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Home , Fuzzball (string theory)

Empty Black Holes, Firewalls, and the Origin of Bekenstein-Hawking Entropy

Mehdi Saravani,1, 2, ∗ Niayesh Afshordi,1, 2, † and Robert B. Mann2, 1, ‡ 1Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, ON, N2L 2Y5, Canada 2Department of Physics and Astronomy, University of Waterloo, Waterloo, ON, N2L 3G1, Canada We propose a novel solution for the endpoint of , in which ends (and is orbifolded) at a microscopic distance from event horizons. This model is motivated by the emergence of singular event horizons in the gravitational aether theory, a semi-classical solution to the cosmological constant problem(s), and thus suggests a catastrophic breakdown of general relativity close to black hole event horizons. A similar picture emerges in fuzzball models of black holes in theory, as well as the recent firewall proposal to resolve the information paradox. We then demonstrate that positing a surface fluid in thermal equilibrium with , with vanishing energy density (but non-vanishing pressure) at the new boundary of spacetime, which is required by Israel junction conditions, yields a thermodynamic entropy that is identical to the Bekenstein-Hawking area law, SBH , for charged rotating black holes. To our knowledge, this is the first derivation of black hole entropy which only employs local thermodynamics. Furthermore, a model for the microscopic degrees of freedom of the surface fluid (which constitute the micro-states of the black hole) is suggested, which has a finite, but Lorentz-violating, quantum field theory. Finally, we comment on the effects of physical boundary on Hawking radiation, and show that relaxing the assumption of equilibrium with Hawking radiation sets SBH as an upper limit for Black Hole entropy.

I. INTRODUCTION ingly, SE = SBH for Euclideanized GR black holes, and thus the non-perturbative decay of the semi-classical so- lutions can be quite fast (i.e. much faster than the Hawk- General relativity (GR) predicts that the endpoint of ing evaporation time) [1, 2]. While the nature of the end- gravitational collapse of (nonrotating and neutral) mat- state of gravitational collapse depends on the full phase ter is a (Schwarzschild) Kerr-Newman black hole. While space of the theory of quantum , in the context of there might be a metric singularity at the horizon of the , it has been argued that fuzzball solutions black hole, there is no real curvature singularity. How- provide the correct multiplicity and asymptotics to rep- ever, there is a real curvature singularity at the centre resent a microscopic description of GR black hole macro- of a black hole. Since the curvature-invariants remain states (e.g. see [3] and references therein). While fuzzball small at black hole horizons, it is widely believed that solutions approximate GR black holes at large distances, black hole solutions of GR are good approximations to they diverge from the classical solution (or each other) the “real” geometry of spacetime all the way to the sin- at/around the classical horizon. In particular, fuzzball gularity, except perhaps for a neighbourhood of the sin- solutions do not have any event horizons or singularities, gularity, or at time-scales comparable to the Hawking although they contain ergo-regions, which could produce evaporation time, where/when quantum mechanical ef- analogue of Hawking radiation. Moreover, the spacetime fects become important. “ends” at a minimal spatial area comparable to that of Nevertheless, there are a number of arguments against the classical . The latter is the most signif- the validity of the semi-classical nature of GR black hole icant macroscopic difference between the fuzzballs, and solutions inside event horizons. If we consider GR as an their semi-classical counter-parts, which we will capture effective field theory, its expected cut-off will be around below in our construction. arXiv:1212.4176v2 [hep-th] 2 Dec 2013 the Planck energy, MP . Quantum loop corrections Very recently, a similar picture has emerged from a re- 2 should therefore be suppressed by powers of O(R/MP ), consideration of the black hole information paradox [4]: which are negligible around horizons of macroscopic states that fall into black hole horizon are entangled with black holes. However, non-perturbative quantum effects those of Hawking radiation, but unitarity implies that the can be big: while the tunnelling rate is suppressed by end state of black hole evaporation (= early+late Hawk- exp(−SE)  1, where SE is the Euclidean action of the ing radiation) should be a pure state. Authors of [5] instanton connecting GR solutions to other states in the argue that (for reasons very close to Mathur’s arguments full theory of , the number of such states [6] , or earlier arguments, e.g., in [7, 8] and references is estimated to be ∼ exp(SBH )  1, with SBH being therein) the most “conservative” resolution is to replace the Bekenstein-Hawking entropy of black holes. Interest- the horizon by a firewall that “burns” infalling observers. Nevertheless, many counter-arguments (and some retrac- tions) soon followed this proposal (e.g. [9–12]).

