S. Carlip
Near the horizon, the obvious symmetries of a black hole spacetime---the horizon-preserving diffeomorphisms---are enhanced to a larger symmetry group with a BMS${}_3$ algebra. Using dimensional reduction and covariant phase space techniques, I investigate this augmented symmetry, and show that it is strong enough to determine the black hole entropy.
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We consider a BMS-type symmetry action on isolated horizons in asymptotically flat spacetimes. From the viewpoint of the non-relativistic field theory on a horizon membrane, supertranslations shift the field theory spatial momentum. The latter is related by a Ward identity to the particle number symmetry current and is spontaneously broken. The corresponding Goldstone boson shifts the horizon angular momentum and can be detected quantum mechanically. Similarly, area preserving superrotations are spontaneously broken on the horizon membrane and we identify the corresponding gapless modes. In as…
In an earlier short paper [Phys.\ Rev.\ Lett.\ 120 (2018) 101301, arXiv:1702.04439], I argued that the horizon-preserving diffeomorphisms of a generic black hole are enhanced to a larger BMS${}_3$ symmetry, which is powerful enough to determine the Bekenstein-Hawking entropy. Here I provide details and extensions of that argument, including a loosening of horizon boundary conditions and a more thorough treatment of dimensional reduction and meaning of a "near-horizon symmetry."
The Bondi-van der Burg-Metzner-Sachs (BMS) group is the asymptotic symmetry group of asymptotically flat gravity. Recently, Donnay et al. have derived an analogous symmetry group acting on black hole event horizons. For a certain choice of boundary conditions, it is a semidirect product of ${\rm Diff}(S^2)$, the smooth diffeomorphisms of the two-sphere, acting on $C^\infty(S^2)$, the smooth functions on the two-sphere. We observe that the same group appears in fluid dynamics as symmetries of the compressible Euler equations. We relate these two realizations of ${\rm Diff}(S^2)\ltimes C^\infty(…
We revisit the theory of null shells in general relativity, with a particular emphasis on null shells placed at horizons of black holes. We study in detail the considerable freedom that is available in the case that one solders two metrics together across null hypersurfaces (such as Killing horizons) for which the induced metric is invariant under translations along the null generators. In this case the group of soldering transformations turns out to be infinite dimensional, and these solderings create non-trivial horizon shells containing both massless matter and impulsive gravitational wave …
When two spacetimes are stitched across a null-shell placed at the horizon of a black hole BMS-supertranslation like soldering freedom arises if one demands the induced metric on the horizon shell should remain invariant under the translations generated by the null generators of the shell. We revisit this phenomenon at the horizon of rotating shells and obtain BMS like symmetries. We further show that superrotation like soldering symmetries in the form of conformal isometries can emerge whenever the degenerate metric of any null hypersurface admits a dependency on null (degenerate direction) c…
We study horizon shells and soldering freedom for extreme black holes and how supertranslation-like Bondi-Metzner-Sachs (BMS) symmetries appear as soldering transformations. Further, for a null shell placed infinitesimally close to the horizon of an extreme Reissner-Nordstr$\ddot{o}$m (RN) black hole, we show superrotation-like symmetries also arise as soldering freedom. Next, considering the interaction of impulsive gravitational waves supported at the horizon shell with test particles, we study how the "memory" (or the imprints) of BMS-like symmetries gets encoded in the geodesics (test part…
This article reviews one of the most intriguing properties of black hole spacetimes known in the literature -- gravitational memory effect, and its connection with asymptotic symmetries, also termed as Bondi-van der Burg-Metzner-Sachs (BMS) symmetries, emerging near the horizon of black holes. Gravitational memory is a non-oscillatory part of the gravitational wave amplitude which generates a permanent displacement for freely falling test particles or test detectors. We highlight a model scenario where asymptotic symmetries appear as a soldering freedom in the context of stitching of two black…
Near horizon Bondi-Metzner-Sachs (BMS) like symmetry is spontaneously broken by the black hole background itself and hence gives rise to Goldstone mode. The associated Goldstone mode for the near horizon BMS like symmetry of a Schwarzschild black hole was found to behave like inverted harmonic oscillators, which has been further shown to lead to thermodynamic temperature in the semi-classical regime. Here we investigate the generalization of this previous findings for the Kerr black hole. The analysis is being performed for two different situations. Firstly, we analyze Goldstone mode dynamics …
