {\\bar g}}_t$ and ${\\bf g}_t$ is limited to $\\cal N$. Moreover, their \ndegeneracy means that they may be regarded as one-parameter families of \nLorentzian 4-metrics on $\\cal N$. Notice that there exists a set of particular \ncoordinate systems especially well adapted to the geometrical structure of \nquasi-metric space-time, {\\em the global time coordinate systems (GTCSs)}. A \ncoordinate system is a GTCS iff the time coordinate $x^0$ is related to $t$ \nvia the equation $x^0=ct$ in ${\\cal N}$.\n\nIn what follows, we will use index notation where Greek indices are ordinary\nspace-time indices taking values in the range $0..3$, while Latin indices are\nspace indices taking values in the range $1..3$. Any implicit dependence on $t$\nwill be indicated with a separate index, e.g., the family ${\\bf {\\bar g}}_t$ \nhas the space-time coordinates ${\\bar g}_{(t){\\mu}{\\nu}}$. Moreover, Einstein's \nsummation convention will be used throughout. Using said notations, and \nexpressed in a suitable GTCS, we now write down the most general form allowed \nfor the family ${\\bf {\\bar g}}_t$ including both explicit and implicit \ndependences on $t$. That is, a general family ${\\bf {\\bar g}}_t$ can be \nrepresented by the family of line elements valid on the FHSs (this may be \ntaken as a definition)\n\\begin{eqnarray}\n{\\overline {ds}}_t^2={\\bar N}_t^2{\\Big \\{ }\n[{\\bar N}_{(t)}^k{\\bar N}_{(t)}^s{\\tilde h}_{(t)ks}-1](dx^0)^2+\n2{\\frac{t}{t_0}}{\\bar N}_{(t)}^k{\\tilde h}_{(t)ks}dx^sdx^0+\n{\\frac{t^2}{t_0^2}}{\\tilde h}_{(t)ks}dx^kdx^s{\\Big \\} }.\n\\end{eqnarray}\nHere, $t_0$ is some arbitrary reference epoch (usually chosen to be the present\nepoch) setting the scale of the spatial coordinates, ${\\bar N}_t$ is the family\nof lapse functions of the FOs and ${\\frac{t_0}{t}}{\\bar N}^k_{(t)}$ are the \ncomponents of the shift vector family of the FOs in \n$({\\cal N},{\\bf {\\bar g}}_t)$. Also, ${\\bar h}_{(t)ks}dx^kdx^s{\\equiv}\n{\\frac{t^2}{t_0^2}}{\\bar N}_t^2{\\tilde h}_{(t)ks}dx^kdx^s$ is the spatial metric \nfamily intrinsic to the FHSs. \n\nIn the OQG, the form of the metric family (1) was severely restricted by \npostulating that the ``basic'' spatial metric family \n$d{\\tilde{\\sigma}}_t^2{\\equiv}{\\tilde h}_{(t)ik}dx^idx^k$ of the FHSs must be \nset equal to the metric $S_{ik}dx^idx^k$ of the 3-sphere (with radius $ct_0$). \nThe reason for this restriction was to ensure the uniqueness of $t$ by \nrequiring the FHSs to be compact [1]. However, this requirement inevitably \nleads to some form of prior 3-geometry. Then said restriction was also thought \nto prevent the possibility that the prior 3-geometry might interfere with the \ndynamics of ${\\bf {\\bar g}}_t$. On the other hand, except for the explicit \ndependence on $t$, the form of equation (1) may seem completely general. But \nthis is not really so since, as we shall see later, in order to have a viable \ntheory it is necessary that the Ricci survature scalar family ${\\tilde P}_t$, \ncalculated from the spatial geometry $d{\\tilde{\\sigma}}_t^2$ of the FHSs, \nshould take a restricted form. This means that the FHSs are still required to \nbe compact and that there still will be prior 3-geometry. The difference from \nthe original theory is that in the revised theory, the prior 3-geometry is less\nrestrictive and it will be indirectly implemented via certain terms in the \nextended field equations rather than as an explicit restricion of equation (1).\n\nThe families ${\\bf {\\bar g}}_t$ and ${\\bf g}_t$ are related by the (local)\ntransformation ${\\bf {\\bar g}}_t{\\rightarrow}{\\bf g}_t$ as described in\n[1, 2]. A general form for the family ${\\bf g}_t$ is given by the family of \nline elements (using a GTCS)\n\\begin{eqnarray}\n{ds}_t^2=[N_{(t)}^kN_{(t)}^s{\\hat h}_{(t)ks}-N^2](dx^0)^2+\n2{\\frac{t}{t_0}}N_{(t)}^k{\\hat h}_{(t)ks}dx^sdx^0+\n{\\frac{t^2}{t_0^2}}{\\hat h}_{(t)ks}dx^kdx^s,\n\\end{eqnarray}\nwhere the symbols have similar meanings to their (barred) counterparts in \nequation (1) (the counterpart to ${\\bar h}_{(t)ks}$ is\n$h_{(t)ks}{\\equiv}{\\frac{t^2}{t_0^2}}{\\hat h}_{(t)ks}$). Note that the propagation\nof sources (and test particles) is calculated by using the equations of motion \nin $({\\cal N},{\\bf g}_t)$ (see equation (7) below). Moreover, since the proper \ntime as measured along a world line of a FO should not directly depend on the \ncosmic expansion, the lapse function $N$ should not depend explicitly on $t$. \nTherefore, any potential $t$-dependence of $N$ must be eliminated by \nsubstituting $t$ with $x^0\/c$ (using a GTCS) whenever it occurs before using \nthe equations of motion. In the same way, any extra $t$-dependence of \n${\\bf g}_t$ coming from the transformation \n${\\bf {\\bar g}}_t{\\rightarrow}{\\bf g}_t$ must be eliminated. Consequently, any \n$t$-dependence of ${\\hat h}_{(t)ks}$ will stem from that of ${\\tilde h}_{(t)ks}$. \nAlso notice that, if for some reason one wants to use the equations of motion \nin $({\\cal N},{\\bf {\\bar g}}_t)$, any explicit dependence of ${\\bar N}_t$ on \n$t$ must be eliminated as well.\n\nNext, $({\\cal N},{\\bf {\\bar g}}_t)$ and $({\\cal N},{\\bf g}_t)$ are equipped \nwith linear and symmetric connections ${\\ }{\\topstar{\\bf {\\bar {\\nabla}}}}{\\ }$\nand ${\\ }{\\topstar{\\bf {\\nabla}}}{\\ }$, respectively. These connections are\nidentified with the usual Levi-Civita connection for constant $t$, yielding the\nstandard form of the connection coefficients not containing $t$.\nThe rest of the connection coefficients are determined by the condition that,\nthe connections ${\\ }{\\topstar{\\bf {\\bar {\\nabla}}}}{\\ }$\nand ${\\ }{\\topstar{\\bf {\\nabla}}}{\\ }$ should be compatible with the \nnon-degenerate part of ${\\bf {\\bar g}}_t$ and ${\\bf g}_t$, respectively. That \nis, we have the conditions\n\\begin{eqnarray}\n{\\topstar{\\bf {\\bar {\\nabla}}}}_{\\frac{\\partial}{{\\partial}t}}\n{\\bf {\\bar g}}_t=0, \\qquad\n{\\topstar{\\bf {\\bar {\\nabla}}}}_{\\frac{\\partial}{{\\partial}t}}\n{\\bf {\\bar n}}_t=0, \\qquad\n{\\topstar{\\bf {\\nabla}}}_{\\frac{\\partial}{{\\partial}t}}\n{\\bf g}_t=0, \\qquad\n{\\topstar{\\bf {\\nabla}}}_{\\frac{\\partial}{{\\partial}t}}\n{\\bf n}_t=0,\n\\end{eqnarray}\nwhere ${\\bf {\\bar n}}_t$ and ${\\bf n}_t$ are families of unit normal vector \nfields to the FHSs in $({\\cal N},{\\bf {\\bar g}}_t)$ and $({\\cal N},{\\bf g}_t)$,\nrespectively. The conditions shown in equation (3) will hold if we make the \nrequirements (where a comma denotes taking a partial derivative)\n\\begin{eqnarray} \n{\\frac{\\partial}{{\\partial}t}}{\\Big [}{\\bar N}_{(t)}^k{\\bar N}_{(t)}^s\n{\\tilde h}_{(t)ks}{\\Big ]}=0, \\qquad {\\Rightarrow} \\qquad\n{\\bar N}_{(t),t}^s=-{\\frac{1}{2}}{\\bar N}_{(t)}^k\n{\\tilde h}_{(t)}^{is}{\\tilde h}_{(t)ik,t},\n\\end{eqnarray}\nand\n\\begin{eqnarray} \n{\\frac{\\partial}{{\\partial}t}}{\\Big [}N_{(t)}^kN_{(t)}^s\n{\\hat h}_{(t)ks}{\\Big ]}=0, \\qquad {\\Rightarrow} \\qquad\nN_{(t),t}^s=-{\\frac{1}{2}}N_{(t)}^k{\\hat h}_{(t)}^{is}{\\hat h}_{(t)ik,t}.\n\\end{eqnarray}\nGiven the requirements (4) and (5), the conditions shown in equation (3) \nnow yield the in general nonzero extra connection coefficients (using a GTCS)\n\\begin{eqnarray}\n{\\topstar{\\bar {\\Gamma}}}_{t0}^{{\\,}0}={\\frac{{\\bar N}_{t},_t}{{\\bar N}_t}},\n\\quad \n{\\topstar{\\bar {\\Gamma}}}_{tj}^{{\\,}i}=\n{\\Big (}{\\frac{1}{t}}+{\\frac{{\\bar N}_t,_t}{{\\bar N}_t}}{\\Big )}{\\delta}^i_j\n+{\\frac{1}{2}}{\\tilde h}_{(t)}^{is}{\\tilde h}_{(t)sj,t},\n\\quad \n{\\topstar{\\Gamma}}_{tj}^{{\\,}i}={\\frac{1}{t}}{\\delta}^i_j+\n{\\frac{1}{2}}{\\hat h}_{(t)}^{is}{\\hat h}_{(t)sj,t}.\n\\end{eqnarray}\nNote that all connection coefficients are symmetric in the lower indices.\nThe equations of motion in $({\\cal N},{\\bf g}_t)$ are given by [1, 2]\n\\begin{eqnarray}\n{\\frac{d^2x^{\\mu}}{d{\\lambda}^2}}+{\\Big (}\n{\\topstar{\\Gamma}}_{t{\\nu}}^{{\\,}{\\mu}}{\\frac{dt}{d{\\lambda}}}+\n{\\topstar{\\Gamma}}_{{\\beta}{\\nu}}^{{\\,}{\\mu}}{\\frac{dx^{\\beta}}{d{\\lambda}}}\n{\\Big )}{\\frac{dx^{\\nu}}{d{\\lambda}}}\n={\\Big (}{\\frac{d{\\tau}_t}{d{\\lambda}}}{\\Big )}^2a_{(t)}^{\\mu}.\n\\end{eqnarray}\nHere, $d{\\tau}_t$ is the proper time interval as measured along the curve,\n${\\lambda}$ is some general affine parameter, and ${\\bf a}_t$ is the \n4-acceleration measured along the curve.\n\\subsection{The extended field equations}\nOne important postulate of the OQG is that gravitational quantities should be\n``formally'' variable when measured in atomic units. This formal variability \nis also a postulate of revised quasi-metric gravity and applies to all \ndimensionful gravitational quantities. Said formal variability may be viewed as\nan interpretation of equation (1) and is directly connected to the spatial \nscale factor ${\\bar F}_t{\\equiv}{\\bar N}_tct$ of the FHSs [1, 2]. \nIn particular, the formal variability applies to any potential gravitational \ncoupling parameter $G_t$. It is convenient to transfer the formal variability \nof $G_t$ to mass (and charge, if any) so that all formal variability is taken \ninto account and included in the {\\em active stress-energy tensor} ${\\bf T}_t$,\nwhich is the object that couples to space-time geometry via field equations. \nHowever, dimensional analysis yields that the gravitational coupling must be \nnon-universal, i.e., that the electromagnetic active stress-energy tensor \n${\\bf T}_t^{{\\rm (EM)}}$ and the active stress-energy tensor for material \nparticles ${\\bf T}_t^{\\rm (MA)}$ couple to space-time curvature via two different \n(constant) coupling parameters $G^{\\rm B}$ and $G^{\\rm S}$, respectively. This \nnon-universality of the gravitational coupling is required for consistency \nreasons. As a consequence, compared to GR, the non-universal gravitational \ncoupling yields a modification of the right hand side of any quasi-metric \ngravitational field equations. (Said modification was missed in the original \nformulation of quasi-metric gravity.) The quantities $G^{\\rm B}$ and $G^{\\rm S}$\nplay the roles of gravitational constants measured in some local gravitational\nmeasurements at some chosen event at the arbitrary reference epoch $t_0$.\n\nBefore trying to construct quasi-metric field equations, we notice that we\ncannot use curvature tensors calculated from the full connection in \n$({\\cal N},{\\bf {\\bar g}}_t)$ since its dependence on $t$ should not have \nanything directly to do with gravitation. Rather, we must use curvature tensors\ncalculated from the usual Levi-Civita connection in \n$({\\cal M},{\\bf {\\bar g}}_t)$, i.e., such tensors should be calculated from \nequation (1) holding $t$ fixed. When $t$ varies, said curvature tensors \nconstitute tensor families in $({\\cal N},{\\bf {\\bar g}}_t)$. Potential field \nequations in $({\\cal N},{\\bf {\\bar g}}_t)$ may then be found by using \nprojections of said curvature tensor families with respect to the FHSs and \ncoupling said projections to the relevant projections of ${\\bf T}_t^{\\rm (EM)}$ \nand ${\\bf T}_t^{\\rm (MA)}$.\n\nAs mentioned earlier, the form of equation (1), and thus of ${\\bf {\\bar g}}_t$, \nin the OQG was too restricted to admit the existence of a full coupling \nbetween space-time curvature and the active stress-energy tensor ${\\bf T}_t$. \nRather, a subset of the Einstein field equations (with the right hand sides\nmodified) was tailored to ${\\bf {\\bar g}}_t$, yielding {\\em partial} couplings \nto space-time curvature of ${\\bf T}_t^{{\\rm (EM)}}$ and ${\\bf T}_t^{\\rm (MA)}$. That\nis, a postulate of the OQG was the field equations\n\\begin{eqnarray}\n2{\\bar R}_{(t){\\bar {\\perp}}{\\bar {\\perp}}}=\n2(c^{-2}{\\bar a}_{{\\cal F}{\\mid}i}^i+\nc^{-4}{\\bar a}_{{\\cal F}i}{\\bar a}_{\\cal F}^i-\n{\\bar K}_{(t)ik}{\\bar K}_{(t)}^{ik}+\n{\\cal L}_{{\\bf {\\bar n}}_t}{\\bar K}_t) \\nonumber \\\\\n={\\kappa}^{\\rm B}(T^{{\\rm (EM)}}_{(t){\\bar {\\perp}}{\\bar {\\perp}}}\n+{\\hat T}^{{\\rm (EM)}i}_{(t)i})+\n{\\kappa}^{\\rm S}(T^{{\\rm (MA)}}_{(t){\\bar {\\perp}}{\\bar {\\perp}}}\n+{\\hat T}^{{\\rm (MA)}i}_{(t)i}), \\qquad c^{-2}{\\bar a}_{{\\cal F}i}{\\equiv}\n{\\frac{{\\bar N}_t,_i}{{\\bar N}_t}},\n\\end{eqnarray}\n\\begin{eqnarray}\n{\\bar R}_{(t)j{\\bar {\\perp}}}={\\bar K}_{(t)j{\\mid}i}^i-{\\bar K}_t,_j=\n{\\kappa}^{\\rm B}T^{{\\rm (EM)}}_{(t)j{\\bar {\\perp}}}\n+{\\kappa}^{\\rm S}T^{\\rm (MA)}_{(t)j{\\bar {\\perp}}}.\n\\end{eqnarray}\nHere, ${\\bf {\\bar R}}_t$ is the Ricci tensor family corresponding to the metric\nfamily ${\\bf {\\bar g}}_t$ and the symbol '${\\bar {\\perp}}$' denotes a scalar\nproduct with $-{\\bf {\\bar n}}_t$. Moreover, ${\\cal L}_{{\\bf {\\bar n}}_t}$ denotes \na projected Lie derivative in the direction normal to the FHSs, \n${\\bf {\\bar K}}_t$ denotes the extrinsic curvature tensor family (with trace \n${\\bar K}_t$) of the FHSs, a ``hat'' denotes an object projected into the FHSs \nand the symbol '${\\mid}$' denotes a spatial covariant derivative. (Note that \n${\\cal L}_{{\\bf {\\bar n}}_t}$ operates on spatial objects only.) Finally \n${\\kappa}^{\\rm B}{\\equiv}8{\\pi}G^{\\rm B}\/c^4$ and\n${\\kappa}^{\\rm S}{\\equiv}8{\\pi}G^{\\rm S}\/c^4$, where the values of $G^{\\rm B}$ and \n$G^{\\rm S}$ are by convention chosen as those measured in some local \ngravitational measurements at some chosen event at the arbitrary reference \nepoch $t_0$. \n\nEquations (8) and (9) consist of one dynamical scalar equation and one \nconstraint 3-vector equation, respectively. The dynamical field in \n$({\\cal N},{\\bar {\\bf g}}_t)$ is the lapse function family ${\\bar N}_t$.\nThat is, the time evolution of ${\\bf {\\bar K}}_t$ (with $t$ fixed) is \ndetermined by the time evolution of ${\\bar N}_t$ (with $t$ fixed), since we \nhave that $2{\\bar K}_{(t)ij}=-{\\frac{1}{{\\bar N}_t}}\n{\\cal L}_{{\\bar N}_t{\\bf {\\bar n}}_t}{\\bar h}_{(t)ij}$ (see, e.g., [4]). (In addition \nthe matter variables evolve in time according to the local conservation laws in \n$({\\cal N},{\\bar {\\bf g}}_t)$, see equations (22) and (23) below.) \nUnfortunately, the equation set (8), (9) has no (scalar) wave-like solutions \nin vacuum, given the restricions on ${\\bf {\\bar g}}_t$ from the OQG. Thus using\nthe approach of the OQG, no aspects of ${\\bf {\\bar g}}_t$ were left free, \nmeaning that ${\\bf {\\bar g}}_t$ would be fully determinable by the matter \nsources alone. So the OQG is essentially a waveless approximation theory, and \nit must therefore be discarded as a potentially viable candidate for \nquasi-metric gravity.\n\nTo correct said inadequacies of the OQG, it is crucial to find new field \nequations that allow the existence of GR-like gravitational waves. Since it is \nnecessary to have a correspondence between the new field equations and the OQG,\nit is necessary to keep equations (8) and (9) and to extend them with \nadditional field equations. One might expect that an extended set should \nrepresent a full coupling between space-time curvature and ${\\bf T}_t$, in \naddition to being compatible with equation (1). That is, we would expect to \nfind a new spacetime tensor family ${\\bf {\\bar Q}}_t$ defined from its \nprojections ${\\bar Q}_{(t){\\bar {\\perp}}{\\bar {\\perp}}}$,\n${\\bar Q}_{(t)j{\\bar {\\perp}}}={\\bar Q}_{(t){\\bar {\\perp}}j}$ and\n${\\bar Q}_{(t)ik}$ with respect to the FHSs. These projections are expected to \nplay almost the same role as do the projections of the Einstein tensor in \ncanonical GR. However, it is important to notice that unlike \n${\\bf {\\bar R}}_t$ and the Einstein tensor family ${\\bf {\\bar G}}_t$, the \ndefinition of ${\\bf {\\bar Q}}_t$ depends directly on the geometry of the FHSs \nand their extrinsic curvature. This means that the expected expressions for the\nprojections of ${\\bf {\\bar Q}}_t$ will not be valid for any hypersurfaces other\nthan the FHSs. In contrast, in canonical GR, the projections of the Einstein \ntensor ${\\bf {\\bar G}}$ on a Lorentzian manifold with metric ${\\bf {\\bar g}}$ \nis valid for any foliation of ${\\bf {\\bar g}}$ into spatial hypersurfaces. \nThose projections are given by (see, e.g., [4] for a derivation)\n\\begin{eqnarray}\n{\\bar G}_{{\\bar {\\perp}}{\\bar {\\perp}}}={\\frac{1}{2}}({\\bar P}+{\\bar K}^2-\n{\\bar K}_{mn}{\\bar K}^{mn}),\n\\end{eqnarray}\n\\begin{eqnarray}\n{\\bar G}_{{\\bar {\\perp}}j}={\\bar G}_{j{\\bar {\\perp}}}{\\equiv}\n{\\bar R}_{j{\\bar {\\perp}}}=\n({\\bar K}_{{\\ }j}^k-{\\bar K}{\\delta}_{{\\ }j}^k)_{{\\mid}k},\n\\end{eqnarray}\n\\begin{eqnarray}\n{\\bar G}_{ik}=-{\\frac{1}{\\bar N}}\n{\\cal L}_{{\\bar N}{\\bf {\\bar n}}}({\\bar K}_{ik}-\n{\\bar K}{\\bar h}_{ik})+3{\\bar K}{\\bar K}_{ik}-\n2{\\bar K}_{i}^{{\\ }s}{\\bar K}_{sk}-{\\frac{1}{2}}({\\bar K}^2+\n{\\bar K}_{mn}{\\bar K}^{mn}){\\bar h}_{ik} \\nonumber \\\\\n-c^{-2}{\\bar a}_{i{\\mid}k}-c^{-4}{\\bar a}_i{\\bar a}_k+\n(c^{-2}{\\bar a}^s_{{\\ }{\\mid}s}+c^{-4}{\\bar a}^s{\\bar a}_s){\\bar h}_{ik}+\n{\\bar H}_{ik}, \\qquad c^{-2}{\\bar a}_i{\\equiv}\n{\\frac{{\\bar N}_{,i}}{{\\bar N}}},\n\\end{eqnarray}\nwhere ${\\bar h}_{ik}$ are the components of the spatial metric. Moreover, \n${\\bar P}$ and ${\\bar H}_{ik}$ are the spatial Ricci scalar and the components \nof the spatial Einstein tensor, respectively.\n\nWe will now require that ${\\bf {\\bar Q}}_t$ and ${\\bf {\\bar G}}$ should have \nsomewhat similar dynamical structures. That is, ${\\bar Q}_{(t)ik}$ and \n${\\bar G}_{ik}$ should both predict weak GR-like gravitational waves in vacuum \nvia having common (up to signs) second order terms $-{\\frac{1}{\\bar N}}\n{\\cal L}_{{\\bar N}{\\bf {\\bar n}}}{\\bar K}_{ik}$ and ${\\bar H}_{ik}$ in equation (12).\nFurthermore, we must have that ${\\bar Q}_{(t){\\bar {\\perp}}{\\bar {\\perp}}}+\n{\\hat{\\bar Q}}^s_{(t)s}=2{\\bar R}_{(t){\\bar {\\perp}}{\\bar {\\perp}}}$ to fulfil equation\n(8), and ${\\bar Q}_{(t){\\bar {\\perp}}j}={\\bar R}_{(t){\\bar {\\perp}}j}$ to fulfil \nequation (9). Besides, the extended field equations should also yield the same \nsolutions as the OQG for the metrically static vacuum cases (for which the \nextrinsic curvature vanishes identically). Thus for these cases, the equation\n${\\bar Q}_{(t)ik}=0$ should yield the relationship ${\\bar H}_{(t)ik}+\nc^{-2}{\\bar a}_{{\\cal F}i{\\mid}k}+c^{-4}{\\bar a}_{{\\cal F}i}{\\bar a}_{{\\cal F}k}-\n(c^{-2}{\\bar a}_{{\\cal F}{\\mid}s}^s-{\\frac{1}{(ct{\\bar N}_t)^2}}){\\bar h}_{(t)ik}=0$,\nwhich follows directly from the OQG [2]. But the extrinsic curvature also \nvanishes identically for metrically static interiors, so this means that we \nshould have ${\\bar Q}_{(t)ik}=-c^{-2}{\\bar a}_{{\\cal F}i{\\mid}k}-\nc^{-4}{\\bar a}_{{\\cal F}i}{\\bar a}_{{\\cal F}k}+(c^{-2}{\\bar a}_{{\\cal F}{\\mid}s}^s\n-{\\frac{1}{(ct{\\bar N}_t)^2}}){\\bar h}_{(t)ik}-{\\bar H}_{(t)ik}$ and thus\n${\\bar Q}_{(t){\\bar {\\perp}}{\\bar {\\perp}}}=-{\\frac{1}{2}}{\\bar P}_t\n+3c^{-4}{\\bar a}_{{\\cal F}s}{\\bar a}_{\\cal F}^s+{\\frac{3}{(ct{\\bar N}_t)^2}}$\nfor the metrically static cases. (The other sign for ${\\bar Q}_{(t)ik}$ cannot \nbe chosen since we for physical reasons in general must have \n${\\bar Q}_{(t){\\bar {\\perp}}{\\bar {\\perp}}}>0$ and ${\\hat{\\bar Q}}^s_{(t)s}>0$ for \nmetrically static interiors.)\n\nHowever, at this point a crucial problem arises due to the contracted Bianchi\nidentities ${\\bar G}_{(t){\\mu};{\\nu}}^{\\nu}{\\equiv}0$. Projected with respect to \nthe FHSs, these identities read (see, e.g., [4])\n\\begin{eqnarray}\n{\\cal L}_{{\\bf {\\bar n}}_t}{\\bar G}_{(t){\\bar {\\perp}}{\\bar {\\perp}}}=\n{\\bar K}_t{\\bar G}_{(t){\\bar {\\perp}}{\\bar {\\perp}}}+{\\bar K}_{(t)}^{ik}\n{\\bar G}_{(t)ik}-2c^{-2}{\\bar a}_{\\cal F}^i{\\bar G}_{(t){\\bar {\\perp}}i}\n-{\\hat {\\bar G}}^i_{(t){\\bar {\\perp}}{\\mid}i},\n\\end{eqnarray}\n\\begin{eqnarray}\n{\\frac{1}{{\\bar N}_t}}{\\cal L}_{{\\bar N}_t{\\bf {\\bar n}}_t}\n{\\bar G}_{(t)j{\\bar {\\perp}}}={\\bar K}_t{\\bar G}_{(t)j{\\bar {\\perp}}}\n-c^{-2}{\\bar a}_{{\\cal F}j}{\\bar G}_{(t){\\bar {\\perp}}{\\bar {\\perp}}}\n-c^{-2}{\\bar a}_{\\cal F}^i{\\bar G}_{(t)ji}-{\\hat {\\bar G}}^i_{(t)j{\\mid}i}.\n\\end{eqnarray}\nThat is, it turns out that equations (9) and (14), in combination with the \ndeduced expressions for ${\\bar Q}_{(t){\\bar {\\perp}}{\\bar {\\perp}}}$ and \n${\\bar Q}_{(t)ik}$ for metrically static systems, yield the wrong Newtonian \nlimit, so that equation (14) does not correspond with its counterpart Euler \nequation (see below). In fact, the only way to avoid this problem while still \nkeeping the relationship ${\\bar Q}_{(t){\\bar {\\perp}}{\\bar {\\perp}}}+\n{\\hat{\\bar Q}}^s_{(t)s}=2{\\bar R}_{(t){\\bar {\\perp}}{\\bar {\\perp}}}$, is to choose\n${\\bar Q}_{(t){\\bar {\\perp}}{\\bar {\\perp}}}=2{\\bar R}_{(t){\\bar {\\perp}}{\\bar {\\perp}}}$ and\n${\\bar Q}_{(t)ik}=0$. Thus there can be no extra scalar field equation besides \nequation (8) and also no additional spatial tensor equation coupling directly\nto ${\\bf T}_t$. In other words, we have found that {\\em it is not possible \nto construct a viable, fully coupled quasi-metric gravitational theory.} \n\nNevertheless, fortunately it is still possible to have a traceless field \nequation ${\\bar Q}_{(t)ik}=0$ with only indirect dependence on equation (8) \nhaving the desired dynamical properties in addition to being compatible with \nequation (14). The simplest possible choice of such an equation (with a \nminimum number of terms quadratic in extrinsic curvature) is\n\\begin{eqnarray}\n{\\bar Q}_{(t)ik}{\\equiv}{\\frac{1}{{\\bar N}_t}}{\\cal L}_{{\\bar N}_t\n{\\bf {\\bar n}}_t}{\\bar K}_{(t)ik}-{\\frac{1}{3}}({\\cal L}_{{\\bf {\\bar n}}_t}\n{\\bar K}_t){\\bar h}_{(t)ik}+2{\\bar K}_{(t)is}{\\bar K}_{(t)k}^s\n\\nonumber \\\\\n-c^{-2}{\\bar a}_{{\\cal F}i{\\mid}k}-\nc^{-4}{\\bar a}_{{\\cal F}i}{\\bar a}_{{\\cal F}k}\n+(c^{-2}{\\bar a}_{{\\cal F}{\\mid}s}^s\n-{\\frac{1}{(ct{\\bar N}_t)^2}}){\\bar h}_{(t)ik}-{\\bar H}_{(t)ik}=0,\n\\end{eqnarray}\nwhere the requirement on the spatial Ricci curvature scalar ${\\bar P}_t$,\n\\begin{eqnarray}\n{\\bar P}_t=-4c^{-2}{\\bar a}_{{\\cal F}{\\mid}s}^s\n+2c^{-4}{\\bar a}_{{\\cal F}}^s{\\bar a}_{{\\cal F}s}+{\\frac{6}{(ct{\\bar N}_t)^2}},\n\\end{eqnarray}\nensures that equation (15) is indeed traceless. Besides, the components of\nthe spatial Einstein tensor family ${\\bf {\\bar H}}_t$ is given by\n\\begin{eqnarray}\n{\\bar H}_{(t)ik}=-c^{-2}{\\bar a}_{{\\cal F}i{\\mid}k}-\nc^{-4}{\\bar a}_{{\\cal F}i}{\\bar a}_{{\\cal F}k}+\nc^{-2}{\\bar a}_{{\\cal F}{\\mid}s}^s{\\bar h}_{(t)ik}+{\\tilde H}_{(t)ik},\n\\end{eqnarray}\nwhere ${\\bf {\\tilde H}}_t$ is the ``basic'' spatial Einstein tensor family\ncalculated from the line element family $d{\\tilde{\\sigma}}_t^2$. Notice that, \nwhile equation (16) implies that ${\\tilde P}_t={\\frac{6}{(ct_0)^2}}$,\n${\\tilde H}_{(t)ik}$ is not necessarily equal to \n$-{\\frac{1}{(ct_0)^2}}{\\tilde h}_{(t)ik}$. This shows that the prior 3-geometry \nof the extended theory is less restrictive than that present in the \nOQG, as mentioned in the previous section. This is further illustrated by \nwriting equation (15) in the form (using equation (17))\n\\begin{eqnarray}\n{\\frac{1}{{\\bar N}_t}}{\\cal L}_{{\\bar N}_t\n{\\bf {\\bar n}}_t}{\\bar K}_{(t)ik}-{\\frac{1}{3}}({\\cal L}_{{\\bf {\\bar n}}_t}\n{\\bar K}_t){\\bar h}_{(t)ik}+2{\\bar K}_{(t)is}{\\bar K}_{(t)k}^s=\n{\\tilde H}_{(t)ik}+{\\frac{1}{(ct{\\bar N}_t)^2}}{\\bar h}_{(t)ik}.\n\\end{eqnarray}\nIt is also useful to write down the expression for ${\\bar G}_{(t)ik}$ obtained\nfrom equations (12) and (15). We get\n\\begin{eqnarray}\n{\\bar G}_{(t)ik}=-{\\bar Q}_{(t)ik}+{\\bar K}_t{\\bar K}_{(t)ik}-\n2c^{-2}{\\bar a}_{{\\cal F}i{\\mid}k}-\n2c^{-4}{\\bar a}_{{\\cal F}i}{\\bar a}_{{\\cal F}k} \\nonumber \\\\\n+{\\Big [}{\\frac{2}{3}}{\\bar R}_{(t){\\bar {\\perp}}{\\bar {\\perp}}}+{\\frac{1}{6}}\n{\\bar K}_{(t)mn}{\\bar K}_{(t)}^{mn}-{\\frac{1}{2}}{\\bar K}_t^2\n+{\\frac{4}{3}}c^{-2}{\\bar a}_{{\\cal F}{\\mid}s}^s+{\\frac{1}{3}}\nc^{-4}{\\bar a}_{{\\cal F}}^s{\\bar a}_{{\\cal F}s}-{\\frac{1}{(ct{\\bar N}_t)^2}}\n{\\Big ]}{\\bar h}_{(t)ik}.\n\\end{eqnarray}\nEquations (8), (9) and (15) determine ${\\bar Q}_{(t){\\bar {\\perp}}{\\bar {\\perp}}}\n{\\equiv}2{\\bar R}_{(t){\\bar {\\perp}}{\\bar {\\perp}}}$, ${\\bar Q}_{(t){\\bar {\\perp}}j}\n{\\equiv}{\\bar R}_{(t){\\bar {\\perp}}j}{\\equiv}{\\bar G}_{(t){\\bar {\\perp}}j}$ and \n${\\bar Q}_{(t)ik}$, respectively. (However, as mentioned earlier, \n${\\bf {\\bar Q}}_t$ is not a ``genuine'' space-time object since it is defined \nfrom its projections with respect to a particular space-time foliation.)\nThis makes it possible to calculate ${\\bf {\\bar g}}_t$ from the projections of \nthe physical source ${\\bf T}_t$ with respect to the FHSs. Notice that the field\nequations determine the FHSs as well, since no other foliations of \n${\\bf {\\bar g}}_t$ into spatial hypersurfaces should fulfil these equations. \nMoreover, as we shall see later, the new field equation (15) is sufficiently \nflexible to enable the prediction of weak GR-like gravitational waves in \nvacuum.\n\nFrom equations (14)-(19) we are able to calculate the expression (using the \nfact that ${\\hat{\\bar Q}}^s_{(t)j{\\mid}s}=0$)\n\\begin{eqnarray}\n{\\frac{1}{{\\bar N}_t}}{\\cal L}_{{\\bar N}_t{\\bf {\\bar n}}_t}\n{\\bar G}_{(t)j{\\bar {\\perp}}}=2c^{-2}{\\bar a}_{\\cal F}^s{\\Big [}\n{\\tilde H}_{(t)sj}+{\\frac{1}{(ct{\\bar N}_t)^2}}{\\bar h}_{(t)sj}{\\Big ]}-\n{\\frac{2}{3}}{\\Big [}({\\cal L}_{{\\bf {\\bar n}}_t}{\\bar K}_t),_j+\n({\\cal L}_{{\\bf {\\bar n}}_t}{\\bar K}_t)c^{-2}{\\bar a}_{{\\cal F}j}{\\Big ]}\n\\nonumber \\\\\n+{\\frac{1}{2}}({\\bar K}_{(t)mn}{\\bar K}_{(t)}^{mn}),_j+\n{\\bar K}_{(t)mn}{\\bar K}_{(t)}^{mn}c^{-2}{\\bar a}_{{\\cal F}j}-\n{\\bar K}_{(t)j}^s{\\Big [}{\\bar K}_t,_s+{\\bar K}_tc^{-2}{\\bar a}_{{\\cal F}j}\n{\\Big ]}.\n\\end{eqnarray}\nNotice that equation (20) depends only indirectly on \n${\\bar R}_{(t){\\bar {\\perp}}{\\bar {\\perp}}}$, i.e., equation (20) takes the same form\nboth for vacuum and interior to matter sources.\n \nHowever, we also have the original local conservation laws in \n${\\bf {\\bar g}}_t$ of ${\\bf T}_t$. These are unchanged from the OQG, i.e., for \nfixed $t$ we have\n\\begin{eqnarray}\nT_{(t){\\mu};{\\nu}}^{\\nu}=2{\\frac{{\\bar N}_t,_{\\nu}}{{\\bar N}_t}}\nT_{(t){\\mu}}^{\\nu}=2c^{-2}{\\bar a}_{{\\cal F}s}{\\hat T}_{(t){\\mu}}^s\n-2{\\frac{{\\bar N}_t,_{\\bar {\\perp}}}{{\\bar N}_t}}T_{(t){\\bar {\\perp}}{\\mu}}.\n\\end{eqnarray}\nThe conservation laws must take the form (21) to be consistent with classical \nelectrodynamics coupled to quasi-metric gravity [5]. However, equation (21) and\n(22)-(23) below apply to both ${\\bf T}_t^{{\\rm (EM)}}$ and ${\\bf T}_t^{\\rm (MA)}$\nalike. Moreover, if the only $t$-dependence of ${\\bf T}_t$ is via the above\nmentioned formal variability, ${\\bf T}_t$ is locally conserved when $t$ varies \nas well. Notice that, the covariantly conserved quantity following from \nequation (21) is ${\\bar N}_t^{-2}{\\bf T}_t$ rather than ${\\bf T}_t$. But since \n${\\bar N}_t^{-2}{\\bf T}_t$ depends on the distinguished foliation of \nquasi-metric space-time into the FHSs, this means that potential field \nequations cannot be found from an invariant action principle obtaind from any \nLagrangian involving only ${\\bf {\\bar g}}_t$ and its derivatives, with no \ndependence on any particular foliation. We thus have, unlike its counterpart in\nGR, that equation (21) does not automatially follow from the field equations. \nThat is, equation (21) represents real restrictions on what kind of sources can\nbe admitted in the field equations for a given gravitational system. We shall \nsee an example of this in the next section.\n\nBy projecting equation (21) with respect to the FHSs we get (see, e.g., [4] for\ngeneral projection formulae)\n\\begin{eqnarray}\n{\\cal L}_{{\\bf {\\bar n}}_t}T_{(t){\\bar {\\perp}}{\\bar {\\perp}}}=\n{\\Big (}{\\bar K}_t-2{\\frac{{\\bar N}_t,_{\\bar {\\perp}}}{{\\bar N}_t}}\n{\\Big )}T_{(t){\\bar {\\perp}}{\\bar {\\perp}}}\n+{\\bar K}_{(t)ik}{\\hat T}_{(t)}^{ik}-{\\hat T}^i_{(t){\\bar {\\perp}}{\\mid}i},\n\\end{eqnarray}\n\\begin{eqnarray}\n{\\frac{1}{{\\bar N}_t}}{\\cal L}_{{\\bar N}_t{\\bf {\\bar n}}_t}\nT_{(t)j{\\bar {\\perp}}}={\\Big (}{\\bar K}_t\n-2{\\frac{{\\bar N}_t,_{\\bar {\\perp}}}{{\\bar N}_t}}\n{\\Big )}T_{(t)j{\\bar {\\perp}}}-c^{-2}{\\bar a}_{{\\cal F}j}\nT_{(t){\\bar {\\perp}}{\\bar {\\perp}}}\n+c^{-2}{\\bar a}_{{\\cal F}i}{\\hat T}_{(t)j}^i-{\\hat T}^i_{(t)j{\\mid}i}.