∗Electronic address: [email protected] There is also another argument, rooted in the quan- †Electronic address: [email protected] tum nature of gravity, for the breakdown of semi-classical ‡Electronic address: [email protected] spacetime at black hole horizons, which (as shown here) 2 is further validated by the first derivation (to our knowl- area law can arise in a realistic setting. Finally, Section edge) of Bekenstein-Hawking entropy based on local ther- V concludes the paper. modynamics. The argument follows from the behaviour of an incompressible fluid in GR, which has been argued to develop singularities close to event horizons, while si- II. EMPTY BLACK HOLES AND THE multaneously explaining the observed scale of cosmolog- BEKENSTEIN-HAWKING ENTROPY ical dark energy without any fine-tuning [13]. The struc- ture of these black hole solutions is depicted in Fig. (1). In the previous section, we discussed motivations to But why an incompressible fluid? The reason comes posit a minimum area surface close to/at black hole hori- from an attempt to solve the (old) cosmological con- zons. As a result, we may model a black hole as a hole stant problem, which is arguably the most puzzling as- in spacetime – a bubble of nothing – and end spacetime pect of coupling gravity to relativistic quantum mechan- at a microscopic distance from the putative horizon (at ics [14]. Given that the natural expectation value for stretched horizon). It will be the responsibility of a full the vacuum of the standard model of particle physics quantum gravity theory to resolve this singularity. is ∼ 60 orders of heavier than the gravita- First, we will explain the empty black hole model for tional measurements of vacuum density, it is reasonable spherically symmetric black holes (Schwarzschild) and to entertain an alternative theory of gravity where the derive Bekenstein-Hawking entropy. Then, we will ex- standard model vacuum decouples from gravity. Such a tend the model to the most general black holes in four theory could be realized by coupling gravity to the trace- dimensions, i.e. Kerr-Newman black holes. less part of the quantum mechanical energy-momentum tensor. However, the consistency/covariance of gravita- tional field equations then requires introducing an auxil- A. Schwarzschild Black Holes iary fluid, the so-called gravitational aether [15]. The sim- plest model for gravitational aether is an incompressible Once there is a boundary in spacetime, we need to fluid (with vanishing energy density, but non-vanishing specify a boundary condition. We impose radial Z2 sym- pressure), which is currently consistent with all cosmolog- metry at the boundary, as it is a natural boundary condi- ical, astrophysical, and precision tests of gravity [16, 17]: tion for a spherically symmetric solution. This Z2 bound- 3 1 ary condition also appears in the for G = T − T αg + T 0 , black holes [21]. 32πG µν µν 4 α µν µν N Consider a static spherically symmetric spacetime 0 0 0 0 µν Tµν = p (uµuν + gµν ),T ;ν = 0, (1) which, in general, has the following line element where GN is Newton’s constant, Tµν is the matter energy 2 0 2 2 2 dr 2 2 momentum tensor and Tµν is the incompressible gravita- ds = −N (r)dt + + r dΩ , (2) tional aether fluid. In vacuum, the theory reduces to GR f(r) coupled to an incompressible fluid. where dΩ2 ≡ dθ2 + sin2(θ)dφ2 is the line element of 2- Motivated by the existence of singularities or minimum sphere. Here N(r) and f(r) are arbitrary functions sat- area surfaces close to black hole event horizons in the isfying GR field equations in the bulk (for Schwarzschild models mentioned above, we propose a new model for N 2(r) = f(r) = 1 − 2m .) If there were no boundary, black holes in which spacetime ends, and is orbifolded, at r then we would have a horizon at r = r where N(r ) = 0. a microscopic distance from black hole horizons. In this 0 0 However, in this model, spacetime ends at a microscopic model, a black hole is a “bubble of nothing” (reminiscent distance r = r∗ > r from the horizon. of [18]). We then show that putting a (2+1 dimensional) 0 We assume there is a thin layer of fluid – not aether surface fluid with vanishing density (i.e. incompressible) – sitting at the boundary (r = r∗), with the following at the new boundary, which is required by Israel junction energy-momentum tensor conditions, gives a thermodynamic entropy identical to the Bekenstein-Hawking entropy. This work also suggests Tab = (Σ + Π)UaUb + Πhab, (3) an analogy between black holes in 3+1 dimensions and

2+1 dimensional incompressible fluids, which have been where Tab is the surface energy-momentum tensor, Σ is studied earlier in the context of holography [19, 20]. surface energy density, Π is surface pressure, Ua is the The outline of our paper is as follows. Section II is de- fluid 3-velocity, hab is the induced metric on the hyper- voted to our new model of empty black holes (EBHs) and surface r = r∗ and a, b ∈ {t, θ, φ}. In fact, as we will the resulting local derivation of the Bekenstein-Hawking show later, imposing radial Z2 symmetry on the bound- entropy for charged rotating empty black holes. We ary requires the existence of this fluid. then present a toy microscopic description for the sur- For a general hypersurface Sr defined as r = ∗ face fluid with (near-)vanishing density in Section III. In constant > r , the line element on Sr can be written section IV, we comment on the effect of putting a physi- as cal boundary at the stretched horizon of a black hole, and 2 2 2 2 2 discuss how Hawking radiation and Bekenstein-Hawking dl3 = −N (r)dt + r dΩ . (4) 3

FIG. 1: Comparison of the causal diagrams for the static Schwarzschild black hole, and the static black holes in gravitational aether [13]. In both diagrams, the solid lines depict null infinities, while the squiggly lines are singularities, and grr vanishes on dotted lines. However, the latter is a null surface in Schwarzschild BH which coincides with event horizon, while it is time-like in the aether BH and corresponds to a throat or minimal area surface. Moreover, while the singularity is space-like and sits at zero area deep inside the horizon in the Schwarzschild BH, it is null in the Aether BH and sits at finite area, roughly a inside the throat. The latter assumption is the key ingredient for aether pressure to match the observed dark energy pressure for astrophysical BH [13].