We investigate the standard and dual Bondi-Metzner-Sachs (BMS) supertranslation generators on a black hole horizon and draw some conclusions about black hole physics. Recently, it has been shown that in addition to conventional BMS supertranslation symmetries, there exists an additional infinite set of magnetic asymptotic symmetries, dual BMS supertranslations, again parametrized by a function on the two-sphere. We show that the Dirac bracket between these generators exhibits an anomalous central term when one parameter function exhibits a singularity in the complex stereographical coordinates…
Dynamical Selection of Horizon-BMS Goldstone Modes in Evaporating Black Holes
本研究针对蒸发的Vaidya-Schwarzschild黑洞，推导近视界BMS超平移的戈德斯通模式有效作用量，发现黑洞蒸发会动力学筛选视界戈德斯通模式，建立了宏观视界演化与引力软自由度微观组织的直接联系。
Near-Horizon BMS Symmetry and Implications on Black Hole Entropy
该研究探讨适配视界的BMS类超平移在动态史瓦西黑洞热力学中的作用，构造相关微分同胚与戈德斯通模式，推导其有效作用量与守恒荷，修正黑洞热力学描述并得出熵的次 leading阶修正依赖于超平移扇区与黑洞动态演化的结论。
We study the displacement memory effect and its connection with the extended-BMS symmetries near the horizon of black holes. We show there is a permanent shift in the geodesic deviation vector relating two nearby timelike geodesics placed close to the horizon of black holes, upon the passage of gravitational waves. We also relate this memory effect with the asymptotic symmetries near the horizon of asymptotic black hole spacetimes. The shift of the relative position of the detectors is shown to be induced by a combination of BMS generators near the horizon. The displacement memory effect near …
We explore the holographic principle in the context of asymptotically flat spacetimes. In analogy with the AdS/CFT scenario we analyse the asympotically symmetry group of this class of spacetimes, the so called Bondi-Metzner-Sachs (BMS) group. We apply the covariant entropy bound to relate bulk entropy to boundary symmetries and find a quite different picture with respect to the asymptotically AdS case. We then derive the covariant wave equations for fields carrying BMS representations to investigate the nature of the boundary degrees of freedom. We find some similarities with 't Hooft S-matri…
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.…
Black hole entropy, denoted by N, in (semi)classical limit is infinite. This scaling reveals a very important information about the qubit degrees of freedom that carry black hole entropy. Namely, the multiplicity of qubits scales as N, whereas their energy gap and their coupling as 1/N. Such a behavior is indeed exhibited by Bogoliubov-Goldstone degrees of freedom of a quantum-critical state of N soft gravitons (a condensate or a coherent state) describing the black hole quantum portrait. They can be viewed as the Goldstone modes of a broken symmetry acting on the graviton condensate. In this …
BMS transformed Quantum String Dynamics near a Black Hole
研究BMS超平移对近黑洞视界处量子弦动力学的影响，发现角向变形破坏了背景对称性，揭示了BMS诱导变形的非平凡特征。
In this note we first review the degenerate vacua arising from the BMS symmetries. According to the discussion in [1] one can define BMS-analogous supertranslation and superrotation for spacetime with black hole in Gaussian null coordinates. In the leading and subleading orders of near horizon approximation, the infinitely degenerate black hole solutions are derived by considering Einstein equations with or without cosmological constant, and they are related to each other by the diffeomorphism generated by horizon supertranslation. Higher order results and degenerate Rindler horizon solutions …
Near-Horizon Symmetries of Local Black Holes in General Relativity
本文研究了广义相对论中具有球面和环面拓扑的局部黑洞的近视界对称性，计算了自旋-$s$ BMS$_d$ 代数及 Noether-Wald 荷代数，发现环面拓扑会改变代数结构，并指出非零自旋参数下热力学第一定律一般不成立。
We characterise the geometrical nature of smooth supertranslations defined on a generic non-expanding horizon (NEH) embedded in vacuum. To this end we consider the constraints imposed by the vacuum Einstein's equations on the NEH structure, and discuss the transformation properties of their solutions under supertranslations. We present a freely specifiable data set which is both necessary and sufficient to reconstruct the full horizon geometry, and is composed of objects which are invariant under supertranslations. We conclude that smooth supertranslations do not transform the geometry of the …
We discuss BMS supertranslations both at null-infinity and on the horizon for the case of the Schwarzschild black hole. We show that both kinds of supertranslations lead to infinetly many gapless physical excitations. On this basis we construct a quotient algebra using suited superpositions of both kinds of transformations which cannot be compensated by an ordinary BMS-supertranslation and therefore are intrinsically due to the presence of an event horizon. We show that these quotient transformations are physical and generate gapless excitations on the horizon that can account for the gravitat…
