Decoding the Apparent Horizon: A Coarse-Grained Holographic Entropy

TL;DR

This paper establishes that the apparent horizon's area in a black hole corresponds to the maximum coarse-grained holographic entropy consistent with external measurements, linking geometric and informational aspects in holography.

Contribution

It proves the apparent horizon's area equals the maximum holographic entropy compatible with exterior data, clarifying its thermodynamic and informational significance.

Findings

Apparent horizon area equals maximum coarse-grained entropy.

Identifies the boundary dual to this entropy.

Shows the entropy obeys a Second Law of Thermodynamics.

Abstract

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.

Equations

$$S[\rho ]=-\mathrm{tr}(\rho \ln \rho ),$$

$$S^{\mathrm{coarse}}(t)=\underset{{\rho }^{\prime }}{\max }\left(S[{\rho }^{\prime }]:\mathcal{M}(t)\right).$$

$$S_{BH}=\frac{\mathrm{Area}[H]}{4G\hslash },$$

$$S[\rho ]=\frac{\mathrm{Area}[X_{HRT}]}{4G\hslash }.$$

$$S^{\mathrm{coarse}}=\frac{\mathrm{Area}[\mu ]}{4G\hslash },$$

$$S^{(\mathrm{outer})}[\sigma ]\equiv \underset{{\rho }^{\prime }}{\max }\left(S[{\rho }^{\prime }]:O_W[\sigma ]\right),$$

$$S^{(\mathrm{outer})}[\mu ]=\frac{\mathrm{Area}[\mu ]}{4G\hslash }.$$

$$\mathrm{Area}[\mu ]\ge \mathrm{Area}[\Sigma \cap N_{\pm k}[\mu ]]\ge \mathrm{Area}[X_{HRT}].$$

$$\mathrm{Area}[X^{\prime }]\ge \mathrm{Area}[\Sigma \cap N_{\pm k}[X^{\prime }]]\ge \mathrm{Area}[X],$$

$$S^{(\mathrm{simple})}(t_i)=\underset{{\rho }^{\prime }}{\max }\left(S[{\rho }^{\prime }]:⟨E^{†}\mathcal{O}(t)E⟩\mathrm{fixed}\right),$$

$$E=\mathcal{T}\exp [-i\underset{t_i}{\overset{t}{\int }}J(t^{\prime }){\mathcal{O}}_J(t^{\prime })d{t}^{\prime }]$$

$$S^{(\mathrm{simple})}(t_i)\ge S^{(\mathrm{outer})}[\mu ].$$

$$S^{(\mathrm{simple})}(t_i)=\frac{\mathrm{Area}[\mu ]}{4G\hslash }.$$

$${\nabla }_k{\theta }_k=-\frac{1}{D-2}{\theta }_k^2-{\varsigma }_k^2-8\pi G{T}_{kk},$$

$${\nabla }_k{\chi }_i=-{\theta }_k{\chi }_i+\left(\frac{D-3}{D-2}\right){\nabla }_i{\theta }_k-(\nabla \cdot {\varsigma }_k{)}_i+8\pi G{T}_{ik},$$

$${\nabla }_k{\theta }_{\ell }=-\frac{1}{2}\mathcal{R}-2\nabla \cdot \chi -{\theta }_{\ell }{\theta }_k+2{\chi }^2+8\pi G{T}_{\ell k},$$

$$0={\theta }_{\ell }[\mu ]+{\theta }_{\ell ,k}v+\square v+2\chi \cdot \nabla v,$$

Full text

I Introduction

The Second Law of Thermodynamics states that entropy increases with time. One natural notion of entropy is the von Neumann entropy:

[TABLE]

where $\rho$ is the density matrix of a quantum system. However, this quantity is conserved under unitary time evolution, in apparent tension with the Second Law. To obtain an increasing entropy, it is necessary to coarse grain $S$ by “forgetting” certain information, since the vast majority of microscopic data in a thermal system is inaccessible to macroscopic observations. One common coarse-graining method is the maximization of the system’s entropy subject to fixed the values of a set of feasible macroscopic measurements $\mathcal{M}(t)$ at a moment in time Jay57a ; Jay57b ; GelHar06 :

[TABLE]

Assuming that any ordered information inaccessible at early times remains so at later times, $S^{\mathrm{coarse}}$ should increase with time, defining a nontrivial Second Law.