So we obtain this fact, we can show (see Appendix A for details) that

2 2 2 2 imposing radial Z2 symmetry (for time-like boundaries) hab = diag(−N , r , r sin θ), (5) modifies (7) and (8) to and Kab − Khab = −8πTab. (12) 0 1/2 1/2 1/2 2 Kab = diag(−N Nf , rf , rf sin θ), (6) As a result, we get c d p for the extrinsic curvature Kab = hahb ∇cnd of the hy- f(r∗) persurface, where N 0 ≡ dN . 4πΣ = − , (13) dr r∗ We now employ the Israel junction conditions N 0(r∗) pf(r∗) 8πΠ = pf(r∗) + . (14) [hab] = 0 (7) N(r∗) r∗ ∗ Note that for , in the limit r → r0 equation (13) gives Σ = 0. [Kab] − [K]hab = −8πTab, (8) Since the surface fluid is at constant radius it conse- where [A] ≡ A(r+) − A(r−) is the discontinuity of A(r) quently sees the thermal radiation due to its acceleration (Unruh effect [22]; we will further justify this choice in across the hypersurface Sr. Using equations (3), (6), (7) V). Assuming the fluid is in thermal equilibrium with and (8) for a static solution (Ua having only a non-zero temporal component), we obtain the Unruh radiation, its temperature is fixed by the tem- perature of the radiation in the fluid’s vicinity, and so [N] = 0, (9) f 1/2 1 N 0(r∗) 4πΣ = −[ ], (10) T (r∗) = T = pf(r∗). (15) r Unruh 2π N(r∗) N 0 f 1/2 8πΠ = [ f 1/2] + [ ] (11) Note that the fluid pressure (14) and temperature (15) N r ∗ diverge in the limit r → r0. for a hypersurface of radius r. In particular, we could We now have everything to calculate the entropy of write the previous junction conditions at r = r∗. How- the surface fluid. Assuming local thermodynamic equi- ever, Sr∗ is the boundary of spacetime, and the discon- librium (LTE) at zero chemical potential (which is ex- tinuity of functions across Sr∗ is not defined. Despite pected at high temperatures), the entropy per unit area 4 of this fluid is given by that the fluid is rotating with angular frequency Ω in the direction of φ. Σ + Π s = (16) Using (12), we get T 1 yielding Σ = 0, Π = . (25) p ∗ 8π Γ+λ 1 s = , (17) 4 As before, we assume equilibrium, and so the tempera- ture of the surface fluid is fixed by the temperature of which is the same as Bekenstein-Hawking entropy. Unruh radiation. Since the acceleration of the fluid is 1 a = , (26) p ∗ B. Kerr-Newman Black Holes Γ+λ

We can also extend this model to charged rotating we find black holes. As shown in Appendix B (see Eq. B1), 1 the near horizon geometry of a Kerr-Newman black hole T = a/2π = , (27) 2πpΓ λ∗ is + sin2 θ for the Unruh temperature. Note that this angle- ds2 = −Γ λ2dτ 2 + Γ dλ2 + Γ dθ2 + dψ2, dependent temperature is the same as blue-shifted Hawk- + + + Γ + ing temperature of Kerr-Newman black hole for a co- 2 2 2 rotating observer at the position of the boundary. Fi- where Γ+ ≡ r+ + a cos θ and λ = 0 corresponds to the horizon of black hole. Similar to the Schwarzschild case, nally, the entropy per unit area of the fluid will be ∗ we end the spacetime at the stretched horizon (λ = λ > Σ + Π 1 0, taking the limit λ∗ → 0) and impose Z boundary s = = . (28) 2 T 4 condition at the new boundary. The Z2 symmetric boundary requires that the extrin- sic curvature of the boundary should satisfy (12). Ex- III. MICROSCOPIC DERIVATION OF AN pressing the induced metric on the boundary in (τ, θ, ψ) INCOMPRESSIBLE FLUID coordinates we obtain sin2 θ In the previous section, we showed that a stationary h = diag(−Γ λ∗2, Γ , ) (18) solution requires the existence of an incompressible sur- ab + + Γ + face fluid on the boundary of an EBH. In this section we where the normal vector to the hypersurface λ = λ∗ is show that a class of dispersion relations for matter can give rise to a nearly incompressible fluid at high energies p nµ = Γ+(0, 1, 0, 0), (19) (by nearly incompressible, we mean that the pressure of the fluid is much greater than its energy density). µ 1 n = p (0, 1, 0, 0). (20) For a thermal gas of /fermions at temperature Γ+ T , energy density ρ and pressure P are as follows Consequently we find Z d3p ρ(T ) = g E(p)n(p) =< E >, (29) 1 ∂hab p ∗ 3 Kab = √ ( )λ∗ = diag(− Γ+λ , 0, 0), (21) (2π) 2 Γ ∂λ + 1 Z d3p dE 1 dE ab √ 1 P(T ) = g p n(p) = < p >, (30) K = Kabh = ∗ . (22) 3 Γ+λ 3 (2π) dp 3 dp

