The thermodynamics of a black hole in equilibrium implies the breakdown of Einstein equations on a macroscopic near-horizon shell

We study a black hole of mass $M$, enclosed within a spherical box, in equilibrium with its Hawking radiation. We show that the spacetime geometry inside the box is described by the Oppenheimer-Volkoff equations for radiation, except for a {\em thin shell} around the horizon. We use the maximum entropy principle to show that the invariant width of the shell is of order $\sqrt{M}$, its entropy is of order $M$ and its temperature of order $1/\sqrt{M}$ (in Planck units). Thus, the width of the shell is {\em much larger} than the Planck length. Our approach is to insist on thermodynamic consistency when classical general relativity coexists with the Hawking temperature in the description of a gravitating system. No assumptions about an underlying theory are made and no restrictions are placed on the origins of the new physics near the horizon. We only employ classical general relativity and the principles of thermodynamics. Our result is strengthened by an analysis of the trace anomaly associated to the geometry inside the box, i.e., the regime where quantum field effects become significant corresponds to the shells of maximum entropy around the horizon.

The attribution of thermodynamic properties to black holes [1,2] is incompatible with classical general relativity (GR). The derivation of Hawking radiation requires a quantum treatment of matter degrees of freedom. For this reason, the origin of black hole thermodynamics is commonly sought at the quantum gravity level. We focus on the thermodynamic level of description for black holes. Thermodynamics is a theory for macroscopic coarse-grained variables and it can be consistently formulated without any reference to the underlying physics. For this reason, we believe that it is possible to formulate a thermodynamic description of black holes within a classical theory of gravity. In Ref. [3], we showed that the thermodynamics of gravitating systems in equilibrium is holographic at the classical level, in the sense that all thermodynamic properties are fully specified by variables defined on the system's boundary. In Ref. [4], we constructed a consistent thermodynamic description of solutions to Einstein's equations that correspond to radiation in a box. In this work, we employ these solutions in order to describe a black hole inside a box, in thermal equilibrium with its Hawking radiation. We find that the breakdown of classical GR takes place in a thin shell around the horizon. Since the principles of thermodynamics are insensitive to the microscopic dynamics, we identify the shell's physical characteristics by employing the maximum entropy principle. We find that the shell is characterized by high temperature and its invariant thickness is much larger than the Planck scale.
The idea of a thin shell or membrane around the horizon is, at present, widely discussed in relation to the firewall conjecture [5]. Our results imply that, if a firewall exists, its proper width is much larger than the Planck length and thus it can be described in terms of a classical geometry even if the classical Einstein equations fail.
An electromagnetic (EM) analogue will help put our perspective in focus. The quantum EM field has a consistent statistical mechanical description, while the classical EM field has none. Nonetheless, the thermodynamics of the classical EM field is well defined: the equation of state follows from the classical action, and the entropy functional is inferred from the equation of state. The only imprint of quantum theory is the Stefan-Boltzmann constant that appears as a phenomenological parameter in the entropy functional. Similarly, when seeking an integrated description of black hole thermodynamics and GR at the macroscopic level, we expect that quantum effects are incorporated into phenomenological parameters of the thermodynamic potentials.
The equilibrium black hole. A black hole in an asymptotically flat spacetime is not an equilibrium system because it radiates. However, a black hole enclosed within a perfectly reflecting spherical box is an equilibrium system because it involves two competing processes: emission of Hawking quanta, and their re-absorption after reflection from the boundary. One expects that the equilibrium state corresponds to the black hole coexisting with its Hawking radiation. This system has been studied before [6,7], albeit with simplifying assumptions.
Since the Hawking emission of massive particles is exponentially suppressed [2], radiation is well described by the thermodynamic equations for ultra-relativistic particles: ρ = bT 4 , P = 1 3 ρ, s = 4 3 b 1/4 ρ 3/4 , where ρ is the energy density, P is the pressure, T is the temperature and s is the entropy density; b is the Stefan-Boltzmann constant that takes the value π 2 15 for pure EM radiation. (We use Planck units, = c = G = 1.) Particle numbers are not preserved in the processes of black hole formation and evaporation; thus, they do not define thermodynamic variables and the associated chemical potentials vanish.