The most mysterious application of the Second Law is to black holes. Stationary black holes (e.g. Kerr) have entropy, which is proportional to the area of their horizon $H$ Bek72 ; Haw75 :

[TABLE]

as suggested by the Laws of Black Hole Mechanics Haw71 ; BarCar73 ; Bek72 ; Haw75 . However, despite some clues from string theory and other approaches (reviewed in Car08 ), it is still unclear in general what microscopic quantum-gravitational degrees of freedom are counted by this entropy. Dynamically evolving black holes such as those formed from stellar collapse are even more controversial, since there are multiple possible definitions of a horizon, e.g. the event horizon and the apparent horizon HawEll — and correspondingly, multiple area increase theorems Haw71 ; Hay93 ; AshKri02 ; GouJar06 ; BouEng15a ; SanWei16 .

In holographic models of quantum gravity, a black hole is dual to some boundary state $\rho$ whose von Neumann entropy $S[\rho]$ can be computed from a compact extremal (HRT) surface in the bulk, as conjectured in RyuTak06 ; HubRan07 and essentially proven in LewMal13 ; DonLew16 :

[TABLE]

A surface is extremal if its area is unchanged by any first order perturbation to the surface’s location; if there is more than one, $X_{HRT}$ is the one with the minimal area extremal surface (and homologous to the boundary HubRan07 ; HeaTak ). 111Because we restrict attention to the entropy of the whole CFT, in this case the HRT surfaces are compact and do not reach the boundary. This quantity is independent of time, so it is not suitable for describing the entropy increase of a growing black hole. Unitarity of the boundary theory implies that no information is lost, but this is not enough: to account for the increase of black hole entropy, a coarse graining scheme must be specified.

Even though black hole thermodynamics was the original motivation for the holographic principle Tho93 ; Sus95 , no one has yet given a clear explanation of the role of the black hole horizon as a repository of information about the interior. Indeed, it was recently shown EngWal17 that if we know the outcome of all classical measurements $\mathcal{M}(t)$ outside of the event horizon $H$, then $S^{\mathrm{coarse}}<\mathrm{Area}[H]/4G\hbar$: we have access to too much information for our remaining ignorance to be given by the event horizon’s area (thus refuting a broad class of proposals relating entropy to area, including Sor97 ; BiaMye12 ; FreMos13 ; KelWal13 .)

We therefore look for alternatives to the event horizon. An appealing option is the apparent horizon $\mu$, the outermost compact surface (at a moment of time) which is marginally outer trapped HawEll , i.e. the the expansion $\theta_{k}\equiv\nabla_{k}\ln(\mathrm{Area}[n])=0$, where $k$ is a future-outwards null vector, and $n$ is a small pencil of lightrays shot out in the $k$-direction from a small neigborhood of a point on $\mu$. In the case of a black hole that forms from collapse, such marginally trapped surfaces form behind the event horizon, even though the HRT surface is the empty set (so that the boundary state is pure).

In this Letter, we give a geometric proof (using classical GR methods in the bulk) that the area of the apparent horizon $\mu$ does play the role of a coarse-grained entropy:

[TABLE]

where we coarse grain over the region behind the apparent horizon (the “microstates”) while holding all classical measurements in the exterior fixed (i.e. we fix all data in the exterior, but working in the classical regime). This makes it plausible that the interior is encoded holographically by a set of independent qubits, one per $4/\ln 2$ Planck-areas, on the apparent horizon (but not the event horizon!) Bou99d ; Bou02 ; SanWei16a ; NomSal16b . Our classical proof explicitly constructs the entropy-maximizing geometry, which would correspond to maximally scrambling all of these qubits. If our result can be extended to the quantum regime (along the lines of Wal10QST ; BouEng15c ; EngWal14 ; FauLew13 ; DonLew17 it might provide insight into the firewalls paradox AMPS ; AMPSS ; MatPlu11 ; BraPir09 , a puzzle about whether maximally scrambled black holes have an interior. An investigation on areas of non-compact analogues of the apparent horizons will appear in MarWhiTA .