Using (3) with where g is the degeneracy factor, E(p) is dispersion rela- tion (relation between energy and momentum of a parti- p ∗ Ua = Γ+λ (1, 0, 0) (23) cle) and (so that U aU bh = −1), we find 1 ab n(p) = , (31) eE(p)/T ± 1 2 ∗2 sin θ Tab = diag(Σ Γ+λ , ΠΓ+, Π ). (24) where for simplicity we set the chemical potential µ = 0 Γ+ (+ for fermions, − for bosons.) Note that this means that the fluid is comoving with the Once we specify a dispersion relation, we are able to p ∗ black hole, where Γ+λ is the normalization factor. compute the energy density and pressure of a fluid of Recall that we are working in (τ, θ, ψ) coordinates, so these particles. While particles with m at low en- a zero velocity component in the direction of ψ means ergies satisfy the Lorentzian dispersion relation E2 = 5 p2 + m2, as we argue below, there are reasons to be- increases and goes to infinity as temperature goes to in- lieve the energy-momentum relation might be modified finity. Consequently, according to (45), w increases un- at high energies (e.g. [23]). boundedly with temperature. For example, consider the dispersion relation [24] On the other hand, the dispersion relation (41) regu- lates the UV infinities of quantum field theory, since there p2 E2 = , (32) is a maximum momentum for any particle. We show here 1 − p2/Λ2 the argument for a real scalar field φ. In the interaction picture, φ can be expanded in terms of creation and an- which reduces to the Lorentzian dispersion relation (with nihilation operators as m = 0) at low energies (p  Λ), while it deviates from Lorentzian dispersion relation at high energies. Indeed, Z Λ 3 d ~p 1 ip·x † −ip·x energy becomes infinite for a finite value of momentum. φ(x) = (a~pe + a e ), (36) (2π)3 p2E ~p In Fig.2 we depict the equation of state variable w (pres- ~p sure over density) of a fluid obeying the dispersion re- where p·x ≡ −p0t+~p·~x, p0 = E (≡ E(p)) and [a , a†] = lation (41), as a function of temperature. We see that ~p ~p ~q (2π)3δ(3)(~p − ~q). Using the above expansion, we obtain w GF (x − y) = h0|T φ(x)φ(y)|0i Z 4 d p θ(Λ − |~p|) ip·(x−y) 104 = i 4 0 2 2 e . (37) (2π) (p ) − (E~p) + i

1000 Fermi-Dirac For canonical interactions, loop infinities originate from integration over products of Feynman Green’s func- 100 Bose-Einstein tions. However, since there is a cut off for spatial part of momentum, loop corrections of quantum field theory

10 will be finite.

T

4 5 IV. WHY UNRUH TEMPERATURE? 10 100 1000 10 10 L

P Our derivation of EBH entropy crucially relied on as- FIG. 2: Equation of State variable (w = ρ ) as a function of Temperature. suming that the surface fluid is heated up to the Un- ruh temperature [22], set by the acceleration of the ob- servers on the stretched horizon. The original deriva- w grows as temperature increases, and approaches infin- tion of Unruh (Hawking) radiation relies on a (locally) ity in the limit of infinite compression, expected for the Minkowski vacuum at the vicinity of the horizon, and al- surface fluid. tering the dispersion relation (Sec. III) at high momenta Generically, fluid of particles with a dispersion relation will change the properties of the vacuum state. As a re- in which energy as a function of momentum grows faster sult, on small scales one no longer expects the Minkowski than any power law becomes a near incompressible fluid vacuum to be an adequate description. However, further at high temperatures. We will show this by an example. investigation [25, 26] showed that generally the Hawk- Consider the following dispersion relation ing radiation does not heavily depend on the dispersion p2l relation at high momenta (see [26] for conditions and ex- E2 = p2 + . (33) Λ2(l−1) ceptions), and the spectrum at energies below the UV scale, Λ (e.g., Planck energy) remains unchanged to low- At low temperatures (T  Λ), energies of particles are est order in powers of E/Λ. effectively equal to their momentum E ≈ p, and accord- Similarly, one might suspect that the existence of a ing to (38) and (40) physical boundary at the stretched horizon might change the radiation process. Here, we demonstrate that putting P 1 w = ≈ . (34) a boundary at the stretched horizon will not change ρ 3 Hawking radiation to the lowest order. Reduced to its However, at high temperatures (T  Λ) the dispersion bare bones, the Unruh temperature is simply a statement l of Heisenberg uncertainty principle: the temperature of relation changes to E ≈ p and, as a result Λ(l−1) an emitting region should be bigger than the inverse of its l size. Any lower temperature cannot be in local thermal w ≈ . (35) equilibrium (LTE), which was our key assumption pre- 3 ceding Eq. (16). In order to show this rigorously, we first However, if dispersion relation grows faster than any briefly review a simple derivation of Hawking radiation power law for large p, then l increases as temperature similar to [27]. 6

A massless scalar field Φ in Schwarzschild spacetime which results in the following power spectrum satisfies the following wave equation ˜ 2 8πm 1 1 √ |Φk(ω)| = 2 8πmω . (44) √ ∂ −g gµν ∂ Φ = 0. (38) r ω e − 1 −g µ ν This shows that, up to gray body factors and gravita- Using Kruskal coordinates tional redshift, the observer in Schwarzschild coordinates