\n\\end{eqnarray}\nIt can be shown that equations (22)-(23) yield correct weak-field\napproximations to Newtonian order. That is, one may consider the gravitating\nsource as a perfect fluid where ${\\bf {\\bar u}}_t$ is the 4-velocity vector \nfamily in $({\\cal N},{\\bf {\\bar g}}_t)$ of observers co-moving with the fluid. \nIt is useful to set up the general formula for the split-up of \n${\\bf {\\bar u}}_t$ into pieces respectively normal to and intrinsic to the \nFHSs:\n\\begin{eqnarray}\n{\\bf {\\bar u}}_t= {\\topstar{\\bar {\\gamma}}}(c{\\bf {\\bar n}}_t+\n{\\bf {\\bar w}}_t), \\qquad {\\topstar{\\bar {\\gamma}}}{\\equiv}\n(1-{\\frac{{\\bar w}^2}{c^2}})^{-{\\frac{1}{2}}},\n\\end{eqnarray}\nwhere ${\\bf {\\bar w}}_t$ (with norm ${\\bar w}$) is the 3-velocity family with \nrespect to the FOs. Then it is straightforward to show, using equation (24)\nin combination with equations (22)-(23), that to Newtonian accuracy, each of \nequations (22)-(23) corresponds with the counterpart Euler equation valid for \nNewtonian fluid dynamics. The only difference from the corresponding GR \nweak-field case is that the 3-velocity ${\\bf {\\bar w}}_t$ is taken with respect\nto the FOs rather than with respect to some suitable coordinate system. \nHowever, this difference is crucial since weak-field approximations of \nequations (22)-(23) must be consistent with those of equations (13)-(14). (As \nmentioned above, this requirement is the reason why it is not possible to \nconstruct a mathematically consistent, fully coupled theory of quasi-metric \ngravity that may potentially be viable.) Then one finds that to Newtonian \naccuracy, said consistency requirement is fulfilled only for {\\em metrically \nstationary} systems, i.e., for systems with no dependence on the time \ncoordinate $x^0$ (using a GTCS). Furthermore, the FOs may be considered at rest\nwith respect to some suitable coordinate system only for {\\em metrically \nstatic} systems, i.e., for systems where, in addition to said \ntime-independence, ${\\bf {\\bar w}}_t$ vanishes identically. This means that any\nattempt to construct a general weak-field approximation formalism for \nquasi-metric gravity (along the lines of the parametrized post-Newtonian (PPN) \nformalism valid for metric theories of gravity) will meet some extra \ncomplications. That is, since the FOs will not be at rest with respect to the\nPPN coordinate system whenever ${\\bf {\\bar w}}_t$ does not vanish, the \ncontribution to ${\\bf T}_t $ from ${\\bf {\\bar w}}_t$ will be different from\nthe counterpart GR case. This is one of the reasons why quasi-metric gravity \nis unsuitable for a full standard PPN-analysis.\n\nEquation sets (10)-(12) and (8), (9) plus (15) (or (19)), considered as \ndynamical systems for initial value problems, share some similiarities but are\nalso dissimilar in crucial ways. Both sets consist of dynamical equations plus \nconstraints. The constraint equations are determined by initial data on an \ninitial FHS (or, for the set (10)-(12), initial data on an arbitrary spatial \nhypersurface), whereas the dynamical equations are not. Now we see that the \nquantities ${\\bar G}_{{\\bar {\\perp}}{\\bar {\\perp}}}$, ${\\bar G}_{{\\bar {\\perp}}j}$ and \n${\\bar R}_{(t){\\bar {\\perp}}j}{\\equiv}{\\bar G}_{(t){\\bar {\\perp}}j}$ are all determined\nby the initial data, while the quantities \n${\\bar R}_{(t){\\bar {\\perp}}{\\bar {\\perp}}}$, ${\\bar G}_{ik}$ and ${\\bar Q}_{(t)ik}$ are\nnot. Thus for the quasi-metric system, equation (9) represents the constraints \nwhereas equations (8) and (15) (or (19)) represent the dynamical equations. \nThis means that the number of dynamical\/constraint equations is different for \nthe two systems. That is, in revised quasi-metric gravity, equation (8) \nrepresents an independent dynamical equation on its own in contrast to its \ncounterpart valid for the dynamical system (10)-(12). Also there is no \nquasi-metric counterpart to the constraint equation (10). \n\nBesides, another crucial difference between said dynamical systems is that \nequation (15) is not directly coupled to matter sources. Equivalently, as can \neasily be seen by comparing equations (12) and (19); while the former is fully \ndetermined by the spatial projection $T_{ik}$ of the stress-energy tensor and \nthe Einstein field equations, the latter is obviously not fully determined by \nmatter sources. This property of equation (15) means that interior solutions \nwill be less dependent on the source's equation of state than for comparable\nsituations in GR, so that any quasi-metric interior solution should cover a \nwider range of physical conditions than its counterparts in GR.\n\nSince equations (20) and (23) are related via equation (9), we can use equation\n(9) to substitute the dynamical term at the left hand side of equation (20) \nwith the nondynamical terms (containing no second order time derivatives)\nat the right hand side of equation (23) (including gravitational coupling \nconstants). Similarly, we can use equation (8) to substitute the terms at the \nright hand side of equation (20) containing the quantity\n${\\cal L}_{{\\bf {\\bar n}}_t}{\\bar K}_t$, with nondynamical terms. In this way, \nfrom equation (20) we obtain a nondynamical equation having the appearance of \nan extra, secondary constraint equation. For vacuum, this equation reads\n\\begin{eqnarray}\n2c^{-2}{\\bar a}_{\\cal F}^s{\\Big [}{\\tilde H}_{(t)sj}+\n{\\frac{1}{(ct{\\bar N}_t)^2}}{\\bar h}_{(t)sj}{\\Big ]}\n-{\\frac{1}{6}}({\\bar K}_{(t)mn}{\\bar K}_{(t)}^{mn}),_j\n+{\\frac{1}{3}}{\\bar K}_{(t)mn}{\\bar K}_{(t)}^{mn}c^{-2}{\\bar a}_{{\\cal F}j}\n\\nonumber \\\\\n+{\\frac{2}{3}}{\\Big [}c^{-2}{\\bar a}_{{\\cal F}{\\mid}sj}^s+c^{-4}\n({\\bar a}_{{\\cal F}s}{\\bar a}_{\\cal F}^s),_j{\\Big ]}\n+{\\frac{2}{3}}{\\Big [}c^{-2}{\\bar a}_{{\\cal F}{\\mid}s}^s+c^{-4}\n{\\bar a}_{{\\cal F}s}{\\bar a}_{\\cal F}^s{\\Big ]}c^{-2}{\\bar a}_{{\\cal F}j}\n\\nonumber \\\\\n-{\\bar K}_{(t)j}^s{\\Big [}{\\bar K}_t,_s+{\\bar K}_tc^{-2}{\\bar a}_{{\\cal F}j}\n{\\Big ]}=0 & \\text{(vacuum).}\n\\end{eqnarray}\nFor the general case we must include the matter terms and we get a rather\nmore bulky formula than that given by equation (25).\n\nNext we notice that, when specifying initial data ${\\bf {\\bar h}}_t$ and \n${\\bf {\\bar K}}_t$ on an initial FHS, due to equation (16) there is no freedom \nto choose the lapse function family ${\\bar N}_t$ independently. Moreover, due \nto equation (25) (with matter terms included) there is no freedom to choose the\ncomponents ${\\frac{t}{t_0}}{\\bar N}_{(t)}^j$ of the shift vector family \nindependently either. But this means, unlike the GR case, that the quasi-metric\ninitial-value system describes the time evolution of a fixed sequence of \nspatial hypersurfaces, i.e., the FHSs. That is, there is no gauge freedom to \nchoose lapse and shift as for the GR case, where the evolution of an initial \nspatial hypersurface into some fixed final one may be done by foliating \nspace-time in many different ways. We also notice that, if equation (25) holds \non an initial FHSs plus all subsequent FHSs, we have that equation (9) for \nvacuum also holds on all subsequent FHSs as long as equation (8) for vacuum \nholds there. That is, the dynamical equation (8) for vacuum preserves the \nconstraints (9) for vacuum on subsequent FHSs as long as equation (25) holds. \nThis result is a counterpart to a somewhat similar result valid for GR, the \ndifference being that the GR counterpart to equation (20) vanishes identically \nfor vacuum since there are no secondary constraints in GR.\n\nFinally we notice that the quantities ${\\bar N}_t,_t$ and ${\\tilde h}_{(t)ik},_t$ \nplay no dynamical role in the quasi-metric initial-value problem since in \nprinciple, they can be chosen freely on an initial FHS, yet their values at \nsubsequent FHSs cannot be determined from dynamical equations. Rather, to \ncontrol the evolution of ${\\bar N}_t$ and ${\\tilde h}_{(t)ik}$, the values of \nsaid quantities must be determined independently from indirect effects of the \ncosmic expansion on the matter source for each time step. An example of this is \ngiven in section 3.1 below.\n\nIn this section we have described all necessary changes in the basic equations \nof quasi-metric gravity when switching from the OQG to the revised theory.\nThere will be no further modifications. In particular, the transformation \n${\\bf {\\bar g}}_t{\\rightarrow}{\\bf g}_t$ will be defined as before [1, 2]. In \nthis context, we notice that to have the full initial value problem in \n$({\\cal N},{\\bf g}_t)$, the transformation \n${\\bf {\\bar g}}_t{\\rightarrow}{\\bf g}_t$ must be performed at each time step\nso that equation (7) can be used to propagate the sources. \n\\section{Two example solutions}\nIn this section, we find two solutions of the extended field equations for\nsimple systems. Of these, the cosmological solution has been found previously \nfor the OQG and is included here for illustrative purposes. Example solutions \ndo not cover metrically static systems since for such systems, the solutions of\nthe extended field equations and those of the OQG coincide. (This can be seen \ndirectly from equation (18) since ${\\bf {\\bar K}}_t$ vanishes identically for \nmetrically static systems.) See [5, 6] for some spherically symmetric cases.\n\\subsection{Isotropic cosmology}\nIsotropic cosmology in the OQG has been treated in [3]. Now equation (16) \nyields that the solution found there is the unique solution also of the revised\ntheory. That is, introducing a spherical GTCS \n${\\{ }x^0,{\\chi},{\\theta},{\\phi}{\\} }$, for isotropic cosmology equation (16) \nensures that equation (1) takes the form\n\\begin{eqnarray}\n{\\overline {ds}}_t^2={\\bar N}_t^2{\\Big \\{}-(dx^0)^2+(ct)^2\n{\\Big (}d{\\chi}^2+{\\sin}^2{\\chi}d{\\Omega}^2{\\Big )}{\\Big \\}},\n\\end{eqnarray}\nwhere $d{\\Omega}^2{\\equiv}d{\\theta}^2+{\\sin}^2{\\theta}d{\\phi}^2$. The extrinsic\ncurvature tensor and the intrinsic curvature of the FHSs obtained from \nequation (26) are given by\n\\begin{eqnarray}\n{\\bar K}_{(t)ik}={\\frac{{\\bar N}_{t,{\\bar {\\perp}}}}{{\\bar N}_t}}\n{\\bar h}_{(t)ik}, \\qquad\n{\\bar K}_t=3{\\frac{{\\bar N}_{t,{\\bar {\\perp}}}}{{\\bar N}_t}}, \\qquad\n{\\bar H}_{(t)ik}=-{\\frac{1}{(ct{\\bar N}_t)^2}}{\\bar h}_{(t)ik}, \n\\qquad {\\bar P}_t={\\frac{6}{(ct{\\bar N}_t)^2}}.\n\\end{eqnarray}\nNext we assume that the quasi-metric universe is filled with a perfect fluid \nwith active mass density ${\\tilde {\\varrho}}_{\\rm m}$ and corresponding pressure\n${\\tilde p}$, so that\n\\begin{eqnarray}\nT_{(t){\\bar {\\perp}}{\\bar {\\perp}}}={\\tilde {\\varrho}}_{\\rm m}c^2{\\equiv}\n{\\Big (}{\\frac{t_0}{{\\bar N}_tt}}{\\Big )}^2{\\bar {\\varrho}}_{\\rm m}(t)c^2,\n\\qquad T_{(t){\\chi}}^{\\chi}=T_{(t){\\theta}}^{\\theta}=T_{(t){\\phi}}^{\\phi}=\n{\\tilde p}{\\equiv}{\\Big (}{\\frac{t_0}{{\\bar N}_tt}}{\\Big )}^2{\\bar p}(t),\n\\end{eqnarray}\nwhere we have set the arbitrary boundary condition ${\\bar N}_t(t_0)=1$ for the\npresent reference epoch $t_0$. Furthermore we have the relationship\n\\begin{eqnarray}\n{\\bar {\\varrho}}_{\\rm m}=\n\\left\\{\n\\begin{array}{ll}\n{\\frac{t^3}{t_0^3}}{\\bar N}_t^3{\\varrho}_{\\rm m}\n& \\text{for a fluid of material particles,} \\\\ [1.5 ex]\n{\\frac{t^4}{t_0^4}}{\\bar N}_t^4{\\varrho}_{\\rm m} & \\text{for the \nelectromagnetic field,}\n\\end{array}\n\\right.\n\\end{eqnarray}\nbetween the quantity ${\\bar {\\varrho}}_{\\rm m}$ and the directly measurable\npassive (inertial) mass density ${\\varrho}_{\\rm m}$. Now, from equation (22)\nwe find that\n\\begin{eqnarray}\n{\\cal L}_{{\\bf {\\bar n}}_t}T_{(t){\\bar {\\perp}}{\\bar {\\perp}}}=\n{\\frac{{\\bar N}_{t,{\\bar {\\perp}}}}{{\\bar N}_t}}{\\Big (}\nT_{(t){\\bar {\\perp}}{\\bar {\\perp}}}+{\\hat T}_{(t)s}^s{\\Big )}=\n{\\frac{t_0^2}{t^2}}{\\frac{{\\bar N}_{t,{\\bar {\\perp}}}}{{\\bar N}_t^3}}\n{\\Big (}{\\bar {\\varrho}}_{\\rm m}c^2+3{\\bar p}{\\Big )},\n\\end{eqnarray}\nwhile taking the Lie derivative directly of equation (28) we find\n\\begin{eqnarray}\n{\\cal L}_{{\\bf {\\bar n}}_t}T_{(t){\\bar {\\perp}}{\\bar {\\perp}}}=\n2{\\frac{{\\bar N}_{t,{\\bar {\\perp}}}}{{\\bar N}_t}}\nT_{(t){\\bar {\\perp}}{\\bar {\\perp}}}=\n2{\\frac{t_0^2}{t^2}}{\\frac{{\\bar N}_{t,{\\bar {\\perp}}}}{{\\bar N}_t^3}}\n{\\bar {\\varrho}}_{\\rm m}c^2.\n\\end{eqnarray}\nBut then, to be consistent equations (30) and (31) imply that the perfect \nfluid must satisfy the equation of state ${\\varrho}_{\\rm m}=3p\/c^2$, i.e., it \nmust be a null fluid. That is, any material component of the fluid must be \nultrarelativistic, so that any deviation from said equation of state is \nnegligible. This is a good approximation for a hot plasma mainly consisting of \nphotons and neutrinos. The above result was arrived at also for the OQG, see \n[3] for a further discussion. Notice that the null fluid condition follows\nfrom the requirement of isotropy. This means that the assumption of (exact)\nisotropy will no longer hold when the cosmic fluid has cooled so much that the \nenergy density of non-relativistic particles becomes comparable to that of the \nphotons (plus the neutrinos). Rather, cosmologically induced flows will be set \nup and (spatial) metric fluctuations must necessarily occur as seeds for \nstructure formation. The details of this and if the resulting predictions are \nconsistent with observations is a subject for further work.\n\nIt turns out that for the metric family (26) it is sufficient to solve equation\n(8) in order to find a solution ${\\bar N}_t$ (since equation (18) is \nidentically fulfilled). Such a solution with a correct vacuum limit was found \nin [3]. The solution found from equation (8) is given by [3] (expressed by the \n``critical'' density \n${\\bar {\\varrho}}_{\\rm m}^{\\rm cr}(t_0){\\equiv}{\\frac{3}{8{\\pi}t_0^2G^{\\rm S}}}$)\n\\begin{eqnarray}\n{\\bar N}_t={\\exp}{\\Big [}-{\\frac{1}{2}}\n{\\frac{(x^0)^2}{(ct)^2}}{\\frac{{\\topstar{\\bar {\\varrho}}}_{\\rm m}(t)}\n{{\\bar {\\varrho}}^{\\rm cr}_{\\rm m}(t_0)}}\n+{\\frac{1}{2}}{\\frac{{\\topstar{\\bar {\\varrho}}}_{\\rm m}(t_1)}\n{{\\bar {\\varrho}}^{\\rm cr}_{\\rm m}(t_0)}}{\\Big ]}, \\qquad\n{\\topstar{\\bar {\\varrho}}}_{\\rm m}{\\equiv}{\\frac{G^{\\rm B}}{G^{\\rm S}}}\n{\\bar {\\varrho}}^{\\rm (EM)}_{\\rm m}+{\\bar {\\varrho}}^{\\rm (MA)}_{\\rm m}.\n\\end{eqnarray}\nHere, the epoch $t_1$ is interpreted as the epoch when matter creation\nceases, so that essentially ${\\frac{\\partial}{{\\partial}t}}\n{\\topstar{{\\bar {\\varrho}}}}_{\\rm m}=0$ for $t{\\geq}t_1$. Notice that the \nexplicit dependence on $t$ of ${\\bar N}_t$ shown in equation (32) is not \ndetermined from the field equations. Rather, this particular dependence was \nchosen for physical reasons. For a further discussion of the solution (32), \nsee [3].\n\\subsection{Weak gravitational waves in vacuum}\nIf we ignore the global curvature of space, the weak-field (linear) \napproximation of the field equation (15) for vacuum is the same as for GR \nprovided that both ${\\bf {\\bar a}}_{\\cal F}$ and ${\\bar K}_t$ vanish identically.\n(This can be easily seen directly from equations (12) and (15).) Thus the \ncounterpart weak-field GR-solution of locally plane-fronted waves with two \nindependent polarizations will also be an approximate solution of equation \n(15). The only difference from the GR-solution is that the global cosmic \nexpansion is included via the scale factor. Thus the family of line elements \ntakes the form\n\\begin{eqnarray}\n{\\overline {ds}}_t^2=-(dx^0)^2+{\\frac{t^2}{t_0^2}}\n{\\Big [}(E_{ks}+{\\bar {\\varepsilon}}_{(t)ks})dx^kdx^s{\\Big ]},\n\\end{eqnarray}\nwhere $E_{ks}dx^kdx^s$ is the metric of Euclidean space and where the terms\n\\begin{eqnarray}\n{\\bar {\\varepsilon}}_{(t)ks}={\\Re}[{\\bar {\\cal A}}_{(t)ks}\n{\\exp}(i{\\bar {\\vartheta}}_t)],\n\\end{eqnarray}\ndescribe the plane wave perturbation from the Euclidean background. Moreover,\nwe have\n\\begin{eqnarray}\n{\\bar {\\vartheta}}_t{\\equiv}{\\bar k}_{(t)0}(x^0-x^0_1)+{\\bar k}_{(t)i}x^i, \n\\quad {\\bar k}_{(t){\\mu}}={\\bar {\\vartheta}}_{t},_{\\mu}, \\quad\n{\\bar k}_{(t)0},_t=-{\\frac{1}{t}}{\\bar k}_{(t)0}, \\quad \n{\\bar k}_{(t)j},_t=0,\n\\end{eqnarray}\nwhere ${\\bar {\\vartheta}}_t$ is the phase factor and where ${\\bar k}_{(t){\\mu}}$ \ndenotes the components of the wave 4-vector family. (Also, $x^0_1$ is an \narbitrary reference epoch.) Finally, \n${\\bar {\\cal A}}_{(t)ks}={\\frac{t_0}{t}}{\\bar {\\cal A}}_{(t_0)ks}$ is the \n(possibly complex) polarization tensor. As for the counterpart GR case, \nequation (15) for vacuum (ignoring global space curvature) yields that the \nplane wave is null, transverse and traceless. That is, choosing Cartesian\ncoordinates $(x,y,z)$ with the wave travelling in the $z$-direction, equation \n(33) takes the form\n\\begin{eqnarray}\n{\\overline {ds}}_t^2=-(dx^0)^2+{\\frac{t^2}{t_0^2}}{\\Big [}\n(1+{\\bar {\\varepsilon}}_{(t)xx})dx^2+(1-{\\bar {\\varepsilon}}_{(t)xx})dy^2\n+2{\\bar {\\varepsilon}}_{(t)xy}dxdy+dz^2{\\Big ]}.\n\\end{eqnarray}\nSince equations (33) and (36) only describe approximative solutions of equation\n(15), to further investigate the nature of gravitational radiation in \nquasi-metric gravity some exact solutions should be found. Such solutions are \nexpected to differ from their GR counterparts. However, finding exact wave-like\nsolutions of equation (15) may turn out to be difficult, and is beyond the \nscope of the present paper.\n\\section{Conclusion}\nIn this paper, we have relaxed the original restrictions on the quasi-metric \nspace-time geometry $({\\cal N},{\\bf {\\bar g}}_t)$ so that its most general form\nis now given by equation (1). The reason for this revision was to make possible\nthe prediction of (weak) GR-like gravitational waves since such have now been \ndirectly detected. However, we have shown that it is not possible to construct\na quasi-metric gravitational theory where space-time curvature is fully coupled\nto the active stress-energy tensor family ${\\bf T}_t$, and such that the \nresulting field equations will have a sensible Newtonian limit. Nevertheless,\nwe have also shown that the original quasi-metric gravitational field equations\ncan be extended with the extra equation (15) (or equivalently, equation (19)) \nnot being directly coupled to ${\\bf T}_t$ such that there are no obvious \nproblems in the weak-field limit. The extended field equations are, by \nconstruction, sufficiently flexible and designed to predict GR-like \ngravitational waves in vacuum for the weak-field approximation. On the other \nhand, exact gravitational wave solutions are expected to differ from their GR \ncounterparts. \n\nBesides the prediction of gravitational waves, the differences between the \npredictions of the extended quasi-metric gravitational theory and the OQG are \nsmall. In particular, several observations indicating that the cosmic expansion\nis relevant for the solar system (constituting a powerful {\\em empirical} \nmotivation for introducing the QMF) have identical explanations coming from the\nOQG and the extended theory (see [6] and references therein). This means that, \ndisregarding gravitational waves and systems emitting gravitational waves \n(e.g., binary pulsars), the observational status of the extended gravitational \ntheory is the same as for the OQG (i.e., currently nonviable [3]).\n\\\\ [4mm]\n{\\bf References} \\\\ [1mm]\n{\\bf [1]} D. {\\O}stvang, {\\em Grav. {\\&} Cosmol.} {\\bf 11}, 205\n(2005) (gr-qc\/0112025). \\\\\n{\\bf [2]} D. {\\O}stvang, {\\em Doctoral thesis}, (2001) (gr-qc\/0111110). \\\\\n{\\bf [3]} D. {\\O}stvang, {\\em Indian Journal of Physics}, {\\bf 92}, 669\n(2018) (arXiv:1701.09151). \\\\\n{\\bf [4]} K. Kucha{\\v{r}}, {\\em Journ. Math. Phys.} {\\bf 17}, 792 (1976). \\\\\n{\\bf [5]} D. {\\O}stvang, {\\em Grav. {\\&} Cosmol.} {\\bf 12} 262 (2006) \n(gr-qc\/0303107). \\\\\n{\\bf [6]} D. {\\O}stvang, {\\em Grav. {\\&} Cosmol.} {\\bf 13}, 1 (2007)\n(gr-qc\/0201097).\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe Casimir effect \\cite{Casimir} is the most important example of a slew of phenomena \nusually referred to as {\\em fluctuation-induced interactions}, their \nphenomenology extending from cosmology on the one side to \nnanoscience on the other \\cite{Kardar,Dalvit,Mostepanenko,Bordag,Mohideen,Krech,\nFrench-RMP,Parsegian2005}. The general idea tying these diverse phenomena \ntogether is that the confining surfaces constrain the quantum and \nthermal field fluctuations, inducing long-range interactions \nbetween these boundaries \\cite{Kardar,Bordag}. For electromagnetic \nfields, these confinement effects lead to the Casimir-van der \nWaals interactions that can be derived within the specific framework of QED, \nand quantum field theory more generally \\cite{Mohideen}. Inspired by the close \nanalogy \nbetween thermal fluctuations in fluids and quantum fluctuations in \nelectromagnetism, Fisher and deGennes predicted the existence of long-range \nfluctuation forces in other types of critical condensed matter \n systems \\cite{Fisher} and the terms ``Casimir'' or ``Casimir-like effect'' now \ndenote a range of other non-electromagnetic \nfluctuation-induced forces \\cite{Kardar,French-RMP}. \n\nBeyond detailed measurements of the Casimir-van der Waals interactions \n\\cite{Mohideen}, attention has been directed \ntowards Casimir-like forces engendered by density fluctuations in the \nvicinity of the vapor-liquid critical point \\cite{Krech,Krech2,Krech3};\nin binary liquid mixtures near the critical demixing point \n\\cite{Gambassi,Fukuto}; and in thin polymer \\cite{Morariu04,Schaeffer,Morariu03} \nand liquid crystalline films \\cite{LC,Li-kardar}.\nMost recently, several studies have examined fluctuation-induced interactions \nfor the Casimir-Lifshitz force out of thermal equilibrium \\cite{Antezza,Kruger}, \nfor the temporal relaxation of the thermal Casimir or van der Waals force \n\\cite{Dean}, and for nonequilibrium steady states in \nfluids \\cite{Kirkpatrick13,Kirkpatrick14,Ortiz}, where fluctuations \nare anomalously large and long-range.\n\nIt is instructive to \nrecall that the original 1955 derivation of the electromagnetic Casimir-van der \nWaals interactions by Lifshitz \\cite{Lifshitz} was not \nfundamentally rooted in QED but rather in stochastic electrodynamics, first \nformulated by Rytov \\cite{rytov59}. In stochastic \nelectrodynamics, Maxwell's equations are augmented by fluctuating displacement \ncurrent sources \\cite{Rosa}. This leads to two coupled electrodynamic \nLangevin-type equations, for each of the fundamental electrodynamic fields, that are \nthen solved with standard boundary conditions. The interaction force is \nobtained by averaging the Maxwell stress tensor and taking into account the \nstatistical properties of the fluctuating sources \\cite{Narayanaswami}. This \nparadigmatic Lifshitz-route to fluctuation-induced interactions later became \ndisfavored as other formal approaches gained strength \\cite{Mohideen}, \nbut appears to be reborn in recent endeavors regarding non-equilibrium \nfluctuation-induced interactions \\cite{Kirkpatrick13,Kirkpatrick14}. In fact, in the \nDean-Gopinathan method there exists a mapping of the non-equilibrium problem \ncharacterized by dissipative dynamics onto a corresponding static (Lifshitz) \npartition function provided by the Laplace transform of the time-dependent force \nand the static partition function \\cite{Ajay1,Ajay2}.\n\nBased on the success of stochastic electrodynamics, Landau and Lifshitz \nproposed by analogy the stochastic dissipative hydrodynamic equations \n\\cite{LandauLifshitz}, augmenting the linearized Navier-Stokes equations with \nfluctuating heat flow vector and fluctuating stress tensor \\cite{Ortiz,Forster}. \nThis leads to three coupled hydrodynamic equations involving the fundamental \nhydrodynamic fields of mass density, velocity and local temperature, which can \nnow be solved in different contexts. With the assumption of fixed temperature, \nthis system further reduces to a Langevin-type equation for the velocity \nfield, involving the stress tensor fluctuations, and a continuity equation for \nthe mass density field. Since the fundamental hydrodynamic equations are \nnon-linear, the derivation of fluctuating Landau-Lifshitz hydrodynamics already \ninvolves heavy linearity {\\em Ans\\\" atze} and the possible generalization to a \nfull non-linear fluctuating hydrodynamics is not clear \\cite{spohn,Tailleur}.\n\nAlthough fluctuating electrodynamics is based on \n{\\em linear} Maxwell's equations with stresses {\\em quadratic} in the field and\nfluctuating hydrodynamics stems from {\\em non-linear} Navier-Stokes equations \nwith stresses {\\em linear} in velocities, the general similarity between these \napproaches might nevertheless lead one to assume that, in confined geometries, there \nshould exist Casimir-like hydrodynamic fluctuation forces. But this notion is \nat odds with the standard decomposition of the classical partition function into \nmomentum and configurational \nparts. This decomposition has far-reaching consequences, \nwhich were clearly understood as far back as van der Waals' \nthesis \\cite{Rowlinson}. While there is an analogy between the description of \nfluctuations in these two areas of physics, caution should be exercised when \ntrying to translate results from one field directly into the other. We will show \nthat there does exist a type of Casimir effect in the hydrodynamic \ncontext, but that this effect has fundamentally different properties from the \nconventional Casimir effect.\n\nThe first step in bringing together the Casimir force in electrodynamics and \nits putative counterpart in hydrodynamics was made by Jones \\cite{Jones}. \nInspired by the obvious analogy between electrodynamics and hydrodynamics, Jones\ninvestigated the possible existence of a long-ranged, fluctuation-induced, \neffective force generated by confining boundaries in a fluid. He showed that  \nin linearized hydrodynamics the net (mean) stochastic force vanishes, which \nled him to introduce a next-to-leading order formalism. \nThe status of this formalism, however, is not entirely clear, \nbecause there are linearity assumptions rooted deep within fluctuational \nhydrodynamics \\cite{Forster,Ortiz}. Within the context of this next-to-leading \norder formalism, Jones \ndemonstrated that long-range forces could exist in a semi-infinite fluid\nor around an immersed spherical body, and would be strongest in \nincompressible fluids, with much weaker forces in compressible fluids. This \nresult is at odds with the momentum decomposition of the classical partition \nfunction and should be considered an artifact of the next-to-leading order \nanalysis of the stochastic equations governing the hydrodynamic field \nevolution. \n\nChan and  White \\cite{Chan}, therefore, reconsidered the whole calculation. They \nconcentrated on the planar geometry of two hard walls immersed in a fluid and \nargued that hydrodynamic fluctuations could give rise to a repulsive force in \n{\\em incompressible} fluids, but that this force would vanish for classical \n{\\em compressible} fluids. Since an incompressibility {\\em Ansatz} \ndoes not translate directly into the interaction potential in the classical\npartition function \\cite{vankampen}, this fictional case could lead to a \nfluctuation-induced interaction that would not be contrary to the argument based on the momentum \ndecomposition of the classical partition function. The repulsive fluctuation-induced \nforce would also in itself not be that hard to envision since the existence of \na repulsive force in the context of van der Waals  interactions is \nwell-established and was originally proposed in Ref.~\\cite{Dzyaloshinskii}. The \nvanishing of the fluctuation-induced force for classical compressible fluids is based on \na rough argument of analytic continuation of the viscosities into the infinite \nfrequency domain \\cite{Chan}. While this latter argument is appropriate in \nelectrodynamics, because an infinite frequency corresponds to the vacuum, it is \nnot reasonable in hydrodynamics, where the whole basis of the continuum \nhydrodynamic theory breaks down before any such limit could be enforced \n\\cite{Forster}.\n\nTherefore, both approaches to the problem of hydrodynamic Casimir-like interactions \nhave strong limitations and subsequent developments  \nfailed to conclusively prove either point of view \\cite{Ivlev}.\n\nIn this paper, we revisit the question of the existence of long-range, \nfluctuation-induced forces in fluids. We work strictly within the framework of \nlinearized stochastic hydrodynamics and rather than considering the net force, \nwhich is zero trivially, we study the force correlators.\nIn other words, we focus on the question: In what way do boundary conditions \nand statistical properties of the fluctuating hydrodynamic stresses affect the \nstatistical properties (correlators) of the random forces acting on the bounding surfaces?\n\nWe formulate a general approach to this problem by considering a fluid of \narbitrary compressibility and fixed homogeneous temperature, bounded \nbetween two plane-parallel, hard walls with \nno-slip boundary conditions. Thermal fluctuations lead to spatio-temporal \nvariations in the pressure and velocity fields that can be calculated \nusing the linearized, stochastic Navier-Stokes formalism of Landau and Lifshitz\n\\cite{LandauLifshitz}. Within this approach, we derive analytical expressions \nfor the time-dependent correlators  (for both the same-plate and the\ncross-plate correlators)\nof the fluctuation-induced forces acting on the walls. \nIn particular, we express the variance of these forces in terms of frequency \nintegrals that have simple plate-separation dependence in the \nsmall and large plate-separation limits.