Note that although apparent horizons are highly non-unique due to the choice of time slicing, the above construction is valid for each of them.

We also identify the boundary dual to $S^{\mathrm{coarse}}$ of the apparent horizon. This quantity may be computed by maximizing the boundary von Neumann entropy while keeping fixed the outcomes of a set of “simple” experiments performed after a given moment in time. This new entry in the holographic dictionary (which we show is exact to all orders in perturbation theory for near-equilibrium black holes), extends the HRT prescription to a much more general class of bulk surfaces.

Both the bulk and corresponding boundary entropies automatically satisfy the Second Law. This provides the first valid holographic explanation of the Area Increase Law for black holes.

II Outer Entropy

The outer entropy is a coarse-grained entropy that holds fixed the exterior of a codimension-2 surface $\sigma$. We define $O_{W}[\sigma]$, the outer wedge, as the region spacelike outside of $\sigma$ (on the side with the asymptotic boundary). The outer entropy is

[TABLE]

where $\rho^{\prime}$ is any state of the boundary CFT with a classical bulk dual geometry $M^{\prime}$; we choose $\rho^{\prime}$ to maximize the von Neumann entropy $S[\rho^{\prime}]=X_{HRT}[M^{\prime}]$, subject to the constraint that $M^{\prime}$ have the same outer wedge $O_{W}[\sigma]$ as the original classical bulk $M$ dual to $\rho$. Although we have phrased this maximization in terms of the boundary state, note that this can be regarded as a pure bulk construction involving maximizing the area of the HRT surface. The only holographic aspect (in this section) is the identification of an extremal surface lodged inside the black hole with a fine-grained entropy (i.e. the von Neumann entropy). Any theory with such an identification — even one with asymptotically flat boundary conditions (should such a theory exist) — allows the interpretation of $S^{\mathrm{(outer)}}$ as a coarse-grained entropy.

While this coarse-grained entropy can be defined for a general surface $\sigma$, when $\sigma=\mu$, an apparent horizon, we will show that:

[TABLE]

Hence, the area of the apparent horizon has a statistical interpretation as the maximum boundary entropy that is compatible with the geometry of its exterior. This provides a holographic answer to the disputed question: what does the Bekenstein-Hawking entropy of a black hole count? Jac99 ; Sor05 ; JacMar05 ; FreHub05 ; HsuRee08 ; Mar08

Outline of Proof: Let $k$ (respectively $\ell$) be the orthogonal future-directed null vectors pointing outward (respectively inward) from a surface. An extremal surface $X$ satisfies $\theta_{k}=\theta_{\ell}=0$. An HRT surface additionally must be the minimal area surface (homologous to the boundary) on some spatial slice $\Sigma$ Wal12 .

An apparent horizon $\mu$ (an outermost marginally trapped surface) satisfies $\theta_{k}=0$, $\theta_{\ell}\leq 0$, and (generically) $\nabla_{k}\theta_{\ell}<0$ AndMet07 ; Mar14 . We assume that $\mu$ is homologous to the boundary, i.e. there exists a spatial slice $\Sigma$ connecting $\mu$ to the boundary, and moreover that there exists a $\Sigma$ such that the area of any surface circumscribing $\mu$ is larger than the area of $\mu$. These requirements are reasonable for black hole horizons.

In any spacetime, $\mathrm{Area}[X_{HRT}]\leq\mathrm{Area}[\mu]$; this can be proven by a simple focusing argument: in a spacetime satisfying the Null Energy Condition ($T_{vv}\geq 0$ for any null vector $v$), a null surface $N_{\pm k}[\mu]$ shot out along the $\pm k$-direction of $\mu$ has monotonically decreasing area moving away from $\mu$ along $N_{\pm k}$ in the $+k$ or $-k$ directions, where we truncate the surface when generators intersect Pen65 ; HawEll ; Wald . We extend $N_{\pm k}$ along its generators to the slice $\Sigma$ on which $X_{HRT}$ is minimal Wal12 .