1 detects thermal radiation with temperature T = TH = r 2 r 1 v = − 1 e 4m 8πm . 2m Now, let’s assume we cut off the solution Φ =       k t t 1 eik(u−v) at some minimum radius r = 2m(1 + ) × sinh θ(r − 2m) + cosh θ(2m − r) , r min 4m 4m on a v = 0 hypersurface (for simplicity), and replace it 1 with r 2 r 4m u = − 1 e 2m c 1 iku   t   t   Φ0,k(u, v = 0) = e θ(r − rmin). (45) × cosh θ(r − 2m) + sinh θ(2m − r) , r 4m 4m Using where θ(x) is the Heaviside step function, the metric be- 2 2  r  r comes u − v = − 1 e 2m , (46) 2m 3 2 32m − r 2 2 2 2 ds = e 2m (−dv + du ) + r dΩ . (39) on v = 0, we get r θ(r − r ) = θ(u − u ), (47) Considering only the radial mode ∂θΦ = ∂ϕΦ = 0, Equa- min min tion (38) results in 1 rmin rmin 2 4m where umin ≡ ( 2m −1) e . As a result, the outgoing 2  2  −∂v r ∂vΦ + ∂u r ∂uΦ = 0, (40) solutions to (41) with initial conditions (45) will be

Where r is an implicit function of u and v. Substituting c 1 c 1 ik(u−v) 1 Φk(u, v) = Ψ0,k(u−v, v = 0) = e θ(u−v−umin). Φ = r Ψ into (40), we get r r (48)  4  c 2 2 64m − r Expressing Φ in terms of Fourier modes of the −∂ + ∂ − e 2m Ψ = 0. (41) k v u r4 Schwarzschild observer, we get

ik(u−v) Z +∞ Plane-waves e are outgoing solutions to the above ˜ c −iωt c 4 r Φk(ω) = dt e Φk(r, t) 2 64m − 2m equation provided that k  r4 e . Since we are −∞ +∞ 1 considering the outgoing modes outside horizon, the term Z − t  r−t  4 1 4m  r  2 64m − r −iωt iσe 4m e 2m reaches its maximum at r = 2m, which is of = dt e e θ − 1 e − umin r4 r 2m order unity. As a result, for k  1, i.e. wavelengths much −∞ +∞ 1 ik(u−v) 1 Z − t shorter than the size of the black hole, Φk = e −iωt iσe 4m r = dt e e θ(tmax − t), (49) are valid solutions outside the horizon. Note that the r −∞ non-radial modes satisfy a similar equation to (41) with 1 tmax − r a different potential term. For the case of non-radial − 4m r  2 − 4m where e = umin 2m − 1 e = r modes or k 1, the potential term only introduces a r 1 min . min 2 4m ( 2m −1) e gray body factor in the last result [28], which is of no 1 r . r −1 2 e 4m interest in the present calculation. ( 2m ) A static observer in Schwarzschild coordinates (r, t) de- Using ˜ composes this wave into its Fourier modes Φk(ω) using 1 Z +∞ df time coordinate t (which is its proper time, up to a grav- θ(t) = − e−ift, (50) 2πi f + ia itational redshift factor). As a result, −∞