Assuming spherical symmetry, the metric outside the box is a Schwarzschild solution with Arnowitt-Deser-Misner (ADM) mass M . An observer outside the box has access to several macroscopic variables that are constant in absence of external intervention. Such variables are the mass M , the area 4πR 2 of the box, the boundary temperature T R and the boundary pressure P R . The internal energy of a spherically symmetric system coincides with the ADM mass M [3]. A change δR of the boundary radius corresponds to work −P R (4πR 2 )δR/ 1 − 2M/R as measured by a local static observer, or −P R (4πR 2 ) to an observer at infinity. The first law of thermodynamics then becomes δM = T ∞ δS − P R (4πR 2 )δR where T ∞ = T R / 1 − 2M/R is the temperature at infinity. The first law above implies that the thermodynamic state space of the system consists of the variables M and R.
This physical system is characterized by two phases, the radiation phase and the black-hole phase. For fixed R, and for sufficiently small values of M , the box contains only radiation; for higher values of M the box contains a black hole coexisting with its Hawking radiation [12].
The radiation phase was studied in Ref. [4]. Next, we construct the thermodynamics of the black-hole phase through the following steps: (i) we derive the geometry inside the box using classical GR; (ii) since radiation cannot coexist in equilibrium with a horizon in GR, we identify the spacetime region where Einstein' s equations break down; (iii) we find an effective macroscopic description for the physics of this region by using the maximum-entropy principle.
Classical geometry inside the box. The metric inside the box but outside the black hole is a static solution to Einstein's equations with radiation, where dΩ 2 = (dθ 2 + sin 2 θdφ 2 ) and (t, r, θ, φ) are the standard coordinates. The mass function m(r) satisfies dm dr = 4πr 2 ρ, and the energy density ρ(r) satisfies the Oppenheimer-Volkoff (OV) equation ( We change the variables to ξ := ln r R , u := 2m(r) r , and v := 4πr 2 ρ, to obtain Eq. (3) is to be integrated from the boundary (ξ = 0, or r = R) inwards, because the thermodynamic variables M and R are defined at the boundary. We denote the values of u and v at the boundary as u R and v R , respectively. Thus, u R = 2M/R and v R = 4πbR 2 T 4 R . There are two classes of solutions to Eq. (3) that are distinguished by their behavior as r → 0 [4,8]. The first The integration of Eq. (3) from the boundary inwards does not encounter a horizon (u = 1), except for the trivial case of v R = 0 that corresponds to a Schwarzschild horizon and no radiation inside the box [4]. However, there is a sub-class of singular solutions with u ≃ 1 near a surface r = r * . These solutions arise for v R << u R , i.e., for low density at the boundary. We call these geometries Approximate-Horizon (AH) solutions.
Next, we study the properties of the AH solutions. Plots of u and v as a function of r are given in Fig. 1. A typical AH solution is characterized by three regions: In region I, u increases and v decreases with decreasing r. P is the local minimum of v. In region II, u keeps increasing with decreasing r until it reaches a maximum very close to unity at O * (r = r * ≃ 2M ); v also increases with decreasing r in region II and equals 1 2 at O * . It is important to note that the density ρ * ≃ (32πM 2 ) −1 and the local temperature T * = 1 (32πb) 1/4 M −1/2 at O * depend only on M , within an excellent approximation. Region III corresponds to decreasing u; v increases dramatically shortly after O * , but then drops to zero at r = 0.
The evaluation of the AH solutions is described in Appendix A. Every AH solution is characterized by the parameter ǫ * = 1 − u(r * ) << 1 that defines the maximal blue-shift at O * . In Appendix A, we express ǫ * as a function of the boundary variables, ǫ * = 16 This implies that the boundary temperature T R satisfies We also calculate the radial coordinate at O * , r * = 2M 1 + 3ǫ * 8 , and the corresponding value of the mass function m * := m(r * ) = M 1 − 5ǫ * 8 . In the vicinity of O * , the metric Eq. (1) becomes where x = r − r * , and N * = 3 4 √ ǫ * is the lapse function. Breakdown of the OV equation. The regions I and II of an AH solution describe the geometry of the black hole phase at some distance from the horizon. Since the OV equation cannot account for the presence of a horizon, it must break down somewhere in region II, and close to O * . It must be substituted by a different equation that is compatible with the the formation of an horizon. However, any such modification must be very drastic: the OV equation is compatible with a horizon only for matter configurations with negative pressure [9].