[TABLE]

Hence the entropy $S[\rho^{\prime}]$ cannot exceed $\mathrm{Area}[\mu]/4G\hbar$.

To prove that this inequality is saturated, we construct a bulk spacetime $M^{\prime}$ (with the same outer wedge $O_{W}[\mu]$) satisfying Area$[X_{HRT}]=\mathrm{Area}[\mu]$. To specify the interior data in $M^{\prime}$, we impose initial data on $N_{-k}$, the null surface fired from $\mu$ in the $-k^{a}$ direction. We choose our initial data so that the surface $N_{-k}$ is stationary; every cross-section has the same geometry. (The Appendix shows this construction satisfies the constraint equations, so that a spacetime solution $M^{\prime}$ exists, due to $\nabla_{\ell}\theta_{k}<0$.)

By following $N_{-k}$ far enough, we eventually come to an extremal surface $X$ (see Appendix for details). Since $N_{-k}$ is stationary, Area$[X]=\mathrm{Area}[\mu]$. We can complete the spacetime by requiring it to be invariant under a CPT-reflection about $X$ (i.e. we reflect space and time about $X$ while exchanging matter with antimatter). See Fig. 1. The resulting bulk $M^{\prime}$ has two asymptotic boundaries, and therefore represents a pure state (analogous to the thermofield double wormhole construction Mal01 ). When the state $\rho^{\prime}$ is restricted to a single boundary, the entropy $S[\rho^{\prime}]=X_{HRT}[M^{\prime}]$. (Note that the region $O_{W}[X_{HRT}]$ agrees with the original bulk geometry dual to $\rho$ CzeKar12 ; Wal12 ; HeaHub14 ; JafLew15 ; DonHar16 ; FauLew17 .)

Because $N_{-k}$ is stationary and by assumption $\mu$ is minimal on a slice of $O_{W}[\mu]$, we now have an initial data slice $\Sigma$ on which $X$ is the minimal cross-section. Any other extremal surface $X^{\prime}$ has greater area than $X$:

[TABLE]

where the first inequality comes from focusing of a null surface $N_{\pm k}[X^{\prime}]$ shot out from $X^{\prime}$. Hence $X=X_{HRT}$, proving Eq. (7).

III Simple Entropy

Thus far, our coarse-grained entropy has been defined from the bulk point of view. We now identify the boundary dual to the outer entropy, which we call the simple entropy, as it relies on “simple operators”.

In AdS/CFT, single trace operators on the boundary correspond to locally propagating fields in the bulk. More generally, we expect that the product of a small number of single trace operators also propagates locally in the bulk. However, it is known that sufficiently complicated operators (known as precursors PolSus99 ; Fre02 ) can change the deep bulk region acausally; hence to define a coarse graining that is dual to $O_{W}[\mu]$, we must avoid such complicated operations. We therefore define a “simple” experiment as a procedure performed after a moment of time $t_{i}$, in which we measure a local operator $\mathcal{O}(t>t_{i})$ after having turned on a set of local sources $J(t>t_{i})$; we require that these sources propagate causally into the bulk. For classical solutions, we can restrict attention to one-point operators and sources, since the higher-point functions are determined from them. (The “one-point entropy” KelWal13 , proposed as a holographic dual to the area of the event horizon, did not allow sources.) To prevent recurrences, we implicitly include a late time cutoff $t_{f}$ prior to exponentially large values of $t$.

The simple entropy is now defined as the maximum entropy of a state $\rho^{\prime}$ compatible with the outcomes of all such simple experiments (i.e. the maximization is done over a subspace of $\rho$’s that all yield the same outcomes):

[TABLE]

where $\rho^{\prime}$ is defined at $t_{i}$, and

[TABLE]

is the time-ordered insertion of sources $J(t)$ used to prepare the simple experiment by which $\mathcal{O}(t)$ is measured.