+∞ +∞ where a is a small positive number, together with (43), Z Z − t −iωt 1 −iωt iσe 4m Φ˜ k(ω) = dt e Φk(r, t) = dt e e , we get −∞ r −∞ (42) Z +∞     c df 1 −ift 4m r 1 r − t ˜ max where we have used u − v = ( − 1) 2 e 4m e 4m and σ ≡ Φk(ω) = − e 2m −∞ 2πi f + ia r (51) r 1 r k( − 1) 2 e 4m > 0. After a straightforward calculation, 2m ×(−iσ)4im(f−ω)Γ [4im(ω − f)] . we get 4m Closing the contour in the lower half complex plane of Φ˜ (ω) = (−iσ)−4imωΓ(4imω), (43) k r f (note that tmax > 0), we can express (51) in terms of 7 residues of the integrand. Contribution from the pole at f = −ia is 4m (−iσ)−4imωΓ(4imω) = Φ˜ (ω). (52) r k The Gamma function Γ(z) has also simple poles at z = (−1)n −n where n = 0, 1, ... with residues n! ; their contri- bution is n  − tmax  4m e−iωtmax iσe 4m Φ˜ (n)(ω) ≡ − . (53) r n + 4imω n! Despite the fact that the pole f = ω corresponding to n = 0 is on the real line and must be treated more carefully, we argue that either there is no contribution from this FIG. 3: “” of a : This figure demon- pole or that this contribution is not important. First, the strates that even though the average temperature of a firewall could be greater than Λ, where the fluid becomes incompress- contribution from the n = 0 pole is the same as adding a c ible, the “photosphere” might have an effective temperature constant to Φk(r, t). Since the field Φ is massless, adding less than the Lorentz violation scale Λ. As a result the deriva- a constant term has no observational effect and can be tions of Hawking/Unruh temperature goes through. As we ignored. On the other hand, we expect to recover the argue in the text, this implies that Bekenstein-Hawking area result of the previous calculation for Φ˜ k(ω) in the limit law is an upper limit for the entropy of the firewall. tmax → +∞. This fact also shows that the n = 0 pole ˜ c should not contribute to Φk(ω). Finally, we get ∞ will put these ideas together to construct a more physical ˜ c ˜ X ˜ (n) picture. Φk(ω) = Φk(ω) + Φ (ω) n=1 The surface fluid can be thought of as a thin shell  t n of matter with finite (non-zero) thickness concentrated ∞ − max 4m X iσe 4m on the stretched horizon of the EBH (Fig. 3). Upon = Φ˜ (ω) − e−iωtmax . (54) k r (n + 4imω)n! , matter is heated up to the firewall average n=1 temperature Tave. For an “optically” thick object, this The new contributions are suppressed with the power of temperature is higher than the effective temperature of 1 tmax rmin 1 − rmin  2 the “photosphere”, Teff which is where “radiation” leaves σe 4m = k − 1 e 4m ≈ k(e) 2 for small value 2m the surface of the fluid 1. We should note that “photo- of . 1 sphere” here refers to the radius where “optical” thick- For k 2  1, the leading contribution from the correc- ness to infinity equals unity, for the particles that are tions comes from n = 1 term. Comparing this term with mostly responsible for energy transfer. For example, for the thermal part Φ˜ (ω), we get k a explosion, this is mainly due to neutrinos, 2 as become optically thin much farther out, and Φ˜ (1)(ω) e 4mω = k2 e8πmω − 1 . (55) thus have much lower effective temperatures. ˜ 2 2 Φk(ω) 2π 1 + 16m ω We have already shown above that vacuum state de- fined by Kruskal modes is seen by a Schwarzschild ob- This shows that the corrections are not important for server as a thermal state with blue-shifted Hawking (or 2 ω/TH . − ln(k ). However, because of exponentially Unruh) temperature. This argument relies on the fact damping term in thermal power spectrum, the correc- that Lorentz symmetry is still preserved. However, we tions become important for higher frequencies. Conse- have changed the dispersion relation to acquire an in- quently, putting a physical boundary at the stretched compressible fluid on the stretched horizon. The only horizon, only changes the tail of the thermal power spec- consistent way to satisfy all these requirements is to as- trum. sume that Teff is smaller than Lorentz violation energy Let us now recap what we have done so far: We have scale, Λ. With this assumption, the relativistic derivation constructed a model for a general black hole in Section II above for Hawking/Unruh radiation is still valid. which realizes the idea of a firewall on its “horizon”, and On the other hand, Tave must be higher than Λ in order results in a local derivation for Bekenstein-Hawking en- to get an incompressible fluid on the stretched horizon. tropy. The latter is based on the assumption of an Unruh temperature for the firewall, which we justified earlier in this section. However, this model is idealized, as there is a perfect incompressible surface fluid on the stretched 1 This is dictated by radiative transfer equations, which is exactly horizon of EBH. In Section III, we have proposed a mi- the same reason the average temperature of the (∼ 1.5 × croscopic description for incompressible fluid. Here, we 107K) is higher than that of its photosphere (∼ 6 × 103 K). 8

Consequently, we require a classical theory and so in principle an analysis of the collapse of matter in this theory can be carried out. Fig. TUnruh ' Teff  Λ  Tave. (56) (4) compares the expected causal diagrams for collapse of standard black holes and EBH’s. This condition is consistent with what we expect from Clearly a surface fluid with vanishing energy, but non- radiative transfer within the firewall, i.e. T > T to ave eff vanishing pressure (i.e. incompressible) necessitates in- get the outward flux of radiation. voking an exotic phase of matter. Interestingly, we notice Consider next the entropy equation, that a field theory with a Lorentz violating momentum Π cut-off, Λ, which regulates all UV divergences of canon- S = . (57) T ical quantum field theories, approaches this equation of state at high temperatures. In other words, an (admit- Previously, we derived that S = SBH if T = Tave was tedly na¨ıve) UV completion of the quantum field theory, blue-shifted Hawking (or Unruh) temperature. However, also reproduces the correct entropy of the black hole. Fi- note that Teff is the blue-shifted Hawking temperature nally, in the last section we showed that, even if Lorentz and since Tave > Teff , we have symmetry is violated in the firewall, to ensure its incom- pressibility, its photosphere could still be much cooler S < S . (58) BH than the scale Λ, and thus emit the canonical Hawking This shows that Bekenstein-Hawking entropy is an abso- spectra. This sets the Bekenstein-Hawking area law as a lute upper limit for the entropy of matter condensation strict upper limit for the entropy of the firewall. on the firewall. The ratio S = Teff depends on how To our knowledge, our derivation of Bekenstein- SBH Tave sharply the Lorentz violation and rise in optical depth Hawking entropy is the first truly local description of happen. For example, in an extreme case one can imag- black hole micro-states. For example, models for the area ine a very sharp violation of the Lorentz dispersion rela- law based on entanglement entropy, or string theory (e.g., tion and rise in optical depth, where Tave and Teff can be in the fuzzball proposal) are inherently non-local: the to- made very close to Λ. In this case S → 1. tal entanglement entropy depends on an uncertain cut- SBH Finally, we should also mention that Λ is the energy off, and only reproduces the area law up to an order unity scale of Lorentz violation in the matter sector and it factor [30]. Computing the change in the entanglement can be smaller than Planck energy. This implies that entropy can reproduce the correct change in the area law although the fluid is heated up to an internal tempera- (assuming Einstein equations) [31, 32], but does NOT ture above Λ, its temperature can still be smaller than localize the total entropy on the surface. Similarly, the Planck energy. As a result, the classical general relativity counting of string theory fuzzball solutions involves sum- description of the Israel junction conditions may still be ming over states with no classical 4d counterpart (e.g., valid. [1]). Another proposal for the endpoint of gravitational col- lapse, which is similar to our model, is the [33]. V. DISCUSSION AND CONCLUSIONS The exterior of a gravastar is the Schwarzschild geometry, whereas the interior is replaced by a de Sitter spacetime We have shown that a surface fluid at Unruh temper- that is matched to the exterior via a thin shell of stiff fluid ature with vanishing energy density (but non-vanishing (a fluid with p = ρ.) However, the entropy of a gravastar pressure) on the stretched horizon of a black hole – which is much less than the Bekenstein-Hawking entropy. we call an empty black hole – has the same thermody- Additionally, the Bekenstein-Hawking entropy can be namic entropy as Bekenstein-Hawking entropy. The sur- derived as the thermodynamic entropy of a stiff fluid in face fluid is the result of ending (or orbifolding) space- the presence of a black hole [34, 35]. In this model, the time at the stretched horizon and replacing it with a Z2 horizon is replaced by a high density thin shell of stiff symmetric boundary. We therefore conjecture that the fluid. However, this solution suffers from the presence microstates of a black hole are those of the surface fluid of a point-like with negative mass at at the stretched horizon. We emphasize that this descrip- the center. In this context, an interesting question for tion is very similar to the traditional membrane paradigm future study would be to consider the gravitational col- for black hole horizons [21]. However, in our description, lapse of matter (e.g. a ball of dust) in the more general the membrane properties are physical and result from context of gravitational aether to see how the end state condensation of accreted matter onto a physical mem- is (classically) approached. , while they arise only as a mathematical analogy While the list of motivations for truncating the classi- in the traditional membrane paradigm (e.g. the pres- cal spacetime at black horizons has grown manyfold over sure in [29]). Although the EBH model was constructed the past few years, here we note the recurrence of the in vacuum, there is a singularity (or boundary) at the notion of incompressibility, in our exposition. Indeed, horizon. This situation could arise via some unknown incompressibility is no longer considered a mathemati- quantum gravity effects such as a firewall or via gravi- cal novelty (e.g. [19, 20, 36–38]), and has appeared, and tational aether. The latter has the advantage that it is re-appeared in different physical contexts. In particular, 9

