General Theory of Large D Membranes Consistent with Second Law of Thermodynamics
Arunabha Saha
备注：
36+10 pages, Section on matching membrane entropy to Wald entropy changed to include discussions on modified entropy current. Corresponding changes included in abstract and other relevant places in the paper
We write down the most general membrane equations dual to black holes for a general class of gravity theories, up to sub-leading order in $1/D$ in large $D$ limit. We derive a "minimal" entropy current which satisfies a local form of second law from these membrane equations. We find that consistency with second law requires the membrane equations to satisfy certain constraints. We find additional constraints on the membrane equations from the existence of membrane solutions dual to stationary black holes. Finally we observe a tension between second law and matching with Wald entropy for dual stationary black hole configurations, for the minimal entropy current. We propose a simple modification of the membrane entropy current so that it satisfies second law and also the stationary membrane entropy matches the Wald entropy.
We derive the effective equations of the membranes dual to black holes in a particular theory of higher derivative gravity namely Einstein-Gauss-Bonnet (EGB) gravity at sub-leading order in $1/D$ upto linear order in the Gauss-Bonnet (GB) parameter $β$. We find an expression for an entropy current which satisfies a local version of second law onshell in this regime. We also derive the membrane equations upto leading order in $1/D$ but non-perturbatively in $β$ for EGB gravity. In this regime we write down an expression for a world-volume stress tensor of the membrane and also work out the effe…
It has recently been demonstrated that black hole dynamics at large D is dual to the motion of a probe membrane propagating in the background of a spacetime that solves Einstein's equations. The equation of motion of this membrane is determined by the membrane stress tensor. In this paper we `improve' the membrane stress tensor derived in earlier work to ensure that it defines consistent probe membrane dynamics even at finite $D$ while reducing to previous results at large D. Our improved stress tensor is the sum of a Brown York term and a fluid energy momentum tensor. The fluid has an unusual…
We study the membrane paradigm for horizons in Lanczos-Lovelock models of gravity in arbitrary D dimensions and find compact expressions for the pressure p and viscosity coefficients ηand ζof the membrane fluid. We show that the membrane pressure is intimately connected with the Noether charge entropy S_Wald of the horizon when we consider a specific m-th order Lanczos-Lovelock model, through the relation pA/T=(D-2m)/(D-2)S_Wald, where T is the temperature and A is the area of the horizon. Similarly, the viscosity coefficients are expressible in terms of entropy and quasi-local energy associat…
To an outside observer, a black hole's event horizon appears to behave exactly like a dynamical fluid membrane. We extend this membrane paradigm to black holes in general $f(R)$ theories of gravity. We derive the stress tensor and various transport coefficients of the fluid and find that the membrane behaves as a non-Newtonian fluid except for the special case of Einstein gravity. Using Euclidean methods, we study the thermodynamics of the membrane. We speculate on what theories of gravity admit horizons with fluid properties.
We discuss a two-body interaction of membrane fuzzy spheres in a pp-wave matrix model at finite temperature by considering a fuzzy sphere rotates with a constant radius r around the other one sitting at the origin in the SO(6) symmetric space. This system of two fuzzy spheres is supersymmetric at zero temperature and there is no interaction between them. Once the system is coupled to the heat bath, supersymmetries are completely broken and non-trivial interaction appears. We numerically show that the potential between fuzzy spheres is attractive and so the rotating fuzzy sphere tends to fall i…
We study first order fluctuations of a relativistic membrane in the curved background of a black hole. The zeroth-order solution corresponds to a spherical membrane tightly covering the event horizon. We obtain a massive Klein-Gordon equation for the fluctuations of the membrane's radial coordinate on the 2+1 dimensional world-volume. We finally suggest that quantization of the fluctuations can be related to black hole's mass quantization and the corresponding entropy is computed. This entropy is proportional to the membrane area and is related to the one-loop correction to the thermodynamical…
The Bondi-van der Burg-Metzner-Sachs (BMS) group is the asymptotic symmetry group of asymptotically flat spacetime. It is infinite dimensional and entails an infinite number of conservation laws. According to the black hole membrane paradigm, null infinity (in asymptotically flat spacetime) and black hole event horizons behave like fluid membranes. The fluid dynamics of the membrane is governed by an infinite set of symmetries and conservation laws. Our main result is to point out that the infinite set of symmetries and conserved charges of the BMS group and the membrane paradigm are the same.…
It has recently been shown that the dynamics of black holes in large number of dimensions D can be recast as the dynamics of a probe membrane propagating in the background spacetime which solves Einstein equations without matter. The equations of motion of this membrane are simply the statement of conservation of the stress tensor and charge current defined on this membrane. In this paper we obtain the effective equations of motion for stationary membranes in any empty background both in presence and absence of charge. It turns out that the thermodynamic quantities associated with the stationa…
Black rings are the black objects found in $D$ spacetime dimensional gravity when $D \geq 5$. These have event horizon topology $S^{D-3}\times S^1$. In this work the solutions of the large $D$ membrane paradigm dual to stationary black rings in Einstein-Maxwell theory with or without cosmological constant are studied. It is shown that the first order membrane equations can only admit static asymptotically flat black rings, and the equilibrium angular velocity for the asymptotically AdS black rings at large $D$ was obtained. The thermodynamic and dynamic stability of the asymptotically flat bla…
We study spacetime thermodynamics for non-equilibrium processes. We first generalize the formulation of spacetime thermodynamics by using an observer outside the horizon. Then we construct the entropy balance equation of spacetime thermodynamics for non-equilibrium processes in f(R) gravity. The coefficients of the expansion and shear terms are equal to the viscosities of the black hole membrane paradigm, and a new entropy production term appears.