A simple experiment, by definition, can only access the subset of the bulk $F(t_{i})$ that is to the future of the boundary time $t_{i}$. When the spacetime has a black hole, turning on simple sources can shift the location of any event horizon $H$ in the spacetime SheSta14 . However, the event horizon must always remain outside of any marginally trapped surface (assuming the Null Energy Condition) HawEll ; Wald . Therefore, if $\mu$ is a marginally trapped surface on $N(t_{i})$, the boundary of $F(t_{i})$, a simple experiment can access at most the outer wedge $O_{W}[\mu]$. Note that by causality, turning on simple sources cannot modify the fact that $\mu$ is marginally trapped (a similar argument was given for extremal surfaces in EngWal14 ). See Fig. 2(a). It immediately follows that

[TABLE]

If $N(t_{i})$ contains more than one marginally trapped surface, we restrict attention the earliest (i.e. outermost) one. This guarantees that $\mu$ is in fact an apparent horizon. We propose that in this case, the inequality (13) is saturated. In other words the simple entropy is the holographic dual of the area of the apparent horizon.

We now show that this is true for a black hole that is approaching thermal equilibrium after time $t_{i}$. We may use the “HKLL” procedure HamKab05 ; HamKab06 ; HeeMar ; BalKraLaw98 ; BalKraLaw98b ; BanDou98 ; Ben99 to reconstruct the ‘causal wedge” $C_{W}[t_{i}]$ of $t_{i}$, i.e. the subset of $F(t_{i})$ outside of the event horizon BouLei12 ; HubRan12 . If no matter or gravitational radiation were falling across the event horizon $H$, it would be stationary; there would be no separation between $H$ and $\mu$, and we would be done. In order to reconstruct the data in $O_{W}[\mu]$, we must ensure that no matter falls across $H$ after $\mu$.

Since $\mu$ is perturbatively close to the event horizon, Kab11 ; HeeMar allows us to map the matter fields falling across the event horizon to data on the boundary. We can therefore turn these fields “off” by adding suitable sources to the boundary after $t_{i}$. This has the effect of shifting the event horizon to the location of $\mu$, so that $C_{W}[t_{i}]=O_{W}[\mu]$.222When lightrays in $N(t_{i})$ intersect before reaching $\mu$, $O_{W}[\mu]\supset C_{W}[t_{i}]]$ since the past boundaries do not coincide. However, $O_{W}[\mu]$ still lies in the domain of dependence of $C_{W}[t_{i}]]$ allowing reconstruction of the full data. EngWalTA . This shows that we can use HKLL to reconstruct the spacetime data arbitrarily close to $\mu$. (Although to reconstruct points a distance $\epsilon$ from $\mu$, we need to wait a time of order $\ln(\epsilon^{-1})$ for the signal to reach the boundary.) This shows that, order-by-order in small perturbation to a stationary black hole,

[TABLE]

This is a new entry in the holographic dictionary, which we conjecture also holds for finite deviations from thermality.

IV An Explanation for the Second Law

A surface $\mathcal{H}$ foliated by marginally trapped surfaces and satisfying certain regularity conditions obeys an area law: the area of the marginally trapped surfaces increases with evolution along $\mathcal{H}$ Hay93 ; AshKri02 ; GouJar06 ; BouEng15a ; SanWei16 . In the case where the marginally trapped surfaces foliating $\mathcal{H}$ are apparent horizons, $\mathcal{H}$ must be spacelike Hay93 , and are called trapping horizons Hay93 , dynamical horizons AshKri02 ; AshGal05 , or spacelike future holographic screens BouEng15a . The area law for these surfaces says that the area of slices of $\mathcal{H}$ increase going in an outward direction.

The spacelike holographic screen $\mathcal{H}$ is illustrated in Fig. 2(b) in a collapsing black hole, where such objects are ubiquitous. The area increases in outwards evolution along apparent horizon slices of $\mathcal{H}$. The corresponding outer wedges are nested: evolving in the direction of increasing area corresponds to computing the outer entropy of progressively smaller outer wedges. This provides an immediate explanation for why the outer entropy increases along $\mathcal{H}$: evolution along $\mathcal{H}$ is the equivalent of maximizing the von Neumann entropy with progressively fewer constraints.