FIG. 4: Comparison of the causal diagrams for a collapsing Schwarzschild black hole, and our proposed picture for collapsing black holes in gravitational aether or firewall scenarios. In both diagrams, the dotted lines depict classical event horizons, while the squiggly lines are singularities. The black area correspond to the collapsing . While in the Schwarzschild BH, the singularity is space-like and deep inside the horizon, in the aether/firewall case it approaches the horizon and becomes null asymptotically. The accreted material smoothly crosses the Schwarzschild horizon, but it condenses into Planckian densities just inside the horizon of the aether BH. the appearance of an incompressible fluid in both bulk Even more speculative, but most exciting, is the pos- and boundary of gravitational aether black holes is sug- sibility of directly probing quantum gravitational effects gestive of the continuity of the underlying microscopic by precision studies of astrophysical horizons (e.g. [39]). phenomenon. The obvious difference between the two Given that the conditions of big bang is now replicated fluids, however, is that the boundary fluid carries a finite close to black hole horizons (and not deep inside them), entropy, while the gravitational aether has a degenerate the observational constraints on the early universe might phase space (and thus zero or negligible entropy). Nev- now be applied to the microscopic structure of horizons, ertheless, it is not clear whether this difference might be or conversely, black hole observations can potentially con- an artifact of the surface condensation, or rather point strain the nature of cosmological big bang. to different origins of the two fluids. A related puzzle is why Bekenstein-Hawking entropy is localized on the surface (at least at the classical level), while the bulk incompressible fluid can cause non-local interactions. Acknowledgement: The authors would like to thank Furthermore, one might ask what type of horizons are Siavash Aslanbeigi, Eugenio Bianchi, Samir Mathur, expected to develop singularities. Do we expect singu- Paul McFadden, Rafael Sorkin, and Dejan Stojkovic for larities for Rindler or de Sitter horizons? For example invaluable discussions. This work was supported by the it has recently been argued [5] that singularities (or fire- Natural Science and Engineering Research Council of walls) only occur in “old” horizons, at a fraction of their Canada, the University of Waterloo and the Perimeter In- evaporation time, which never happens for de Sitter or stitute for Theoretical Physics. Research at the Perime- Rindler horizons. ter Institute is supported by the Government of Canada 10 through Industry Canada and by the Province of Ontario through the Ministry of Research & Innovation.