\n\nOur results do not depend upon the \nnext-to-leading order formalism of Jones \\cite{Jones}, nor do they depend on the\nunrealistic validity of analytic continuation of the viscosities in the whole \nfrequency domain \\cite{Chan}. We show that, while the mean force vanishes, \nthe variance of the fluctuation-induced normal force is finite and \nindependent of the separation between the bounding surfaces for large \nseparations. We call this the \\emph{secondary Casimir \neffect}, because the primary Casimir effect refers to the average value of \nthe fluctuation-induced force (which is zero here) and not strictly its variance. Both quantities \nhave been investigated in other Casimir-like situations \\cite{Bartolo, Dean} \nand in disordered charged systems \\cite{disorder-PRL,pre2011,epje2012}.\nThe equal-time, \ncross-plate force correlation exhibits long-range behavior that is \nindependent of the fluid viscosity and decays proportional to the inverse plate\nseparation. Finally, we find that the time-dependent correlators exhibit \ndamped oscillatory behavior for small plate separations that becomes\nirregular at large distances.\n\nIn Sec.~\\ref{sec:formalism}, we outline the stochastic formalism of Landau \nand Lifshitz and the strategy of our calculation of hydrodynamic \nfluctuation-induced forces in the general case of compressible fluids. Sections \n\\ref{sec:meanforce} and \\ref{sec:forcevar_t} present the main steps of our \ncalculation. We show results for the equal-time force correlators and the \ntwo-point, \ntime-dependent correlators in Sections \\ref{sec:numerics} and \n\\ref{sec:tnumerics}, respectively. We conclude our discussions in Sec.~\n\\ref{sec:conclusions}.\n\n\\section{Formalism}\n\\label{sec:formalism}\n\nWe consider the hydrodynamic fluctuations in a Newtonian fluid at rest and in\nthe absence of heat transfer. These fluctuations are described by the \nstochastic \nLandau-Lifshitz equations \\cite{LandauLifshitz}\n\\begin{align}\n&\\eta \\nabla^2 s}{{\\mbox{\\boldmath $\\rho $}}{v} + \\left( \\frac{\\eta}{3} + \\zeta\\right) \n\\nabla(\\nabla\\cdot s}{{\\mbox{\\boldmath $\\rho $}}{v}) - \\nabla p {}  \\nonumber\\\\\n&\\qquad\\qquad\\qquad{}- \\rho\\left(\\frac{\\partial\ns}{{\\mbox{\\boldmath $\\rho $}}{v}}{\\partial t} + s}{{\\mbox{\\boldmath $\\rho $}}{v}\\cdot \\nablas}{{\\mbox{\\boldmath $\\rho $}}{v}\\right)  \n=\n-\\nabla\\cdot s}{{\\mbox{\\boldmath $\\rho $}}{S},  \\label{eq:ns1}\\\\\n&\\frac{\\partial \\rho}{\\partial t} + \\nabla \\cdot (\\rho s}{{\\mbox{\\boldmath $\\rho $}}{v}){} = \n0, \\label{eq:ns2}\n\\end{align}\nwhere $s}{{\\mbox{\\boldmath $\\rho $}}{v}=s}{{\\mbox{\\boldmath $\\rho $}}{v}(s}{{\\mbox{\\boldmath $\\rho $}}{r};t)$, $p=p(s}{{\\mbox{\\boldmath $\\rho $}}{r};t)$ and\n$\\rho=\\rho(s}{{\\mbox{\\boldmath $\\rho $}}{r};t)$ are the  velocity, pressure and\ndensity fields and $\\eta$ and $\\zeta$ are the shear and bulk\nviscosity coefficients, respectively \\cite{Note1}. The\nrandomly fluctuating microscopic degrees of freedom are driven by the\nrandom stress tensor $s}{{\\mbox{\\boldmath $\\rho $}}{S}=s}{{\\mbox{\\boldmath $\\rho $}}{S}(s}{{\\mbox{\\boldmath $\\rho $}}{r};t)$, which is assumed to \nhave a Gaussian distribution with zero mean $\\left\\langle S_{ij}(s}{{\\mbox{\\boldmath $\\rho $}}{r}; t) \n\\right\\rangle = {}  0$ and the two-point correlator \n\\begin{align}\n&\\left\\langle\nS_{kl}(s}{{\\mbox{\\boldmath $\\rho $}}{r};t)\\,S_{mn}(s}{{\\mbox{\\boldmath $\\rho $}}{r}\n^{\\prime};t^{\\prime}) \\right\\rangle= {} \n2k_{\\mathrm{B}}T\\delta(s}{{\\mbox{\\boldmath $\\rho $}}{r}-s}{{\\mbox{\\boldmath $\\rho $}}{r}^{\\prime})\\delta(t -\nt^{\\prime}) \\nonumber\\\\\n&\\qquad \\times \\Big[\n \\eta\\big(\\delta_{km}\\delta_{ln} +\n\\delta_{kn}{} \\delta_{lm}\\big)\n- \\left( \\frac{2\\eta}{3}-\\zeta \\right) \\delta_{kl}\n\\delta_{mn} \\Big],  \\label{eq:gaussprop2} \n\\end{align}\nwhere the subindices ($i, j, k, \\ldots$) denote the Cartesian components $(x, \ny, z)$,  $k_{\\mathrm{B}}$ is Boltzmann's constant and $\\langle\\cdots \\rangle$ \ndenote an equilibrium ensemble average at temperature $T$.  We do not consider \nany possible relaxation effects, which would formally correspond to \nfrequency-dependent viscosities, but these effects can be easily incorporated \n\\cite{LandauLifshitz}. Denoting the frequency Fourier \ntransform by a tilde, i.e., \n\\begin{equation}\n\\widetilde{f}(\\omega) = \\int \\mathrm{d}t\\,e^{i\\omega t} f(t),\n\\end{equation}\nwe have $\\langle \\widetilde{S}_{ij}(s}{{\\mbox{\\boldmath $\\rho $}}{r};\\omega) \\rangle \n=   0$ and \n\\begin{align}\n&\\left\\langle\n\\widetilde{S}_{kl}(s}{{\\mbox{\\boldmath $\\rho $}}{r};\\omega)\\,\\widetilde{S}_{mn}^{\\ast}(s}{{\\mbox{\\boldmath $\\rho $}}{r}\n^{\\prime};\\omega^{\\prime}) \\right\\rangle= {} \n4\\pi k_{\\mathrm{B}}T\\delta(s}{{\\mbox{\\boldmath $\\rho $}}{r}-s}{{\\mbox{\\boldmath $\\rho $}}{r}^{\\prime})\\delta(\\omega -\n\\omega^{\\prime}) \\nonumber\\\\\n&\\qquad \\times \\Big[\n \\eta\\big(\\delta_{km}\\delta_{ln} +\n\\delta_{kn}\\delta_{lm}{} \\big)\n- \\left( \\frac{2\\eta}{3}-\\zeta \\right) \\delta_{kl}\n\\delta_{mn} \\Big],  \\label{eq:gausswprop2} \n\\end{align}\nwhich hold independent of the boundary conditions imposed on the fluid system.\n\nBefore proceeding further, we should note that this form of fluctuating \nhydrodynamics is analogous to the\nRytov fluctuating electrodynamics \\cite{Lifshitz}, where the basic equations \nfor the electric and magnetic fields are \n\\begin{align}\n\\nabla \\times s}{{\\mbox{\\boldmath $\\rho $}}{E}(s}{{\\mbox{\\boldmath $\\rho $}}{r}, t) = {} & -  \\frac{\\partial \ns}{{\\mbox{\\boldmath $\\rho $}}{B}(s}{{\\mbox{\\boldmath $\\rho $}}{\nr}, t)}{\\partial t}, \\label{bcfhgjsk}\\\\\n\\nabla \\times s}{{\\mbox{\\boldmath $\\rho $}}{H}(s}{{\\mbox{\\boldmath $\\rho $}}{r}, t) = {} & \\frac{\\partial \ns}{{\\mbox{\\boldmath $\\rho $}}{D}(s}{{\\mbox{\\boldmath $\\rho $}}{ r}, \nt)}{\\partial t} + \\frac{\\partial s}{{\\mbox{\\boldmath $\\rho $}}{K}(s}{{\\mbox{\\boldmath $\\rho $}}{r}, t)}{\\partial t}, \n\\end{align}\nsupplemented by $\\nabla \\cdot s}{{\\mbox{\\boldmath $\\rho $}}{D}(s}{{\\mbox{\\boldmath $\\rho $}}{r}, t) = 0$ and  $\\nabla \\cdot \ns}{{\\mbox{\\boldmath $\\rho $}}{B}(s}{{\\mbox{\\boldmath $\\rho $}}{r}, t) = 0$ and appropriate boundary conditions. In this \ncase, the fluctuating random polarization, $s}{{\\mbox{\\boldmath $\\rho $}}{K}(s}{{\\mbox{\\boldmath $\\rho $}}{r}; t)$, \nhas Gaussian properties with $\\big\\langle \\widetilde{K}_{i}(s}{{\\mbox{\\boldmath $\\rho $}}{r};\\omega) \n\\big\\rangle =  0$, and \n\\begin{equation}\n\\left\\langle  \n\\widetilde{K}_{i}(s}{{\\mbox{\\boldmath $\\rho $}}{r};\\omega)\\,\\widetilde{K}_{j}^{\\ast}(s}{{\\mbox{\\boldmath $\\rho $}}{r}\n^{\\prime};\\omega^{\\prime}) \\right\\rangle = {} \n{k_{\\mathrm{B}}T}\\frac{\\varepsilon_{\\mathrm{I}}(\\omega)}{\\omega}\\delta_{ij} \n\\delta(s}{{\\mbox{\\boldmath $\\rho $}}{r}-s}{{\\mbox{\\boldmath $\\rho $}}{r}^{\\prime})\\delta(\\omega -\n\\omega^{\\prime}),\n\\label{axrmeoi}\n\\end{equation}\nwhere we have assumed a dispersive dielectric response function \n$\\varepsilon(\\omega) = \n\\varepsilon_{\\mathrm{R}}(\\omega) + \ni \\varepsilon_{\\mathrm{I}}(\\omega)$. We can immediately see the similarity \nbetween Eqs.~\\eqref{eq:ns1}-\\eqref{eq:gausswprop2} and \nEqs.~\\eqref{bcfhgjsk}-\\eqref{axrmeoi}.\n\nThus, the stochastic approach to hydrodynamics is very close to \nLifshitz's \noriginal analysis of the electromagnetic \nproblem \\cite{Lifshitz}, provided one fully \ntakes into account the basic differences between the Maxwell equations and the \nNavier-Stokes equations \\cite{Chan}: The former are linear in the fields with \nstresses quadratic in the fields, while the latter are non-linear in the fields \nwith stresses linear in the fields. This difference leads to \nsome important distinctions and precludes directly applying results from \nelectrodynamics to the hydrodynamic domain.\n\n\\subsection{Linearized stochastic hydrodynamics}\n\\label{sec:linhydro}\n\nFor vanishing random stress tensor, the equilibrium solution of \nEqs.~\\eqref{eq:ns1} and \\eqref{eq:ns2} is $s}{{\\mbox{\\boldmath $\\rho $}}{v}=s}{{\\mbox{\\boldmath $\\rho $}}{0}$, $p=p_0$ and \n$\\rho \n=\n\\rho_0$, corresponding to a fluid at rest at constant temperature, $T$, with\nuniform pressure, $p_0$, and density, $\\rho_0$. The random stress tensor,\n$s}{{\\mbox{\\boldmath $\\rho $}}{S}$, is of order $k_{\\mathrm{B}}T$ and, consequently, \nmacroscopically small. Therefore, the corresponding fluctuations in the \nvelocity, pressure and density fields are also macroscopically small. \nTherefore we introduce a linearized treatment of the Landau-Lifshitz\nequations, by setting $s}{{\\mbox{\\boldmath $\\rho $}}{v} = s}{{\\mbox{\\boldmath $\\rho $}}{v}^{(1)}$, $p\n= p_0+p^{(1)}$ and $\\rho = \\rho_0+\\rho^{(1)}$, where the superscript\n$(1)$ denotes a term of order $s}{{\\mbox{\\boldmath $\\rho $}}{S}$.\n\nWe assume local equilibrium, which enables us to relate the\ndensity and pressure as \n\\begin{equation}\\label{eq:prho}\np^{(1)} = c_0^2\\rho^{(1)},\\qquad \\mathrm{with}\\qquad \nc_0^2 = \\left(\\frac{\\partial p }{\\partial \\rho}\\right)_0. \n\\end{equation}\nHere $c_0$ is the adiabatic speed of sound, so that $\\rho_0 c_0^2$ equals \nthe inverse adiabatic compressibility (Newton-Laplace equation). \nEqs.~\\eqref{eq:ns1} and \n\\eqref{eq:ns2} can be linearized as  \n\\begin{align}\n&\\eta \\nabla^2 s}{{\\mbox{\\boldmath $\\rho $}}{v}^{(1)} + \\left( \\frac{\\eta}{3} + \\zeta\\right)\n\\nabla\\left(\\nabla\\cdot s}{{\\mbox{\\boldmath $\\rho $}}{v}^{(1)}\\right)    \\nonumber\\\\\n&\\qquad\\qquad\\qquad {}- \\nabla \np^{(1)}- \\rho_0\\frac{\\partial\ns}{{\\mbox{\\boldmath $\\rho $}}{v}^{(1)}}{\\partial t}   = \n-\\nabla\\cdot s}{{\\mbox{\\boldmath $\\rho $}}{S},  \\label{eq:lns1}\\\\\n&\\frac{\\partial \\rho^{(1)}}{\\partial t} + \\rho_0\\nabla \\cdot s}{{\\mbox{\\boldmath $\\rho $}}{v}^{(1)} \n{} \n= 0. \\label{eq:lns2}\n\\end{align}\nor, in the frequency domain and using Eq.~\\eqref{eq:prho}   \\cite{Note2},\n\\begin{align}\n&\\eta \\nabla^2\\widetilde{s}{{\\mbox{\\boldmath $\\rho $}}{v}}^{(1)} +\n\\left(\\frac{\\eta}{3} + \\zeta\\right)\\nabla \\left(\\nabla \\cdot\n\\widetilde{s}{{\\mbox{\\boldmath $\\rho $}}{v}}^{(1)}\\right) \n{}  \\nonumber\\\\\n &\\qquad\\qquad\\qquad {} {} -c_0^2\\nabla \\widetilde{\\rho}^{\\,(1)}+ i\\omega \\rho_0 \n\\widetilde{s}{{\\mbox{\\boldmath $\\rho $}}{v}}^{(1)} = \n-\\nabla \\cdot \\widetilde{s}{{\\mbox{\\boldmath $\\rho $}}{S}}, \\label{eq:linLL1}\\\\\n&\\nabla \\cdot \\widetilde{s}{{\\mbox{\\boldmath $\\rho $}}{v}}^{(1)} -\\frac{i\\omega}{\\rho_0}\n\\widetilde{\\rho}^{\\,(1)} {} =  0.\\label{eq:linLL2}\n\\end{align}\n\nWe now introduce transverse and longitudinal components of the velocity \nfluctuations $s}{{\\mbox{\\boldmath $\\rho $}}{v}^{(1)}$, which \nwe denote $s}{{\\mbox{\\boldmath $\\rho $}}{v}^\\mathrm{T}$ and $s}{{\\mbox{\\boldmath $\\rho $}}{v}^\\mathrm{L}$, respectively. We \nhave dropped the superscript $(1)$ for notational simplicity, i.e., \n$s}{{\\mbox{\\boldmath $\\rho $}}{v}^{(1)} = s}{{\\mbox{\\boldmath $\\rho $}}{v}^\\mathrm{T}+s}{{\\mbox{\\boldmath $\\rho $}}{v}^\\mathrm{L}$, with \n\\begin{equation}\n\\nabla\\cdot s}{{\\mbox{\\boldmath $\\rho $}}{v}^\\mathrm{T} = 0 \\qquad \\mathrm{and}\\qquad \\nabla\\times \ns}{{\\mbox{\\boldmath $\\rho $}}{v}^\\mathrm{L} = 0.\n\\label{eq:def_LT}\n\\end{equation}\nThe random force density vector $s}{{\\mbox{\\boldmath $\\rho $}}{\\Sigma} = -\\nabla \\cdot s}{{\\mbox{\\boldmath $\\rho $}}{S}$ can \nbe \ndecomposed into transverse and \nlongitudinal components as well, using $s}{{\\mbox{\\boldmath $\\rho $}}{\\Sigma} = \ns}{{\\mbox{\\boldmath $\\rho $}}{\\Sigma}^\\mathrm{T} + s}{{\\mbox{\\boldmath $\\rho $}}{\\Sigma}^\\mathrm{L}$, where\n\\begin{equation}\n\\nabla\\cdot s}{{\\mbox{\\boldmath $\\rho $}}{\\Sigma}^\\mathrm{T} = 0 \\qquad \\mathrm{and}\\qquad \n\\nabla\\times \ns}{{\\mbox{\\boldmath $\\rho $}}{\\Sigma}^\\mathrm{L} = 0.\n\\end{equation}\nThese random force density vector components have zero mean and zero cross \ncorrelations. Their self-correlations follow \nfrom Eq.~\\eqref{eq:gaussprop2} as\n\\begin{align}\n\\left\\langle\n\\widetilde{\\Sigma}^{\\mathrm{L}}_i(s}{{\\mbox{\\boldmath $\\rho $}}{r};\\omega)\\,\\widetilde{\\Sigma}^{\n\\mathrm{L}}_j(s}{{\\mbox{\\boldmath $\\rho $}}{r}^{\\prime};\\omega^{\\prime}) \\right\\rangle= {} & \n4 \\pi k_{\\mathrm{B}}T\n\\left( \\frac{4\\eta}{3}+\\zeta \\right) \n\\nabla_i\\nabla_j^\\prime \n\\nonumber\\\\\n & \\quad \\times \\delta(s}{{\\mbox{\\boldmath $\\rho $}}{r}-s}{{\\mbox{\\boldmath $\\rho $}}{r}^{\\prime})\\delta(\\omega +\n\\omega^{\\prime}), \\label{eq:svecLw-2}\\\\\n\\left\\langle\n\\widetilde{\\Sigma}^{\\mathrm{T}}_i(s}{{\\mbox{\\boldmath $\\rho $}}{r};\\omega)\\,\\widetilde{\\Sigma}^{\n\\mathrm{T}}_j(s}{{\\mbox{\\boldmath $\\rho $}}{r}^{\\prime};\\omega^{\\prime}) \\right\\rangle= {} & \n4 \\pi k_{\\mathrm{B}}T\n\\eta\\left(\n\\nabla_k\\nabla_k^\\prime \\delta_{ij} -\n\\nabla_i\\nabla_j^\\prime\n\\right)\\nonumber\\\\\n & \\quad \\times \\delta(s}{{\\mbox{\\boldmath $\\rho $}}{r}-s}{{\\mbox{\\boldmath $\\rho $}}{r}^{\\prime})\n\\delta(\\omega +\n\\omega^{\\prime}). \\label{eq:svecTw-2}\n\\end{align}\nThe stochastic Landau-Lifshitz equations can thus be written as\n\\begin{align}\n&\\eta \\nabla^2\\widetilde{s}{{\\mbox{\\boldmath $\\rho $}}{v}}^{\\mathrm{L}}  + \\left(\\frac{\\eta}{3} + \n\\zeta\\right)\\nabla\\left(\\nabla \\cdot\n\\widetilde{s}{{\\mbox{\\boldmath $\\rho $}}{v}}^{\\mathrm{L}}\\right) {}  \\nonumber\\\\\n&\\qquad\\qquad\\qquad-c_0^2\\nabla \\widetilde{\\rho}^{\\,(1)}+ i\\omega \\rho_0 \\widetilde{s}{{\\mbox{\\boldmath $\\rho $}}{v}}^{\\mathrm{L}} {}  = \n s}{{\\mbox{\\boldmath $\\rho $}}{\\widetilde{\\Sigma}}^\\mathrm{L}, \\label{eq:lllong}\\\\\n&\\eta \\nabla^2\\widetilde{s}{{\\mbox{\\boldmath $\\rho $}}{v}}^{\\mathrm{T}}+ i\\omega \\rho_0 \n\\widetilde{s}{{\\mbox{\\boldmath $\\rho $}}{v}}^{\\mathrm{T}}{}  = \ns}{{\\mbox{\\boldmath $\\rho $}}{\\widetilde{\\Sigma}}^\\mathrm{T}, \\label{eq:lltrans}\\\\\n&\\nabla \\cdot \\widetilde{s}{{\\mbox{\\boldmath $\\rho $}}{v}}^{\\mathrm{L}} -\\frac{i\\omega}{\\rho_0}\\widetilde{\\rho}^{\\,(1)}{}  =  \n0.\\label{eq:lllong2}\n\\end{align}\n\nWe may simplify Eqs.~\\eqref{eq:lllong}-\\eqref{eq:lllong2} by \nusing the vector identity\n\\begin{equation}\\label{eq:vecid}\n\\nabla_j\\nabla_j \\widetilde{v}^{\\mathrm{L}}_i =  \\nabla_j\\nabla_j \\widetilde{v}^{\\mathrm{L}}_i + \n\\nabla_j\\left(\\nabla_i \\widetilde{v}^{\\mathrm{L}}_j -\\nabla_j\n\\widetilde{v}^{\\mathrm{L}}_i\\right)= \\nabla_i \\nabla_j\\widetilde{v}^{\\mathrm{L}}_j\n\\end{equation}\nfor the curl-free longitudinal component and by substituting \nEq.~\\eqref{eq:lllong2} into Eq.~\\eqref{eq:lllong} to obtain\n\\begin{align}\n&\\left[\\frac{4\\eta}{3} + \n\\zeta \n+ \\frac{i\\rho_0c_0^2}{\\omega}\\right]\\nabla^2\n\\widetilde{s}{{\\mbox{\\boldmath $\\rho $}}{v}}^{\\mathrm{L}}+ i\\omega \\rho_0 \n\\widetilde{s}{{\\mbox{\\boldmath $\\rho $}}{v}}^{\\mathrm{L}} = \n{} \n \\widetilde{s}{{\\mbox{\\boldmath $\\rho $}}{\\Sigma}}^\\mathrm{L}, \\label{eq:ll}\\\\\n&\\eta \\nabla^2\\widetilde{s}{{\\mbox{\\boldmath $\\rho $}}{v}}^{\\mathrm{T}}+ i\\omega \\rho_0 \n\\widetilde{s}{{\\mbox{\\boldmath $\\rho $}}{v}}^{\\mathrm{T}} = {} \n\\widetilde{s}{{\\mbox{\\boldmath $\\rho $}}{\\Sigma}}^\\mathrm{T}. \\label{eq:tt}\n\\end{align}\n\nWe have now decoupled the transverse and longitudinal components of the \nvelocity fluctuations. Eqs.~\\eqref{eq:ll} and \\eqref{eq:tt} are\nnothing but the Langevin equations for each component of the velocity field in \nthe frequency domain. In fact, Eq.~\\eqref{eq:ll} is a scalar equation for \nthe longitudinal component of the velocity fluctuation \\cite{Forster}. \n\nThe density field fluctuations can be obtained from the longitudinal component \nof the velocity field fluctuations, \n\\begin{equation}\n\\widetilde{\\rho}^{\\,(1)}(s}{{\\mbox{\\boldmath $\\rho $}}{r};\\omega) = -\\frac{i\\rho_0}{\\omega}\\nabla \\cdot \n\\widetilde{s}{{\\mbox{\\boldmath $\\rho $}}{v}}^{\\mathrm{L}}(s}{{\\mbox{\\boldmath $\\rho $}}{r};\\omega). \\label{eq:rhovl}\n\\end{equation}\n\n\\section{Mean interaction force}\n\\label{sec:meanforce}\n\nIn order to obtain the net effective interaction force between the fluid's \nconfining boundaries, we integrate the fluctuating hydrodynamic stress tensor, \n$\\sigma_{ij} \n= \n\\sigma_{ij}(s}{{\\mbox{\\boldmath $\\rho $}}{r};t)$, over the bounding surfaces, \n$\\Gamma$, i.e., \n\\begin{equation}\n\\big\\langle{\\cal F}_i(t)\\big\\rangle = \\int_{\\Gamma}\n\\big\\langle\\sigma_{ij}(s}{{\\mbox{\\boldmath $\\rho $}}{r};t)\\big\\rangle\\, \\mathrm{d} A_j, \n\\end{equation}\nwhere the fluctuating hydrodynamic stress tensor, which is \\cite{LandauLifshitz}\n\\begin{equation}\n\\sigma_{ij} = \\eta \\big[\\nabla_i v_j + \\nabla_j v_i\\big] -\n\\left[\\left(\\frac{2\\eta}{3} - \\zeta\\right) \\nabla_k v_k +\np\\right]\\delta_{ij}+S_{ij},\n\\end{equation}\n can be written up to first order in field fluctuations as \n $\\sigma_{ij} = -p_0\\delta_{ij}+\\sigma_{ij}^{(1)}$, with \n\\begin{align}\\label{eq:sigma}\n\\sigma_{ij}^{(1)} = {} & \\eta \\big[\\nabla_i v^{(1)}_j + \\nabla_j v^{(1)}_i\\big]\n\\nonumber\\\\\n{} & \\quad \n-\n\\left[\\left(\\frac{2\\eta}{3} - \\zeta\\right) \\nabla_k v_k^{(1)} \n+c_0^2\\rho^{(1)}\\right]\\delta_{ij}+S_{ij}. \n\\end{align}\n\n\nThe stress tensor is linear in the fluid fluctuations, which are \nthemselves linear in the random stress tensor and, thus, their ensemble \naverages vanish  $\\big\\langle s}{{\\mbox{\\boldmath $\\rho $}}{v}^{\\mathrm{T}}(s}{{\\mbox{\\boldmath $\\rho $}}{r};t) \\big\\rangle \n= \\big\\langle \ns}{{\\mbox{\\boldmath $\\rho $}}{v}^{\\mathrm{L}}(s}{{\\mbox{\\boldmath $\\rho $}}{r};t) \\big\\rangle = 0$ and $\\big\\langle \n\\rho^{(1)}(s}{{\\mbox{\\boldmath $\\rho $}}{r};t)  \\big\\rangle =  0$. \nAs a result, the  net force acting on the fluid boundaries must vanish, \nirrespective of the geometry of the fluid system, i.e., \n\\begin{equation}\n\\big\\langle{\\cal F}_i(t)\\big\\rangle = 0. \n\\end{equation}\n\nIn what follows, we limit our discussion to the plane-parallel geometry of two \nrigid walls of arbitrarily large surface area, $A$. We assume that the walls \nare located along the $z$ axis at $z=0$ and $z=L$ at a separation distance of \n$L$ and that the fluid velocity satisfies no-slip boundary conditions on the \nwalls. \n\n\\section{\\label{sec:forcev}Two-point, time-dependent correlations of the force}\n\\label{sec:forcevar_t}\n\nAlthough, as we have already noted, the mean inter-plate force due to \nhydrodynamic fluctuations in the fluid layer must vanish, its variance or \ncorrelation functions need not and do not. In \nthis Section, we study the two-point, time-dependent correlators, including the \nvariance, \nof the forces that act on the boundaries in the two-wall geometry.  \nIn this plane-parallel geometry, we are primarily  \nconcerned with the force perpendicular to the plane boundaries, in which case \nthe two-point, time-dependent force correlator is given \nby\n\\begin{align}\\label{eq:forcecorr}\n{\\cal C} {} &(z,z';t,t') =  \\big\\langle {\\cal F}_z(z;t) \n{\\cal F}_z(z';t') \n\\big\\rangle \\nonumber\\\\\n{} & \\quad= \\iint_A\n\\left\\langle\n\\sigma_{zz}(s}{{\\mbox{\\boldmath $\\rho $}}{r};t)\\sigma_{zz}(s}{{\\mbox{\\boldmath $\\rho $}}{r}^{\\prime};t')\\right\\rangle\\,\n\\mathrm{d} x \\, \\mathrm{d} y\\, \\mathrm{d} x'\\, \\mathrm{d} y', \n\\end{align}\nwhere the integrals run over the surface areas $A$ of the two walls that are \nlocated\nat $z=0$ and $z=L$.  Throughout this paper, we use an uppercase ${\\cal C}$ to \ndenote  correlation \nfunctions of the normal forces acting on the fluid boundaries and a lowercase \n$c$ to refer \nto correlation functions \nof the fluctuating hydrodynamic fields. We express the former quantity in \nterms of the latter ones (see Appendix \\ref{app:f3f3simple}). In the present \ncase, the correlators of \nthe velocity and density fluctuations are given \nby \n\\begin{align} \\label{eq:vvstart}\nc_{ij}^{\\mathrm{T}\\mathrm{T}}(s}{{\\mbox{\\boldmath $\\rho $}}{r},s}{{\\mbox{\\boldmath $\\rho $}}{r}';t,t') = {} & \\big\\langle \nv_i^{\\mathrm{T}}(s}{{\\mbox{\\boldmath $\\rho $}}{r};t) \nv_j^{\\mathrm{T}}(s}{{\\mbox{\\boldmath $\\rho $}}{r}^\\prime;t') \\big\\rangle,\\\\ \nc_{ij}^{\\mathrm{L}\\mathrm{L}}(s}{{\\mbox{\\boldmath $\\rho $}}{r},s}{{\\mbox{\\boldmath $\\rho $}}{r}';t,t') = {} & \\big\\langle \nv_i^{\\mathrm{L}}(s}{{\\mbox{\\boldmath $\\rho $}}{r};t) \nv_j^{\\mathrm{L}}(s}{{\\mbox{\\boldmath $\\rho $}}{r}^\\prime;t') \\big\\rangle, \\\\\nc^{\\rho\\rho}(s}{{\\mbox{\\boldmath $\\rho $}}{r},s}{{\\mbox{\\boldmath $\\rho $}}{r}';t,t') = {} & \\big\\langle \n\\rho^{(1)}(s}{{\\mbox{\\boldmath $\\rho $}}{r};t)  \n\\rho^{(1)}(s}{{\\mbox{\\boldmath $\\rho $}}{r}^\\prime;t')  \n\\big\\rangle.\n\\end{align}\nThe correlation function of the transverse and longitudinal \ncomponents of the velocity\nvanishes by construction. Furthermore, the transverse velocity and density \nfluctuations  are  independent fields, with vanishing correlation function. \nTherefore, the only \nother correlation function we need is the density-velocity cross correlator,\n\\begin{equation}\\label{eq:vpstart}\nc_i^{\\mathrm{L}\\rho}(s}{{\\mbox{\\boldmath $\\rho $}}{r},s}{{\\mbox{\\boldmath $\\rho $}}{r}';t,t')= \\big\\langle \nv_i^{\\mathrm{L}}(s}{{\\mbox{\\boldmath $\\rho $}}{r};t) \n\\rho^{(1)}(s}{{\\mbox{\\boldmath $\\rho $}}{r}^\\prime;t')\\big\\rangle.  \n\\end{equation}\n\nNot all of these correlators contribute to the time-dependent correlator of the \nforces between the two hard boundaries. \nIn Appendices \\ref{app:vtzero} and  \\ref{app:pvzero}, we show that the \ncontributions to the normal \nforce correlator generated by the correlation function of \nthe transverse velocity field and also by the correlation function between the \nvelocity and density fields vanish for our geometry.  \nTherefore, applying the formulae of the previous Section, we can write the \ntime-dependent force correlator as the sum of three terms (see  Appendix \n\\ref{app:f3f3simple} for details),\n\\begin{equation}\\label{eq:czzpt}\n{\\cal C}(z,z';t,t')  =  \\sum_{i = 0}^2{\\cal P}_i(z,z';t,t').\n\\end{equation}\nThe first term,\n\\begin{equation}\\label{eq:p0delta}\n{\\cal P}_0(z,z';t,t') \\equiv 2k_{\\mathrm{B}}T \\eta \\chi A \n\\delta(z-z')\\delta(t-t'),\n\\end{equation}\nstems directly from the integration of the random \nstress correlator, $\\langle\nS_{zz}(s}{{\\mbox{\\boldmath $\\rho $}}{r};t)S_{zz}(s}{{\\mbox{\\boldmath $\\rho $}}{r}^{\\prime};t')\\rangle$, over the bounding \nsurfaces; this term vanishes unless $z= z'$ and $t= t'$, in which case it \nreduces to an irrelevant constant that will be dropped in the rest of our\nanalysis. The two other terms are\n\\begin{align}\n{\\cal P}_1(z, z'; t,t') \\equiv {} &\\left(\\frac{4\\eta}{3} + \n\\zeta\\right)^2\\iint_A\\mathrm{d} x \\, \\mathrm{d} y\\, \\mathrm{d} x'\\, \n\\mathrm{d} y' \\nonumber\\\\\n{} & \\qquad \\times \n\\nabla_z\\nabla_z'c^{\\mathrm{L}\\mathrm{L}}_{zz}(s}{{\\mbox{\\boldmath $\\rho $}}{r},s}{{\\mbox{\\boldmath $\\rho $}}{r}';t,\nt'),\\label{eq:p1t}\\\\\n{\\cal P}_2(z, z'; t,t') \\equiv {} &c_0^4\\iint_A\\mathrm{d} x \\, \\mathrm{d} y\\, \n\\mathrm{d} x'\\, \\mathrm{d} y' c^{\\rho\\rho}(s}{{\\mbox{\\boldmath $\\rho $}}{r},s}{{\\mbox{\\boldmath $\\rho $}}{r}';t,\nt'). \\label{eq:p2t}\n\\end{align}\nWe note that, in the above, we have used Eq.~\\eqref{eq:rhovl}, which relates \nthe density fluctuations to the fluctuations of the longitudinal components of \nthe velocity.\n\nWith this expression in hand, we can see that we need to determine the \ncorrelation functions of the density fields and the longitudinal component of \nthe velocity fields. We proceed via the following steps \\cite{LandauLifshitz}:\n\\begin{enumerate}\n\\item Obtain the Green functions of Eq.~\\eqref{eq:ll};\n\\item Express the  fluctuating fields and their correlation functions in\nterms of the Green functions above;\n\\item Integrate the resulting expressions over the boundaries of the\nfluid according to Eqs.~\\eqref{eq:p1t} and \\eqref{eq:p2t}. \n\\end{enumerate}\n\n\\subsection{Green functions}\n\nIn the present model with no-slip walls, the velocity and, therefore, the \ncorresponding Green function\nshould vanish at the boundaries. Translational invariance in the two \n(transverse) directions\nperpendicular to the $z$-axis prompts us \nto search for Green functions of\nthe form\n\\begin{equation}\\label{eq:Gdef}\n\\widetilde{G}(s}{{\\mbox{\\boldmath $\\rho $}}{r},s}{{\\mbox{\\boldmath $\\rho $}}{r}^{\\prime\\prime};\\omega) =\n\\frac{1}{(2\\pi)^2}\\int \\mathrm{d}^2 s}{{\\mbox{\\boldmath $\\rho $}}{k} \\, e^{is}{{\\mbox{\\boldmath $\\rho $}}{k}\\cdot\n(s}{{\\mbox{\\boldmath $\\rho $}}{s}-s}{{\\mbox{\\boldmath $\\rho $}}{s}^{\\prime\\prime})}\\widetilde{G}(z,z^{\n\\prime\\prime}\n;s}{{\\mbox{\\boldmath $\\rho $}}{k};\\omega),\n\\end{equation}\nwhere $s}{{\\mbox{\\boldmath $\\rho $}}{r} = (s}{{\\mbox{\\boldmath $\\rho $}}{s}, z)$ with $s}{{\\mbox{\\boldmath $\\rho $}}{s} = (x,y)$ and $s}{{\\mbox{\\boldmath $\\rho $}}{k} \n= (k_x,k_y)$. \nThe longitudinal Green function corresponding to Eq.~\\eqref{eq:ll} is a \nsolution of the following equation: \n\\begin{equation}\n\\label{eq:gl}\n\\left[\\nabla_z^2 - m^2\\right]\\widetilde{G}^{\\,\\mathrm{L}}(z,z^{\n\\prime\\prime}\n;s}{{\\mbox{\\boldmath $\\rho $}}{k};\\omega)\n= \\frac{i\\lambda^2}{\\omega\\rho_0}\\delta(z-z^{\\prime\\prime}),\n\\end{equation}\nwhere $m^2 = s}{{\\mbox{\\boldmath $\\rho $}}{k}^2+\\lambda^2$ and we \nhave defined the longitudinal decay constant $\\lambda$ as\n\\begin{equation}\n\\lambda^2 = -\\frac{i\\omega^2\\rho_0}{\\left(4\\eta\/3 + \\zeta\\right)\\omega + i \n\\rho_0c_0^2}.\n\\label{eq:lambda}\n\\end{equation}\n\nThe solution of Eq.~\\eqref{eq:gl} is well known \n\\cite{Erbas,Kim,Schwinger}, and with no-slip boundary conditions at $z = 0$ and \n$z = L$, the Green function is obtained as \n\\begin{align}\n\\widetilde{G}^{\\,\\mathrm{L}}(z,z^{\n\\prime\\prime}\n;s}{{\\mbox{\\boldmath $\\rho $}}{k};\\omega) = &\\, \n{} g_1^{\\,\\mathrm{L}}e^{-mz}+g_2^{\\,\\mathrm{L}}e^{m(z-L)} \\nonumber\\\\\n& \\qquad{} - \n\\frac{i\\lambda^2 }{2m \n\\omega\\rho_0}e^{-m|z-z''|},\\label{eq:greenl}\n\\end{align}\nwhere\n\\begin{align}\ng_1^{\\,\\mathrm{L}} = & {} \\frac{i\\lambda^2}{2m\\omega \\rho_0} \n\\operatorname{csch}(mL)\\sinh(m(L-z'')), \\\\\ng_2^{\\,\\mathrm{L}} = & {} \\frac{i\\lambda^2}{2m\\omega \\rho_0}  \\operatorname{csch}(mL)\\sinh(m \nz''),\n\\end{align}\nare constants of integration that satisfy the no-slip boundary conditions.\n\n\\subsection{Characteristic scales and dimensionless parameters}\n\\label{subsec:para}\n\nWe simplify the following analysis by introducing dimensionless parameters that \ncharacterize the fluid and the plane-parallel  geometry of our system. There \nare \ntwo length scales that can be used for this purpose: The macroscopic plate \nseparation, $L$, and the microscopic \nscale at which the continuum hydrodynamic description breaks down, which we \ndenote $a $. There are two characteristic vorticity frequencies associated \nwith each of these length scales \\cite{Erbas},\n\\begin{equation}\\label{eq:omega0def}\n\\omega_0 = \\frac{\\eta}{L^2\\rho_0} \\qquad \\mathrm{and}\\qquad \\omega_\\infty = \n\\frac{\\eta}{a ^2\\rho_0}.\n\\end{equation}\nThe inverse frequencies, $\\omega_0^{-1}$ and $\\omega_\\infty^{-1}$, correspond \nto the time that vorticity requires to diffuse a certain distance, in this case \n$L$ or $a $, respectively. We also define the dimensionless parameter \n$\\gamma$, which is given by\n\\begin{equation}\n\\gamma = \\frac{c_0^2}{L^2\\omega_0^2} = \\left(\\frac{L\\rho_0 \nc_0}{\\eta}\\right)^2.\\label{eq:gammadef}\n\\end{equation}\nThis parameter is the squared ratio of the vorticity time scale and the typical \ncompression time scale in which a propagating sound wave travels a distance $L$ \n\\cite{Erbas}.\n\nTo facilitate our later discussions, we introduce the dimensionless ratios\n\\begin{equation}\\label{eq:udef}\nu = \\frac{\\omega}{\\omega_0\\gamma} \\qquad \\mathrm{and} \\qquad u_\\infty = \n\\frac{\\omega_\\infty}{\\omega_0\\gamma},\n\\end{equation}\nand define the function\n\\begin{equation}\\label{eq:fudef}\nf_m(u) = \\frac{u^{2-m}}{1+\\chi^2u^2},\n\\end{equation}\nwhere \n\\begin{equation}\n\\chi = 4\/3+\\zeta\/\\eta.\\label{eq:chi}\n\\end{equation}\nWe can now express the real and imaginary parts \nof the longitudinal decay \nconstant, $\\lambda$, as\n\\begin{align}\n\\ell_+ = {} & \\lambda_{\\mathrm{R}}L = \n\\frac{\\omega_0\\gamma \nL}{c_0}\\frac{|u|}{\\sqrt{2}}\\sqrt{\\left[1-\\sqrt{f_2(u)}\\right]\\sqrt{f_2(u)} } ,\\\\\n\\ell_- = {} & \\lambda_{\\mathrm{I}}L =-\n\\frac{\\omega_0\\gamma \nL}{c_0}\\frac{u}{\\sqrt{2}}\\sqrt{\\left[1+\\sqrt{f_2(u)}\\right]\\sqrt{f_2(u)} } .