The cause for the breakdown of the OV equation in region II is the extreme blue-shift ǫ −1/2 * . At extreme blue-shifts, the description of matter in terms of hydrodynamic variables (e.g, energy density) fails because the hydrodynamic description is not fundamentally continuous but presupposes a degree of coarse-graining.
In Minkowski spacetime, the energy density ρ is defined as ρ = U/L 3 , where U is the energy in a cube of size L. L defines the degree of spatial coarse-graining and it cannot be arbitrarily small [13]. The energy density can be treated as a continuous field only when measured at scales much larger than L. The hydrodynamic description fails when the fluid dynamics generate length-scales of order L. Then, either the consideration of fluctuations or a microscopic treatment is necessary.
In curved spacetimes, the coarse-graining scale L is defined with respect to the local rest frame, so it represents a proper length. By Eq. (5), the coarse-graining scale L corresponds to a radial distance ∆r the hydrodynamic fluctuations obscure the difference of O * from a genuine horizon [14]. For another justification of Eq. (6), see Appendix C.
Maximum-entropy principle. The fundamental thermodynamic variables of the system are the ADM mass M and the box radius R. However, the solutions to Einstein equations depend on three independent parameters, which can be chosen as the mass M , the box radius R, and the blue-shift parameter ǫ * . By Eq. (4), the dependence on ǫ * is equivalent to a dependence on the boundary temperature T R . The equilibrium configuration is determined by the maximum-entropy principle: the value assumed by any unconstrained parameter in a thermodynamic system is the one that maximizes the entropy subject to the system's constraints [10].
The thermodynamic constraints for an isolated box is the constancy of M and R; the blue-shift parameter ǫ * is unconstrained. Hence, the equilibrium configuration for fixed M and R corresponds to the value of ǫ * that maximizes the entropy functional. We expect that the entropy functional has one local maximum for each phase.
The radiation phase maximum has the larger value of ǫ * . For √ ǫ * >> L/M , the OV equation holds everywhere and we recover the thermodynamics of self-gravitating radiation [4]. Smaller values of ǫ * correspond to the black hole phase. For √ ǫ * ∼ L/M , the OV equation breaks down near the surface O * . This breakdown is accompanied by a formation of a horizon H near O * , at r = r H < r * . The violation of the OH equations is restricted to a thin shell around O * with a radial width δr := r * − r H of order ǫ * M . All properties of the shell depend on ǫ * , and they are fully specified once ǫ * is fixed by the maximum-entropy principle.
We model the spacetime geometry of the black-hole phase as follows. In the region between the bounding box and the surface O * , the metric is described by an AH solution. A horizon is formed at r = r H < r * and a thin shell where the OV equation does not apply extends from r H to r * . We write r H = 2M (1 − λǫ * ), where λ > 5 8 is of order unity. The simplification involved in this model is that we assume the breakdown of the OV equation to occur sharply at O * , rather than considering a gradual degradation. This approximation should not affect the order-of-magnitude estimate of the shell's properties.
The total entropy enclosed in a box is a sum of three terms, S tot = S H + S sh + S rad , where (iii) The entropy of radiation S rad is the volume integral of the entropy density s in the regions I and II. It is evaluated to (see, Appendix D) In an isolated box, the values of M and R are constrained, while ǫ * may fluctuate. Hence, the equilibrium configuration is defined as the maximum of the total entropy S tot with respect to ǫ * . The maximum occurs for ]. By Eq. (4), the corresponding boundary temperature is The boundary temperature should coincide with the Hawking temperature T ∞ = 1 8πM , blue-shifted by a factor √ 1 − u R . It is a non-trivial check of our model that the R dependence of Eq. (8) is compatible with such an identification for B = (8λ − 1 2 )πM . Then, the entropy functional for small ǫ * is expressed solely in terms of known parameters, and the equilibrium configuration corresponds to Eq. (10) implies that N * T * = T ∞ , i.e., Tolman's law is satisfied for the Hawking temperature at infinity. This agrees with the results of Refs. [3,11], where Tolman's law is derived solely from the maximum-entropy principle and it is independent of the dynamics of GR.