The membrane paradigm is the remarkable view that, to an external observer, a black hole appears to behave exactly like a dynamical fluid membrane, obeying such pre-relativistic equations as Ohm's law and the Navier-Stokes equation. It has traditionally been derived by manipulating the equations of motion. Here we provide an action formulation of this picture, clarifying what underlies the paradigm, and simplifying the derivations. Within this framework, we derive previous membrane results, and extend them to dyonic black hole solutions. We discuss how it is that an action can produce dissipat…
We analyze the statistical mechanics of a free-standing quantum crystalline membrane within the framework of a systematic perturbative renormalization group (RG). A power-counting analysis shows that the leading singularities of correlation functions can be analyzed within an effective renormalizable model in which the kinetic energy of in-plane phonons and subleading geometrical nonlinearities in the expansion of the strain tensor are neglected. For membranes at zero temperature, governed by zero-point motion, the RG equations of the effective model provide a systematic derivation of logarith…
We propose a quantum description of black holes. The degrees of freedom to be quantized are identified with the microscopic degrees of freedom of the horizon, and their dynamics is governed by the action of the relatistic bosonic membrane in $D=4$. We find that a consistent and plausible description emerges, both at the classical and at the quantum level. We present results for the level structure of black holes. We find a ``principal series'' of levels, corresponding to quantization of the area of the horizon. From each level of this principal series starts a quasi-continuum of levels due to …
The character of elastic forces of relativistic membranes and $p$-branes encoded in their nonlinear equations is studied. The toroidal brane equations are reduced to the classical equations of anharmonic elastic media described by monomial potentials. Integrability of the equations is discussed and some of their exact solutions are constructed.
We discuss recent work showing that in certain cases the membrane paradigm equations governing the dynamics of black hole horizons can be recast as relativistic conservation law equations. In the context of gauge/gravity dualities, these equations are interpreted as defining the viscous hydrodynamics of a holographically dual relativistic field theory. Using this approach, one can derive the viscous transport coefficients and the form of the entropy current for field theories dual to gravity plus matter fields.
We describe the horizon of a quantum black hole in terms of a dynamical surface which defines the boundary of space-time as seen by external static observers, and we define a path integral in the presence of this dynamical boundary. Using renormalization group arguments, we find that the dynamics of the horizon is governed by the action of the relativistic bosonic membrane. {}From the thermodynamical properties of this bosonic membrane we derive the entropy and the temperature of black holes, and we find agreement with the standard results. With this formalism we can also discuss the correctio…
A study of the thermodynamics in IIA Matrix String Theory is presented. The free string limit is calculated and seen to exactly reproduce the usual result. When energies are enough to excite non-perturbative objects like D-particles and specially membranes, the situation changes because they add a large number of degrees of freedom that do not appear at low energies. There seems to be a negative specific heat (even in the Microcanonical Ensemble) that moves the asymptotic temperature to zero. Besides, the mechanism of interaction and attachment of open strings to D-particles and D-membranes is…
Following the Membrane Paradigm, we show that the stretched horizon of a black hole retains information about particles thrown into the hole for a time of order the scrambling time m ln(m/M_P), after the particles cross the horizon. One can, for example, read off the proper time at which a particle anti-particle pair thrown into the hole, annihilates behind the horizon, if this time is less than the scrambling time. If we believe that the Schwarzschild geometry exterior to the horizon is a robust thermodynamic feature of the quantum black hole, independent of whether it is newly formed, or has…
This thesis addresses some classical and semi-classical aspects of black holes using an effective membrane representation of the event horizon. The classical "membrane paradigm" equations are derived from an underlying action formulation, even though the equations are dissipative. It is shown how the membrane action can also account for the hole's entropy, including the numerical factor. Two short derivations of Hawking radiation are then presented, the first based on particles in a dynamical geometry and the second on the membrane. The calculations obey conservation laws and lead to a correct…
Small perturbations of a black brane are interpreted as small deviations from thermodynamic equilibrium in a dual theory with the AdS/CFT correspondence. In this paper, we calculate hydrodynamics of the dual Yang-Mills theory in the gravity side using membrane paradigm. This method is different from the usual AdS/CFT correspondence and evaluate classical solutions not at boundaries but at the place slightly away from a horizon. There are sound modes or shear modes for gravity perturbation. For sound modes, such calculation at the horizon has not yet been done. Then, we find that boundary stres…
We examine whether the cosmologies with varying speed of light (VSL) are compatible with the second law of thermodynamics. We find that the VSL cosmology with varying fundamental constant is severely constrained by the second law of thermodynamics, whereas the bimetric cosmological models are less constrained.