\n\\end{align}\n\nThe vorticity frequency scale $\\omega_0$ marks the boundary \nbetween the low-frequency \npropagative regime, for which $\\omega < \\omega_0\\gamma$ (or $u<1$) and sound \nwaves \npropagate with speed $c \\sim |\\lambda_{\\mathrm{I}}^{-1}|$, and the \nhigh-frequency \ndiffusive regime, for which $\\omega > \\omega_0\\gamma$ (or $u>1$) and viscosity \neffects damp \ncompression perturbations \\cite{Erbas}.\nFurthermore, the dimensionless ratio $u_\\infty$ can be expressed in terms of a \nnew length scale $\\delta $:\n\\begin{equation}\n\\label{eq:u_inf}\nu_\\infty = \\frac{\\delta^2}{a ^2} \\qquad \\mathrm{where}\\qquad \\delta  = \n\\frac{\\eta}{\\rho_0c_0} = \\frac{c_0}{\\omega_0 \\gamma}.\n\\end{equation}\nThis length scale characterizes the boundary between the propagative and \ndiffusive regimes at $\\omega_0 \\gamma$. We can also define a characteristic \ntime scale, \n\\begin{equation}\nt_0 = \\delta\/c_0, \n\\label{eq:t0}\n\\end{equation}\nassociated \nwith this boundary. Finally, then, we can write $\\ell_+$ and $\\ell_-$ as \n\\begin{align}\n\\ell_+ = {} & \n\\frac{\nL}{\\delta }\\frac{|u|}{\\sqrt{2}}\\sqrt{\\left[1-\\sqrt{f_2(u)}\\right]\\sqrt{f_2(u)} \n} , \n\\label{eq:lambdar}\\\\\n\\ell_- = {} & -\\frac{\nL}{\\delta }\\frac{u}{\\sqrt{2}}\\sqrt{\\left[1+\\sqrt{f_2(u)}\\right]\\sqrt{f_2(u)} \n} . \n\\label{eq:lambdai}\n\\end{align}\n\nFor any reasonable choice of realistic parameters for a fluid far from the \ncritical point, we have $u\\ll1$, i.e., we work in the propagative regime. In \nthis case, the plate separation of a realistic experiment satisfies $L\/\\delta \n\\gg~1$. \nFor liquids close to the critical point, or polymers in solution, however, the \ncrossover frequency can be much lower and, therefore, we can have $u\\gg1$. In \nthis \ncase, the system is in the diffusive regime and the crossover length scale, \n$\\delta $, may be macroscopic. \n\n\\subsection{Correlation functions}\n\\label{sec:corrfns}\n \nNow that we have explicit expressions for the Green function solutions in \nhand, we turn to the correlation functions $c_{zz}^{\\mathrm{L}\\mathrm{L}}$ and\n$c^{\\rho\\rho}$, which enter in Eqs.~\\eqref{eq:czzpt}-\\eqref{eq:p2t}, and express \nthese \ncorrelation functions in terms of the \ncorresponding Green functions. Here, we simply sketch the derivation for \n$c_{zz}^{\\mathrm{L}\\mathrm{L}}$, as an example, and leave the details of the \ncorresponding calculation of $c^{\\rho\\rho}$ to Appendix\n\\ref{app:corrfn}.\n\nThe longitudinal velocity fluctuations are given in terms of the longitudinal \nGreen function as\n\\begin{equation}\\label{eq:vtt}\nv_i^{\n\\mathrm{L}}(s}{{\\mbox{\\boldmath $\\rho $}}{r};t) = \\int \\mathrm{d}t^{\\prime\\prime}  \\int \n\\mathrm{d}^3s}{{\\mbox{\\boldmath $\\rho $}}{r}^{\\prime\\prime} \n\\,\nG^{\\,\\mathrm{L}}(s}{{\\mbox{\\boldmath $\\rho $}}{r},s}{{\\mbox{\\boldmath $\\rho $}}{r}^{\\prime\\prime} ;t-t^{\\prime\\prime} \n)\\Sigma^{\n\\mathrm{L}}_i(s}{{\\mbox{\\boldmath $\\rho $}}{r}^{\\prime\\prime} ;t^{\\prime\\prime} ).\n\\end{equation}\nWe require the correlation function\n\\begin{align}\\label{eq:vvtrep}\n\\big\\langle  \nv_i^{\\mathrm{L}}(s}{{\\mbox{\\boldmath $\\rho $}}{r};t) & {}\nv_j^{\\mathrm{L}}(s}{{\\mbox{\\boldmath $\\rho $}}{r}^\\prime;t') \\big\\rangle  = \\int \n\\mathrm{d}t^{\\prime\\prime} \\int \n\\mathrm{d}t^{\\prime\\prime\\prime}\\int \n\\mathrm{d}^3s}{{\\mbox{\\boldmath $\\rho $}}{r}^{\\prime\\prime} \\,\\int \n\\mathrm{d}^3s}{{\\mbox{\\boldmath $\\rho $}}{r}^{\\prime\\prime\\prime} \\nonumber\\\\\n& {} \\times\nG^{\\,\\mathrm{L}}(s}{{\\mbox{\\boldmath $\\rho $}}{r},s}{{\\mbox{\\boldmath $\\rho $}}{r}^{\\prime\\prime};t-t^{\\prime\\prime})G^{\\,\n\\mathrm { L\n}}(s}{{\\mbox{\\boldmath $\\rho $}}{r}^\\prime,s}{{\\mbox{\\boldmath $\\rho $}}{r}^{\\prime\\prime\\prime};t'-t^{\n\\prime\\prime\\prime } )\\nonumber\\\\\n& {} \\quad \\times\\left\\langle\n\\Sigma^{\\mathrm{L}}_i(s}{{\\mbox{\\boldmath $\\rho $}}{r}^{\\prime\\prime};t^{\\prime\\prime})\\Sigma^{\\mathrm\n{ L } } _j(s}{{\\mbox{\\boldmath $\\rho $}} {\nr}^{\\prime\\prime\\prime};t^{\n\\prime\\prime\\prime })\\right\\rangle.\n\\end{align}\nRecalling the stochastic properties of the random stress tensor, \nEq.~\\eqref{eq:svecLw-2}, we obtain\n\\begin{align}\\label{eq:vvtrep2}\nc_{ij}^{\\mathrm{L}\\mathrm{L}}{} & (s}{{\\mbox{\\boldmath $\\rho $}}{r},s}{{\\mbox{\\boldmath $\\rho $}}{r}';t,t') \n= 2k_{\\mathrm{B}}T\\eta\\chi  \\int \n\\mathrm{d}t'' \\int \n\\mathrm{d}^3s}{{\\mbox{\\boldmath $\\rho $}}{r}''  \\nonumber\\\\\n& {} \\quad \\times\n\\nabla''_iG^{\\,\\mathrm{L}}(s}{{\\mbox{\\boldmath $\\rho $}}{r},s}{{\\mbox{\\boldmath $\\rho $}}{r}'';t-t''\n)\\nabla''_jG^{\\,\n\\mathrm { L\n}}(s}{{\\mbox{\\boldmath $\\rho $}}{r}',s}{{\\mbox{\\boldmath $\\rho $}}{r}'';t'-t'' ).\n\\end{align}\n\nWe now introduce a Fourier representation of the Green functions\n\\begin{align}\\label{eq:cvv}\nc_{ij}^{\\mathrm{L}\\mathrm{L}} {} &(s}{{\\mbox{\\boldmath $\\rho $}}{r},s}{{\\mbox{\\boldmath $\\rho $}}{r}';t,t') \n= 2k_{\\mathrm{B}}T\\eta\\chi  \\int \n\\mathrm{d}t'' \\int \n\\mathrm{d}^3s}{{\\mbox{\\boldmath $\\rho $}}{r}'' \\nonumber\\\\\n& {} \\qquad \\times \\int\\frac{\\mathrm{d}\\omega}{2\\pi} e^{-i \\omega (t-t'')}\n\\nabla''_i\\widetilde{G}^{\\,\\mathrm{L}}(s}{{\\mbox{\\boldmath $\\rho $}}{r},s}{{\\mbox{\\boldmath $\\rho $}}{r}\n'';\\omega)\\nonumber\\\\\n& {} \\qquad \\times \\int\\frac{\\mathrm{d}\\omega'}{2\\pi} e^{-i \\omega' \n(t'-t'')}\\nabla_j''\\widetilde{G}^{\\,\n\\mathrm { L\n}}(s}{{\\mbox{\\boldmath $\\rho $}}{r}',s}{{\\mbox{\\boldmath $\\rho $}}{r}'';\\omega' ).\n\\end{align}\nThe integral over $t''$ generates a Dirac delta function for the frequencies, \n$\\delta(\\omega+\\omega')$, and therefore one of the frequency integrals becomes \ntrivial:\n\\begin{align}\nc_{ij}^{\\mathrm{L}\\mathrm{L}} {} &(s}{{\\mbox{\\boldmath $\\rho $}}{r},s}{{\\mbox{\\boldmath $\\rho $}}{r}';t,t') \n= 2k_{\\mathrm{B}}T\\eta\\chi\\int\\frac{\\mathrm{d}\\omega'}{2\\pi}e^{i \\omega' \n(t-t')} \\nonumber\\\\\n& {} \\quad \\times   \\int \n\\mathrm{d}^3s}{{\\mbox{\\boldmath $\\rho $}}{r}''\n\\nabla''_i\\widetilde{G}^{\\,\\mathrm{L}}(s}{{\\mbox{\\boldmath $\\rho $}}{r},s}{{\\mbox{\\boldmath $\\rho $}}{r}\n'';-\\omega') \\nabla_j''\\widetilde{G}^{\\,\n\\mathrm { L\n}}(s}{{\\mbox{\\boldmath $\\rho $}}{r}',s}{{\\mbox{\\boldmath $\\rho $}}{r}'';\\omega' ).\n\\end{align}\n\nIn principle, we could substitute our explicit expression for the Green \nfunction, Eq.~\\eqref{eq:greenl}, into this \ncorrelation function and attempt to directly calculate the integrals at this \nstage. We will see, however, that this is not the most straightforward \napproach: Spatial integrations over the fluid boundary will simplify our task \nconsiderably. We also take advantage of the fact that we only require the \ncomponents of the velocity fields perpendicular to the plane boundaries. \nTherefore, we set $i = j= z$ in our expression for the \ncorrelation function,\n$c_{ij}^{\\mathrm{L}\\mathrm{L}}(s}{{\\mbox{\\boldmath $\\rho $}}{r},s}{{\\mbox{\\boldmath $\\rho $}}{r}';t,t')$, and use the \ntranslational-invariant structure of the Green function, \nEq.~\\eqref{eq:Gdef}, to write\n\\begin{align}\n c_{zz}^{\\mathrm{L}\\mathrm{L}} {} &(s}{{\\mbox{\\boldmath $\\rho $}}{r},s}{{\\mbox{\\boldmath $\\rho $}}{r}';t,t') =\n2k_{\\mathrm{B}}T\\eta\\chi \n\\int\\frac{\\mathrm{d}\\omega'}{2\\pi}e^{i \n\\omega' \n(t-t')} \\nonumber\\\\\n& \\times \\int \n\\mathrm{d}^3s}{{\\mbox{\\boldmath $\\rho $}}{r}''\\int \\frac{\\mathrm{d}^2 \ns}{{\\mbox{\\boldmath $\\rho $}}{k}}{(2\\pi)^2}\ne^{is}{{\\mbox{\\boldmath $\\rho $}}{k}\\cdot\n(s}{{\\mbox{\\boldmath $\\rho $}}{s}-s}{{\\mbox{\\boldmath $\\rho $}}{s}'')}\\nabla''_z \n\\widetilde{G}^{\\,\\mathrm{L}}(z,z'' \n;s}{{\\mbox{\\boldmath $\\rho $}}{k};-\\omega') \\nonumber\\\\\n& \\times \\int \\frac{\\mathrm{d}^2 \ns}{{\\mbox{\\boldmath $\\rho $}}{k}'}{(2\\pi)^2}\\,e^{is}{{\\mbox{\\boldmath $\\rho $}}{k}'\\cdot\n(s}{{\\mbox{\\boldmath $\\rho $}}{s}'-s}{{\\mbox{\\boldmath $\\rho $}}{s}'')} \\nabla''_z \n\\widetilde{G}^{\\,\\mathrm{L}}(z',\nz'';s}{{\\mbox{\\boldmath $\\rho $}}{k}';\\omega').\n\\end{align}\n\nThe double integral over $s}{{\\mbox{\\boldmath $\\rho $}}{s}''$ generates a \nwavenumber Dirac delta function, $\\delta (s}{{\\mbox{\\boldmath $\\rho $}}{k}+s}{{\\mbox{\\boldmath $\\rho $}}{k}')$, that \nenables us to carry out one of the wavenumber integrals immediately and, thus, \nobtain\n\\begin{align}\n c_{zz}^{\\mathrm{L}\\mathrm{L}} {} &(s}{{\\mbox{\\boldmath $\\rho $}}{r},s}{{\\mbox{\\boldmath $\\rho $}}{r}';t,t') =\n2k_{\\mathrm{B}}T\\eta\\chi\n\\int\\frac{\\mathrm{d}\\omega'}{2\\pi}e^{i\\omega' (t-t')} \\nonumber\\\\\n& \\times \\int \n\\mathrm{d}z''\\int \\frac{\\mathrm{d}^2 \ns}{{\\mbox{\\boldmath $\\rho $}}{k}}{(2\\pi)^2}\ne^{is}{{\\mbox{\\boldmath $\\rho $}}{k}\\cdot\n(s}{{\\mbox{\\boldmath $\\rho $}}{s}-s}{{\\mbox{\\boldmath $\\rho $}}{s}')}\\nabla''_z \n\\widetilde{G}^{\\,\\mathrm{L}}(z,z'' \n;s}{{\\mbox{\\boldmath $\\rho $}}{k};-\\omega') \\nonumber\\\\\n& \\qquad \\times  \\nabla''_z \n\\widetilde{G}^{\\,\\mathrm{L}}(z',\nz'';-s}{{\\mbox{\\boldmath $\\rho $}}{k};\\omega')\n.\\label{eq:ttcorr}\n\\end{align}\n\nAnalogous arguments apply to the density\ncorrelation function, which is (see Appendix \\ref{app:corrfn})\n\\begin{align}\n{} & c^{\\rho\\rho}(s}{{\\mbox{\\boldmath $\\rho $}}{r},s}{{\\mbox{\\boldmath $\\rho $}}{r}';t,t') =  \\frac{k_{\\mathrm{B}}T}{\\pi} \n\\rho_0^2 \\eta\\chi \n\\int \\frac{\\mathrm{d}\\omega'}{\\omega'^2} e^{i\\omega'(t-t')}\\int \n\\mathrm{d}z'' \\nonumber\\\\\n{} & \\quad \\times \\int \\frac{\\mathrm{d}^2 \ns}{{\\mbox{\\boldmath $\\rho $}}{k}}{(2\\pi)^2}\\,e^{is}{{\\mbox{\\boldmath $\\rho $}}{k}\\cdot\n(s}{{\\mbox{\\boldmath $\\rho $}}{s}-s}{{\\mbox{\\boldmath $\\rho $}}{s}')}\\left(\\nabla_z \n\\nabla''_z+s}{{\\mbox{\\boldmath $\\rho $}}{k}^2\\right)  \n\\widetilde{G}^{\\,\\mathrm{L}}(z,z''; s}{{\\mbox{\\boldmath $\\rho $}}{k} ;-\\omega') \\nonumber \\\\\n{} & \\qquad \\times \\left(\\nabla_z' \n\\nabla''_z+s}{{\\mbox{\\boldmath $\\rho $}}{k}^2\\right)  \n\\widetilde{G}^{\\,\\mathrm{L}}(z',z''; -s}{{\\mbox{\\boldmath $\\rho $}}{k} ;\\omega').\\label{eq:ppcorr}\n\\end{align}\n\n\\subsection{Spatial integration over surface boundaries}\n\\label{sec:singleforce}\n\nOur final step is to integrate the correlation functions, \nEqs.~\\eqref{eq:ttcorr} and \\eqref{eq:ppcorr}, over the \nboundaries of the fluid according to Eqs.~\\eqref{eq:czzpt}-\\eqref{eq:p2t}. \nThese integrals \ngive our final result for the time-dependent correlators of the force \nacting on the fluid boundaries.\n\nThe double integrals over $(x,y)$ and $(x^\\prime,y^\\prime)$ in \nEqs.~\\eqref{eq:czzpt}-\\eqref{eq:p2t} lead to a Dirac delta function over the \ntransverse wavenumbers, $(2\\pi)^2 A \\delta(s}{{\\mbox{\\boldmath $\\rho $}}{k})$. Thus, we can write \nthese equations in terms of \nthe Green function as\n\\begin{align}\n& {\\cal P}_{1}(z, z'; t,t') =\\frac{k_{\\mathrm{B}}T}{\\pi} \\eta^3\\chi^3 A \n\\int\n\\mathrm{d}\\omega'\\cos[\\omega'(t-t')]\n\\nonumber \\\\\n& \\times \\int\n\\mathrm{d}z''\n\\nabla_z\n\\nabla^{\n\\prime\\prime}_z \\widetilde{G}^{\\,\\mathrm{L}}(z,z^{\n\\prime\\prime}\n;s}{{\\mbox{\\boldmath $\\rho $}}{0};-\\omega')\\nabla_z' \\nabla''_z \\widetilde{G}^{\\,\\mathrm{L}}(z',z''\n;s}{{\\mbox{\\boldmath $\\rho $}}{0};\\omega'),\\label{eq:p1zero}\\\\\n& {\\cal P}_2(z, z'; t,t') =\\frac{k_{\\mathrm{B}}T}{\\pi} \\rho_0^2\\eta\\chi c_0^4 \nA\\int\n\\frac{\\mathrm{d}\\omega'}{\\omega^{\\prime\\,2}}\\cos[\\omega'(t-t')] \n \\nonumber\\\\\n& \\times \\int\n\\mathrm{d}z''\n \\nabla_z\n\\nabla^{\n\\prime\\prime}_z \\widetilde{G}^{\\,\\mathrm{L}}(z,z^{\n\\prime\\prime}\n;s}{{\\mbox{\\boldmath $\\rho $}}{0};-\\omega')\\nabla_z' \\nabla''_z \\widetilde{G}^{\\,\\mathrm{L}}(z',z''\n;s}{{\\mbox{\\boldmath $\\rho $}}{0};\\omega').\\label{eq:p2zero}\n\\end{align}\nThese frequency integrals run over the frequency range\n$\\omega \\in [-\\omega_\\infty,\\omega_\\infty]$ and the spatial integral is over \n$z\\in [0,L]$. In writing the above relations, we have used the fact that the \nintegrands involved in calculating ${\\cal P}_1$ and ${\\cal P}_2$ (see \nEqs.~\\eqref{eq:ttcorr} and \\eqref{eq:ppcorr}) have odd imaginary parts, which \nthus \nvanish, leading to the factor $\\cos[\\omega'(t-t')]$ from the real part of the \nexponential factor $e^{i\\omega'(t-t')}$. We also \nnote that $\\widetilde{G}^{\\,\\mathrm{L}\\,\\ast}(z',z'';s}{{\\mbox{\\boldmath $\\rho $}}{0};\\omega') = \n\\widetilde{G}^{\\,\\mathrm{L}}(z',z'';s}{{\\mbox{\\boldmath $\\rho $}}{0};-\\omega')$, which follows from \nthe reality of $G^{\\,\\mathrm{L}}(z',z'';s}{{\\mbox{\\boldmath $\\rho $}}{0};t)$. Therefore, as \nexpected, the final correlators are purely real.\n\nCarrying out the derivatives and the remaining spatial integral is\nfairly straightforward. The results are \n\\begin{align}\n{\\cal P}_{1}(z, z^\\prime; t,t') = {} & \\frac{k_{\\mathrm{B}}T}{\\pi} \n\\frac{\\rho_0 c_0^2\\,  A }{L}\\chi^3 \\Big[{\\cal W}_0(z,z';\\tau)\\nonumber\\\\\n{} & \\quad\\qquad  +L {\\cal V}_0(z,z';\\tau) \\delta \n(z-z')\\Big],\\label{eq:p1zero123}\\\\\n{\\cal P}_2(z, z^\\prime; t,t') = {} &  \\frac{k_{\\mathrm{B}}T}{\\pi} \n\\frac{\\rho_0 \nc_0^2\\, \n A }{L}\\chi\\Big[{\\cal W}_2(z,z';\\tau)\\nonumber\\\\\n{} & \\quad\\qquad+\nL {\\cal V}_2(z,z';\\tau) \\delta (z-z')\\Big].\\label{eq:p2zero123}\n\\end{align}\nThe relevant frequency integrals are given by  (see \nAppendix \\ref{app:wmderiv})\n\\begin{align}\n{} &{\\cal W}_m (0,0;\\tau)\n=  2\\int_0^{u_\\infty} \n\\mathrm{d}u \\, f_m(u)\\cos[u\\tau] \\nonumber\\\\\n{} & \\quad \\times \\frac{1}{\\cosh[2\\ell_+]\n-\\cos[2\\ell_-]} \n\\bigg[\\left(\\frac{\\ell^2}{2\\ell_-} \n-2\\ell_-\\right) \\sin[2\\ell_-]\n\\nonumber\\\\\n{} &\\qquad \\qquad  +\\left(\\frac{\\ell^2}{2\\ell_+} \n-2\\ell_+\\right) \\sinh[2\\ell_+]\\bigg],\\label{eq:wmt}\\\\\n{} &{\\cal W}_m (0,L;\\tau)   = 2\\int_0^{u_\\infty} \n\\mathrm{d}u\\,f_m(u)\\cos[u\\tau] \\nonumber\\\\\n{} & \\quad\\times \\frac{1}{\\cosh[2\\ell_+]\n-\\cos[2\\ell_-]} \n\\bigg[\\bigg(\\frac{\\ell^2}{\\ell_-} - 4\\ell_-\\bigg)\\cosh[\\ell_+]\n\\sin[\\ell_-] \\nonumber\\\\\n{} & \\qquad\\qquad  + \\bigg(\\frac{\\ell^2}{\\ell_+} - 4\\ell_+\\bigg)\\cos[\\ell_-]\n\\sinh[\\ell_+]\\bigg],\n\\label{eq:wmcrosst} \n\\end{align}\nand\n\\begin{equation}\n{\\cal V}_m(0,0;\\tau) = \n                     2{\\displaystyle \\int_0^{u_\\infty} }\n\\mathrm{d}u \\, f_m(u)\\cos[u\\tau].\n\\label{eq:vmt0}\n\\end{equation}\nIn these equations $\\ell^2 = \\ell_+^2+\\ell_-^2$, where we have defined $\\ell_+$ \nand \n$\\ell_-$ in Eqs.~\\eqref{eq:lambdar} and \\eqref{eq:lambdai} and the function \n$f_m(u)$ in Eq.~\\eqref{eq:fudef}. The \ndimensionless time parameter is\n\\begin{equation}\n\\tau = (t-t')\/t_0, \n\\end{equation}\nwith $t_0$ being the \ncharacteristic microscopic timescale defined in Eq.~\\eqref{eq:t0}. We have used \nthe \nsymmetry of the integrand  \nto integrate over the positive real axis up to the dimensionless \nmicroscopic cutoff, $u_\\infty$, of Eqs.~\\eqref{eq:udef} and \\eqref{eq:u_inf}. \nTo simplify these expressions further, we note that\n\\begin{equation}\n{\\cal V}_2(0,0;\\tau)+\\chi^2{\\cal V}_0(0,0;\\tau) \n= \\frac{2}{\\tau}\\sin[u_\\infty \\tau].\\label{eq:vmt}\n\\end{equation}\n\nPutting together all of these results, from Eqs.~\\eqref{eq:czzpt} and \n \\eqref{eq:p1zero}-\\eqref{eq:vmt}, we find\n \\begin{align}\n {\\cal C}(0,0;t,t')\n=  {} &   \\frac{k_{\\mathrm{B}}T}{\\pi}    \n\\frac{\\rho_0 c_0^2  A}{L} \\chi\\bigg[\\frac{2L}{\\tau}\\sin[u_\\infty \\tau]\\delta(0) \n \\nonumber\\\\\n {} & \\quad +\\chi^2 {\\cal W}_0(0,0;\\tau)\n + {\\cal \nW}_2(0,0;\\tau) \\bigg],\n\\label{eq:final1}\n\\\\\n {\\cal C}(0,L;t,t')\n= {}&  \\frac{k_{\\mathrm{B}}T}{\\pi}    \n\\frac{\\rho_0 c_0^2  A}{L} \\chi \\nonumber\\\\\n{} & \\quad \\times \\Big[ \\chi^2\n {\\cal W}_0(0,L;\\tau) + {\\cal \nW}_2(0,L;\\tau)\\Big].\n\\label{eq:final2}\n\\end{align}\n\nThese are our final results: The same-plate and the \ncross-plate correlators of the normal force, expressed in terms of the four \nfrequency \nintegrals ${\\cal W}_m(0,0;\\tau)$ and ${\\cal W}_m(0,L;\\tau)$ where $m=0, 2$. \nThus, while the average fluctuation-induced force between the bounding surfaces \nvanishes identically (see Sec.~\\ref{sec:meanforce}), the correlation \nfunctions of the force show a pronounced dependence on the inter-plate \nseparation and the time difference.\n\nThe same-plate correlator of the normal force, $ {\\cal C}(0,0;t,t')$ \nin Eq.~\\eqref{eq:final1}, contains three terms.\nThe first contribution is local in space and therefore proportional to the Dirac \ndelta \nfunction. Comparing this first term to ${\\cal \nP}_0(z,z'; t, t')$ in Eq.~\\eqref{eq:p0delta} indicates\nthat this contribution to $ {\\cal C}(0,0;t,t')$ is related to the integral of \nthe random stress correlator, Eq.~\\eqref{eq:gaussprop2}, across \nthe bounding surfaces, but with the hydrodynamic coupling nevertheless\nfully taken into account. Incorporating the hydrodynamic coupling leads to\nnon-locality in time, while \nlocality in space is preserved at leading order. \nIn contrast, ${\\cal P}_0(z,z'; t, t')$ is local both in \ntime and in space, because this term follows directly from the correlator of \nthe random stress tensor without any hydrodynamic coupling and has, \ntherefore, been dropped from our present analysis. \nThe other two terms in $ {\\cal C}(0,0;t,t')$ are different in nature. They are \nnon-local both in time \nand in space. They correspond to self-correlations mediated by the hydrodynamic \ninteraction between the boundaries, leading to separation-dependent \ncontributions to the same-plate, normal force correlator. \nThese two terms present a non-trivial generalization of the normal force \ncorrelator that hydrodynamically\ncouples the boundaries. We now define these contributions to be the {\\em \nexcess} correlator,\n\\begin{align}\n\\Delta{\\cal C} (0,0; t, t')\n\\equiv  {} & \\frac{k_{\\mathrm{B}}T}{\\pi}   \\frac{\\rho_0 c_0^2\\,  A }{L} \\chi \n\\nonumber\\\\\n{} & \\times \n\\bigg[\\chi^2\n {\\cal W}_0(0,0; \\tau) + {\\cal W}_2(0,0; \\tau)\\bigg],\n \\label{eq:DeltaC_t}\n\\end{align}\nwhich we investigate in detail in the following sections. \n\nThe cross-plate correlator of the normal force in Eq.~\\eqref{eq:final2} \ndoes not contain any local terms. In fact, it is purely\nnon-local and does not include any hydrodynamic self-interactions. The \ncross-plate correlator is due entirely to hydrodynamic interactions \nacross the fluid between the boundaries, and thus \nnaturally depends on the boundary separation. \n\nIn summary, for the same-plate force correlator, we have identified a trivial \nterm that is local in space and non-trivial terms that are non-local in space \nand correspond to \nself-correlations mediated by the hydrodynamic coupling between the \nboundaries. This leads to the separation-dependent excess same-plate force \ncorrelator. On the \nother hand, the cross-plate force correlator contains no local terms, as \nexpected, and \nstems entirely from hydrodynamic interactions between the bounding surfaces.\n\n\\section{Results for equal-time force correlators}\n\\label{sec:numerics}\n\n\\begin{table}[t!]\n\\caption{\\label{tab:parms}Representative ranges of physical parameters in a \nrealistic fluid; see the text for definitions.\n\\\\}\n\\begin{ruledtabular}\n\\begin{tabular}{ccc}\n\\vspace*{-5pt}\\\\\nParameter & Description & Range \\\\\n\\vspace*{-5pt}\\\\\n\\hline\n\\vspace*{-5pt}\\\\\n$L $ & Plate separation &  10$^{-6}$ to 10$^{-3}$ m \\\\\n$\\delta $ & Propagative-diffusive boundary & 10$^{-9}$ to 10$^{-6}$ m \\\\\n$a $ & Microscopic cutoff & 10$^{-9}$ m \\\\\n$\\eta$ & Shear viscosity & 10$^{-4}$ to 1 Pa$\\cdot$s \\\\\n$\\zeta$ & Bulk viscosity & 10$^{-4}$ to 1 Pa$\\cdot$s \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\\begin{comment}\n\\delta(acetone) = 0.000316\/(785*1170) = 3.4 * 10^-10m\n\\delta(glycerol) = 0.950\/(1260*1920) = 3.9 * 10^-7m\n\\delta(water)=0.001\/(1000*1482)=6.75 *10^-10m\n\\delta(hexane)=0.000297\/(679.5*1203) = 3.63* 10^-10\n\\delta(chloroform)=0.00053\/(1465*984) = 3.68 *10^-10\n\\delta(caster oil)=0.650\/(956.1*1474) = 4.61* 10^-7\n\\delta(ethylene glycol)=0.0162\/(1097*1660) = 8.896* 10^-9\n\\end{comment}\n\nOur task now is to explore and evaluate the frequency integrals that appear in \nEqs.~\\eqref{eq:final1}-\\eqref{eq:DeltaC_t}. We start by considering the \nequal-time correlators that follow from these equations by setting $\\tau=0$ or, \nequivalently,  $t=t'$, i.e., \n\\begin{align}\n\\Delta{\\cal C} (0,0)\n=  {}&  \\frac{k_{\\mathrm{B}}T}{\\pi}   \\frac{\\rho_0 c_0^2\\,  A }{L} \\chi \n\\bigg[\\chi^2\n {\\cal W}_0(0,0) + {\\cal W}_2(0,0)\\bigg],\n \\label{eq:DeltaC} \\\\\n{\\cal C}(0,L) = {} & \\frac{k_{\\mathrm{B}}T}{\\pi}   \n\\frac{\\rho_0 c_0^2\\,  A }{L} \\chi \n\\bigg[ \\chi^2\n {\\cal W}_0(0,L) + {\\cal W}_2(0,L)\\bigg],\\label{eq:ffp5678}\n\\end{align}\nwhere we have set ${\\cal C}(z,z')\\equiv {\\cal C}(z,z';t,t) $ and $ {\\cal \nW}_m(z,z')\\equiv {\\cal W}_m(z,z';\\tau=0)$. We note that $\\Delta{\\cal C} (0,0)$ \nis, in fact, the excess \\emph{force variance}.\n\nThe dimensionless integrals, ${\\cal W}_m(z,z')$, are functions of \njust three dimensionless ratios: The ratio of the fluid viscosities, \n$\\zeta\/\\eta$, which enters through $\\chi$, defined in Eq.~\\eqref{eq:chi}; the \nratio of the plate separation to the propagative-diffusive boundary length \nscale $L\/\\delta $; and the ratio of the propagative-diffusive boundary length \nscale to the microscopic cutoff scale, $u_\\infty = (\\delta \/a )^2$. We tabulate \nour choices for these parameters, which correspond to a range of reasonable \nphysical values, in Table \\ref{tab:parms}.\n\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=7.5cm]{fig1a.eps} (a)\n\\\\\n\\vspace{.3cm}\n\\includegraphics[width=7.5cm]{fig1b.eps} (b)\n\\caption{\\label{fig:fig1} \n(Color online) (a) Equal-time, excess same-plate force correlator \n(or force variance), $\\Delta {\\cal C}(0,0)$, as defined in \nEq.~\\eqref{eq:DeltaC}, plotted as a function of $L\/\\delta$ for fixed  $u_\\infty \n= (\\delta\/a)^2 = 1$ and \n$\\zeta\/\\eta=1, 3, 5$ and 10 (top to bottom). (b) Same as (a) but here we plot \nthe  equal-time, cross-plate force correlator ${\\cal C}(0,L)$ as defined in Eq.~\n(\\ref{eq:ffp5678}) (dashed curves) and compare it with the equal-time, excess \nsame-plate force correlator (solid curves). We plot the force correlators in \nunits of \n$(k_{\\mathrm{B}}T\/\\pi)\\cdot(\\rho_0 c_0^2\\, A)$.  \n}\n\\end{figure}\n\nWe evaluate the frequency integrals $ {\\cal W}_m(z,z')$ numerically and plot \nthe excess force variance, $\\Delta{\\cal C} (0,0)$, as a function \nof $L\/\\delta$ for $\\zeta\/\\eta=$1, 3, 5 and 10 in Figs.~\\ref{fig:fig1}(a) and \n\\ref{fig:fig1}(b) (solid lines). The cross-plate correlator, ${\\cal C} (0,L)$, \nis shown by dashed lines in Fig.~\\ref{fig:fig1}(b), where, for the sake of \ncomparison, the curves for $\\Delta{\\cal C} (0,0)$ are replotted. In the \nfigures, \nwe plot the force correlators in units of $(k_{\\mathrm{B}}T\/\\pi)\\cdot(\\rho_0 \nc_0^2\\, A)$.  \n\nFigs.~\\ref{fig:fig1}(a) and \\ref{fig:fig1}(b) show that both $\\Delta{\\cal C} \n(0,0)$ and ${\\cal C} (0,L)$ become negative at \n{\\em small separations}, $L\/\\delta\\ll 1$, and eventually diverge when \n$L\/\\delta\\rightarrow 0$. In this limit, the curves for both these correlators \noverlap and, thus, they are approximated by the same limiting form. At {\\em \nlarge separations}, $L\/\\delta\\gg 1$, the cross-plate correlator tends to zero \nwhile the excess same-plate correlator tends to a constant depending on the \nviscosity parameters.\nTherefore, the cross-plate correlator remains negative over the whole range of \nseparations, indicating that the two bounding surfaces are subjected to {\\em \ncounter-phase correlations}. The excess same-plate correlator, on the other \nhand,  can be negative (for intermediate to large values of $\\zeta\/\\eta$) or \npositive (for sufficiently small $\\zeta\/\\eta$). Thus, when the two correlators \nare compared, as in Fig.~\\ref{fig:fig1}(b), one can see that, at small to \nintermediate separations, the cross-plate correlator (dashed curves) is  \nlarger in magnitude than the excess same-plate correlator (solid curves); \nwhile, \nat large separations, it can become smaller than the latter. The difference \nbetween these two quantities decreases with increasing $\\zeta\/\\eta$. \n\n\\subsection{Analytic limits}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=7.5cm]{fig2_comp_a2.eps} (a)\n\\\\\n\\vspace{.3cm}\n\\includegraphics[width=7.5cm]{fig2_comp_b2.eps} (b)\n\\caption{\\label{fig:c0LsmallL} \n(Color online) (a) Ratio of full numerical results to analytic limiting \nbehavior, Eq.~\\eqref{eq:smallL}, for $\\Delta{\\cal C}(0,0)$ as a\nfunction of $L\/\\delta $ for $L\/\\delta \\leq 1$ and $\\zeta\/\\eta=1, 3, 5, 10$ and \n20. We fix \n$u_\\infty = (\\delta \/a )^2 = 1$. (b) Same as (a) but for ${\\cal \nC}(0,L)$. }\n\\end{figure}\n\nBeyond these numerical results, we can analytically calculate the small and \nlarge plate-separation limits and the limits of vanishing and infinite \nspeed of sound (Burger's and incompressible limits, respectively). In order to \nstudy \nthe small and large\nplate-separation cases, we first note that the \nfrequency integrals of Eqs.~\\eqref{eq:wmt} and \\eqref{eq:wmcrosst} depend on \n$L\/\\delta $ through $\\ell_+$ and $\\ell_-$, which are both linear in this ratio \n(see Eqs.~\\eqref{eq:lambdar} and \\eqref{eq:lambdai}). \n\n\nThus, in the {\\em small separation limit}, $L\/\\delta \\ll 1$, we can expand the \nfrequency integrands as Taylor series in $L\/\\delta $ for both ${\\cal W}_m \n(0,0)$ \nand ${\\cal W}_m (0,L)$ and keep terms up to linear order in $L\/\\delta $. \nThe resulting integrals are trivial, giving \n\\begin{equation}\n\\Delta{\\cal C} (0,0)\n\\,{\\simeq }\\,  {\\cal C} (0,L) \n\\,{\\simeq }\\, \n-\\frac{k_{\\mathrm{B}}T}{\\pi} \n\\frac{2 \\left(4\\eta\/3+\\zeta\\right)\\eta  A}{\\rho_0 a^2 L}.\\label{eq:smallL}\n\\end{equation}\nThese expressions agree with the full numerical results for $L\/\\delta \\ll 1$, as\nwe illustrate in\nFig.~\\ref{fig:c0LsmallL}. This figure shows the ratio of the full numerical \nresult to the analytic approximation of \nEq.~\\eqref{eq:smallL} for  $\\Delta{\\cal \nC}(0,0)$ (panel a) and ${\\cal C}(0,L)$ (panel b). \nThe plots show that the ratio in both cases tends to unity as $L\/\\delta$ \nbecomes sufficiently small, but the domain of validity of the \nanalytic approximation depends strongly on the ratio $\\zeta\/\\eta$ and \nincreases with increasing $\\zeta\/\\eta$.\n\nIn the {\\em large separation limit}, $L\/\\delta\\gg 1$, the \nforce variance reduces to the semi-infinite fluid result (see Appendix \n\\ref{app:othergeom}),\n\\begin{align}\\label{eq:c00si}\n{} & \\Delta{\\cal C}(0, 0) \\stackrel{L\/\\delta\\rightarrow \\infty}{=}\n\\frac{k_{\\mathrm{B}}T}{\\pi} \n\\frac{2\\rho_0^2c_0^3A}{\\eta\\chi}\\bigg[\\frac{\n2\\sqrt { z_\\infty-1 } } { z_\\infty}\\nonumber\\\\\n{} & \\quad +\\frac{8\\sqrt{2}}{3}\n\\frac{z_\\infty(3-z_\\infty)}{\\sqrt{z_\\infty-1}}\\sin^4\\left(\\frac{1}{2}\n\\arctan(z_\\infty^2-1)\\right)\\bigg],\n\\end{align}\nwhere $z_\\infty=\\sqrt{1+x_\\infty^2}$ and $x_\\infty = \\chi u_\\infty = \n\\chi\\eta^2\/(a^2\\rho_0^2c_0^2)$.\n\nThe corresponding equal-time, cross-plate  correlator \ntends to zero in the large\nplate-separation limit as \n\\begin{equation}\n{\\cal C}(0,L)\n\\,{\\simeq}\\,\n-\\frac{k_{\\mathrm{B}}T}{\\pi} \n\\frac{\\pi\\rho_0c_0^2  A}{L}.\\label{eq:largeL}\n\\end{equation}\nThis limiting behavior is independent of the ratio $\\zeta\/\\eta$, as is clearly \ndemonstrated by the plots of the ratio of the full numerical result to the \nanalytic approximation of \nEq.~\\eqref{eq:largeL} in Fig.~\\ref{fig:c0LlargeL}. However, the exact value \nof $L\/\\delta$ beyond which \nEq.~\\eqref{eq:largeL} is a reasonable approximation does depend on $\\zeta\/\\eta$.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=7.5cm]{fig3_comp2.eps}\n\\caption{\\label{fig:c0LlargeL} \n(Color online) Ratio of full numerical results to analytic limiting \nbehavior, Eq.~\\eqref{eq:largeL}, for\n${\\cal C}(0,L)$ as a\nfunction of $L\/\\delta $ for $L\/\\delta \\gg 1$ and $\\zeta\/\\eta=1, 3, 5, 10$ and \n20. We fix \n$u_\\infty = (\\delta \/a )^2 = 1$.}\n\\end{figure}\n\nIn the {\\em incompressible fluid limit}, we consider the \nleading contributions for $c_0\\rightarrow \\infty$, giving\n\\begin{equation}\n\\Delta{\\cal C} (0,0) \\stackrel{c_0\\rightarrow \n\\infty}{=} {\\cal C} (0,L) \\stackrel{c_0\\rightarrow \n\\infty}{=}  \n-\\frac{k_{\\mathrm{B}}T}{\\pi} \n\\frac{2 \\left(4\\eta\/3+\\zeta\\right)\\eta  A}{\\rho_0 a^2 L}.\\label{eq:c0infty}\n\\end{equation}\nThere is a correspondence between the incompressible fluid limit and the small \nplate-separation limiting result \nof Eq.~\\eqref{eq:smallL}: At small separations, the fluid behaves as if it were \nincompressible. \n\nIn the limit of vanishing adiabatic speed of sound (``Burger's limit''), \n$c_0\\rightarrow 0$, \non the other hand, \nboth $C(0,0)$ and $C(0,L)$ tend to zero as $c_0^2$.\n\n\\section{Results for time-dependent correlators}\n\\label{sec:tnumerics}\n\nWe now turn to the two-point, time-dependent correlators of the normal forces \nacting on\nthe walls, which we compute numerically using Eqs.