The equilibrium configuration Eq. (10) must also satisfy the condition L √ ǫ * M for the existence of a black hole phase. By Eq. (10), L √ M , i.e., the coarsegraining scale L defines an upper bound to the mass of a black hole that can be nucleated in a box. This bound is not particularly restrictive: it is satisfied even by supermassive black holes for L in the atomic scale.
Properties of the shell. The width δr of the shell in the equilibrium configuration is i.e., it is of the order of the Planck length. However, the proper width l of the shell is by no means Planckian. Eq. (5) implies that l ∼ δr/ √ ǫ * ∼ √ M , an estimation that is verified by a simple interpolation of the metric inside the shell-see, Appendix E.
The entropy of the shell in the equilibrium configuration is S sh = (8λ − 1 2 ) √ 2πb 9π M . We estimate the internal energy E of the shell by treating the shell as a single thermodynamic object of temperature T sh = (∂S sh /∂E) −1 . In thermal equilibrium, T sh should coincide with the local temperature T * of radiation. Then we find that E ∼ √ M , modulo a constant of order unity. Conclusions. We showed that the horizon of an equilibrium black hole is surrounded by a thin shell where the Einstein equations break down. The existence of the shell follows from the requirement that classical GR coexists with the quantum effect of Hawking radiation in a consistent thermodynamic description. The shell has proper width l ∼ √ M , temperature T sh ∼ 1/ √ M , internal energy E ∼ √ M and entropy S sh ∼ M . The proper width of the shell is much larger than the Planck length. Hence, the breakdown of the equations of GR is fundamentally not due to quantum gravity effects, but due to the quantum properties of matter (radiation). The shell's properties are independent of the box radius R. This strongly suggests that these properties persist even when the box is removed and the system evolves slowly out of equilibrium, i.e., to Schwarzschild black holes.
We emphasize the robustness of our conclusions. We made no assumptions about the quantum characteristics of the underlying theory (unitarity, CTP symmetry, holography), and placed no restrictions on the origin of the new physics near the horizon. In deriving the properties of the shell, we used only thermodynamic principles and classical GR.
unity, almost all the mass is contained in the black hole of radius 2M , hence, the Bekenstein-Hawking formula for the black hole entropy applies, SBH = 4πM 2 . The black hole phase is entropically favored if SBH > S rad , i.e., for [13] For thermal radiation at temperature T , the requirement that the energy fluctuations are much smaller than the mean energy in a volume L 3 implies that LT >> 1. At higher temperatures, the Compton wave-length of the electron defines an absolute lower limit to L. We present an approximate analytic expression for the AH solutions that is valid in the regions α and β of Fig.  1.
An AH solution is characterized by v R << u R . In the region I, u increases with decreasing r and v decreases with decreasing r. Hence, the condition v << u applies to all points in region I. By continuity, the condition v << u also applies in a part of region II.
In what follows, we denote derivative with respect to ξ by a prime.
For v << u, Eq. (3) becomes Hence, The solution of Eq. (13) with the boundary condition Eq. (12) implies that u(ξ) = u R e −ξ . Substituting into Eq. (14), we derive Next, we study the AH solution in the regime where 1 − u(ξ) << 1. For sufficiently small v R , this condition applies to the whole of region II and to a segment of region I.
Comparing Eqs. (14) and (20) near P , we find that which relates the parameter ǫ * defined on the approximate horizon O * to the boundary variables u R and v R . Eq. (23) implies the following relation between the boundary temperature T R and the parameter ǫ * .
Using Eq. (22) for a choice of the reference point ξ = ξ r lying in the domain of validity of Eq. (15), Setting ξ = ξ * in Eq. (25), we obtain Using the radial coordinate r = Re ξ , we identify the radial coordinate r * at the approximate horizon to leading order in ǫ * The corresponding value of the mass function m * = m(r * ) is In the vicinity of O * , the metric Eq. (1) becomes The acceleration a i = ∇ i log N is purely radial, with Hence, the proper acceleration a = √ a µ a µ near O * is

B. Pressure and density near the approximate horizon
It is conceivable that the equation of state for radiation needs is modified near O * in order to incorporate quantum effects, such as vacuum polarization. However, such modifications are unlikely to lead to the negative pressures that are necessary for the formation of a horizon. For a solar mass black hole, ρ * ∼ 10 16 ρ H2O , where ρ H2O is the density of water. Hence, ρ * is of the same order of magnitude with the density at the center of a neutron star. The corresponding local temperature T * is of the order of 10 12o K, which is a typical temperature for quark-gluon plasma. No existing model of strong/nuclear interactions suggests the possibility of negative pressure in these regimes. For supermassive black holes, ρ * ∼ 10 2 ρ H2O ; negative pressures are even more implausible in this regime. For this reason, we expect that quantum effects at high densities may cause quantitative changes in the thermodynamics of self-gravitating radiation, but they are not strong enough to generate a black hole phase. In further support of this assertion, we note that any contribution from quantum effects would have strong and complex dependence on the mass M , involving masses and thresholds from high energy physics. The resulting thermodynamics would not manifest the simplicity and universality of the Bekenstein-Hawking entropy.