From a boundary perspective, the simple entropy increases for much the same reason, since as $t_{i}$ is increased, there are fewer simple experiments available. It may seem odd that the simple entropy also allows measurements to be made at times after $t_{i}$, but this is equivalent to saying that, for a coarse-graining scheme to have a Second Law, information cannot be discarded if it is going to become available later. (Our very late time cutoff $t_{f}$, which is held constant as $t_{i}$ is increased, prevents us from having to worry about recurrences.)

Acknowledgments: It is a pleasure to thank R. Bousso, X. Dong, G. Horowitz, J. Maldacena, D. Marolf, F. Pretorius, J. Santos, D. Stanford, H. Verlinde, S. Weinberg, B. White, and E. Witten for helpful discussions. The work of NE is supported in part by NSF grant PHY-1620059, while AW was supported by the Institute for Advanced Study, the Raymond and Beverly Sackler Foundation Fund, and NSF grant PHY-1314311.

V Appendix: Constraint Equations

Since we are imposing data on $N_{-k}$, we need to use the “characteristic initial data formalism” Ren90 ; BraDro95 ; ChoCru10 ; Luk12 ; Chr12 ; ChrPae12 ; ChrPae14 , which guarantees the existence of a solution333Luk Luk12 only guarantees a local solution, but then presumably it is possible to deform the characterstic Cauchy slice into a nearby spacelike slice, guaranteeing existence and uniqueness of $M^{\prime}$ Wald .) if we satisfy the following constraint equations on $N_{-k}$ (one for each spacetime dimension $D$):

[TABLE]

as well as the corresponding junction conditions which require $\theta_{k}$, $\chi_{i}$, and $\theta_{\ell}$ to be continuous. Here $\varsigma_{k}$ is the shear tensor, which is free data on $N_{-k}$; $\mathcal{R}$ is the intrinsic Ricci curvature of cross-sections of $N_{-k}$; $\chi_{i}$ is a $D-2$ component twist 1-form gauge field that tells you how much a normal vector gets boosted when transported in the transverse $i$-direction; $T_{ab}$ is the stress tensor. All quantities are defined on constant $v$-slices, where $v$ is an affine parameter defined on each null geodesic of $N_{-k}$, normalized so that $\nabla_{k}=\nabla_{v}$, and $k\cdot\ell=-1$.

We can solve these constraint equations for stationary $N_{-k}$ by stipulating that $\varsigma=\theta_{k}=T_{kk}=T_{ki}=0$, while $\mathcal{R}$, $\chi_{i}$, $T_{\ell k}$ are constant along $v$. The marginality condition $\theta_{k}[\mu]=0$ ensures continuity of $\theta_{\ell}$ and $\theta_{k}$ on the junction between $N_{-k}$ and $O_{W}[\mu]$. The shear is generically discontinuous across the junction, but that is not a problem for local evolution of the Einstein equation LukRod12 ; LukRod13 . We assume without proof that evolution is possible with AdS boundary conditions.

The above conditions on the stress tensor can be satisfied by reasonable matter fields. For a minimally coupled scalar field $\phi$, take $\phi=$ constant in the $k$-direction; for a Maxwell field $A_{a}$, impose $\nabla_{k}A_{i}=0$ in the gauge $A_{k}=0$. In the Maxwell case there is one additional constraint equation for $\nabla_{k}F_{\ell k}$ that is satisfied if the current $j_{k}=0$.

Because $\mu$ is a apparent horizon, generically $\nabla_{k}\theta_{\ell}<0$ on $N_{-k}$ and $\theta_{\ell}[\mu]<0$. It follows that there exists an extremal cross-section $X$ of $N_{-k}$ with $\theta_{\ell}=0$ (and $\theta_{k}=0$). We can solve for the location of $X$:

[TABLE]

where $v$ is a function of the transverse directions. There is a unique solution to this equation, with $v<0$ (see AndMar05 ).

To complete our spacetime $M^{\prime}$, we invoked CPT-conjugation across the extremal surface $X$. The junction conditions are satisfied at $X$ because $\theta_{\ell}=\theta_{k}=0$ while $\chi_{i}$, $F_{\ell k}$, $A_{i}$ and $\phi$ are even under CPT; for more general matter fields, we expect that CPT-invariance ensures that this gluing is always possible.