~\\eqref{eq:final1} and \n\\eqref{eq:final2} for the same-plate and the cross-plate correlators, ${\\cal  \nC}(0,0;t,t')$ and ${\\cal  C}(0,L;t,t')$, respectively. \n\nWe plot the behavior of the excess force correlator, Eq.~\\eqref{eq:DeltaC_t}, \nas a function of the rescaled time difference, $\\tau=(t-t')\/t_0$, for\nrescaled inter-plate separations $L\/\\delta=1, 4$ and 10 in \nFig.~\\ref{fig:fig2}(a). In Fig.~\\ref{fig:fig2}(b), we show the time-dependent \nbehavior of the same quantity for $\\zeta\/\\eta=1, 3, 5$ and 10.\n\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=7.5cm]{fig4a.eps}(a)\n\\\\\n\\vspace{.3cm}\n\\includegraphics[width=7.5cm]{fig4b.eps}(b)\n\\caption{\\label{fig:fig2}\n(Color online) (a) Time-dependent, excess same-plate force correlator,\n$\\Delta {\\cal C}(0,0;t, t')$, as defined in Eq.~ \\eqref{eq:final1} plotted as \na function of the rescaled time difference $\\tau=(t-t')\/t_0$ for fixed  \n$u_\\infty = (\\delta\/a)^2 = 1$, $\\zeta\/\\eta=3$ and $L\/\\delta=1, 4$ and 10 as \nindicated on the graph. (b) Same as (a) but here we show the results for fixed \n$u_\\infty = (\\delta\/a)^2 = 1$, $L\/\\delta=4$ and $\\zeta\/\\eta=1, 3, 5$ and 10.  \n}\n\\end{figure}\n\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=7.5cm]{fig5.eps}\n\\caption{\\label{fig:c00t}\n(Color online) Time-dependent, excess same-plate force correlator, \n$\\Delta {\\cal C}(0,0;t, t')$ (solid curves), is compared with the \ntime-dependent cross-plate correlator, ${\\cal C}(0,L;t, t')$ (dashed curves), \nfor fixed  $u_\\infty = (\\delta\/a)^2 = 1$, $\\zeta\/\\eta=3$ and at two different \nrescaled inter-plate separations $L\/\\delta=4$ and 10 as indicated on the graph. \n}\n\\end{figure}\n\nAs seen in these figures, $\\Delta {\\cal C}(0,0;t, t')$ exhibits a \ndamped {\\em oscillatory} behavior in $\\tau$. For $L\/\\delta \\lesssim 3$, these \noscillations are well  described by a function of the form \n$\\alpha\\sin(u_\\infty\\tau)\/\\tau$, where \n$\\alpha$ is a function of the viscosity ratio, $\\zeta\/\\eta$, the \ndimensionless \ncutoff, $u_\\infty$, and the rescaled plate separation, $L\/\\delta$. For \n$L\/\\delta \n\\gtrsim 3$, this simple behavior breaks down, although $\\Delta {\\cal C}(0,0;t, \nt')$ remains oscillatory with an amplitude that gradually decreases for large \n$\\tau$. For the example of water at room \ntemperature, with the plate separation $L\/\\delta = 1$, $\\zeta\/\\eta = 3$ \nand cutoff $u_\\infty \n= 1$, we find $\\alpha = -8.5(3)$. \n\nThe cross-plate force correlator shows a similar \ntime-dependent behavior as the excess same-plate correlator, and the onset \nof irregular oscillations occurs for similar values of $L\/\\delta$. We\ncompare the same-plate (solid curves) and cross-plate (dashed curves) \ncorrelators in Fig.~\\ref{fig:c00t}. \n\nWe plot the difference between the excess same-plate correlator and the \ncross-plate \ncorrelator, defined as $\\delta {\\cal C} \\equiv \\Delta {\\cal C}(0,0;t, t')-{\\cal \nC}(0,L;t, t')$, in Fig.~\\ref{fig:fig4}. This plot shows that \nthe two correlators exhibit similar period of oscillations for a wide range of \nviscosities, especially at small to intermediate  inter-plate separations. In \nthe special case of equal-time correlators with $\\tau=0$, one can see a \nnon-monotonic behavior for the two correlators in Fig.~\\ref{fig:fig4}(a): At \nsmall inter-plate \nseparations, $\\delta {\\cal C}|_{\\tau=0}$ is positive and increases by \nincreasing $L\/\\delta$, but this trend changes at around $L\/\\delta\\simeq 3$, and \nthen tends to zero for large $L\/\\delta$.\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=7.5cm]{fig6a.eps}(a)\n\\\\\n\\vspace{.3cm}\n\\includegraphics[width=7.5cm]{fig6b.eps}(b)\n\\caption{\\label{fig:fig4} \n(Color online) (a) The difference between the\nexcess same-plate and the cross-plate force correlators defined as $\\delta \n{\\cal C} \\equiv \\Delta {\\cal C}(0,0;t, t')-{\\cal C}(0,L;t, t')$, is plotted as \na function of the rescaled time difference $\\tau=(t-t')\/t_0$ for fixed  \n$u_\\infty = (\\delta\/a)^2 = 1$ and (a) fixed $\\zeta\/\\eta=3$ and $L\/\\delta=1, 4$ \nand 10 and (b) fixed $L\/\\delta=4$ and $\\zeta\/\\eta=1, 3, 5$ and 10 as indicated \non the graphs. \n}\n\\end{figure}\n\n\\section{Conclusion and Discussion}\n\\label{sec:conclusions}\n\nWe have revisited the problem of long-range, fluctuation-induced \n(or  Casimir-like) hydrodynamic interactions within the context of \nLandau-Lifshitz's \nlinear, stochastic hydrodynamics in a classical, compressible, viscous fluid \nconfined between two rigid, planar walls with no-slip boundary conditions.  We \nshow conclusively that, at this level \nand within the pertinent approximations, there is {\\em no standard or primary \nCasimir effect} manifest in the average value of the interaction \nforce between the fluid boundaries. Nevertheless, we show that there does exist \na {\\em secondary Casimir \neffect} in the {\\em variance of the normal force} as well as in the \n{\\em cross-correlation function} of the normal force between the bounding \nsurfaces. Fluctuations in such effective fluctuation-induced forces have been \ninvestigated in other Casimir-like contexts \\cite{Bartolo, Dean} \nand in disordered charged systems \\cite{disorder-PRL,pre2011,epje2012}. \n\nWe derive general expressions for  the two-point, time-dependent, force \ncorrelations and, thus, show that: \n\\begin{enumerate}\n\\item The variance of the fluctuation-induced force is finite and \nindependent of the separation between the bounding surfaces for large \nseparations; \n\\item The equal-time, cross-plate force correlation exhibits a \nlong-range decay with the inverse plate separation that is independent of the \nfluid viscosities;\n\\item The time-dependent force correlations exhibit a damped oscillatory \nbehavior for \nsmall and intermediate inter-plate separations that grows more irregular at \nlarge separations.\n\\end{enumerate}\n\nOur calculation is based on the Landau-Lifshitz linear stochastic hydrodynamics \nand, therefore, does not include putative non-linear effects \\cite{Jones}. If \nsuch \neffects could be brought into the fold, they would have to be considered \nconsistently for all variables. Moreover, we find that incorporating \ncompressibility \ndoes not completely obliterate all fluctuation effects, contrary to \nprevious attempts, based on contour integration in the complex plane, that\nrequired the limiting behavior of hydrodynamics at infinite frequencies \n\\cite{Chan}. In fact, our calculation explicitly includes the scale at \nwhich the macroscopic \nhydrodynamics breaks down. The limit of vanishing compressibility is \nnon-trivial and has to be taken carefully, because it can never be \nderived from a realistic inter-particle potential with infinite stiffness \n\\cite{vankampen}.  \n\nWe interpret the non-zero hydrodynamic force correlations predicted in this work\nas a modification of  the thermal stochastic force correlations that act on a \nBrownian particle in a fluid. Since the force correlator depends on the \nseparation between the particles, the bath-mediated force fluctuations between \nthe particles would modify the particles' Langevin dynamics and  \nshould thus be eminently detectable \\cite{Sakimoto}. The separation dependence \nof \nthe normal force cross-correlation function represents an interesting case of \ncolloidal bodies which do not interact directly, but are driven by correlated \nnoise sources that can provide an alternative mechanism which can produce \nnon-trivial, ordered steady states \\cite{Maritan}.  We have considered infinite \nbounding surfaces, so our results are not strictly applicable to the case of \nfinite particles, but our calculation can be straightforwardly generalized to \ninclude a spherical geometry, which would also admit an analytic, albeit \nmuch more complicated, solution.\n\nFor experimental verification of our results, we again note that one would\nhave to generalize our calculation to the case of two spheres in a fluctuating \nhydrodynamic medium. This is different from the existing analysis of \nfluctuations of two unconnected, but hydrodynamically interacting spheres \n\\cite{Netz}, a problem in some sense dual to ours. In order to exploit this \nconnection, our first step will be to calculate the \ncross-correlation function for two spherical particles. \n\n\\begin{acknowledgments}\nThis work has been partially funded by the U.S.~Department of Energy. \nC.M.~was supported in part by the \nU.S.~National Science Foundation under Grant NSF PHY10-034278. \nA.N.~acknowledges partial support from the \nRoyal Society, the Royal Academy of Engineering, and the British Academy. \nWe acknowledge illuminating discussions with M.~Kardar in the KITP program on \n{\\em The Theory and Practice of Fluctuation-Induced Interactions} (2008). \nR.P.~would like to thank \nJoel Cohen for his careful reading of the manuscript and for his comments.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\t\n\t%\n\tSingle image super-resolution (SR) is an algorithm to reconstruct a high-resolution (HR) image from a single low-resolution (LR) image \\cite{park2003super}.\n\tIt allows a system to overcome limitations of LR imaging sensors or from image processing steps in multimedia systems.\n\tSeveral SR algorithms \\cite{martin1995high,wang2006improved,thornton2006sub,zhang2010super,zou2012very} have been proposed and applied in the fields of computer vision, image processing, surveillance systems, etc.\n\tHowever, SR is still challenging due to its ill-posedness, which means that multiple HR images are solutions for a single LR image.\n\tFurthermore, the reconstructed HR image should be close to the real one and, at the same time, visually pleasant.\n\t\n\t%\n\tIn recent years, various deep learning-based SR algorithms have been proposed in literature.\n\tConvolutional neural network architectures are adopted in many deep learning-based SR methods following the super-resolution convolutional neural network (SRCNN) \\cite{dong2014learning}, which showed better performance than the classical SR methods.\n\tThey typically consist of two parts, feature extraction part and upscaling part.\n\tWith improving these parts in various ways, recent deep learning-based SR algorithms have achieved significant enhancement in terms of distortion-based quality such as root mean squared error (RMSE) or peak signal-to-noise ratio (PSNR) \\cite{kim2018deep,ledig2017photo,haris2018deep,lai2017deep,kim2016accurate,lim2017enhanced}.\n\t\n\t%\n\tHowever, it has been recently shown that there exists the trade-off relationship between distortion and perception for image restoration problems including SR \\cite{blau2017perception}.\n\tIn other words, as the mean distortion decreases, the probability for correctly discriminating the output image from the real one increases.\n\tGenerative adversarial networks (GANs) are a way to approach the perception-distortion bound.\n\tThis is achieved by controlling relative contributions of the two types of losses popularly employed in the GAN-based SR methods, which are a content loss and an adversarial loss \\cite{ledig2017photo}.\n\n\tFor the content loss, a reconstruction loss such as the L1 or L2 loss is used.\n\tHowever, optimizing to the content loss usually leads to unnatural blurry reconstruction, which can improve the distortion-based performance, but decreases the perceptual quality.\n\tOn the other hand, focusing on the adversarial loss leads to perceptually better reconstruction, which tends to decrease the distortion-based quality.\n\t\n\t%\n\tOne of the keys to improve both the distortion and perception is to consider perceptual part in the content loss.\n\tIn this matter, consideration of proper high frequency components would be helpful, because many perceptual quality metrics consider the frequency domain to measure the perceptual quality \\cite{ma2017learning,mittal2013making}.\n\tNot only traditional SR algorithms such as \\cite{yang2010image,kim2017blind} but also deep learning-based methods \\cite{lai2017fast,gharbi2017deep} focus on restoration of high frequency components.\n\tHowever, there exists little attempt to consider the frequency domain to compare the real and fake (i.e., super-resolved) images in GAN-based SR.\n\t\n\t%\n\tIn this study, we propose a novel GAN model for SR considering the trade-off relationship between perception and distortion.\n\tBased on good distortion-based performance of our base model, i.e., the deep residual network using enhanced upscale modules (EUSR) \\cite{kim2018deep}, the proposed GAN model is trained to improve both the perception and distortion.\n\tTogether with the conventional content loss for deep networks, we consider additional loss functions, namely, the discrete cosine transform (DCT) loss and differential content loss.\n\tThese loss functions directly consider the high frequency parts of the super-resolved images, which are related to the perception of image quality by the human visual system.\n\tThe proposed model was ranked in the 2nd place among 13 participants in \\textit{Region 1} of the PIRM Challenge \\cite{pirmpaper} on perceptual super-resolution at ECCV 2018.\n\t\n\t%\n\tThe rest of the paper is organized as follows.\n\tWe first describe the base model of the proposed method and the proposed loss functions in Section \\ref{sec2}.\n\tThen, in Section \\ref{sec3}, we explain the experiments conducted for this study.\n\tThe results and analysis are given in Section \\ref{sec4}.\n\tFinally, we conclude the study in Section \\ref{sec5}.\n\t\n\t\n\n\n\t\\section{Proposed Method}\n\t\\label{sec2}\n\t\n\t\\subsection{Super-resolution with enhanced upscaling modules}\n\t\n\t\\begin{figure}[h]\n\t\t\\includegraphics[width=\\linewidth]{fig\/eusr_model.pdf}\n\t\t\\caption{Overall structure of the EUSR model \\cite{kim2018deep}.}\n\t\t\\label{fig:eusr}\n\t\\end{figure}\n\t\n\tAs the generator in the proposed model, we employ the recently developed EUSR model \\cite{kim2018deep}.\n\tIts overall structure is shown in \\figurename~\\ref{fig:eusr}.\n\tIt is a multi-scale approach performing reconstruction in three different scales ($\\times2$, $\\times4$, and $\\times8$) simultaneously.\n\tLow-level features for each scale are extracted from the input LR image by two residual blocks (RBs).\n\tAnd, higher-level features are extracted by the residual module (RM), which consists of several local RBs, one convolution layer, and global skip connection.\n\tThen, for each scale, the extracted features are upscaled by enhanced upscaling modules (EUMs).\n\tThis model showed good performance for some benchmark datasets in the NTIRE 2018 Challenge \\cite{Timofte_2018_CVPR_Workshops} in terms of PSNR and structural similarity (SSIM) \\cite{wang2004image}.\n\tWe set the number of RBs in each RM to 80, which is larger than that used in \\cite{kim2018deep} (i.e., 48) in order to enhance the learning capability of the network.\n\t\n\t%\n\tThe discriminator network in the proposed method is based on that of the super-resolution using a generative adversarial network (SRGAN) model \\cite{ledig2017photo}.\n\tThe network consists of 10 convolutional layers followed by leaky ReLU activations and batch normalization units.\n\tThe resulting feature maps are processed by two dense layers and a final sigmoid activation function in order to determine the probability whether the input image is real (HR) or fake (super-resolved).\n\t\n\t\n\t\n\t\\subsection{Loss functions}\n\t\n\tIn addition to the conventional loss functions for GAN models for SR, i.e., content loss (${ l }_{ c }$) and adversarial loss (${ l }_{ D }$), we consider two more content-related losses to train the proposed model.\n\tThey are the DCT loss (${ l }_{ dct }$) and differential content loss (${ l }_{ d }$), which are named as perceptual content losses (PCL) in this study.\n\tTherefore, we use four loss functions in total in order to improve both the perceptual quality and distortion-based quality.\n\tThe details of the loss functions are described below.\n\t\n\t\n\t\n\t\\begin{itemize}\n\t\t\\item \\textbf{Content loss (${ l }_{ c }$)} : The content loss is a pixel-based reconstruction loss function. The L1-norm and L2-norm are generally used for SR. We employ the L1-norm between the HR image and SR image: \n\t\t\\begin{equation}\\label{eq:contentloss}\n\t\t{ l }_{ c }=\\frac { 1 }{ WH } \\sum _{ w }^{  }{ \\sum _{ h }^{  }{ \\left| { I }_{ w,h }^{ HR }-{ I }_{ w,h }^{ SR } \\right|  }  } , \n\t\t\\end{equation}\n\t\twhere $W$ and $H$ are the width and height of the image, respectively. And, ${ I }_{ w,h }^{ HR }$  and ${ I }_{ w,h }^{ SR }$ are the pixel values of the HR and SR images, respectively, where $w$ and $h$ are the horizontal and vertical pixel indexes, respectively.\\\\\n\t\t\n\t\t\\item \\textbf{Differential content loss (${ l }_{ d }$)} : The differential content loss evaluates the difference between the SR and HR images in a deeper level. It can help to reduce the over-smoothness and improve the performance of reconstruction particularly for high frequency components. We also employ the L1-norm for the differential content loss: \n\t\t\\begin{equation}\\label{eq:differentialloss}\n\t\t{ l }_{ d }=\\frac { 1 }{ WH } \\left( \\sum _{ w }^{  }{ \\left| { { d }_{ x }I }_{ w }^{ HR }-{ d }_{ x }{ I }_{ w }^{ SR } \\right|  } +\\sum _{ h }^{  }{ \\left| { { d }_{ y }I }_{ h }^{ HR }-{ d }_{ y }{ I }_{ h }^{ SR } \\right|  }  \\right) ,\n\t\t\\end{equation}\n\t\twhere ${d}_{x}$ and ${d}_{y}$ are horizontal and vertical differential operators, respectively.\\\\\n\t\t\n\t\t\\item \\textbf{DCT loss (${ l }_{ dct }$)} : The DCT loss evaluates the difference between DCT coefficients of the HR and SR images. This enables to explicitly compare the two images in the frequency domain for performance improvement. In other words, while different SR images can have the same value of ${l}_{c}$, the DCT loss forces the model to generate the one having a frequency distribution as similar to the HR image as possible. The L2-norm is employed for the DCT loss function:\n\t\t\\begin{equation}\\label{eq:dctloss}\n\t\t{ l }_{ dct }=\\frac { 1 }{ WH } \\sum _{ w }^{  }{ \\sum _{ h }^{  }{ { \\left\\| { DCT }({ I }^{ HR })_{ w,h }-{ DCT }({ I }^{ SR })_{ w,h } \\right\\|  }^{ 2 } }  } ,\n\t\t\\end{equation}\n\t\twhere $DCT(I)$ means the DCT coefficients of image $I$.\\\\\n\t\t\n\t\t\\item \\textbf{Adversarial loss (${ l }_{ D }$)} : The adversarial loss is used to enhance the perceptual quality. It is calculated as \n\t\t\\begin{equation}\\label{eq:advloss}\n\t\t{ l }_{ D }=-\\log { (D({ I }^{ SR } } |{ I }^{ HR }))\n\t\t\\end{equation}\n\t\twhere $D$ is the probability of the discriminator calculated by a sigmoid cross-entropy of logits from the discriminator \\cite{ledig2017photo}, which represents the probability that the input image is a real image.\n\t\t\n\t\\end{itemize}\n\t\n\t\n\t\n\t\n\t\n\n\n\t\\section{Experiments}\n\t\\label{sec3}\n\t\n\t\\subsection{Datasets}\n\tWe use the DIV2K dataset \\cite{div2kdb} for training of the proposed model in this experiment, which consists of 1000 2K resolution RGB images.\n\tLR training images are obtained by downscaling the original images using bicubic interpolation.\n\tFor testing, we evaluate the performance of the SR models on several datasets, i.e., Set5 \\cite{bevilacqua2012low}, Set14 \\cite{zeyde2010single}, BSD100 \\cite{martin2001database}, and PIRM self-validation set \\cite{pirmpaper}.\n\tSet5 and Set14 consist of 5 and 14 images, respectively.\n\tAnd, BSD100 and PIRM self-validation set include 100 challenging images.\n\tAll testing experiments are performed with a scale factor of $\\times4$, which is the target scale of the PIRM Challenge on perceptual super-resolution.\n\t\n\t\n\t\\subsection{Implementation details}\n\t%\n\tFor the EUSR-based generator in the proposed model, we employ 80 and two local RBs in each RM and the upscaling part, respectively.\n\tWe first pre-train the EUSR model as a baseline on the training set of the DIV2K dataset \\cite{div2kdb}.\n\tIn the pre-training phase, we use only the content loss (${l}_{c}$) as the loss function.\n\t\n\t%\n\tFor each training step, we feed two randomly cropped image patches having a size of 48$\\times$48 from LR images into the networks.\n\tThe patches are transformed by random rotation by three angles (90$^{\\circ}$, 180$^{\\circ}$, and 270$^{\\circ}$) or horizontal flips.\n\tThe Adam optimization method \\cite{kingma2015adam} with $\\beta1 = 0.9$, $\\beta2 = 0.999$, and $\\epsilon = {10}^{-8}$ is used for both pre-training and training phases.\n\tThe initial learning rate is set to ${10}^{-5}$ and the learning rate is reduced by a half for every ${2\\times10}^{5}$ steps.\n\tA total of 500,000 training steps are executed.\n\tThe networks are implemented using the Tensorflow framework.\n\tIt roughly takes two days with NVIDIA GeForce GTX 1080 GPU to train the networks.\n\t\n\t\n\t\\subsection{Performance measures}\n\t\n\tAs proposed in \\cite{blau2017perception}, we measure the performance of the SR methods using distortion-based quality and perception-based quality.\n\tFirst, we measure the distortion-based quality of the SR images using RMSE, PSNR, and SSIM \\cite{wang2004image}, which are calculated by comparing the SR and HR images.\n\tIn addition, we measure the perceptual quality of the SR image by \\cite{blau2017perception}\n\t\\begin{equation}\\label{eq:perceptualindex}\n\tPerceptual\\ index ({ I }_{ SR }) = \\frac { 1 }{ 2 } \\left( \\left( 10-Ma({ I }_{ SR }) \\right) +NIQE({ I }_{ SR } \\right)).\n\t\\end{equation}\n\twhere ${I}_{SR}$ is a SR image, $Ma(\\cdot)$ means the quality score measure proposed in \\cite{ma2017learning}, and $NIQE(\\cdot)$ means the quality score by the natural image quality evaluator (NIQE) metric \\cite{mittal2013making}.\n\tThis perceptual index is also adopted to measure the performance of the SR methods in the  PIRM Challenge on perceptual super-resolution \\cite{pirmpaper}.\n\tThe lower the perceptual index is, the better the perceptual quality is.\n\tWe compute all metrics after discarding the 4-pixel border and on the Y-channel of YCbCr channels converted from RGB channels as in \\cite{ledig2017photo}.\n\t\n\t\n\t\\section{Results}\n\t\\label{sec4}\n\t\n\t\\begin{table}[h]\n\t\t\\centering\n\t\t\\caption{\\label{tb:result1} Performance of the SR methods in terms of the distortion (i.e., RMSE, PSNR, and SSIM) and perception (i.e., perceptual index) for Set5 \\cite{bevilacqua2012low}, Set14 \\cite{zeyde2010single}, and BSD100 \\cite{martin2001database}. The methods are sorted in an ascending order in terms of the perceptual index.}\n\t\t{\n\t\t\t\\begin{tabularx}{0.8\\columnwidth}{@{\\extracolsep{\\fill}}lcccc}\n\t\t\t\t\\textbf{Set5} &RMSE&PSNR&SSIM&Perceptual Index\\\\ \\hline\t\t\t\n\t\t\t\tSRGAN\t&\t9.1402 \t&\t29.5687 \t&\t0.8358 \t&\t3.4199 \t\\\\\n\t\t\t\tHR\t&\t\\textbf{---} \t&\t\\textbf{---}  &\t\\textbf{---} \t&\t3.6237 \t\\\\\n\t\t\t\tEUSR-PCL\t&\t7.1542 \t&\t31.5679 \t&\t0.8743 \t&\t4.5686 \t\\\\\t\t\t\n\t\t\t\tSRResNet\t&\t8.0195 \t&\t30.5012 \t&\t0.8689 \t&\t5.2848 \t\\\\\n\t\t\t\tEUSR\t&\t6.4439 \t&\t32.5213 \t&\t0.8972 \t&\t5.9667 \t\\\\\n\t\t\t\tMS-LapSRN\t&\t7.1376 \t&\t31.7181 \t&\t0.8878 \t&\t6.0969 \t\\\\\n\t\t\t\tD-DBPN\t&\t6.5736 \t&\t32.3974 \t&\t0.8960 \t&\t6.1735 \t\\\\\n\t\t\t\tBicubic\t&\t11.8227 \t&\t28.4178 \t&\t0.8097 \t&\t7.3851 \t\\\\\\\\\n\t\t\t\\end{tabularx}\n\t\t\t\n\t\t\t\\begin{tabularx}{0.8\\columnwidth}{@{\\extracolsep{\\fill}}lcccc}\n\t\t\t\t\\textbf{Set14} &RMSE&PSNR&SSIM&Perceptual Index\\\\ \\hline\t\t\t\n\t\t\t\tSRGAN\t&\t14.5572 \t&\t26.1138 \t&\t0.6957 \t&\t2.8816 \t\\\\\n\t\t\t\tHR\t&\t\\textbf{---} \t&\t\\textbf{---} \t&\t\\textbf{---} \t&\t3.4825 \t\\\\\n\t\t\t\tEUSR-PCL\t&\t11.5799 \t&\t28.2363 \t&\t0.7567 \t&\t3.5524 \t\\\\\n\t\t\t\tSRResNet\t&\t12.6528 \t&\t27.2718 \t&\t0.7419 \t&\t4.9652 \t\\\\\n\t\t\t\tEUSR\t&\t10.9577 \t&\t28.8080 \t&\t0.7875 \t&\t5.3028 \t\\\\\n\t\t\t\tMS-LapSRN\t&\t10.9974 \t&\t28.7636 \t&\t0.7863 \t&\t5.5108 \t\\\\\n\t\t\t\tD-DBPN\t&\t11.6467 \t&\t28.2595 \t&\t0.7756 \t&\t5.7191 \t\\\\\n\t\t\t\tBicubic\t&\t14.1889 \t&\t26.0906 \t&\t0.7050 \t&\t7.0514 \t\\\\\\\\\n\t\t\t\\end{tabularx}\t\n\t\t\t\n\t\t\t\\begin{tabularx}{0.8\\columnwidth}{@{\\extracolsep{\\fill}}lcccc}\n\t\t\t\t\\textbf{BSD100} &RMSE&PSNR&SSIM&Perceptual Index\\\\ \\hline\t\t\t\n\t\t\t\tHR\t&\t\t\\textbf{---} \t&\t\t\\textbf{---} \t&\t\t\\textbf{---} \t&\t2.2974 \t\\\\\n\t\t\t\tSRGAN\t&\t16.3332 \t&\t25.1762 \t&\t0.6408 \t&\t2.3513 \t\\\\\n\t\t\t\tEUSR-PCL\t&\t13.0691 \t&\t27.1131 \t&\t0.7043 \t&\t3.2417 \t\\\\\n\t\t\t\tSRResNet\t&\t14.1260 \t&\t26.3218 \t&\t0.6940 \t&\t5.1833 \t\\\\\n\t\t\t\tEUSR\t&\t12.3966 \t&\t27.7129 \t&\t0.7418 \t&\t5.2552 \t\\\\\n\t\t\t\tD-DBPN\t&\t12.4434 \t&\t27.6711 \t&\t0.7397 \t&\t5.4331 \t\\\\\n\t\t\t\tMS-LapSRN\t&\t12.7599 \t&\t27.4153 \t&\t0.7306 \t&\t5.6138 \t\\\\\n\t\t\t\tBicubic\t&\t14.5413 \t&\t25.9566 \t&\t0.6693 \t&\t6.9948 \t\\\\\\\\\n\t\t\t\t\n\t\t\t\\end{tabularx}\t\n\t\t\t\n\t\t}\n\t\\end{table}\n\t\n\t\n\t%\n\tWe evaluate the performance of the proposed method and the state-of-the-art SR algorithms, i.e., the generative adversarial network for image super-resolution (SRGAN) \\cite{ledig2017photo}, the SRResNet (SRGAN model without the adversarial loss) \\cite{ledig2017photo}, the dense deep back-projection networks (D-DBPN) \\cite{haris2018deep}, and the multi-scale deep Laplacian pyramid super-resolution network (MS-LapSRN) \\cite{lai2017deep}.\n\tAnd, the bicubic upscaling method and pre-trained EUSR model are also included.\n\tOur proposed model, named as deep residual network using enhanced upscale modules with perceptual content losses (EUSR-PCL), and SRGAN are adversarial networks, and the others are non-adversarial models.\n\tNote that, the SRResNet and SRGAN have variants that are optimized in terms of MSE or in the feature space of a VGG net \\cite{simonyan2014very}.\n\tWe consider SRResNet-VGG$_{2,2}$ and SRGAN-VGG$_{5,4}$ in this study, which show better perceptual quality among their variants.\n\tFor the Set5, Set14, and BSD100 datasets, the SR images of the SR methods are either obtained from their supplementary materials  (SRGAN\\footnote{\\label{srganweb}\\url{https:\/\/twitter.app.box.com\/s\/lcue6vlrd01ljkdtdkhmfvk7vtjhetog}}, SRResNet\\textsuperscript{\\ref{srganweb}}, and MS-LapSRN\\footnote{\\label{lapsrnweb}\\url{http:\/\/vllab.ucmerced.edu\/wlai24\/LapSRN\/}}) or reproduced from their pre-trained model (D-DBPN\\footnote{\\label{dbpnweb}\\url{https:\/\/drive.google.com\/drive\/folders\/1ahbeoEHkjxoo4NV1wReOmpoRWbl448z-?usp=sharing}}).\n\tFor the PIRM set, the SR images of D-DBPN and EUSR are generated using their own pre-trained models.\n\t\n\t\n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\begin{tabular}{cccc}\n\t\t\tHR&Bicubic&MS-LapSRN&D-DBPN\\\\\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/Set5\/butterfly_HR_all.png}&\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/Set5\/butterfly_bicubic_all.png}&\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/Set5\/butterfly_MSLapSRN_all.png}&\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/Set5\/butterfly_DDBPN_all.png}\\\\\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/Set5\/butterfly_HR_crop.png}&\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/Set5\/butterfly_bicubic_crop.png}&\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/Set5\/butterfly_MSLapSRN_crop.png}&\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/Set5\/butterfly_DDBPN_crop.png}\\\\\n\t\t\tSRResNet&SRGAN&EUSR&EUSR-PCL\\\\\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/Set5\/butterfly_SRResNet-VGG22_all.png}&\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/Set5\/butterfly_SRGAN-VGG54_all.png}&\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/Set5\/butterfly_EUSR_all.png}&\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/Set5\/butterfly_EUSR-PCL_all.png}\\\\\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/Set5\/butterfly_SRResNet-VGG22_crop.png}&\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/Set5\/butterfly_SRGAN-VGG54_crop.png}&\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/Set5\/butterfly_EUSR_crop.png}&\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/Set5\/butterfly_EUSR-PCL_crop.png}\\\\\n\t\t\\end{tabular}\n\t\t\\caption{Examples of the HR image and SR images of the seven methods for \\textit{butterfly} from the Set5 dataset\\cite{bevilacqua2012low}.}\n\t\t\\label{fig:result:set5}\n\t\\end{figure}\n\t\n\t\n\t\\begin{figure}[h!]