Recently it has been shown that there is asymptotic BMS-like symmetry associated with the near-horizon geometry of black holes in three and four dimensions. In this paper, we show that the presence of such BMS-like symmetry is a ubiquitous feature for black holes in generic dimensions. For black holes in $D$ dimensions, the symmetry contains 2 supertranslations and $D-2$ generalized superrotations. The superrotations are found to generate a generalized Witt-like algebra that was previously noticed in a rather different construction. In the case of stationary and axisymmetric black holes, we ca…
It is well-known that blackhole and cosmological horizons in equilibrium situations are well-modeled by non-expanding horizons (NEHs). In the first part of the paper we introduce multipole moments to characterize their geometry, removing the restriction to axisymmetric situations made in the existing literature. We then show that the symmetry group $\mathfrak{G}$ of NEHs is a 1-dimensional extension of the BMS group $\mathfrak{B}$. These symmetries are used in a companion paper to define charges and fluxes on NHEs, as well as perturbed NEHs. They have physically attractive properties. Finally,…
Expanding around null hypersurfaces, such as generic Kerr black hole horizons, using co-rotating Kruskal-Israel-like coordinates we study the associated surface charges, their symmetries and the corresponding phase space within Einstein gravity. Our surface charges are not integrable in general. Their integrable part generates an algebra including superrotations and a BMS_3-type algebra that we dub "T-Witt algebra". The non-integrable part accounts for the flux passing through the null hypersurface. We put our results in the context of earlier constructions of near horizon symmetries, soft hai…
The asymptotically flat structure of $\mathcal{N}=(2,0)$ supergravity in three spacetime dimensions is explored. The asymptotic symmetries are spanned by an extension of the super-BMS$_3$ algebra, with two independent $\hat{u}(1)$ currents of electric and magnetic type. These currents are associated to $U(1)$ fields being even and odd under parity, respectively. Remarkably, although the $U(1)$ fields do not generate a backreaction on the metric, they provide nontrivial Sugawara-like contributions to the BMS$_3$ generators, and hence to the energy and the angular momentum. The entropy of flat c…
We show that the geometry of a black hole horizon can be described as a Carrollian geometry emerging from an ultra-relativistic limit where the near-horizon radial coordinate plays the role of a virtual velocity of light tending to zero. We prove that the laws governing the dynamics of a black hole horizon, the null Raychaudhuri and Damour equations, are Carrollian conservation laws obtained by taking the ultra-relativistic limit of the conservation of an energy-momentum tensor; we also discuss their physical interpretation. We show that the vector fields preserving the Carrollian geometry of …
Lectures on the Bondi--Metzner--Sachs group and related topics in infrared physics
本文基于BMS群的定义与性质，探讨其在引力理论、量子场论及量子引力中的应用，包括软引力子定理、记忆效应和Hadamard态构造等核心贡献。
We provide boundary conditions for three-dimensional gravity including boosted Rindler spacetimes, representing the near-horizon geometry of non-extremal black holes or flat space cosmologies. These boundary conditions force us to make some unusual choices, like integrating the canonical boundary currents over retarded time and periodically identifying the latter. The asymptotic symmetry algebra turns out to be a Witt algebra plus a twisted u(1) current algebra with vanishing level, corresponding to a twisted warped CFT that is qualitatively different from the ones studied so far in the litera…
The entropy of dynamical de Sitter horizons
将动力学黑洞熵的新公式推广到渐近德西特时空的动力学宇宙学事件视界，引入修正项推广事件视界第一定律至非平稳时期，并推导任意截面间的非平稳物理过程第一定律。
Starting from the surface term of gravitational action, one can construct a Virasoro algebra with central extension, with which the horizon entropy can be derived by using Cardy formula. This approach gives a new routine to calculate and interpret the horizon entropy. In this paper, we generalize this approach to a more general case, the isolated horizon, which contains non-stationary spacetimes beyond stationary ones. By imposing appropriate boundary conditions near the horizon, the full set of diffeomorphism is restricted to a subset where the corresponding Noether charges form a Virasoro al…
We describe a simple way of obtaining horizon entropy using the approach based on the Virasoro algebra and central charge. We show that the Virasoro algebra defined by the Noether currents corresponding to the surface term of gravitational action, for the diffeomorphisms which leave the horizon structure unaltered, has a central extension that directly leads to the horizon entropy. In this approach there are no ambiguities in the calculation of the central charge. We explain why this approach is physically well motivated and could provide greater insight into the nature of horizon entropy.