C. Another justification for condition (6) In a hydrodynamic system, local densities and temperature are meaningfully defined only if they vary at scales significantly larger than the coarse-graining scale L; the variation within a shell of volume L 3 must be a small fraction of the value. Tolman's law implies that the product of the local temperature T and the lapse function N is constant. Using Eq. (30), we find that near O * , By Eq. (29), the coordinate distance ∆r corresponding to proper length L near O * is ∆r = L √ ǫ * . When the variation of temperature in a cell of proper length L is of the same order of magnitude as the temperature, the hydrodynamic description breaks down. The relevant condition is |∇ r T /T |∆r ∼ 1, which implies Eq. (6).

D. Evaluating the radiation entropy
The entropy of radiation in the regions I and II of an AH solution is For solutions to the Oppenheimer-Volkoff equation, the integrand in Eq.(34) is a total derivative, i.e., Hence, S rad = S 1 − S * where depends on field values at the boundary r = R, and depends on the field values at r = r * . Using Eq. (23) to eliminate v R from S 1 , we obtain (38) Eq. (7) applies in the regime K << 1, where so that the second term inside the parenthesis in the r.h.s. of Eq. (38) is negligible for the value of ǫ * that maximizes the total entropy, Eq. (10).

E. The geometry of the shell
The shell at the horizon can still be described in terms of a classical geometry. We consider a spherically symmetric metric, with a mass function that interpolates between the horizon H at r = r H and the approximate horizon O * at r = r * . We assume a power-law interpolation, where k and a are positive constants. We require that m(r), Eq. (40) is joined with an AH solution at O, such that the metric and its first derivatives are continuous. This implies that m(r * ) = m * , Eq. (28) and that m ′ (r * ) = 1 2 . The horizon is defined by the condition 2m(r H ) = r H . We further require that m ′ (r H ) = 0, i.e., that there is no matter on the horizon; this implies that a > 0.
Then, we obtain Hence, the width of the shell is δr = r * − r H = 2M ǫ * a −1 .
The proper length l of the shell is where C(a) = Comparing Eq. (40) with the OV equation, we can estimate an effective "equation of state" that parameterizes the properties of the shell. The OV equation in the shell is well approximated by dP dz ≃ − (ρ + P ) 2z (1 + 32πM 2 P ), where z = r − r H . Numerical solution of Eq. (45) leads to an effective equation of state, i.e., a relation between ρ and P , as shown in Fig. 2. We note that for a ≥ 1, the effective equation of state is reasonably well approximated by a linear relation of the form P = −wρ, where w > 0. Near the horizon (z = 0), a linear equation of state with negative pressure is a good approximation for all a Substituting Eq. (46) into the continuity equation we derive the lapse function near the horizon N ∼ ρ 1/a ∼ √ r − r H . N can be expressed as where x = r−rH 8M and κ is the surface gravity of the horizon. Then, the geometry near the horizon is of the Rindler type with acceleration κ. We have no analytic expression for κ, but we expect it to be of order 1/M . Indeed, if Eq. (46) were a good approximation to the effective equation of state throughout the shell, we would obtain κ = 3 √ a 16M .
F. Additional tables We list all properties of the shell around the horizon as calculated in the main text and in the Appendix. We choose as free parameter the interpolating exponent a, defined in Eq. (40). The parameter λ employed in the main text is related to a through the relation λ = 5 8 + a −1 . We present the analytic expression and an evaluation for a = 1 that corresponds to linear interpolation for density and for b = π 2 15 (only EM radiation).