\n\t\t\\centering\n\t\t\\begin{tabular}{cccc}\n\t\t\tHR&Bicubic&MS-LapSRN&D-DBPN\\\\\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/BSD100\/86000_HR_all.png}&\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/BSD100\/86000_bicubic_all.png}&\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/BSD100\/86000_MSLapSRN_all.png}&\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/BSD100\/86000_DDBPN_all.png}\\\\\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/BSD100\/86000_HR_crop.png}&\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/BSD100\/86000_bicubic_crop.png}&\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/BSD100\/86000_MSLapSRN_crop.png}&\t\t\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/BSD100\/86000_DDBPN_crop.png}\\\\\\\\\n\t\t\tSRResNet&SRGAN&EUSR&EUSR-PCL\\\\\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/BSD100\/86000_SRResNet-VGG22_all.png}&\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/BSD100\/86000_SRGAN-VGG54_all.png}&\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/BSD100\/86000_EUSR_all.png}&\t\t\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/BSD100\/86000_EUSR-PCL_all.png}\\\\\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/BSD100\/86000_SRResNet-VGG22_crop.png}&\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/BSD100\/86000_SRGAN-VGG54_crop.png}&\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/BSD100\/86000_EUSR_crop.png}&\t\t\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/BSD100\/86000_EUSR-PCL_crop.png}\\\\\n\t\t\\end{tabular}\n\t\t\\caption{Examples of the HR image and SR images of the seven methods for \\textit{86000} from the BSD100 dataset \\cite{martin2001database}.}\n\t\t\\label{fig:result:bsd100}\n\t\\end{figure}\n\t\n\t\n\t%\n\tTable \\ref{tb:result1} shows the performance of the considered SR methods for the Set5, Set14, and BSD100 datasets.\n\tOur proposed model is ranked second among the SR methods in terms of the perceptual quality.\n\tThe perceptual index of the proposed method is between those of SRGAN and SRResNet, which are an adversarial network and the best model among non-adversarial models, respectively.\n\tConsidering the PSNR and SSIM results, EUSR-PCL shows better performance than both SRGAN and SRResNet.\n\tWhen we compare our model with other non-adversarial networks, i.e., EUSR, MS-LapSRN, and D-DBPN, our model shows slightly lower PSNR results, while the perceptual quality is significantly improved.\n\tThese results show that our model achieves proper balance between the distortion and perception aspects.\n\t\n\t\n\t\n\t%\n\tFigs. \\ref{fig:result:set5} and \\ref{fig:result:bsd100} show example images produced by the SR methods for qualitative evaluation.\n\tIn \\figurename~\\ref{fig:result:set5}, except the bicubic interpolation method, most of the methods restore high frequency details in the HR image to some extents.\n\tIf the details of the SR images are examined, however, the models show different qualitative results.\n\tThe SR images of the bottom row (i.e., SRResNet, SRGAN, EUSR, and EUSR-PCL) show relatively better perceptual quality with less blurring.\n\tHowever, the reconstructed details are different depending on the methods.\n\tThe images by SRGAN contain noise, although the method shows the best perceptual quality for the Set5 dataset in Table \\ref{tb:result1}.\n\tOur model shows lower performance than SRGAN in terms of perception, but the noise is less visible.\n\tIn \\figurename~\\ref{fig:result:bsd100}, it is also found that the details of the SR image of EUSR-PCL are perceptually better than those of SRGAN, although SRGAN shows better perceptual quality than EUSR-PCL for the BSD100 dataset in Table \\ref{tb:result1}.\n\tThese results imply that a proper balance between perception and distortion is important and our proposed model performs well for that.\n\t\n\t\n\t%\n\tThe results for the PIRM dataset \\cite{pirmpaper} are summarized in Table \\ref{tb:result2}.\n\tIn this case, we also consider variants of the EUSR-PCL model in order to examine the contributions of the losses.\n\tIn the table, EUSR-PCL indicates the proposed model that considers all loss functions described in Section \\ref{sec2}.\n\tThe EUSR-PCL (${l}_{c}$) is the basic GAN model based on EUSR.\n\tEUSR-PCL (${l}_{c}+{l}_{dct}$) is the EUSR-PCL model considering the content loss and DCT loss, and EUSR-PCL (${l}_{c}+{l}_{d}$) is the model with the content loss and differential content loss.\n\tIn all cases, the adversarial loss is included.\n\tIt is observed that the performance of EUSR-PCL is the best in terms of perception among all methods in the table.\n\tAlthough the PSNR values of the EUSR-PCL variants are slightly lower than EUSR and D-DBPN, their perceptual quality scores are better.\n\tComparing EUSR-PCL and its variants, we can find the effectiveness of the perceptual content losses.\n\tWhen the two perceptual content losses are included, we can obtain the best performance in terms of both the perception and distortion.\n\t\n\t%\n\t\\figurename~\\ref{fig:result:pirm} shows example SR images for the PIRM dataset.\n\tThe images obtained by the EUSR-PCL models at the bottom row have better perceptual quality and are less blurry than those of the other methods.\n\tAs mentioned above, these models show lower PSNR values, but the reconstructed images are better in terms of perception.\n\tWhen we compare the results of the variants of EUSR-PCL, there exist slight differences in the result images, in particular in the details.\n\tFor instance, EUSR-PCL (${l}_{c}+{l}_{dct}$) generates a more noisy SR image than EUSR-PCL.\n\tAlthough the differences between their quality scores are not large in Table \\ref{tb:result2}, the result images show noticeable perceptual differences.\n\tThis demonstrates that the improvement of the perceptual quality of SR is important, and the proposed method achieves good performance for perceptual SR.\n\t\n\t\\begin{table}[t]\n\t\t\\centering\n\t\t\\caption{\\label{tb:result2} Performance of the SR methods in terms of the distortion (i.e., RMSE, PSNR, and SSIM) and perception (i.e., perceptual index) for PIRM \\cite{pirmpaper}. The methods are sorted in an ascending order in terms of the perceptual index.}\n\t\t{\n\t\t\t\\begin{tabularx}{0.8\\columnwidth}{@{\\extracolsep{\\fill}}lcccc}\n\t\t\t\t\\textbf{PIRM}&RMSE&PSNR&SSIM&Perceptual Index\\\\ \\hline\t\t\n\t\t\t\tHR\t&\t\\textbf{---} \t&\t\\textbf{---} \t&\t\\textbf{---} \t&\t2.2818 \\\\\n\t\t\t\tEUSR-PCL\t&\t11.5847 \t&\t27.9049 \t&\t0.7459 \t&\t2.8180 \t\\\\\n\t\t\t\tEUSR-PCL (${l}_{c}+{l}_{dct}$)\t&\t11.6559 \t&\t27.8668 \t&\t0.7456 \t&\t2.8364 \t\\\\\n\t\t\t\tEUSR-PCL (${l}_{c}$)\t&\t12.0131 \t&\t27.7260 \t&\t0.7472 \t&\t2.8665 \t\\\\\n\t\t\t\tEUSR-PCL (${l}_{c}+{l}_{d}$)\t&\t11.8854 \t&\t27.7629 \t&\t0.7442 \t&\t2.8824 \t\\\\\t\t\t\n\t\t\t\tEUSR\t&\t10.8990 \t&\t28.5736 \t&\t0.7812 \t&\t4.9840 \t\\\\\n\t\t\t\tD-DBPN\t&\t10.9339 \t&\t28.5401 \t&\t0.7794 \t&\t5.1423 \t\\\\\n\t\t\t\tBicubic\t&\t13.2923 \t&\t26.5006 \t&\t0.6980 \t&\t6.8050 \t\\\\\\\\\n\t\t\t\\end{tabularx}\n\t\t}\n\t\\end{table}\n\t\n\t\n\t\n\t\\begin{figure}[t]\n\t\t\\small\n\t\t\\centering\n\t\t\\begin{tabular}{cccc}\n\t\t\tHR&Bicubic&D-DBPN&EUSR\\\\\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/PIRM\/68_HR_all.png}&\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/PIRM\/68_bicubic_all.png}&\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/PIRM\/68_DDBPN_all.png}&\t\t\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/PIRM\/68_EUSR_all.png}\\\\\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/PIRM\/68_HR_crop.png}&\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/PIRM\/68_bicubic_crop.png}&\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/PIRM\/68_DDBPN_crop.png}&\t\t\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/PIRM\/68_EUSR_crop.png}\\\\\\\\\n\t\t\t\n\t\t\t\\shortstack{EUSR-PCL\\\\(${ l }_{ c }$)}&\\shortstack{EUSR-PCL\\\\(${ l }_{ c } + { l }_{ d }$)}&\\shortstack{EUSR-PCL\\\\(${ l }_{ c } + { l }_{ dct }$)}&\\shortstack{EUSR-PCL\\\\~ }\\\\\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/PIRM\/68_EUSR-PCL-GAN_all.png}&\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/PIRM\/68_EUSR-PCL-DMSE_all.png}&\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/PIRM\/68_EUSR-PCL-DCT_all.png}&\t\t\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/PIRM\/68_EUSR-PCL_all.png}\\\\\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/PIRM\/68_EUSR-PCL-GAN_crop.png}&\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/PIRM\/68_EUSR-PCL-DMSE_crop.png}&\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/PIRM\/68_EUSR-PCL-DCT_crop.png}&\t\t\n\t\t\t\\includegraphics[width=0.22\\linewidth]{fig\/pirm_examples\/PIRM\/68_EUSR-PCL_crop.png}\\\\\n\t\t\\end{tabular}\n\t\t\\caption{Examples of the HR image and SR images of the seven methods for \\textit{6} from the PIRM self-validation set \\cite{pirmpaper}.}\n\t\t\\label{fig:result:pirm}\n\t\\end{figure}\n\t\n\t\n\t\n\n\n\t\\section{Conclusion}\n\t\\label{sec5}\n\tIn this study, we focused on developing the perceptual content losses and proposed the GAN model in order to properly consider the trade-off problem between perception and distortion.\n\tWe proposed two perceptual content loss functions, i.e., the DCT loss and the differential content loss, used to train the EUSR-based GAN model.\n\tThe results showed that the proposed method is effective in SR applications with consideration of both the perception and distortion aspects.\n\t\n\t\n\t\n\n\n\t\\section*{Acknowledgment}\n\tThis research was supported by the MSIT (Ministry of Science and ICT), Korea, under the ``ICT Consilience Creative Program'' (IITP-2018-2017-0-01015) supervised by the IITP (Institute for Information \\& communications Technology Promotion) and also supported by the IITP grant funded by the Korea government (MSIT) (R7124-16-0004, Development of Intelligent Interaction Technology Based on Context Awareness and Human Intention Understanding).\n\t\n\t\n\t\n\t%\n\n\t%\n\n\n\t%\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nThe recent discoveries of the massive neutron stars PSR J$0348+0432$~\\cite{science} and PSR J$1614-2230$~\\cite{Nature} have \nbrought new challenges for theories of dense matter\nbeyond the nuclear saturation density. Recently the radio timing measurements of the pulsar PSR J$0348+0432$ and its white dwarf \ncompanion have confirmed the mass of the pulsar to be in the\nrange of $1.97 - 2.05$ M$_{\\odot}$ at $68.27\\%$ or $1.90 - 2.18$ M$_{\\odot}$ at $99.73\\%$ confidence~\\cite{science}. This \nis only the second neutron star(NS) with a precisely determined mass around 2M$_{\\odot}$, after PSR J$1614-2230$\nand has a 3$\\sigma$ lower mass limit $0.05$ M$_{\\odot}$ higher than the latter. It therefore\n provides the tightest reliable lower bound on the maximum mass of neutron\nstars.\n\nCompact stars provide the perfect astrophysical environment for testing theories of cold \nand dense matter. Densities at the core of neutron stars can reach values of several times of\n$10^{15} gm \\,\\, cm^{-3}$. At such high densities, the energies of the particles are high enough to favour the appearance of exotic particles\nin the core. Since the lifetime of neutron stars are much\ngreater than those associated with the weak interaction, strangeness\nconservation can be violated in the core due to the weak interaction. This would result in the appearance\nof strange particles such as hyperons. The appearance of such particles produces new degrees of freedom, which results in a\nsofter equation of state (EoS) in the neutron star interior. \n\nThe observable properties of compact stars depend crucially on the EoS.\nAccording to the existing models of dense matter the presence of strangeness in the neutron star interior\nleads to a considerable softening of the EoS, resulting in a reduction of\nthe maximum mass of the neutron star~\\cite{2,3,4,5}.\nTherefore many existing theories involving hyperons cannot explain\nthe large pulsar masses~\\cite{6}. Most relativistic models obtain maximum neutron\nstar masses in the range $1.4-1.8M_{\\odot}$~\\cite{7,8,9,10,11,12,13,14}, when hyperons are included.\nSome authors have tackled this problem by including a strong vector repulsion in the strange sector or by pushing \n the threshold for the appearance of \nhyperons to higher densities~\\cite{14,15,16,17,18,19,20,21}.\n\nIn\nseveral studies the maximum neutron star masses were generally\nfound to be lower than $1.6M_\\odot$~\\cite{3,4,5,22,23,24,25,26} which is in contradiction with observed pulsar masses. \nHowever, neutron stars with maximum mass\nlarger than $2M_{\\odot}$ have been obtained theoretically. \n Bednarek {\\em et al. }~\\cite{27} achieved a stiffening \nof the EoS by using a non-linear relativistic mean\nfield (RMF) model with quartic terms involving the strange vector meson. \n Lastowiecki {\\em et al. }~\\cite{28} obtained massive stars\n including a quark matter core. Taurines {\\em et al. }~\\cite{29} achieved large neutron star masses including\nhyperons by considering a model with density dependent coupling constants. The coupling constants were varied nonlinearly with the\nscalar field. Bonanno and Sedrakian~\\cite{30} also modeled massive neutron stars including hyperons and quark\ncore using a fairly stiff EoS and vector repulsion among quarks.\nAuthors in ref.~\\cite{Gupta} incorporated higher order couplings in the RMF theory in addition to kaonic interactions \nto obtain the maximum neutron star mass. Agrawal {\\em et al. }\\cite{Agrawal1} have optimized the parameters of the extended RMF model using a selected set of\nglobal observables which includes binding energies and charge radii for nuclei along several isotopic\nand isotonic chains and the iso-scalar giant monopole resonance energies for the $^{90}$Zr and $^{208}$Pb\nnuclei. \nWeissenborn {\\em et al. }~\\cite{30a} investigated the vector meson-hyperon coupling, \ngoing from SU(6) quark model to a broader SU(3), and concluded that the maximum mass of a neutron star decreases \nlinearly with the strangeness content of the\nneutron star core independent of the nuclear EoS.\nOn the other hand, H. Dapo {\\em et al. }~\\cite{5} found that for several different bare\nhyperon-nucleon potentials and a wide range of nuclear matter parameters the hyperons in neutron\nstars are always present. \n\n\n The parameters of the RMF model are fitted to the saturation properties of the infinite nuclear matter and\/or the properties \nof finite nuclei. \nAs a result extrapolation to higher densities and asymmetry involve uncertainties. Three of these properties of the \ninfinite nuclear matter are more precisely known: (a) the saturation density,\n(b) the binding energy and (c) the asymmetry energy, compared to\nthe remaining ones - the effective nucleon mass and the compression modulus of the nuclear matter. \nThe uncertainty in the dense matter EoS is basically\nrelated to the uncertainty in these two saturation properties. It has been seen that to reproduce the giant monopole resonance (GMR)\n in $^{208}$Pb, accurately fitted non-relativistic and relativistic models predict compression\nmodulus in the symmetric nuclear matter ($K$) that differ by\nabout $25\\%$. The reason for this discrepancy being the density dependence of the symmetry energy. \nMoreover, the alluded correlation between $K$ and the density dependence of the symmetry energy \nresults in an underestimation of the frequency of oscillations of\nneutrons against protons, the so-called isovector giant\ndipole resonance (IVGDR) in $^{208}$Pb. FSUGold is a recently proposed accurately calibrated relativistic \nparameterization. It simultaneously describes\nthe GMR in $^{90}$Zr and $^{208}$Pb and the IVGDR in $^{208}$Pb without compromising the success in reproducing the ground-state\nobservables~\\cite{30b}. The main virtue of this parameterization is the softening of both the EoS of symmetric nuclear matter and the symmetry energy. \nThis softening appears to be required for an\naccurate description of different collective modes having different neutron-to-proton ratios.\nAs a result, the FSUGold effective interaction predicts neutron star radii that are too large and a maximum stellar\nmass that is too small~\\cite{31}. \n\n The Indiana University-Florida State\nUniversity (IUFSU) interaction, is a new relativistic parameter set, derived from FSUGold. It is simultaneously constrained by\nthe properties of finite nuclei, their collective excitations and the neutron star properties by adjusting\ntwo of the parameters of the theory - the neutron skin thickness\nof $^{208}Pb$ and the maximum neutron star mass~\\cite{32}. As a result the new effective interaction\nsoftens the EoS at intermediate densities and stiffens the\nEoS at high density. As it stands now, the\nnew IUFSU interaction reproduces \nthe binding energies and charge radii of closed-shell\nnuclei, various nuclear giant (monopole and dipole)\nresonances, the low-density behavior of pure neutron\nmatter, the high-density behavior of the symmetric nuclear\nmatter and the mass-radius relationship of neutron stars. Whether this new EoS can accommodate \nthe hyperons inside the compact stars, with the severe constraints imposed by the recent observations of $\\sim 2M_\\odot$ pulsars,\nneeds to be explored. In this work we plan to make a detailed study of such a possibility. For this purpose\nwe have extended the IUFSU interaction by including \nthe full baryon octet.  A new EoS is constructed to investigate the neutron star properties with  \nhyperons. \n\n\n\\begin{table*}[t] \n\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\\hline\n\nModel &$g_{\\sigma n}^2$ & $g_{\\omega n}^2$& $g_{\\rho n}^2$ & $\\kappa$  &   $\\lambda$   &   $\\zeta$   &   $\\Lambda_v $\\\\[0.1cm]\n&          &              &              &                 (MeV)          &                  &              &    \\\\\\hline\nFSU   &   112.1996   &   204.5469   &   138.4701   & 1.4203   &   0.023762   &   0.06   &   0.030\\\\[0.1cm]\nIUFSU &   99.4266   &   169.8349   &   184.6877    & 3.3808   &   0.000296   &   0.03   &   0.046\\\\[0.1cm]\n\\hline\n\\end{tabular}\n\n\\caption{ Parameter sets for the two models discussed in the text. \nThe nucleon mass and the meson masses are kept fixed at $m_n$ = 939 MeV, $m_\\sigma$ = 491.5 MeV,\n$m_\\omega$ = 782.5 MeV, $m_\\rho$ = 763 MeV and $m_\\phi$ = 1020 MeV in both the models.\n\\label{Para}}\n\n\\end{table*}\n\nThe paper is organized as follows. In section 2, we briefly discuss the\nmodel used and the resulting EoS. In the next section we use this EoS \nto look at static and rotating star properties. We give a brief summary in section 4. \n\n\n\\section{ IUFSU with hyperons}\n One of the possible approaches to describe neutron star matter is to adopt\nan RMF model subject to $\\beta$ equilibrium and charge neutrality. For our\ninvestigation of nucleons and hyperons in the compact star matter we choose\nthe full standard baryon octet as well as electrons and muons. Contribution from neutrinos\nare not taken into account assuming that they can escape freely from the system. In\nthis model, baryon-baryon interaction is mediated by the exchange of scalar\n($\\sigma$), vector ($\\omega$), isovector ($\\rho$) and the strange vector ($\\phi$) mesons. \nThe Lagrangian density we consider is given by~\\cite{32}\n\\begin{widetext}\n\\begin{eqnarray}\n\\mathcal{L} &=& \\sum_{B}\\bar{\\psi}_{B}[i\\gamma^{\\mu}\\partial_{\\mu}\n- m_{B}+g_{\\sigma B}\\sigma - g_{\\omega B}\\gamma^{\\mu}\\omega_{\\mu}  - g_{\\phi B}\\gamma^{\\mu}\\phi_{\\mu}\n- \\frac{g_{\\rho B}}{2}\\gamma^{\\mu}\\vec{\\tau}\\cdot\\vec{\\rho}^{\\mu}]{\\psi}_{B} +\n\\frac{1}{2}\\partial_{\\mu}\\sigma\\partial^{\\mu}\\sigma \n- \\frac{1}{2} m_\\sigma^2\\sigma^2 \\nonumber\\\\\n&& - \\frac{\\kappa}{3!}(g_{\\sigma N}\\sigma)^3  -\\frac{\\lambda}{4!}(g_{\\sigma N}\\sigma)^4 - \\frac{1}{4}F_{\\mu\\nu}F^{\\mu\\nu} +\n\\frac{1}{2}m_\\omega^2\\omega_\\mu\\omega^\\mu  + \\frac{\\zeta}{4!}(g^{2}_{\\omega N}\\omega_\\mu\\omega^\\mu)^2 \n+\\frac{1}{2}m_\\rho^2\\vec{\\rho}_{\\mu}\\cdot\\vec{\\rho}^{\\mu} - \\frac{1}{4}\\vec{G}_{\\mu\\nu}\\vec{G}^{\\mu\\nu}\\nonumber\\\\ \n&&+ \\Lambda_{v}(g^{2}_{\\rho N}\\vec{\\rho}_{\\mu}\\cdot\\vec{\\rho}^{\\mu})(g^{2}_{\\omega N}\\omega_\\mu\\omega^\\mu)  \n+ \\frac{1}{2}m_\\phi^2\\phi_\\mu\\phi^\\mu -\\frac{1}{4}H_{\\mu\\nu}H^{\\mu\\nu} \n+\\sum_{l}\\bar{\\psi}_{l}[i\\gamma^{\\mu}\\partial_{\\mu} - m_{l}]{\\psi}_{l}\n\\end{eqnarray}\n\\end{widetext}\n\n\n\\noindent where the symbol B stands for the baryon octet ($p$, $n$, $\\Lambda$, $\\Sigma^{+}$, $\\Sigma^{0}$, $\\Sigma^{-}$,\n$\\Xi^{-}$, $\\Xi^{0}$) and $l$ represents $e^{-}$ and $\\mu^{-}$. The masses $m_B$, $m_\\sigma$, \n$m_\\omega$, $m_\\rho$ and $m_\\phi$ are respectively for baryon, $\\sigma$, $\\omega$, $\\rho$ and $\\phi$ mesons. The antisymmetric \ntensors of vector mesons take the forms ${F}\n_{\\mu\\nu}$ =  $\\partial_{\\mu}\\omega_{\\nu} - \\partial_{\\nu}\\omega_{\\mu}$,\n${G}_{\\mu\\nu}$ =  $\\partial_{\\mu}\\vec{\\rho}_{\\nu} - \\partial_{\\nu}\\vec{\\rho}_{\\mu} + g[\\vec{\\rho}_{\\mu},{\\vec\\rho}_{\\nu}]$ and\n${H}_{\\mu\\nu}$ =  $\\partial_{\\mu}{\\phi}_{\\nu} - \\partial_{\\nu}{\\phi}_{\\mu}$. The isoscalar meson \nself-interactions (via $\\kappa$, $\\lambda$ and $\\zeta$ terms) are necessary for the appropriate EoS of the\nsymmetric nuclear matter~\\cite {33}. The new additional isoscalar-isovector coupling ($\\Lambda_v$) term \nis used to modify the density dependence of the symmetry energy and the neutron-skin thickness of heavy \nnuclei~\\cite {31,32}. The meson-baryon coupling constants are given by\n $g_{\\sigma B}$, $g_{\\omega B}$, $g_{\\rho B}$ and $g_{\\phi B}$.\n\n\nAll the nucleon-meson parameters used in this work are shown in Table  \\ref{Para}. The saturation properties of the symmetric \nnuclear matter produced by IUFSU are: saturation density $n_0=0.155$ $fm^{-3}$, binding energy per nucleon  \n$\\varepsilon_0= -16.40$ MeV and compression modulus $K = 231.2$ MeV.\n\n\n\nThe hyperon-meson couplings are taken from the SU(6) quark model~\\cite{34,35} as,\n\\begin{center}\n$g_{\\rho\\Lambda}$ = 0, $g_{\\rho\\Sigma}$ = $2g_{\\rho\\Xi}$ = $2g_{\\rho N}$\n\\end{center}\n\\begin{center}\n$g_{\\omega\\Lambda}$ = $g_{\\omega\\Sigma}$ = $2g_{\\omega\\Xi}$ = $\\frac{2}{3}g_{\\omega N}$\n\\end{center}\n\\begin{center}\n $2g_{\\phi\\Lambda}$ = $2g_{\\phi\\Sigma}$ = $g_{\\phi\\Xi}$ = $\\frac{-2\\sqrt{2}}{3} g_{\\omega N}$\n\\end{center}\n\n\nThe scalar couplings are determined by fitting the hyperonic potential,\n\\begin{eqnarray}\nU^{(N)}_Y = g_{\\omega Y}\\omega_0 + g_{\\sigma Y}\\sigma_0\n\\end{eqnarray}\nwhere Y stands for the hyperon and $\\sigma_0$, $\\omega_0$ are the values of the scalar and vector meson fields\nat saturation density~\\cite{8}. The values of $U^{(N)}_Y$ are taken from the available hypernuclear data.\nThe best known hyperonic potential is that of $\\Lambda$, having a value of about $U^{(N)}_\\Lambda$ = -30 MeV~\\cite{36a}.\nIn case of $\\Sigma$ and $\\Xi$ hyperons, the potential depths are not as clearly known as in the case of $\\Lambda$.\nHowever, analyses of laboratory experiments indicate that at\nnuclear densities the $\\Lambda$-nucleon potential is attractive but the $\\Sigma^-$ -nucleon potential\nis repulsive~\\cite{potential}.\nTherefore, we have varied both $U^{(N)}_\\Sigma$ and $U^{(N)}_\\Xi$ in the range of -40 MeV to +40 MeV to investigate the \nproperties of neutron star matter.\n\n\n\n \n For neutron star matter, with baryons and charged\nleptons, the $\\beta$-equilibrium conditions are guaranteed with\nthe following relations between chemical potentials for different\nparticles:\n \\begin{eqnarray}\n  \\mu_p &=& \\mu_{\\Sigma^{+}} = \\mu_n - \\mu_e \\nonumber \\\\\n \\mu_{\\Lambda} &=& \\mu_{\\Sigma^{0}} = \\mu_{\\Xi^{0}} = \\mu_n \\nonumber\\\\\n \\mu_{\\Sigma^{-}} &=& \\mu_{\\Xi^{-}} = \\mu_n+\\mu_e \\nonumber\\\\\n \\mu_{\\mu} &=& \\mu_e \n\\end{eqnarray}\nand the charge neutrality condition is fulfilled by\n \\begin{equation}\n  n_p + n_{\\Sigma^{+}} = n_e+n_{\\mu^{-}}+n_{\\Sigma^{-}}+n_{\\Xi^{-}}\n \\end{equation}\n where $n_i$ is the number density of the {\\it i'th} particle. The effective chemical\npotentials of baryons and leptons can be given by\n\\begin{equation}\n \\mu_B = \\sqrt{{k_{F}^{B}}^2+{m_{B}^{\\ast^2}}}+g_{\\omega B}\\omega+g_{\\rho B}\\tau_{3B}\\rho\n\\end{equation} \n\n\\vspace {1cm}\n\n\n \\begin{equation}\n \\mu_l = \\sqrt {{K_F^{l}}^2+m_l^2}\n \\end{equation}\nwhere $m_{B}^{\\ast} = m_B-g_{\\sigma B}\\sigma$ is the baryon effective mass and $K_F^l$ is the Fermi momentum of the lepton (e, $\\mu$).\n The EoS of neutron star matter can be given by,\n \\begin{figure*}[t]\n \\begin{minipage}[b]{.5\\textwidth}%\n\\includegraphics[width = 2.0in,height = 3.7in, angle = 270]{fig1a.eps}\n\\end{minipage}\\hfill\n\\begin{minipage}[b]{.5\\textwidth}\n\\includegraphics[width = 2.1in,height = 3.4in, angle = 270]{fig1b.eps}\n\\end{minipage}\n\n\n\n\\caption{ (color online) a) EoS obtained with varying  $U^{(N)}_\\Sigma$ at fixed $U^{(N)}_\\Xi$. \n The upper branch shows the EoS for a system containing nucleons, leptons and all the non \n strange mesons. The middle branch shows the EoS for a system containing the whole baryon octet, the leptons \n and $\\sigma$, $\\omega$, $\\rho$ and $\\phi$ mesons. The lower branch shows the EoS\n for the particles contained in the middle branch except $\\phi$. \n b) EoS obtained with varying  $U^{(N)}_\\Xi$ at fixed $U^{(N)}_\\Sigma$. The compositions \n of the upper, middle and lower branches are same as those of a) respectively. \n \\label{eos}}\n\\end{figure*}\n\n\n\\begin{widetext}\n\\begin{eqnarray}\n{\\varepsilon} &=& \\frac{1}{2}m_\\sigma^2 \\sigma^2\n+ \\frac{\\kappa}{6} g_{\\sigma N}^3 \\sigma^3 + \\frac{\\lambda}{24}  g_{\\sigma N}^4 \\sigma^4 + \\frac{1}{2} m_\\omega^2 \\omega^2 \n+ \\frac{\\zeta}{8} \ng_{\\omega N}^4 \\omega^4 \n+ \\frac{1}{2} m_\\rho^2 \\rho^2 + 3\\Lambda_v g_{\\rho N}^2 g_{\\omega N}^2{\\omega}^2{\\rho}^2 \\nonumber\\\\\n&& + \\frac{1}{2}m_\\phi^2 \\phi^2 + \\sum_B\\frac{\\gamma_B}{(2\\pi)^3} \n\\int_0^{k_{F}^B} \\sqrt{k^2+m^{* 2}_B} \\ d^3 k \n + \\frac{1}{\\pi^2}\\sum_l\\int_0^{K_F^l} \\sqrt{k^2+m^2_l} \\ k^2 dk\n\\end{eqnarray}\n\n\\begin{eqnarray}\nP &=& - \\frac{1}{2}m_\\sigma^2 \\sigma^2\n- \\frac{\\kappa}{6} g_{\\sigma N}^3 \\sigma^3 - \\frac{\\lambda}{24} g_{\\sigma N}^4  \\sigma^4 + \\frac{1}{2} m_\\omega^2 \\omega^2 +  \\frac{\\zeta}{24}\ng_{\\omega N}^4 \\omega^4 + \\Lambda_v g_{\\rho N}^2 g_{\\omega N}^2{\\omega}^2{\\rho}^2 \n+ \\frac{1}{2} m_\\rho^2 \\rho^2 \\nonumber\\\\\n&& + \\frac{1}{2}m_\\phi^2 \\phi^2 + \\frac{1}{3}\\sum_B \\frac{\\gamma_B}{(2\\pi)^3} \n\\int_0^{k_F^B}\\frac{k^2 \\  d^3 k}{(k^2+m^{* 2}_B)^{1\/2}}\n + \\frac{1}{3} \\sum_{l} \\frac{1}{\\pi^2}\n\\int_0^{K_F^l}\\frac{k^4 \\ dk}{(k^2+m^2_l)^{1\/2}}~\\nonumber\\\\\n\\label{pr}\n\\end {eqnarray}\n\\end{widetext}\nwhere $\\varepsilon$ and $P$ stand for energy density and pressure respectively and $\\gamma_B$ is the baryon spin-isospin degeneracy factor.\n\n\n \\begin{figure}[htb]\n\\hskip 1.5cm\n \\begin{minipage}{.01in}\n \\rotatebox{90}{\\bf particle fractions}\n \\end{minipage}%\n \\begin{minipage}{\\dimexpr\\linewidth-2.50cm\\relax}\n {\\includegraphics[width = 2.8in,height = 1.7in]{fig2a.eps}}\\\\\n \\vspace{-6.6mm}\\bigskip{}\n {\\includegraphics[width = 2.8in, height = 1.7in]{fig2b.eps}}\\\\\n \\vspace*{-6.5mm}\\bigskip{}\n {\\includegraphics[width = 2.8in, height = 1.7in]{fig2c.eps}}\\\\\n \\vspace*{-6.5mm}\\bigskip{}\n {\\includegraphics[width = 2.8in, height = 1.7in]{fig2d.eps}}\\\\\n \\vspace*{0.1cm}\\hspace*{2.0cm}{\\bm {$n_B$ $(fm^{-3})$}}\n \\end{minipage}\n \n \\caption{ (color online) Particle fractions for different $\\Sigma$ potential depths: a)\n for ``$\\sigma\\omega\\rho$''\n  with $U^{(N)}_\\Sigma = -30$ MeV, b) for ``$\\sigma\\omega\\rho$''\n  with $U^{(N)}_\\Sigma = +30$ MeV, c) for ``$\\sigma\\omega\\rho\\phi$''\n  with $U^{(N)}_\\Sigma = -30$ MeV, d) for ``$\\sigma\\omega\\rho\\phi$''\n  with $U^{(N)}_\\Sigma = +30$ MeV. $U^{(N)}_\\Xi$ is fixed at -18 MeV in each case.\\label{pf1}}\n \\end{figure}\n\n\nIn fig. \\ref{eos} we plot the EoS for different values of the hyperonic potentials.\n The upper branch is for the usual nuclear matter which\n  does not contain any strange particle. \n \nThe middle and lower branches \nare for full baryon octet, leptons and $\\sigma$, $\\omega$, $\\rho$ mesons.  In addition, the middle branch contains the $\\phi$ meson. \n In the left \npanel, {\\it i.e.} in fig. \\ref{eos}a, we keep $U^{(N)}_\\Xi$ fixed at -18 MeV,  \nthis value is generally adopted from hypernuclear experimental data~\\cite{pot}. For the middle and lower branches \n we vary the $\\Sigma$ potential from -40 MeV to +40 MeV in steps of 20 MeV. The lower branch shows that for an attractive $\\Sigma$ potential \nthe EoS gets stiffer as $U_{\\Sigma}^{(N)}$ increases. However as $U_{\\Sigma}^{(N)}$ becomes \npositive the EoS seems to become independent of $U_{\\Sigma}^{(N)}$. We see from fig. \\ref{eos}a that for \n$U_{\\Sigma}^{(N)} > 0$ MeV the EoS remains identical to that for $U_{\\Sigma}^{(N)} = 0$ MeV.