For the 2-charge extremal holes in string theory we show that the Bekenstein entropy obtained from the area of the stretched horizon has a statistical interpretation as a `coarse graining entropy': different microstates give geometries that differ near r=0 and the stretched horizon cuts off the metric at r=b where these geometries start to differ.
We impose a certain class of boundary conditions on Killing horizon and show for Lagrangians with arbitrary curvature dependence that one can identify a Virasoro algebra with nontrivial central charge and calculable Hamiltonian eigenvalue. Entropy can then be calculated from Cardy formula.
The Bekenstein-Hawking entropy of black holes in Einstein's theory of gravity is equal to a quarter of the horizon area in units of Newton's constant. Wald has proposed that in general theories of gravity the entropy of stationary black holes with bifurcate Killing horizons is a Noether charge which is in general different from the Bekenstein-Hawking entropy. We show that the Noether charge entropy is equal to a quarter of the horizon area in units of the effective gravitational coupling on the horizon defined by the coefficient of the kinetic term of specific graviton polarizations on the hor…
We study the thermodynamic properties of horizons using the dynamical description of the gravitational degrees of freedom at a horizon found in a previous work. We use the action of the horizon degrees of freedom to posit an ansatz Liouville action in the near-horizon region, and calculate the horizon entropy using the Cardy formula. We also couple the gravitational degrees of freedom at the horizon to a classical scalar background, and show that Hawking radiation is produced.
Based on the entropy relations, we derive thermodynamic bound for entropy and area of horizons of Schwarzschild-dS black hole, including the event horizon, Cauchy horizon and negative horizon (i.e. the horizon with negative value), which are all geometrical bound and made up of the cosmological radius. Consider the first derivative of entropy relations together, we get the first law of thermodynamics for all horizons. We also obtain the Smarr relation of horizons by using the scaling discussion. For thermodynamics of all horizons, the cosmological constant is treated as a thermodynamical varia…
The entropy S of the horizon $theta = pi/2$ of the Hawking wormhole written in spherical Rindler coordinates is computed in this letter. Using Padmanabhan's prescription,we found that the surface gravity of the horizon equals the proper acceleration of the Rindler observer. S is a monotonic function of the radial coordinate $xi$ and vanishes when $xi$ equals the Planck length. In addition, its expression is similar with the Kaul - Majumdar one for the black hole entropy, including logarithmic corrections in quantum gravity scenarios.
We generalize the Noether current method proposed by Majhi and Padmanabhan recently to $f(R)$ gravity and conformal gravity and analyze the near horizon symmetry of black holes in these models. It is shown that the entropy obtained with Cardy's formula agrees with Wald entropy.
We treat D-dimensional black holes with Killing horizon for extended Gauss-Bonnet gravity. We use Carlip method and impose boundary conditions on horizon what enables us to identify Virasoro algebra and evaluate its central charge and Hamiltonian eigenvalue. The Cardy formula allows then to calculate the number of states and thus provides for microscopic interpretation of entropy.
We focus on the entropy relations of black holes in three, four and higher dimensions. These entropy relations include entropy product, "part" entropy product and entropy sum. We also discuss their differences and similarities, in order to make a further study on understanding the origin of black hole entropy at the microscopic level.
Near an event horizon, the action of general relativity acquires a new asymptotic conformal symmetry. Using two-dimensional dilaton gravity as a test case, I show that this symmetry results in a chiral Virasoro algebra with a calculable classical central charge, and that Cardy's formula for the density of states reproduces the Bekenstein-Hawking entropy. This result lends support to the notion that the universal nature of black hole entropy is controlled by conformal symmetry near the horizon.