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Appendix A: Israel Junction Condition at Z2 In addition, if Kab remains bounded, then Symmetric Boundary Z  lim dλ Zab = 0. In order to get junction conditions at a boundary with →0 − radial Z2 symmetry, we will use the same technique as [40]. The metric (2) can be written in terms of proper In (A1) coordinates gab = hab. Consequently, (A5) re- radial distance as sults in 1 2 2 2 2 2 2 ds = −N (λ)dt + dλ + r (λ)dΩ , (A1) −8π(Tab − habT ) = Kab. (A6) 2 where The above equation has been derived in a particular co- ordinate for boundary. However, it is a tensorial equation Z r dr0 λ(r) = . and accordingly, it is valid in any coordinate for bound- p 0 r∗ f(r ) ary. Taking trace of (A6) and expressing T in terms of ab K ≡ Kabh , we get Radial Z2 symmetry thus implies that for points at λ = 0, the metric can be locally extended to negative values of Kab − Khab = −8πTab λ, such that We argued that for the special case of spherical sym- N(λ) = N(−λ) (A2) metry, (A2) and (A3) are the conditions required to have r(λ) = r(−λ). (A3) Z2 symmetric boundary. This definition can be extended to more general . Also, if there is a thin layer of fluid between λ = 0 and Let’s start with an intuitive definition for Z2 symmetry. λ = , radial Z2 symmetry will require the existence of A spacetime (M, g) has (local) Z2 symmetry with respect a fluid with the same energy momentum tensor between to a hypersurface S, which divides spacetime into two λ = 0 and λ = −. parts (M+, g+) and (M−, g−), if local observers on S 3 + − If Kab and Rab represent the extrinsic curvature and cannot distinguish between M and M . This means if intrinsic curvature of a surface with constant λ (which they move perpendicular to S (along normal vector n to means surface with constant r), respectively, then in (A1) S) into M+ or M−, they will see the same geometry. In coordinates [40] mathematical language, it means

2 2 2 + − gab = hab = diag(−N , r , r sin θ) Ln gµν = L−n gµν , (A7) 1 ∂g K = ab , (A4) ab 2 ∂λ where Ln is Lie derivative with respect to n. In particu- lar, it results ∂Kab Rab = − + Zab, ∂λ + − Kab = −Kab, where and Israel junction condition (8) for Z2 symmetric hy- 3 c persurface (with space-like normal vector) gives Zab = Rab − KKab + 2KaKcb.

Using the Einstein field equation 2(Kab − Khab) = −8πTab. (A8) 1 However, if S is the boundary of spacetime, the sit- R = 8π(T − T g ), µν µν 2 µν uation is a bit different, since S does not divide space- time into two parts. In this case, Z2 symmetric bound- and integrating through the layer (from λ = − to λ = ), ary means that we glue a copy of spacetime M to itself we obtain through S (it means that S acts like a mirror.) Now, S has divided the whole spacetime (M + M) into two Z  Z  1 Kab()−Kab(−)− dλZab = −8π dλ(Tab− gabT ). parts and condition (A7) has been satisfied. However, − − 2 we must multiply the right hand side of (A8) by a factor (A5) of two, because there is also a copy of the surface fluid Imposing Z2 symmetry and taking the limit  → 0 gives on the other side (as we showed concretely for spherically symmetric spacetimes.) Z  1 Z  Tab ≡ dλTab = dλTab. As a result, we obtain 0 2 − Kab − Khab = −8πTab, (A9) Also, (A2),(A3) and (A4) give for Z symmetric boundaries. Indeed, equation (A9) can ∗ 2 lim [Kab() − Kab(−)] = 2Kab at r = r . →0 be used as a definition for Z2 symmetric boundary. 12

Appendix B: Near Horizon Geometry of In order to derive the near horizon geometry of the Kerr-Newman Black Hole Kerr-Newman metric we define a new variable

The Kerr-Newman metric describes the geometry of Z r dr0 r r − r λ = ≈ 2 + + O((r − r )3/2). a black hole with J and charge Q. ∆(r0) r − r + This metric can be written as r+ + −

2 2 2 2 2 ds = gttdt + grrdr + gθθdθ + gφφdφ + 2gtφdtdφ Small values of λ correspond to radii close to the horizon 2 r+. Replacing r in the metric with the new coordinate λ gtφ 2 2 2 gtφ 2 = (gtt − )dt + grrdr + gθθdθ + gφφ(dφ + dt) and, keeping the leading terms for λ  1, we obtain gφφ gφφ where (r − r )2 ∆ = (r − r )(r − r ) ≈ + − λ2, 2 + − 4 2mr − rQ gtt = −(1 − ), 2 2 2 2 2 2 Γ Γ = r + a cos θ ≈ r+ + a cos θ ≡ Γ+, 2 2 2 Γ gtφ (r+ − r−) Γ+ λ grr = , gtt − ≈ − 2 2 , ∆ gφφ 4 (r+ + a ) gθθ = Γ, 4Γ+ 2 2 2 ! g ≈ , (2mr − r )a sin θ rr 2 2 2 2 2 Q (r+ − r−) λ gφφ = sin θ r + a + , Γ gθθ ≈ Γ+, g a (2mr − r2 )a sin2 θ tφ ≈ − ≡ −Ω, g = − Q , g r2 + a2 tφ Γ φφ + 2 2 2 2 (r+ + a ) sin θ with gφφ ≈ . Γ+ 2 2 2 2 2 Defining new variables ψ ≡ (φ − Ωt)(r+ + a ) and τ ≡ ∆ = r − 2mr + a + rQ, r+−r− 2(r2 +a2) t, we get Γ = r2 + a2 cos2 θ, +

J Q2 2 and a = and rQ = . The horizon of Kerr-Newman sin θ m 4π0 2 2 2 2 2 2 ds = −Γ+λ dτ + Γ+dλ + Γ+dθ + dψ . (B1) metric is at ∆ = 0 Γ+ q 2 2 2 2 2 2 r − 2mr + a + rQ = 0 → r± = m ± m − a − rQ.