\nHowever, once we add $\\phi$ meson to the system, the EoS continues to get stiffer as $U_{\\Sigma}^{(N)}$ moves to more positive side (middle branch of fig. \\ref{eos}a). \n\n  \\begin{figure}[t]\n \\begin{minipage}{.01in}\n \\rotatebox{90}{\\bf particle fractions}\n \\end{minipage}%\n \\begin{minipage}{\\dimexpr\\linewidth-2.50cm\\relax}\n {\\includegraphics[width = 2.8in,height = 1.7in]{fig3a.eps}}\\\\\n \\vspace*{-6.4mm}\\bigskip{}\n {\\includegraphics[width = 2.8in, height = 1.7in]{fig3b.eps}}\\\\\n \\vspace*{-6.3mm}\\bigskip{}\n {\\includegraphics[width = 2.8in, height = 1.7in]{fig3c.eps}}\\\\\n \\vspace*{-6.3mm}\\bigskip{}\n {\\includegraphics[width = 2.8in, height = 1.7in]{fig3d.eps}}\\\\\n \\vspace*{0.1cm}\\hspace*{2.0cm}{\\bm {$n_B$ $(fm^{-3})$}}\n \\end{minipage}\n \n  \\caption{ (color online) Particle fractions for different $\\Xi$ potential depths: a) for ``$\\sigma\\omega\\rho$''\n  with $U^{(N)}_\\Xi = -30$ MeV, b) for ``$\\sigma\\omega\\rho$''\n  with $U^{(N)}_\\Xi = +30$ MeV, c) for ``$\\sigma\\omega\\rho\\phi$''\n with $U^{(N)}_\\Xi = -30$ MeV, d) for ``$\\sigma\\omega\\rho\\phi$''\n with $U^{(N)}_\\Xi = +30$ MeV. $U^{(N)}_\\Sigma$ is fixed at +30 MeV in each case.\\label{pf2}}\n \\end{figure}\n\n\n\n We then fix $U^{(N)}_\\Sigma$ and vary $U^{(N)}_\\Xi$. \n This is represented in fig. \\ref{eos}b, where we have fixed the value of $U^{(N)}_\\Sigma = +30$ MeV \n (adopted from hypernuclear experimental data~\\cite{pot}). We vary $U^{(N)}_\\Xi$ from -40 MeV to +40 MeV. \nWe see that for the lower branch, {\\it i.e} the case without the $\\phi$ meson, the EoS \ngets stiffer with the increase in $\\Xi$ potential up to $U^{(N)}_\\Xi = 0$ MeV. \nHowever, for positive values of $U^{(N)}_\\Xi$ the EoS remains unchanged. \nAdding an extra repulsion to the system by including the $\\phi$ meson changes\nthe scenario altogether. The EoS becomes totally independent of the $\\Xi$ potential (middle branch of fig. \\ref{eos}b). \nFrom figures 1a and 1b one can generally conclude that the inclusion \nof $\\phi$ meson makes the EoS stiffer, however, hyperonic EoS is much softer than the usual \nnuclear matter EoS. \n\n\\begin{figure*} [t]\n\\begin{minipage}[t]{.5\\textwidth}\n\\includegraphics[width = 2.0in,height = 3.8in, angle = 270]{fig4a.eps}\n\\end{minipage}\\hfill\n\\begin{minipage}[t]{.5\\textwidth}\n\\includegraphics[width = 2.0in,height = 3.8in, angle = 270]{fig4b.eps}\n\\end{minipage}\n\n\\caption{ (color online) Mass-radius curves for static star fixing the a) $\\Xi$ potential depth \nat $U^{(N)}_\\Xi = +40$ MeV and varying the $U^{(N)}_\\Sigma$.\nb) $\\Sigma$ potential depth \nat $U^{(N)}_\\Sigma = +40$ MeV and varying the $U^{(N)}_\\Xi$. The uppermost curve in each case corresponds to the pure nuclear matter.}\n\\label{static}\n\\end{figure*}\n\n\nIn fig. \\ref{pf1} we have plotted the particle fractions for an attractive \n $\\Sigma$ potential $U^{(N)}_\\Sigma = -30$ MeV and a repulsive potential $U^{(N)}_\\Sigma = +30$ MeV keeping $U^{(N)}_\\Xi$ fixed \n at -18 MeV, with and without $\\phi$ in each case.\nFrom fig. \\ref{pf1}a, when $\\phi$ is not present, we see that all the hyperons contribute \nto the particle fractions for an\nattractive $\\Sigma$ potential whereas for repulsive $U^{(N)}_\\Sigma$ there is no $\\Sigma$\npresent in the matter (fig. \\ref{pf1}b). The appearance of $\\Lambda$ is also pushed to higher \ndensity compared to the case of an attractive potential. \nWhen $\\phi$ is included in the system\n $\\Sigma^0$ and $\\Sigma^-$ appear with $\\Lambda$ for $U^{(N)}_\\Sigma = -30$ MeV (fig. \\ref{pf1}c). However, for $U^{(N)}_\\Sigma = +30$ MeV\n(fig. \\ref{pf1}d), the threshold of $\\Sigma^-$ is pushed to higher density compared to the case of $U^{(N)}_\\Sigma = -30$ MeV, $\\Sigma^0$ disappears\nand $\\Xi^-$ appears in the system.\nWe also note that in the case of attractive $\\Sigma$ potential, $\\Sigma^-$ is always the first hyperon to appear in the system.\nFor repulsive $U^{(N)}_\\Sigma$, $\\Xi^-$ appears before others in the ``$\\sigma\\omega\\rho$'' case\nand $\\Lambda$ is the the first hyperon to appear in case of ``$\\sigma\\omega\\rho\\phi$''.\n\n\n\n\n From fig. \\ref{pf1} we see that for negative values of  $U^{(N)}_\\Sigma$, the $\\Sigma$'s are\nbound in matter and the effective mesonic interaction would be more attractive as the potential gets deeper. \nAs a result, the EoS gets softer with more attractive $U^{(N)}_\\Sigma$ (see fig. \\ref{eos}a). For \n$U^{(N)}_\\Sigma\\geq 0$, $\\Sigma$'s are no longer bound to matter and the effective\nmesonic interaction becomes more and more repulsive with increasing\n$U^{(N)}_\\Sigma$. This should, in principle, stiffen the EoS. However, for the\n``$\\sigma\\omega\\rho$'' case, up to neutron star\ndensities, {\\it i.e} about $n_B \\lesssim (4-7)n_0$, $\\Sigma$'s are not present in the \nmatter when the potential is repulsive and hence the EoS up to these densities\nbecomes insensitive to $U^{(N)}_\\Sigma$.\n\n\\begin{figure*}[t]\n\\begin{minipage}[t]{.5\\textwidth}\n\\includegraphics[width = 2.0in,height = 3.8in, angle = 270]{fig5a.eps}\n\\end{minipage}\\hfill\n\\begin{minipage}[t]{.5\\textwidth}\n\\includegraphics[width = 2.0in,height = 3.8in, angle = 270]{fig5b.eps}\n\\end{minipage}\n\\caption{ (color online) Mass-radius curves for rotating stars for two cases: a) $U^{(N)}_\\Xi = +40$ MeV\nand $-40 MeV \\leq U^{(N)}_\\Sigma \\leq +40$ MeV and b) $U^{(N)}_\\Sigma = +40$ MeV\nand $-40 MeV \\leq U^{(N)}_\\Xi \\leq +40$ MeV. The uppermost curve in each case corresponds to the pure nuclear matter.\\label{MR3}}\n\\end{figure*}\n\nIn fig. \\ref{pf2} the particle fractions are plotted for an attractive $\\Xi$ potential $U^{(N)}_\\Xi = -30$ MeV \nand a repulsive potential $U^{(N)}_\\Xi = +30$ MeV keeping $U^{(N)}_\\Sigma$ fixed at +30 MeV.\nWe see that in the first case {\\it i.e.} when $\\phi$ is not present and the potential is attractive \n (fig. \\ref{pf2}a), all the hyperons except $\\Sigma$'s are present in the system\nand the $\\Lambda$ hyperon dominates. When the $\\Xi$ potential becomes positive (fig. \\ref{pf2}b) $\\Xi^0$\ndisappears and the threshold for appearance of $\\Xi^-$ shifts to much higher density. However $\\Sigma^-$ is present in matter in this\npotential and it appears before $\\Xi^-$. When $\\phi$ is introduced in the system, for an attractive $\\Xi$ potential (fig. \\ref{pf2}c), again $\\Sigma^-$ and $\\Xi^-$ \nare present along with $\\Lambda$. However, the difference from fig. \\ref{pf2}b {\\it i.e} ``$\\sigma\\omega\\rho$'' case\nand $U^{(N)}_\\Xi\\geq$0 is that, here $\\Xi^-$ appears much before $\\Sigma^-$.\nIn the last case (fig. \\ref{pf2}d), we see that as a result of the combined effects of inclusion of $\\phi$ and repulsive potentials,\nonly the $\\Lambda$ and $\\Sigma^-$ are present in the system.\n From both figures \\ref{pf1} and \\ref{pf2}, we see that, inclusion of \n $\\phi$ meson decreases the density of hyperons. Since $\\phi$ is a strange particle, further strangeness is suppressed and \n as a result the hyperon densities are reduced compared to the ``$\\sigma\\omega\\rho$'' case. \n\n\n\n\\section {static and rotating stars}\n\n\n\n\nIn this section we are going to discuss the properties of static and rotating axisymmetric stars using the EoS\nwhich we have studied in the last section. The EoS without $\\phi$ meson is softer compared \nto that with $\\phi$ meson. So we do not discuss the EoS without $\\phi$ as it results in less maximum mass. \n\n\n\nThe stationary, axisymmetric space-time used to model the compact stars are defined through the metric\n\\begin{eqnarray}\n ds^2 = -e^{\\gamma+\\rho} dt^2 + e^ {2\\alpha}(dr^2+r^2d\\theta^2)\\nonumber\\\\\n + e^{\\gamma-\\rho}r^2 sin^2{\\theta}(d\\phi-\\omega dt)^2 \n\\end{eqnarray}\n\\noindent where $\\alpha$, $\\gamma$ , $\\rho$ and $\\omega$ are the gravitational potentials which\ndepend on r and $\\theta$ only.\n\n\n\nIn this work we adopt the procedure of Komatsu {\\em et al. }~\\cite{37} to look into the observable properties of static and rotating\nstars. Einstein's equations for the three gravitational potentials $\\gamma$, $\\rho$ and $\\omega$ can be solved using Green's \nfunction technique. The fourth potential $\\alpha$ can be determined using these three potentials. Once these potentials are \ndetermined one can calculate all the observable quantities using those. The solution of the potentials and\nhence the determination of physical quantities is numerically quite an involved process. For this purpose the \n``rns'' code~\\cite{39} is used in this work. This code, developed by Stergoilas, is very efficient in calculating the rotating star \nobservables.\n\n\nWe discuss the properties of static stars first. In fig. \\ref{static} we have plotted the mass-radius curves \nof static stars \nusing the EoS with ``$\\sigma\\omega\\rho\\phi$''. A plot for the pure nuclear matter\ncase is also given for comparison (uppermost curve of both the panels). The maximum mass of pure nuclear matter \nstar in the static case is $1.92 M_\\odot$ with a radius of $11.24$ km. We have found that the mass of hyperonic star\nbecomes maximum for $U_\\Sigma^{N} = +40$ MeV and $U_\\Xi^{N} \\geq 0$ MeV. Hence in fig. \\ref{static} and\nfig. \\ref{MR3} we have shown the effect of these potentials on the maximum mass of neutron stars by fixing one of the \npotentials at +40 MeV and varying the other. The left panel, {\\it i.e.} \nfig. \\ref{static}a, corresponds to $U_\\Xi = +40$ MeV and $U_{\\Sigma}$ varying from -40 MeV to +40 MeV. In \nthe right panel, {\\it i.e.} in fig. \\ref{static}b,  it is the other way round. \nFrom fig. \\ref{static}a one can see that the maximum mass of the star increases with $U_\\Sigma^{(N)}$. \nFor $U_\\Sigma^{(N)} = +40$ MeV the maximum mass is $1.62 M_\\odot$ with a radius of $10.82$ km. The central \nenergy density of such a star is $\\epsilon_c = 2.46 \\times 10^{15} gm \\,\\, cm^{-3}$.  This is a reflection of the \nEoS shown in fig. \\ref{eos}a, which shows that the EoS becomes stiffer with increase in $U_\\Sigma^{(N)}$. However, \nas seen from fig. \\ref{static}b, the maximum mass of static stars is insensitive \nto $U_\\Xi^{(N)}$, which should be obvious from fig. \\ref{eos}b as the EoS is independent of the cascade potential.\nFurthermore, from fig. \\ref{pf2}d one can see that there is no cascade present in the medium. So the \ninsensitivity of the EoS and hence the maximum mass, towards the cascade potential is expected. One should note that the maximum \nmass we obtain for the static stars is less than the observed mass of PSR J$0348+0432$ . So the static stars with hyperons \nin the IUFSU \nparameter set can not incorporate a maximum mass $\\sim 2M_\\odot$. This result is consistent with the findings in Ref.~\\cite{Agrawal}.\nHowever, since both of the observed\n$\\sim 2M_\\odot$ stars are pulsars, it would be\na better idea to compare the observations with results from the rotating stars, which we do in the next part. \n\n\n\n\n\\begin{figure}[h]\n\\resizebox{8cm}{!}{\n \\includegraphics{fig6.eps}}\n\\caption{ (color online) Particle densities varying with radius along the equator.\nThe potential depths for which particle densities are plotted are $U^{(N)}_\\Xi = 0$ and $U^{(N)}_\\Sigma = +40$ MeV.\n \\label{pd}}\n\\end{figure}\n\nIn fig. \\ref{MR3} we plot the mass-radius curves for stars rotating with Keplerian velocities, for two cases. In \nfig. \\ref{MR3}a we fix the cascade potential \nat $U^{(N)}_\\Xi = +40$MeV and vary $U^{(N)}_\\Sigma$ from $-40$MeV to $+40$MeV. In fig. \\ref{MR3}b it \nis the other way round. The pure nuclear matter case is also shown in the uppermost curve. The maximum mass for the pure \nnucleonic star is $2.29 M_\\odot$ with a radius of $15.31$ km. We see that\nthe maximum mass obtained for a rotating star with hyperonic core is $1.93M_{\\odot}$ with a radius of $14.7$ km in\nthe Keplerian limit with angular velocity $\\Omega = 0.86\\times 10^4 s^{-1}$,\n for $U^{(N)}_\\Sigma = +40$ MeV and \n$U^{(N)}_\\Xi \\geq 0$. \nAs in the case of static sequence, we see that the maximum mass for the rotating case also increases with $U^{(N)}_\\Sigma$ as we go \ntowards more positive values of this potential. At $U^{(N)}_\\Sigma = -40$ MeV we get a maximum mass of $1.79 M_{\\odot}$\nwhereas for $U^{(N)}_\\Sigma = +40$ MeV the maximum mass is $1.93 M_{\\odot}$. \nThe effect of $U^{(N)}_\\Xi$ is much less significant on the maximum mass.\nFrom $U^{(N)}_\\Xi = -40$ MeV \nto $U^{(N)}_\\Xi = +40$ MeV mass is changed only by $\\bigtriangleup M = 0.03 M_{\\odot}$. \n\n\nIn order to have a look at the composition of the maximum mass star, we have plotted the particle densities as a function \nof radius along the equator in fig. \\ref{pd}. For $U^{(N)}_\\Xi = 0$ and $U^{(N)}_\\Sigma = +40$ MeV, \n we see that a fair amount of hyperons are present in the core. There are $\\Lambda$, $\\Sigma^-$ and $\\Xi^-$\n present. Another interesting observation is that near the core, \nthe density of $\\Lambda$ is much more compared to that of protons and it continues up to a distance of about 5 km from the center. \n\n\n\n\n\\section{Summary and conclusions}\n\nTo summarize, we have studied the static and rotating axisymmetric stars with hyperons using IUFSU model. The original FSUGold \nparameter set has been very successful in describing the properties of finite nuclei. With the discovery of highly massive neutron stars \nthe reliability of this model was questioned. It was then revised in the form of IUFSU to accommodate such highly massive stars\nleaving the low density finite nuclear properties unchanged. In this\nwork we have studied this new parameter set in the context of the possibility of having a hyperonic core in such massive stars.\n\nWe have included the full octet of baryons in IUFSU. The EoS gets softened due to the inclusion of hyperons\nwhereas the inclusion of the $\\phi$ meson makes the EoS stiffer. We have also investigated the influence of \n$\\Sigma$ and $\\Xi$ potentials on the EoS.\n\n For static stars with hyperonic core we get a maximum mass of \n $1.62 M_\\odot$. So IUFSU with hyperons cannot reproduce the observed \nmass of static stars. However, as the observed $\\sim 2M_\\odot$ neutron stars are both pulsars, we \ncompare the results in the rotating limit. In the Keplerian limit we get a maximum mass of $1.93 M_{\\odot}$, which is within the\n3$\\sigma$ limit of the mass of PSR J$0348+0432$ and 1$\\sigma$ limit of the earlier observation of PSR J$1614-2230$. \nWe have looked at the particle densities inside the star \nhaving the maximum mass and found that a considerable amount of hyperons are present near the core.\nTherefore, our results are consistent with the recent observations of highly massive pulsars confirming the presence of hyperons\nin the core of such massive neutron stars.\n\nTo conclude, IUFSU model, which reproduces the properties of finite nuclei quite successfully also reproduces the recent observations \nof $\\sim 2M_{\\odot}$ stars, in case of stars having exotic core and rotating in the Keplerian limit. It will be interesting to see \nwhether such a star can hold a quark core. Related work is in progress. \n\n\\section{Acknowledgement}\nThis work is funded by the University Grants Commission (RFSMS, DSKPDF and DRS) and  Department of Science and Technology, \nGovernment  of India.  \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction.}\nThe recent characterization of graphene sheets made up of a single layer of\ncarbon atoms\\cite{Netal04,Netal05b} has caused great interest. Their unusual\nelectronic band structure, and the possibility of tuning the number of\nelectrons lead to a number of interesting features, both from a fundamental\nperspective and because of its potential applications\\cite{GN07,NGPNG07}.\n\nThe low energy electronic states of graphene are well described, in\nthe continuum limit, by two decoupled two dimensional Dirac\nequations. The kinetic energy depends linearly on the lattice\nmomentum. The perturbations due to some types of disorder, like\ntopological lattice defects\\cite{GGV92,GGV93b},\nstrains\\cite{Metal06,MG06}, and curvature\\cite{NK07} enter as an\neffective gauge field. Curvature, strains and topological lattice\ndefects are expected to exist in graphene, as experiments show a\nsignificant corrugation both in suspended samples\\cite{Metal07}, in\nsamples deposited on a substrate\\cite{Setal07b,CF07}, and also in\nsamples grown on metallic surfaces\\cite{Vetal08}.\n\nThe statistical properties of the two dimensional Dirac equation\nin a random gauge field have been extensively\nstudied\\cite{LFSG94,CMW96,CCFGM97,HD02,K07}, in relation with the\nInteger Quantum Hall effect. It has been shown that the density of\nstates develops a peak at zero energy when the disorder strength\nexceeds a certain threshold. Furthermore, beyond a second\nthreshold, there is a transition to a glassy phase\\cite{CD00}\nwhere the local density of states is dominated by rare\nregions.\\cite{HD02}.\n\nIn the following, we will apply the analysis in\\cite{HD02} to the\nspecific case of graphene, where there are two Dirac equations\ncoupled to the same random gauge field. The model will be detailed\nin the next Section. We analyze in the following Section the\nstatistical properties of the gauge field. The main results for the\ndensity of states are presented in Section IV.  Given a divergent\ndensity of states at the energy of the Dirac point, we consider the\ninstabilities which may be induced by interactions. Alternative\napproaches to the interplay between gauge fields and interactions\nare given in\\cite{W99,SGV05,HJV08}, although they did not consider\ndiverging densities of states. Sections VI analyze the related\nproblem of the structural changes which can be induced by the same\nrandom strains which give rise to the gauge field, following the\nanalysis in\\cite{CD98b}.  Section VII discusses a buckling\ntransition in suspended graphene. Section VIII estimates the effects\nof ripples on density fluctuations in the quantum Hall regime and\ncompares with recent data \\cite{Metal07b}. The main results of the\npaper are summarized in Section VI.\n\n\n\\section{The model.}\nWe analyze the gauge field induced by the height fluctuations of a\ngraphene layer on a rough substrate. In such case one expects that the shape of the\ngraphene layer is determined by a competition between the interaction\nof the layer with the rough substrate, which tends to impose a preferred\nheight, and the elastic properties of\nthe layer. A simple Hamiltonian which models these effects is:\n\\begin{eqnarray}\n{\\cal H} &= &{\\cal H}_{subs} + {\\cal H}_{lattice} + {\\cal H}_{elec} \\nonumber\n\\\\\n{\\cal H}_{subs} &= &\\frac{g}{2} \\int d^2 \\vec{r} \\left[ h ( \\vec{r} ) - h_0 (\n  \\vec{r} )  \\right]^2 \\nonumber \\\\\n{\\cal H}_{elastic} &= &\\frac{\\kappa}{2} \\int d^2 \\vec{r} \\left[\n    \\nabla^2 h ( \\vec{r} )\n\\right]^2  + \\nonumber \\\\ &+ &\\int d^2 \\vec{r} \\left\\{  \\frac{\\lambda}{2} \\left[\n\\sum_i u_{ii} ( \\vec{r} ) \\right]^2 +\n\\mu \\sum_{ij} \\left[ u_{ij} ( \\vec{r} ) \\right]^2 \\right\\} \\nonumber \\\\\n{\\cal H}_{elec} &= &v_{\\rm F} \\int d^2 \\vec{r}  \\bar{\\Psi}_1 ( \\vec{r} ) \\left\\{\n    \\sigma_x \\left[ - i \\partial_x - A_x ( \\vec{r} ) \\right] +\n  \\right. \\nonumber \\\\ &+ &\\left. \\sigma_y \\left[\n      - i \\partial_y - A_ y ( \\vec{r} ) \\right]\\right\\}\n    \\Psi_1 ( \\vec{r} )  - \\nonumber \\\\\n&- &v_{\\rm F} \\int d^2 \\vec{r}  \\bar{\\Psi}_2 ( \\vec{r} ) \\left\\{\n    \\sigma_x \\left[ - i \\partial_x + A_x ( \\vec{r} ) \\right] +\n  \\right. \\nonumber \\\\ &+ &\\left. \\sigma_y \\left[\n      - i \\partial_y + A_ y ( \\vec{r} ) \\right]\\right\\}\n    \\Psi_2 ( \\vec{r} )\n\\label{hamil}\n\\end{eqnarray}\nwhere $h ( \\vec{r})$ is the height of the graphene layer, $h_0( \\vec{r} )$ is the preferred\nheight which can be assumed to follow closely the substrate height, $\\Psi_1 ( \\vec{r} )$ and\n$\\Psi_2 ( \\vec{r} )$ are the two inequivalent Dirac (iso)-spinors\nwhich can be defined in the graphene lattice, $u_{ij} ( \\vec{r} )$\nis the strain tensor associated to the deformation of the graphene layer, given by:\n\\begin{eqnarray}\nu_{xx} &= &\\frac{\\partial u_x}{\\partial x} + \\mbox{\\small $\\frac{1}{2}$}\\left(\n\\frac{\\partial\n    h}{\\partial x} \\right)^2 \\nonumber \\\\\nu_{yy} &= &\\frac{\\partial u_y}{\\partial y} + \\mbox{\\small $\\frac{1}{2}$}\\left(\n\\frac{\\partial\n    h}{\\partial y} \\right)^2 \\nonumber \\\\\nu_{xy} &= &\\frac{1}{2} \\left( \\frac{\\partial u_x}{\\partial y} +\n    \\frac{\\partial u_y}{\\partial x} \\right) + \\mbox{\\small $\\frac{1}{2}$}\\frac{\\partial h}{\\partial x}\n    \\frac{\\partial h}{\\partial y}\n\\label{strain}\n\\end{eqnarray}\nThe gauge vector acting on the electrons in eq.(\\ref{hamil}) is\nrelated to the strain tensor by\\cite{SA02b,M07}:\n\\begin{eqnarray}\nA_x ( \\vec{r} ) &= &\\frac{\\beta}{a} \\left[ u_{xx} ( \\vec{r} ) - u_{yy} ( \\vec{r} )\n\\right]\n\\nonumber \\\\\nA_y ( \\vec{r} ) &= &-2 \\frac{\\beta}{a} u_{xy} ( \\vec{r} )\n\\label{gauge}\n\\end{eqnarray}\nwhere $a \\approx 1.4$\\AA \\, is the length of the bond between\nneighboring carbon atoms, and $\\beta=C \\tilde \\beta$ where $C$ is a constant of\norder unity and $\\tilde \\beta = -\\partial \\log ( t ) \/\n\\partial \\log ( a ) \\sim 2-3$ is a dimensionless parameter which\ncharacterizes the coupling between the Dirac electrons and lattice deformations. Besides the symmetry arguments\nin\\cite{M07}, we also assume that the coupling between the electrons and lattice deformations is through the\nmodulation of the hopping between nearest neighbor $\\pi$ orbitals, $t \\approx 3$eV\\cite{G81,HKSS88}. The rest of\nthe parameters which determine the hamiltonian in eq.(\\ref{hamil}) are the electron Fermi velocity, $v_{\\rm F} = 3 t a\n\/ 2$, the bending rigidity, $\\kappa \\sim 1$eV, the in-plane elastic constants, $\\lambda , \\mu \\sim 1$eV\n\\AA$^{-2}$. To model the interaction between the graphene layer and the substrate we use a simple quadratic\nexpansion around the height $h_0(\\vec r)$ which minimizes the energy in the absence of elastic and electronic\nenergy, parameterized by a coupling $g$. The value of this parameter is less understood. Estimates based on the\nanalysis of the electrostatic potential between graphene and SiO$_2$\\cite{Setal07b} suggest that $g \\sim\n10^{-2}-10^{-1}$meV \\AA$^{-4}$. By comparing $g$ and $\\kappa$, one finds that the pinning by the substrate\ndominates for length scales greater than $l_{p} \\sim (\\kappa \/ g)^{1\/4} \\sim 10$\\AA. The coupling $g$ being\nstrongly relevant, for $l \\gg l_p$ it can be considered as effectively infinite and the graphene layer rigidly\npinned to the substrate, $h ( \\vec{r} ) \\approx h_0(\\vec{r})$ for $l \\gg l_p$. Note that we assume here that effect\nof direct pinning of the in plane modes by the substrate are small and can be neglected.\n\n\\section{Effective gauge field.}\n\nExperiments\\cite{Setal07,CF07} suggest that the height of the\ngraphene layer shows fluctuations of order $h \\sim 10$\\AA \\, over\nscales $l \\sim 100$\\AA. Similar fluctuations have been observed in\nsuspended graphene sheets\\cite{Metal07}. We will assume that the\neffects of the height fluctuations can be described statistically\nover distances larger than $l_0 \\sim 100$\\AA. We will then relate the\ncorrelations of the effective random gauge field to the (four point)\ncorrelations of the (random) height profile, $h ( \\vec{r} )$. The calculation\nis valid whether this profile arises from interaction with\na static rough substrate (in which case for $l \\gg l_p$ it directly relates\nto substrate correlations) or from any other mechanism such as in\nsuspended graphene.\n\nAn estimate of the magnitude of the effective random gauge field\ncan be obtained by noting that the height change between\nneighboring lattice points is $\\sim a\\nabla h$ hence the distance\nchange is $\\sim a(\\nabla h)^2$ and the modulation in $t$ is\n$\\delta t\\sim \\beta t(\\nabla h)^2$. Hence the modulation in $A$ is\n$\\sim \\delta t\/v_F\\sim \\beta (\\nabla h)^2\/a$ which yields an\nestimate for the variance of the random effective magnetic field\n$B=[{\\bm \\nabla} \\times {\\bm A}]_z$:\n\\begin{eqnarray}\n\\langle B(q) B(q') \\rangle && = C_B(q) (2 \\pi)^2 \\delta^2(q+q') \\label{defB} \\\\\n\\pi \\sigma &&= \\lim_{q \\to 0} q^{-2} C_B(q)\n\\nonumber\\\\&&\\sim \\left( \\frac{\\beta}{a} \\right)^2 \\int_{| \\vec{r} |\n  \\le l_0}  d^2 \\vec{r} \\left(\n  \\frac{h}{l_0} \\right)^4 \\sim \\frac{\\beta^2 h^4}{a^2 l_0^2}\n  \\label{estimate}\n\\end{eqnarray}\nwhere $h \\sim 10$\\AA \\, is the typical scale of the height\nfluctuations, as discussed earlier. Typical parameters allow for\n$\\sigma=O(1)$, within range of the transitions that we consider\nbelow.\n\n\nIn order to perform a more detailed calculation of the effective\ngauge field acting on the electrons, we first compute the in plane\ndisplacement field, $\\vec{u} ( \\vec{r} )$ obtained by minimizing the\nelastic energy for a given realization of $h ( \\vec{r} )$, and then\nwe estimate the strain tensor $u_{ij} ( \\vec{r} )$. We define:\n\\begin{equation}\nf_{ij} ( \\vec{r} ) = \\frac{\\beta}{a} \\frac{\\partial h}{\\partial x_i}\n  \\frac{\\partial h}{\\partial x_j}\n\\end{equation}\nIn terms of these quantities, the procedure described above gives for the\neffective magnetic field acting on the electrons:\n\\begin{eqnarray}\nB ( \\vec{k} ) &= & i k_y \\frac{( 3 k_x^2  -  k_y^2 ) ( \\lambda + \\mu\n)}{( \\lambda + 2 \\mu )\n  k^4} \\times \\nonumber \\\\\n&\\times & \\left[ k_y^2 f_{xx} ( \\vec{k} ) + k_x^2 f_{yy} ( \\vec{k} ) - 2 k_x k_y\n  f_{xy} ( \\vec{k} ) \\right]\n\\label{bmag}\n\\end{eqnarray}\nWe assume\nthat the average properties of the height modulations\nare described by translationally invariant correlation functions, in Fourier:\n\\begin{equation}\n\\left\\langle f_{ij} ( \\vec q ) f_{kl} ( \\vec q ) \\right\\rangle =\n{\\cal F}_{ijkl}(q)  \\label{corr}\n\\end{equation}\nand are of short range character, i.e. with a finite limit for\n$q l_0 \\ll 1$:\n\\begin{equation}\n{\\cal F}_{ijkl}(q)|_{q \\to 0} = f  \\delta_{ij} \\delta_{kl} +  f' \\left(\n\\delta_{ik} \\delta_{jl} + \\delta_{il} \\delta_{jl} \\right)\n\\label{tensor}\n\\end{equation}\na tensor compatible with the hexagonal symmetry of the lattice\nparameterized by two dimensionless constants $f$ and $f'$.\nUsing eqs.(\\ref{bmag}), (\\ref{corr}), and (\\ref{tensor}), we find\nfor the correlations (\\ref{defB}) of the effective magnetic field at small $q$:\n\\begin{equation}\nC_B(q)\n= q^2 \\left(\n  \\frac{\\lambda + \\mu}{\\lambda + 2 \\mu} \\right)^2 \\sin^2 ( 3 \\theta )\n  ( f + 2 f' )  \\label{gauge_B}\n\\end{equation}\nwhere $q_x+i q_y = q e^{i \\theta}$,\nwhere the angle $\\theta$ is measured from a given lattice\naxis. The angular dependence of the correlation is consistent with\nthe lattice symmetry; as we show below, only its angular average\nis relevant for the transitions.\n\nIn Eq. (\\ref{corr}) we have assumed that the 2 point function of\n$(\\bm{\\nabla}h)^2$ field has a finite $q=0$ limit. The\nexact bound $\\frac{\\beta}{a} |\\langle \\partial_i h(\\vec r) \\partial_j h(\\vec r') \\rangle|\n\\leq |{\\cal F}_{iijj}(\\vec r-\\vec r')|^{1\/2}$ implies that the\nroughness $h\\sim r^{\\zeta}$ of the graphene sheet (hence of the substrate if adsorbed)\ncan be at most $\\zeta<1\/2$ in the general case for Eq. (\\ref{corr}) to hold. In a model\nwith Gaussian distributed $h$ the condition is $\\zeta<1\/4$ and higher roughness would result in long range (LR)\ncorrelations in the disorder. Such LR correlations would\npresumably arise when quenching thermal fluctuations of a freely\nfluctuating membrane (which has $\\zeta=0.59$ \\cite{radz}) although a precise estimate\nthen requires taking into account non gaussian fluctuations, a non\ntrivial calculation. Here we restrict to SR disorder and\nsubstrates such that Eq. (7) holds.\n\n\n\\section{Electronic density of states.}\nWe analyze the electronic density of states near the Dirac point,\n$E=0$, using the techniques discussed in\\cite{HD02}. The main\ndifference with the cases considered there is the existence of two\nDirac equations coupled to the same gauge field, with couplings of\nequal absolute value but opposite sign, see ${\\cal H}_{elec}$ in\neq.(\\ref{hamil}).\n\nThe bosonized version of the problem also contains two fields,\nwhich become two sets of coupled fields when the replica trick is\nused to integrate over the disorder. Finally, we make the same\nvariational ansatz as in\\cite{HD02}. The simplest observable is\nthe total density of states (DOS), which is self averaging and is\njust twice the DOS of a single Dirac equation (single layer\nproblem as defined in \\cite{HD02}) and behaves as:\n\\begin{equation}\n\\rho ( E ) \\sim E^{2\/z-1}\n\\label{dos}\n\\end{equation}\nwith:\n\\begin{equation}\nz = \\left\\{ \\begin{array}{lr} 2 - K +  \\sigma K^2 &\\sigma <\n2\/K^2 \\\\  K \\left( \\sqrt{8 \\sigma} - 1 \\right) &\\sigma > 2\/K^2  \\end{array} \\right.\n      \\label{exponent}\n\\end{equation}\nwhere $K$ is a parameter which describes the kinetic energy of the\nfield in the bosonized version of the model, and, for the non\ninteracting case which corresponds to the hamiltonian in\neq.(\\ref{hamil}) takes the value $K=1$. The parameter $\\sigma$\ndetermining the exponent in (\\ref{exponent}) is found to be given\nby the angle average of (\\ref{gauge}):\n\\begin{equation}\n \\sigma = \\frac{1}{2 \\pi} \\left(\n  \\frac{\\lambda + \\mu}{\\lambda + 2 \\mu} \\right)^2\n  ( f + 2 f' )  \\label{sigma}\n\\end{equation}\ni.e. the strength of the random gauge field, consistent with the\norder of magnitude estimate (\\ref{estimate}).