In the Carlip-Majhi-Padmanabhan approach, we calculate microscopic entropy of trapping (apparent) horizon of the FRW metric. We solve Killing equations for $t,r$ part of the metric without fixing {\it a piori} the form of the scaling factor $a(t)$ which is determined from the requirement of consistency of Killing equations. Further restrictions on the form of the Killing vector follow from the requirement that Killing vector is null at the trapping horizon at all $t$. Applying the technique used to calculate microscopic entropy of Killing horizons, we calculate microscopic entropy of trapping …
Equilibrium states of black holes can be modelled by isolated horizons. If the intrinsic geometry is spherical, they are called type I while if it is axi-symmetric, they are called type II. The detailed theory of geometry of \emph{quantum} type I horizons and the calculation of their entropy can be generalized to type II, thereby including arbitrary distortions and rotations. The leading term in entropy of large horizons is again given by 1/4th of the horizon area for the \emph{same} value of the Barbero-Immirzi parameter as in the type I case. Ideas and constructions underlying this extension…
The quantum theory of near horizon regions of spacetimes with classical spatially flat, homogeneous and isotropic Friedman-Robertson-Walker geometry can be approximately described by a two dimensional conformal field theory. The central charge of this theory and expectation value of its Hamiltonian are both proportional to the horizon area in units of Newton's constant. The statistical entropy of horizon states, which can be calculated using two dimensional state counting methods, is proportional to the horizon area and depends on a numerical constant of order unity which is determined by Plan…
Null infinity as a Killing horizon
本文通过定义渐近（共形）Killing 视界上的类 BMS 对称性，提出统一渐近平坦时空与宇宙学视界渐近对称性的一般框架，并发现共形推广会产生包含超伸缩的更大对称群。
We treat spherically symmetric black holes in Gauss-Bonnet gravity by imposing boundary conditions on fluctuating metric on the horizon. Obtained effective two-dimensional theory admits Virasoro algebra near the horizon. This enables, with the help of Cardy formula, evaluation of the number of states. Obtained results coincide with the known macroscopic expression for the entropy of black holes in Gauss-Bonnet gravity.
When a black hole forms from collapse in a holographic theory, the information in the black hole interior remains encoded in the boundary. We prove that the area of the black hole's apparent horizon is precisely the entropy associated to coarse graining over the information in its interior, subject to knowing the exterior geometry. This is the maximum holographic entanglement entropy that is compatible with all classical measurements conducted outside of the apparent horizon. We identify the boundary dual to this entropy and explain why it obeys a Second Law of Thermodynamics.
A detailed analysis of the spherically symmetric isolated horizon system is performed in terms of the connection formulation of general relativity. The system is shown to admit a manifestly SU(2) invariant formulation where the (effective) horizon degrees of freedom are described by an SU(2) Chern-Simons theory. This leads to a more transparent description of the quantum theory in the context of loop quantum gravity and modifications of the form of the horizon entropy.
Although the laws of thermodynamics are well established for black hole horizons, much less has been said in the literature to support the extension of these laws to more general settings such as an asymptotic de Sitter horizon or a Rindler horizon (the event horizon of an asymptotic uniformly accelerated observer). In the present paper we review the results that have been previously established and argue that the laws of black hole thermodynamics, as well as their underlying statistical mechanical content, extend quite generally to what we call here "causal horizons". The root of this general…
We review our recent proposal of a method to extend the quantization of spherically symmetric isolated horizons, a seminal result of loop quantum gravity, to a phase space containing horizons of arbitrary geometry. Although the details of the quantization remain formally unchanged, the physical interpretation of the results can be quite different. We highlight several such differences, with particular emphasis on the physical interpretation of black hole entropy in loop quantum gravity.