We provide a proof of the necessary and sufficient condition on the profile of the temperature, chemical potential, and angular velocity for a charged perfect fluid in dynamic equilibrium to be in thermodynamic equilibrium not only in fixed but also in dynamical electromagnetic and gravitational fields. In passing, we also present the corresponding expression for the first law of thermodynamics for such a charged star.
We investigate the nonequilibrium back reaction on the Schwarzschild black hole from the radiation field. The back reactions are characterized by the membrane close to the black hole. When the membrane is thin, we found that larger temperature difference can lead to more significant negative surface tension, larger thermodynamic dissipation cost and back reaction in energy and entropy as well as larger black hole area. This may be relevant to the primordial black holes in early universe. Moreover, our nonequilibrium model can resolve the inconsistency issue of the black hole back reaction unde…
We discuss the matching of the BPS part of the spectrum for (super)membrane, which gives the possibility of getting membrane's results via string calculations. In the small coupling limit of M--theory the entropy of the system coincides with the standard entropy of type IIB string theory (including the logarithmic correction term). The thermodynamic behavior at large coupling constant is computed by considering M--theory on a manifold with topology ${\mathbb T}^2\times{\mathbb R}^9$. We argue that the finite temperature partition functions (brane Laurent series for $p \neq 1$) associated with …
The second law and the generalized second law of thermodynamics in cosmology in the framework of the modified Gauss-Bonnet theory of gravity are investigated. The conditions upon which these laws hold are derived and discussed.
We investigate the thermodynamic stability of neutral black objects with (at least) two angular momenta. We use the quasilocal formalism to compute the grand canonical potential and show that the doubly spinning black ring is thermodynamically unstable. We consider the thermodynamic instabilities of ultra-spinning black objects and point out a subtle relation between the microcanonical and grand canonical ensembles. We also find the location of the black string/membrane phases of doubly spinning black objects.
In an abstract setting of a general classical mechanical system as a model for the universe we set up a general formalism for a law behind the second law of thermodynamics, i.e. really for "initial conditions". We propose a unification with the other laws by requiring similar symmetry and locality properties.
It has recently been demonstrated that black hole dynamics in a large number of dimensions $D$ reduces to the dynamics of a codimension one membrane propagating in flat space. In this paper we define a stress tensor and charge current on this membrane and explicitly determine these currents at low orders in the expansion in $\frac{1}{D}$. We demonstrate that dynamical membrane equations of motion derived in earlier work are simply conservation equations for our stress tensor and charge current. Through the paper we focus on solutions of the membrane equations which vary on a time scale of orde…
We investigate the validity of the generalized second law of thermodynamics on the $(n-1)$-dimensional brane embedded in the $(n+1)$-dimensional bulk. We examine the evolution of the apparent horizon entropy extracted through relation between gravitational equation and the first law of thermodynamics together with the matter field entropy inside the apparent horizon. We find that the apparent horizon entropy extracted through connection between gravity and the first law of thermodynamics satisfies the generalized second law of thermodynamics. This result holds regardless of whether there is th…
In this paper we will investigate the validity of the Generalized Second Law of thermodynamics for the Quintom model of dark energy. Reviewing briefly the quintom scenario of dark energy, we will study the conditions of validity of the generalized second law of thermodynamics in three cases: quintessence dominated, phantom dominated and transition from quintessence to phantom will be discussed.
Relations between the tunneling rate and the unified first law of thermodynamics at the apparent horizon of the FRW universe are investigated. The tunneling rate arises as a consequence of the unified first law of thermodynamics in such a dynamical system. The analysis shows obviously how the tunneling is intimately connected with the unified first law of thermodynamics through the principle of conservation of energy.