\n\n\n\nThe change in the dependence of the exponent $z$ on the strength\nof the gauge field, $\\sigma$, in eq.(\\ref{exponent}) is associated\nwith a phase transition in the disordered bosonic model. For\n$\\sigma > \\sigma_c = 2\/K^2$ the local DOS (averaged over regions\nof size up to $L\\sim |E|^{-1\/z}$) exhibits strong fluctuations and\nnon gaussian tails (i.e its disorder average being different from\nits typical value) due to the dominance of rare regions. Note that\nthe divergence of the DOS at $E=0$ occurs at $\\sigma = 1\/K^2 <\n\\sigma_c$, i.e. before the freezing transition in the one layer\nproblem as disorder is increased.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[scale=0.25]{rho1.eps}\n\\end{center}\n\\caption{Sketch of the DOS in a finite size $L$ region for $z<2$.\nThe thick line is a typical value, while the thin line represent the\nsize of fluctuations that are enhanced below the energy $L^{-z}$.\nFor $z>2$ the DOS increases at small $E$ and its typical value\nsaturates at $L^{-2+z}$. For $z>3$ (frozen regime) the fluctuations\nbecome so strong that the average DOS grows as $L^{-2+{\\bar z}}$\nwhere ${\\bar z}=1+\\sigma >z$. Such finite size fluctuations should\nbe observable in tunneling experiments.}\n\\end{figure}\n\n\nThe effect on the DOS of an additional smooth random scalar\npotential with variance $\\delta$, corresponding to local\nfluctuations of the chemical potential, induced by e.g. the\nsubstrate, has been discussed in \\cite{HD02}. It leads to:\n\\begin{equation}\n \\rho(E) = E^{2\/z-1} {\\cal R}(E\/\\delta^{z\/z'})  \\label{w}\n\\end{equation}\nwhere the exponent $z'$ is given by:\n\\begin{equation}\nz' = \\left\\{ \\begin{array}{lr} 2 - 2 K +  4 \\sigma K^2 &\\sigma <\n\\frac{1}{2\n      K^2} \\\\  2 K \\left( \\sqrt{8 \\sigma} - 1 \\right) &\\sigma >\n      \\frac{1}{2\n      K^2} \\end{array} \\right.\n      \\label{exponent1}\n\\end{equation}\nand exhibits a transition at $\\sigma'_c=1\/(2 K^2)=\\sigma_c\/4$.\nThis leads to a finite and non zero DOS at zero energy:\n\\begin{equation}\n \\rho(E) \\sim \\delta^{(2-z)\/z'}  \\label{w1}\n\\end{equation}\na behavior which thus exhibits two distinct freezing transitions.\nThe divergence of the DOS at $\\sigma=1$ (for $K=1$) is in between\nthese transitions.\n\n\nAlthough we will not study this aspect in detail here, it is also\ninteresting to note that since the two Dirac equations describing\nthe two valleys (the two Fermi points) feel opposite random gauge\nfields, mutual correlations of the local DOS in the two valleys as\nmeasured by $\\langle \\rho_1(E,r) \\rho_2(E,r) \\rangle^c$ are\nstrong. They are found to exhibit a transition at a different\nvalue of disorder $\\sigma=1\/(2 K^2)$ as can be seen by a study\nanalogous to the two layer model of Section IV B of \\cite{HD02}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[scale=1]{phase_diagram.ps}\n\\end{center}\n\\caption{(Color online). Critical temperature as function of\nchemical potential. The parameters used are $W_0 = 200$meV, $l_0 =\n10 a$, $\\sigma = 1.4 (z=2.4)$, and $U = 1$eV. The value of $\\sigma$\nimplies an average height fluctuation $h \\approx 3.4$\\AA. The blue\ndiamonds give the critical temperature when the transition is\ndiscontinuous. The green triangles are the values of the gap\n$\\Delta$, in Kelvin, at the transition temperature, in the region\nwhere $\\Delta$ jumps discontinuously from zero to a finite value.}\n\\label{phase_diagram}\n\\end{figure}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[scale=0.4]{phase_separation.eps}\n\\end{center}\n\\caption{(Color online). Approximate phase diagram, as function of\nelectron density and temperature, obtained with the same parameters\nused in Fig.[\\protect{\\ref{phase_diagram}}]. The existence of a\nfirst order transition leads to a region where electronic phase\nseparation is induced.} \\label{phase_separation}\n\\end{figure}\n\n\n\n\\section{Interaction effects and electronic instabilities.}\nFor sufficiently large disorder, $\\sigma > 1$, the density of\nstates, and, as a consequence, the electronic compressibility,\ndiverges at $E=0$. The electron-electron interaction, or the\ninteraction of the electrons with other degrees of freedom, will\nlead to instabilities, which suppress the compressibility.\n\nWithin mean field theory, the effects of interactions on the\nelectronic band structure can be described as an external\npotential which must be calculated self consistently. A simple\nsuch potential which opens a gap at $E=0$ is the shift of the\nenergy on one sublattice of the honeycomb structure with respect\nto the other. This shift can be associated to a spin or to a\ncharge density wave, or it can be induced by phonons\\cite{FL07} or\nby short range electron-electron\ninteractions\\cite{ST92,GGV01,PAB04,H06}. In the continuum model\ndescribed here, it enters as a mass term:\n\\begin{eqnarray}\n{\\cal H}_{elec}^{tot} &= &{\\cal H}_{elec} + {\\cal H}_\\Delta \\nonumber \\\\\n{\\cal H}_\\Delta &= &\\Delta \\sum_i \\int d^2 \\vec{r} \\left[ \\bar{\\Psi}_i (\n\\vec{r} ) \\sigma_z \\Psi_i ( \\vec{r} ) \\right]\n\\end{eqnarray}\nwhere ${\\cal H}_{elec}$ is defined in eq.(\\ref{hamil}).\nThe total electronic hamiltonian satisfies:\n\\begin{equation}\n\\left( {\\cal H}_{elec} + {\\cal H}_\\Delta \\right)^2 = \\left( {\\cal H}_{elec}\n\\right)^2 + \\Delta^2 {\\cal I}\n\\end{equation}\nwhere ${\\cal I}$ is the four dimensional unit matrix, independent\nof spatial position, which acts on the space spanned by the four\ncomponent electronic (iso)-spinors. The eigenvalues of the\nelectronic hamiltonian satisfy ${\\epsilon_n^{tot}}^2 =\n\\epsilon_n^2 + \\Delta^2$, where $\\epsilon_n$ is an eigenvalue of\n${\\cal H}_{elec}$. As a result, the density of states associated\nto ${\\cal H}_{elec}^{tot}$, $\\rho_\\Delta ( E )$,  satisfies:\n\\begin{equation}\n\\rho_\\Delta ( E ) = \\frac{E}{\\sqrt{E^2 - \\Delta^2}} \\rho (\n\\sqrt{E^2 - \\Delta^2})\n\\end{equation}\nand, using the expression in eq.(\\ref{dos}), we find:\n\\begin{equation}\n\\rho_\\Delta ( E ) = \\left\\{ \\begin{array}{lr} 0 &| E | < \\Delta \\\\\n    \\frac{1}{l_0^2} \\frac{E \\left( \\sqrt{E^2 - \\Delta^2}\n    \\right)^{2\/z-2}}{W_0^{2\/z}} &W_0 > |E|>\\Delta \\end{array} \\right.\n\\label{dos_D}\n\\end{equation}\nwhere the energy scale $W_0 = v_{\\rm F} \/ l_0$ is inserted so that for\n$E \\gg W_0 \\gg \\Delta$ the value of $\\rho_\\Delta ( E )$ crosses\nover into the density of states of the clean system, $\\rho ( E )\n\\sim | E | \/ v_{\\rm F}^2$.\n\nThe self consistent value of $\\Delta$ is determined by the\ncompetition between the cost in energy associated to the formation\nof the gap, and the decrease in electronic energy due to the\nreduction in the density of states at the Fermi level. Near the\ntransition, $\\Delta$ is small compared to the other energy scales\nof the model, and the energy required to create the charge or spin\ndensity wave can be expanded as function of $\\Delta$. For\nsimplicity, we assume that the ordered phase is a spin density\nwave induced by the on-site Hubbard repulsion, $U$ which breaks\nthe sublattice symmetry and produces a gap $\\Delta=U S\/2$ where\n$S$ is the resulting polarization per site. The total energy is\nthe sum of the kinetic energy and the gain in interaction energy\nobtained by inducing the polarization:\n\\begin{eqnarray}\nE_{tot} &= &E_{elec} + E_{SDW} \\nonumber \\\\\n E_{SDW} &= &  \\frac{\\Delta^2}{U} \\nonumber \\\\\nF_{el}&=&-4Ta^2\\int_{\\Delta}^{W_0}[\\ln\n(1+\\mbox{e}^{(-E-E_F)\/T})+\\nonumber\\\\&&\\ln\n(1+\\mbox{e}^{(E-E_F)\/T})]\\cdot\\rho_{\\Delta}(E)dE\n\\end{eqnarray}\nwhere $4$ allows for spin and valley degeneracy and we allow for a\npossibly non zero Fermi energy $E_F$. The induced gap is given by\nminimizing the total free energy and with a change of integration\nvariable:\n\\begin{eqnarray}\\label{sce}\n1&=&2a^2U\n\\int_0^{W_0}\\frac{\\sinh(\\sqrt{E^2+\\Delta^2}\/T)}{\\cosh(E_F\/T)+\\cosh(\\sqrt{E^2+\\Delta^2}\/T)}\n\\times\\nonumber\\\\&& \\frac{\\rho(E)dE}{\\sqrt{E^2+\\Delta^2})}\n\\end{eqnarray}\n\n\nWe consider first the case $E_F = T = 0$ where $F_{el}\\rightarrow\nE_{el}=-4a^2\\int_{\\Delta}^{W_0}E\\rho_{\\Delta}(E)dE$. The integrand\n can be expanded for $E \\gg\n\\Delta$, where it goes as $\\Delta^2 E^{(2\/z)-2}$. As a result, we\nobtain a contribution to the electronic energy $\\delta_1 E_{elec}\n( \\Delta ) \\sim - ( a \/ l_0 )^2 \\Delta^2 \/ W_0$. There is also a\ncontribution from the region $E \\sim \\Delta$. This term in\n$E_{elec} ( \\Delta )$ can be written as $\\delta_2 E_{elec} (\n\\Delta ) \\sim - ( a \/ l_0 )^2 \\Delta^{(2\/z)+1} W_0^{-2\/z}$.\n\nThe relative strength of the two terms discussed above leads to\nthe existence of three regimes: i) $2\/z-1 > 0$. The electronic\nenergy is determined by $\\delta_1 E_{elec} ( \\Delta )$. Both the\nmagnetic and electronic energy go as $\\sim \\Delta^2$, and, for $U\n\\ll v_{\\rm F} \/ a$ the minimum energy is at $\\Delta=0$. ii) For $2\/z-1 =\n0$, we find $\\delta_1 E_{elec} \\sim -2( a \/ l_0 )^2\\Delta^2 \/ W_0\n\\log ( W_0 \/ \\Delta )$. The magnetic energy is greater by a\nlogarithmic factor, and there is an ordered phase, with $\\Delta\n\\sim W_0 e^{- ( W_0 l_0^2 ) \/ ( 2U a^2 )}$. The problem becomes\nequivalent to the Peierls analysis of the instability of a one\ndimensional metal. iii) For $2\/z-1 < 0$, the leading contribution\nis $\\delta_2 E_{elec}$. There is a magnetic phase with a gap:\n\\begin{equation}\n\\Delta_c \\sim W_0  \\left( \\frac{a^2 U}{l_0^2 W_0}\n\\right)^{\\frac{z}{z-2}}\n\\end{equation}\n\nWe now analyze the way in which the magnetic phase which always\nexists for $2\/z-1 <0$ is modified when $E_F , T \\ne 0$. In\nparticular the order of the transition is determined by the sign\nof the $a_4$ coefficient in the free energy expansion\n$F=a_2\\Delta^2+a_4\\Delta^4$. Taking a $\\partial_{\\Delta^2}|_0$ on\nthe right hand side of Eq. (\\ref{sce}) yields $a_4$, hence the\nsimultaneous conditions $a_2=0$ and $a_4=0$ determines a critical $E_F$,\nwith $E_F^c=\\alpha(z)T_c$ whenever $T_c \\ll W_0$,\nsuch that the transition changes from second order\nfor $E_F<E_F^c$, to first order in the region $E_F>E_F^c$ where we have $a_4<0$.\nWe find numerically that\n$\\cosh \\alpha(z)$ varies between 3.4 at $z=2$ and 2 at $z\\rightarrow\n\\infty$.\n\nA typical phase diagram with $z>2$ is shown in\nFig.[\\ref{phase_diagram}], where the value of the gap $\\Delta_c$ at\nthe critical temperature, in the region where the transition is\nfirst order is also shown. When the line of first order transitions\nis crossed, the electron density jumps discontinuously. For\nsufficiently large values of $E_F$ we find that $\\Delta_c (E_F )\n> E_F$. When the transition line is crossed in this region, the\nelectron density in the ordered phase is zero. The phase diagram as\nfunction of temperature and electron density is shown in\nFig.[\\ref{phase_separation}].\n\n\\section{Formation of lattice defects.}\n\\subsection{Unbinding of dilocations.}\n\nAs discussed earlier, ripples, e.g. due pinning to a rough substrate, induce in plane strains. If these strains\nare sufficiently large, it will become favorable to relax them by creating lattice dislocations. It is\nconvenient to view the out of plane deformations as inducing quenched random stresses coupling linearly to the\nin plane strain tensor $\\tilde u_{ij}$ via an energy density $\\sum_{ij} \\sigma_{ij} \\tilde u_{ij}$. One can then\napply the result of Ref. \\onlinecite{CD98b} for the threshold beyond which random stresses generate\ndislocations.\n\nUsing eq.(\\ref{hamil}), the random stress tensor field which renormalizes the fugacity of dislocations is:\n\\begin{eqnarray}\n\\sigma_{xx} &= &\\frac{\\lambda}{2} \\left[ \\left( \\frac{\\partial h}{\\partial x} \\right)^2 + \\left( \\frac{\\partial\nh}{\\partial y} \\right)^2 \\right] + \\mu \\left( \\frac{\\partial h}{\\partial x}\n\\right)^2 \\nonumber \\\\\n\\sigma_{yy} &= &\\frac{\\lambda}{2} \\left[ \\left( \\frac{\\partial h}{\\partial x} \\right)^2 + \\left( \\frac{\\partial\nh}{\\partial y} \\right)^2 \\right] + \\mu \\left( \\frac{\\partial h}{\\partial y}\n\\right)^2 \\nonumber \\\\\n\\sigma_{xy} &= &\\mu \\frac{\\partial h}{\\partial x} \\frac{\\partial h}{\\partial y}\n\\end{eqnarray}\nWe assume that the correlations of this field are given by:\n\\begin{equation}\n\\left\\langle \\sigma_{ij} ( \\vec q ) \\sigma_{kl} ( - \\vec q ) \\right\\rangle|_{q \\to 0} = \\left[ \\sigma_{\\lambda}\n\\delta_{ij} \\delta_{kl} +\n \\sigma_{\\mu} \\left( \\delta_{ik} \\delta_{lj} + \\delta_{il} \\delta_{jk}\n\\right) \\right]\n\\end{equation}\nwhere the parameters $\\sigma_{\\mu}$ and $\\sigma_{\\lambda}+ 2 \\sigma_\\mu$ measure the strength of random shear\nstresses and compressional stresses, respectively.\n\nIn presence of random stresses an isolated dislocation in a region of size $L$ feels a random potential whose\nminima grow typically $\\sim - \\ln L$. The logarithmic elastic energy cost of creating a dislocation can then be\novercome, and thus dislocations will proliferate at $T=0$, when:\n\\begin{equation}\n\\tilde \\sigma = \\frac{\\lambda ( \\lambda + 2 \\mu ) \\sigma_{\\mu} + \\mu^2 (\\sigma_{\\lambda} + 2 \\sigma_\\mu) }{\\mu^2\n( \\lambda +  \\mu )^2} \\ge \\tilde \\sigma_c = \\frac{a^2}{16 \\pi} \\label{sigma_dis}\n\\end{equation}\nwhere we have neglected the effect of screening of the elastic\ncoefficients by disorder, which have been shown to be small\n\\footnote{We have also taken into account the factor 2 misprint in\n$\\sigma$ as defined below Eq. (5) in \\cite{CD98b}}). To the same\naccuracy this formula holds for all $T < T_m\/2$ where $T_m= K_0\na^2\/(16 \\pi)$ is the KTHNY melting temperature of a pure 2d crystal,\nwith $K_0=4 \\mu (\\mu+\\lambda)\/(\\mu+2 \\lambda)$, while the threshold\ndecreases as $\\tilde \\sigma_c(T)=4 \\tilde \\sigma_c\n\\frac{T}{T_m}(1-\\frac{T}{T_m})$ at higher $T$. For $\\tilde \\sigma >\n\\tilde \\sigma_c$ and at $T=0$ the scale $L$ above which dislocation\nfirst appear can be estimated as in \\cite{BHPLD} and corresponds to\nthe total energy cost $\\frac{K_0 a^2}{8 \\pi} (1 - \\sqrt{\\tilde\n\\sigma\/\\tilde \\sigma_c})  \\ln (L\/l_0) +E_c$ becoming negative. We\nhave taken into account the dislocation core energy $E_c = E_c^0 +\n\\frac{K_0 a^2}{8 \\pi} \\ln (l_0\/a)$ at scale $l_0$ ($E_c^0$ denotes\nthe bare core energy). Because of logarithms this scale can be large\nhence it can alternatively be viewed as defining an effective size\ndependent threshold $\\tilde \\sigma_c(L)$. The dislocation density\nabove this scale can be estimated by arguments similar to\n\\cite{pldtg}.\n\nThe quantities $\\sigma_{\\lambda}$ and $\\sigma_{\\mu}$ can be written in terms of the correlations of the function\n$f_{ij}$, given in eqs.(\\ref{corr}) and (\\ref{gauge}):\n\\begin{eqnarray}\n\\sigma_{\\lambda} &= &\\frac{a^2}{\\beta^2} \\left[ \\mu^2 ( f + 2 f' ) + \\lambda ( \\lambda + 2 \\mu ) ( f +\nf' ) \\right]\\nonumber \\\\\n\\sigma_{\\mu} &= &\\frac{a^2}{\\beta^2} \\mu^2 f' \\label{corr2}\n\\end{eqnarray}\nInserting this result in eq.(\\ref{sigma_dis}), and assuming that $\\beta , \\lambda \/ \\mu \\sim O ( 1 )$, we find\nthat dislocations will proliferate when the height correlations are such that $h^2 \/ ( l_0 a ) \\gtrsim 1$, which\nis the same combination of scales which determines the existence of a divergence in the electronic density of\nstates.\n\n\\subsection{Buckling into the third dimension.}\nAn effect not taken into account above is that dislocations may buckle in the third dimension to lower their\nenergy. For a free membrane (in the absence of a substrate) this occurs for scales larger than the buckling\nradius $R_b$, and below that scale the membrane remains flat and Coulomb gas logarithmic scaling holds. In\nprinciple, for a free membrane in presence of internal in plane random stresses, if $R_b$ is large enough\n(values such as $R_b \\sim 10^2 \\kappa\/(K_0 a)$ are quoted in Ref. \\cite{nelson_seung}), i.e. if $R_b\n> l_0 \\gg a$, the above energy estimate setting $L=R_b$ can be used to determine the disorder threshold at which\nbuckled dislocations would occur. However, if one takes into account the pinning of the height field to the\nsubstrate, the energy calculation of Ref. \\cite{nelson_seung}) remains valid for scales smaller than $l_p$, but\nmust be reexamined for scales larger than $l_p$, a problem left for future study.\n\n\\subsection{Gauge fields associated to dislocations.}\nNote, finally, that dislocation cores act on the electrons outside\nthe core as vortices \\cite{GGV01} of flux $\\Phi = \\epsilon \\Phi_0 \/\n3$, where $\\Phi_0$ is the quantum unit of flux ($=2 \\pi$ in our\nunits), and $\\epsilon = \\pm 1$. Hence, the existence of dislocations\nwill increase the random field due to elastic strains considered so\nfar. Given a set of dislocations at position $\\vec r_n$ and Burgers\ncharges $\\vec b_n$ the resulting effective magnetic field\n can be written $B(\\vec\nr) = (\\Phi_0\/3) n(\\vec r)$ where $n(\\vec r)=\\sum_n \\epsilon_n\n\\delta(\\vec r - \\vec r_n)$ and the signs are given by $\\epsilon = 2\n\\vec b \\cdot a_1 \\text{mod} 2 \\pi$. If positions and signs were\nchosen uncorrelated (such as in a quench from infinite temperature)\nit would result in a LR correlated random gauge field, i.e $C_B(q)\n\\sim \\Phi_0^2 d^{-2}$ at small $q$ in (\\ref{defB}), where $d$ is the\nmean distance between defects\\footnote{Strictly, $B$ is zero outside\nthe core and $\\vec A$ is a pure (singular) gauge, with correlations\nwhich diverge\\cite{K07} as $| \\vec{q} |^{-2}$. A coarse grained\nfield in the whole space can be computed\\cite{GGV01}- including the\ncores - by assigning minimal flux to each vortex (note, however,\nthat each of these fluxes can be increased by an integer flux\nquantum without changing the eigenenergies while increasing the\ndegeneracy).}\n\nThis procedure however leads to Burgers charge fluctuations growing\nas $\\pm \\sim L$ in an area $L^2$ hence a very large elastic energy,\n$L \\ln L$. If the system can relax, this energy is screened and the\nresult is a finite parameter $\\sigma$ as defined in\n(\\ref{estimate}). In cases where the dislocation density is not very\nsmall it can be estimated from a Debye-H\\\"uckel theory. One\nnon-equilibrium example is a quench of a pure crystal from a\n(moderate) temperature $T_Q>T_m$ to low temperature in which case\n$\\langle n(\\vec q) n(-\\vec q) \\rangle = T_Q q^2\/(E_c^0 q^2 + K_0\na^2)$ hence $\\sigma=T_Q \\Phi_0^2\/(9 \\pi K_0 a^2)$. Further\nrelaxation of $\\sim \\ln L$ energy would then occur. Another example\nis the distribution of dislocations induced by the ripples as in\n(\\ref{sigma_dis}), with $\\tilde \\sigma\n> \\tilde \\sigma_c$. Then one estimates \\footnote{Due to the relation between flux and Burgers vector the\nproblem becomes isotropic and one may neglect the vector nature of\nthe charges} $\\langle n(\\vec q) n(-\\vec q) \\rangle = \\frac{1}{4} q^2\n\\tilde \\sigma K_0^2 a^2\/(E_c^0 q^2 + K_0 a^2)^2$ hence $\\sigma=\n\\tilde \\sigma \\Phi_0^2\/(36 \\pi a^2)$. Very near the transition\nDebye-H\\\"uckel does not apply as $\\sigma$ vanishes at $\\sigma_c$\nproportionally to the density of dislocations.\n\n\\section{ripples in suspended graphene}\n\nFinally we discuss a possible source for ripples in suspended\ngraphene\\cite{Metal07,Betal08,DSBA08}. Upon etching a preexisting\nrough substrate, the rippled graphene sheet would tend to relax to a\nflat configuration with higher projected area. This however may be\nprecluded if the sheet is pinned at its boundaries. Indeed it is\nknown that fixed connectivity membranes exhibit a buckled state when\nconstrained at their boundaries by a fixed frame of projected area\n$A_f$ smaller than the equilibrium area of the unconstrained\nmembrane $A$. As discussed in \\cite{buckling1} it results in an\nadditional compressional energy term of the form $\\tau \\int d^2 \\vec\nr \\sum_i u_{ii}$ and hence implies that the energy of flexural modes\nbecomes, to lowest order $\\frac{1}{2} \\int d^2 \\vec r [ \\kappa\n(\\nabla^2 h)^2 + \\tau (\\nabla h)^2]$. In the buckled phase, $A_f<A$,\n$\\tau <0$, an instability thus develops at scales larger than $\\xi_h\n\\sim (\\kappa\/|\\tau|)^{1\/2}$. This phase can be described as a non\nhomogeneous mixture of pure flat phases with different orientations.\nFor arbitrary boundary conditions, it is expected to be non trivial\nsince, contrary to a one dimensional rod, in a polymerized membrane\nthe in-plane modes cannot fully relax the flexural constraints, i.e.\nthe transverse part of the flexural strain tensor, $P^T_{i\nj}(\\nabla) (\\partial_i h \\partial_j h)$, cannot be relaxed by the\nin-plane strain field. While demonstrating the instability is\nsimple, the full calculation of the resulting shape requires\nconsideration of non linear terms and is difficult. The problem of\nrelaxation from a randomly rippled configuration with a fixed frame\nconstraint deserves further study, in particular the question of\nwhether there is some memory of the initial ripple pattern.\n\nNote that one may consider, alternatively, unconstrained boundary and apply a tension $- f \\int d^2 \\vec r\n\\partial_i u_{i}$. The buckling transition \\cite{buckling1} has been mostly studied on the side where the sheet\nis stretched (the effective $\\tau_R \\to 0^+$). It was found that $(A_f-A)\/A \\sim \\tau_R \\sim |f|^{1\/\\delta}\n\\text{sign}(f)$ and the correlation lengths $\\xi_h \\sim \\xi_u \\sim |f|^{-\\nu\/\\delta}$ for flexural and phonon\nmodes at small $f$. While entropic effects produce non trivial values for these exponents, these may be\nobservable only at large scales, and at intermediate scales mean field values $\\delta=1$, $\\nu=1\/2$ (discussed\nabove) are appropriate. It would thus be interesting to study the other side of this transition.\n\n\n\\section{Broadening of Landau levels in presence of a real magnetic field}\n\n In presence of a real magnetic field $B$ we expect that the\nLandau levels will be broadened by the effective field $B_{rip}$\ndue to the ripples. Alternatively, the local electronic density corresponding to $N$ full Landau\nlevels is fluctuating according to $n({\\bf r})=[B+B_{rip}({\\bf r})] N\/\\phi_0$,\nwhere $\\phi_0$ is the flux quantum. Such density fluctuations were\nrecently measured \\cite{Metal07b} showing $\\delta n=\\pm 2.3\\cdot\n10^{11}$cm$^{-2}$ at a field of $11T$ for $N=2,6,10$. In this\nsection we estimate the contribution of the random gauge field due to ripples to these density\nfluctuations.\n\nConsider the density $n$ measured on a length scale $L$, and its probability\ndistribution, near an average density\n$BN\/\\phi_0$.\nWe assume first\n$l_0>l_B\\sqrt{N}$, where $l_B=\\sqrt{\\phi_0\/B}$ is the cyclotron\nradius and $l_B\\sqrt{N}$ estimates the size of an orbit in the N-th Landau\nlevel. Each Landau orbit has then a random shift $\\pm\nB_{rip}({\\bf r})$ where the $\\pm$ corresponds to the K and K' valleys\nthat feel opposite gauge fields. The density distribution is\n\\begin{equation} P(n)=\\int_{r<L} \\sum_{\\pm}\\delta(n-BN\/\\phi_0\\pm\nB_{rip}(r)N\/\\phi_0)d^2r\/2L^2 \\end{equation}\nThe average is $\\langle n\\rangle=BN\/\\phi_0$ while the variance is:\n\\begin{equation} \\label{variance} \\langle \\delta\nn^2\\rangle=\\frac{N^2}{\\phi_0^2 L^4}\\langle\n[\\int_{r<L}B_{rip}({\\bf r})d^2r]^2\\rangle\n=\\frac{N^2}{4\\pi^2L^4}\\langle [\\oint A(u)du]^2\\rangle\n\\end{equation}\nwhere $B_{rip}=[\\bm{\\nabla}\\times A]_z\\phi_0\/2\\pi$ and $A$ is the random\ngauge field considered in the previous Sections (with the appropriate\nchange in units) and the contour encloses the area of measurement.\nTo estimate the variance in (\\ref{variance}) we use\nEq. (\\ref{estimate}) with a cutoff $\\exp{(-q^2l_0^2\/2)}$.\nIn real space it corresponds to  $\\langle\nA_i({\\bf r})A_j({\\bf r})\\rangle \\sim (\\sigma\/2l_0^2)\\exp{[-({\\bf r}-{\\bf r}')^2\/2l_0^2]}$\nwhere we neglect \\footnote{a similar estimate can\nbe done using the magnetic field and yields the same result}\nthe transversality constraint on $A$. It yields:\n\\begin{equation}\\label{dn}\n\\langle \\delta n^2\\rangle_L \\approx \\frac{N^2\\sigma}{4\\pi l_0 L^3}\n\\end{equation}\n\nThe experimentally more relevant case is $l_0<l_B\\sqrt{N}$. In\nthis case we argue that ${\\bf A}({\\bf r})$ can be replaced by its\naverage within a Landau state, so that an average ripple field is\n\\begin{equation}\nB_{av}({\\bf r})=\\frac{1}{2\\pi Nl_B^2}\\int d^2r_0\nB_{rip}({\\bf r}_0)\\mbox{e}^{-({\\bf r}-{\\bf r}_0)^2\/2Nl_B^2}\n\\end{equation}\nand its Fourier transform is\n$B_{av}(q)=B_{rip}(q)\\exp(-q^2Nl_B^2\/2)$. This replaces\n$l_0\\rightarrow l_B\\sqrt{N}$ in Eq. (\\ref{dn}), and identifying\n$L$ with the tip size $l_{tip}$ in the experiment \\cite{Metal07b},\nwe obtain,\n\\begin{equation}\\label{dn1}\n\\langle \\delta n^2\\rangle_L \\approx N^{3\/2}\\frac{\\sigma}{4\\pi l_B\nl_{tip}^3}\n\\end{equation}\nUsing \\cite{Metal07b} $l_0\\approx 100$nm  and $l_B\\approx 10$nm,\nEq. (\\ref{dn1}) yields numbers consistent with the experiment,\nexcept for the $N$ dependence. Note that an even weaker dependence\nin $N$ ($\\delta n \\sim N^{1\/4}$) is obtained if one assumes that\n$L=l_B \\sqrt{N}$ is the only averaging scale.\n\nIt is also interesting to estimate the energy broadening $\\delta\n\\epsilon_N$ of the Landau levels, which in the absence of ripples\nhave energies $\\epsilon_N = v_F\\sqrt{2e} \\sqrt{B N'}$ with\n$N=4N'+2$. The field associated with the ripples changes locally the\nenergy of the Landau levels, which become $\\epsilon_N = v_F\\sqrt{2e}\n\\sqrt{( B \\pm \\delta B_{rip}) N'}$, where $\\delta B_{rip}$ is the\naverage value of $B_{rip}$ in the region occupied by the Landau\nlevel. A similar calculation then yields the estimate for $N>2$\n\\begin{equation}\n\\delta \\epsilon_{N'}\\approx v_F\\left(\\frac{\\sigma}{32\\pi l_0\nl_B}\\right)^{1\/2}N^{-1\/4}\n\\end{equation}\nin the regime $l_B\\sqrt{N}>l_0$.\n\nFinally we note, that the $N'=0$ level has no broadening at all.\nThis remarkable result is obtained by factorizing the $N'=0$\neigenstates of the free Dirac system in a magnetic field with the\nwell known zero energy solutions of the random gauge problem\n\\cite{LFSG94,HD02}. This set has the proper Landau degeneracy and\nis therefore an exact solution for the zero energy Landau level\nwith random gauge.\n\n\n\n\n\n\\section{Conclusions.}\nWe have analyzed the effect of random gauge fields on the electronic\nstructure of corrugated graphene. We find that the local density of states\ndiverges at the Dirac energy $E=0$, as $\\rho ( E ) \\propto E^{2\/z-1}$, with\n$z>2$, for sufficiently strong disorder. The scale of height fluctuations,\n$h$ should satisfy $\\beta h^2 \/ ( l a ) \\gtrsim 1$, where $\\beta \\sim 1-2 $ gives the\ncoupling between the electrons and the lattice strains, $l$ is the typical\nspatial scale of the disorder, and $a$ is the lattice constant.\n\nA divergence in the density of non interacting density of states\nimplies the existence of instabilities in the presence of\nelectron-electron interactions. We have analyzed the possibility\nthat a gap will open at low temperatures, depleting the low energy\ndensity of states. We have found a first order transition to an\nordered state at large $E_F$. This discontinuous transition, in\nturn, implies electronic phase separation.\n\nWhen the strains which induce the gauge potential are sufficiently\nstrong, they can lead to an instability, and the formation of\nlattice dislocations. This change takes place for $C ( \\lambda ,\n\\mu ) h^2 \/ ( l a ) \\gtrsim 1$, where $C ( \\lambda , \\mu ) \\sim 1$\nis a dimensionless parameter which depends on the elastic\nconstants of the material.\n\nWe have described the main features of the buckling instability\nwhich may arise in suspended systems under compression. Finally, we\nanalyze the changes induced in the Landau levels induced by a\nmagnetic field by the gauge potential associated to ripples and show\ncorrespondence with experimental data \\cite{Metal07b.}\n\nOur analysis is consistent with previous work on the changes in the\nelectronic density of states in graphene in the presence of\nripples\\cite{GKV07} (see also\\cite{WBKL07}). A transition to a state\nmagnetically ordered in highly disordered systems agrees with the observation\nof magnetism in irradiated graphite samples\\cite{Eetal03}. The existence\nof charge inhomogeneities, due to electronic phase separation, can help to\nexplain the observation of charge puddles when the Fermi energy is close to\nthe Dirac energy\\cite{Metal07b}.\n\\section{Acknowledgments.} This work was supported by MEC (Spain) through grant\nFIS2005-05478-C02-01, the Comunidad de Madrid, through the program\nCITECNOMIK, CM2006-S-0505-ESP-0337, the European Union Contract\n12881 (NEST), ANR program 05-BLAN-0099-01 and the DIP German Israeli\nprogram. B.H. and F. G. thank the Ecole Normale Sup\\'{e}rieure for\nhospitality and for support during part of this work.\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}