It may be a common understanding at present that, once event horizons are in thermal equilibrium, the entropy-area law holds inevitably. However, no rigorous verification is given to such a very strong universality of the law in multi-horizon spacetimes. In this article, based on thermodynamically consistent and rigorous discussion, we investigate thermodynamics of Schwarzschild-deSitter spacetime in which the temperatures of two horizons are different. We recognize that three independent state variables exist in thermodynamics of the horizons. One of the three variables represents the effect …
It is often argued that {\it all the information of a gravitational theory is encoded in the surface term of the action}; which means one can find several physical quantities just from the surface term without incorporating the bulk part of the action. This has been observed in various instances; e.g. derivation of the Einstein's equations, surface term calculated on the horizon leads to entropy, etc. Here I investigate the role of it in the context of entropy function and entropy of extremal near horizon black holes. Considering only the Gibbons-Hawking-York (GHY) surface term to define an en…
We show that microscopic entropy formula based on Virasoro algebra follows from properties of stationary Killing horizons for Lagrangians with arbitrary dependence on Riemann tensor. The properties used are consequence of regularity of invariants of Riemann tensor on the horizon. Eventual generalisation of these results to Lagrangians with derivatives of Riemann tensor, as suggested by an example treated in the paper, relies on assuming regularity of invariants involving derivatives of Riemann tensor. This assumption however leads also to new interesting restrictions on metric functions near h…
Apparent Horizons Associated with Dynamical Black Hole Entropy
该研究通过将动态黑洞熵与 E-MOTS 联系起来，证明了其在不同静止区域中与 Wald 熵的一致性，从而增强了动态熵公式的可靠性。
Black holes whose near-horizon geometries are locally, but not necessarily globally, AdS$_3$ (three-dimensional anti-de Sitter space) are considered. Using the fact that quantum gravity on AdS$_3$ is a conformal field theory, we microscopically compute the black hole entropy from the asymptotic growth of states. Precise numerical agreement with the Bekenstein-Hawking area formula for the entropy is found. The result pertains to any consistent quantum theory of gravity, and does not use string theory or supersymmetry.
Using a unified approach we study the entropy of extremal black holes through the entropy of an electrically charged thin shell. We encounter three cases in which a shell can be taken to its own gravitational or horizon radius and become an extremal spacetime. In case 1, we use a non extremal shell, calculate all the thermodynamics quantities including the entropy, take it to the horizon radius, and then take the extremal limit. In case 2, we take the extremal limit and the horizon radius limit simultaneously, i.e., as the shell approaches its horizon radius it also approaches extremality. In …
We present a statistical mechanical calculation of the thermodynamical properties of (non rotating) isolated horizons. The introduction of Planck scale allows for the definition of an universal horizon temperature (independent of the mass of the black hole) and a well-defined notion of energy (as measured by suitable local observers) proportional to the horizon area in Planck units. The microcanonical and canonical ensembles associated with the system are introduced. Black hole entropy and other thermodynamical quantities can be consistently computed in both ensembles and results are in agreem…
The Clausius relation between entropy change and heat flux has previously been used to derive Einstein's field equations as an equation of state. In that derivation the entropy is proportional to the area of a local causal horizon, and the heat is the energy flux across the horizon, defined relative to an approximate boost Killing vector. We examine here whether a similar derivation can be given for extensions beyond Einstein gravity to include higher derivative and higher curvature terms. We review previous proposals which, in our opinion, are problematic or incomplete. Refining one of these,…
The standard approach of counting the number of eigenmodes of $N$ scalar fields near the horizon is used as a basis to provide a simple statistical mechanical derivation of the black hole entropy in two and four dimensions. The Bekenstein formula $S={A\ov 4G\h}$ and the two-dimensional entropy $S=2M/ł\h $ are naturally obtained (up to a numerical constant of order 1). This approach provides an explanation on why the black hole entropy is of order $1/\h $ and why it is independent of the number of field-theoretical degrees of freedom.
How Much Geometry Does Horizon Entropy Determine? A Discourse on Entropy, Gravity, and Observable Quantities
本文在Jacobson的热力学推导基础上，对比GEVAG与EGC框架的异同，研究其对贝肯斯坦界的影响，最终提出实验室测量的$G$本质的核心问题。
Recent numerical simulations of gravitational collapse show that there exists a special foliation of the spacetime where matter and entropy accumulate directly on the inside of the horizon surface. In this foliation, the time coincides with the proper time of the asymptotic static observer (ASO) and for spherical symmetry, this corresponds to isotropic coordinates. In this gauge, the three-volume in the interior shrinks to zero and only the horizon area remains at the end of collapse. In a different foliation, matter and entropy accumulate in the volume. The entropy is however independent of t…
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