[Paper Review] Bousso - A Covariant Entropy Conjecture

Original paper is here. 

Pervios paper is here.

Wikipikea is here.

Next paper is here, this reach to Null Energy Condition. 1, 2, 3.

"next paper": QNEC -> Engropy conjecture

"1": QNEC <-(holographic)-> entanglement wedge nesting (EWN.)

1. Introduction

1.1 Bekenstein's bound

Bekenstein has proposed the existence of a universal bound on the entropy $S$ of any thermodynamic system of total energy $M$: $$S\le 2\pi RM.$$ $R$ is defined as teh circumferential radius, $$R=\sqrt{\frac{A}{4\pi}},$$ where $A$ is the area of the smallest sphere circumscribing the system.

For a system contained in a spherical volume, gravitational stability requires that $M\le R/2$. Thus above equation implies $$S\le \frac{A}{4}.$$

Bekenstein specified conditions for the validity of these bounds. The system must be of constant, finite size and must have limited self-gravity, i.e., gravity must not be the dominant force in the system. This excludes, for example, gravitationally collapsing objects, and sufficiently large regions of cosmological space-times. Another important condition is that no matter components with negative energy density are available. This is because the bound relies on the gravitational collapse of systems with excessive entropy, and is intimately connected with the idea that information requires energy.

1.2 Outline

Notation and conventions

We will ban formal definitions into footnotes, whenever they refer to concepts which are intuitively clear. We work with a manifold $M$ of $3+1$ space-time dimensions, since the generalization to $D$ dimensions is obvious. The terms light-like and null are used interchangeably. Any three-dimensional submanifold $H \subset M$ is called a hypersurface of $M$. If two of its dimensions are everywhere spacelike and the remaining dimension is everywhere timelike (null, spacelike), $H$ will be called a timelike (null, spacelike) hypersurface. By a surface we always refer to a two-dimensional spacelike submanifold $B \subset M$. By a light-ray we never mean an actual electromagnetic wave or photon, but simply a null geodesic. We use the terms congruence of null geodesics, null congruence, and family of light-rays interchangeably. A light-sheet of a surface $B$ will be defined in Sec. 3.1 as a null hypersurface bounded by $B$ and generated by a null congruence with non-increasing expansion. A number of definitions relating to Bekenstein’s bound are found on 4.1. We set $\hbar = c = G = k = 1$.

2. The conjecture

In constructing a 'covariant' entropy bound, one first has to decide entropy/area bound. Given a two-dimensional surface $B$ of area $A$, on which hypersurface $H$ should we evaluate the entropy $S$? $B$ must be a boundary of $H$. 

Generalized version of Bekenstein’s entropy/area bound: Let $A$ be the area of any closed two-dimensional surface $B$, and let $S$ be the entropy on the spatial region $V$ enclosed by $B$. Then $S \le A/4$.

We must use null hypersurfaces bounded by $B$. The natural way to construct such hypersurfaces is to start at the surface B and to follow a family of light-rays (technically, a “congruence of null geodesics”) projecting out orthogonally from $B$.

We start at $B$, and follow one of the four families of orthogonal light-rays, as long as the cross-sectional area is decreasing or constant.Expansion of the orthogonal null congruence must be non-positive, in the direction away from the surface $B$. 

We will require the dominant energy condition to hold: for all timelike $v_a$, $T ^{ab}v_av_b \ge 0$ and $T^{ab}v_a$ is a non-spacelike vector. We should also require that the space-time is inextendible and contains no null or timelike (“naked”) singularities. 

Covariant Entropy Conjecture. Let $M$ be a four-dimensional space-time on which Einstein's equation is satisfied with the dominant energy condition holding for matter. Let $A$ be the are of a connected two-dimensional spatial surface $B$ contained in $M$. Let $L$ be a hypersurface bounded by $B$ and generated by one of the four null congruences orthogonal to $B$. Let $S$ be the total entropy contained on $L$. If the expansion of the congruence is non-positive (measured in the direction away from $B$) at every point on $L$, then $S\le A/4$.

3. Discussion

3.1 The recipe

1. Pick any two-dimensional surface $B$ in the space-time $M$.

2. There will be four families of light-rays projecting orthogonally away from $B$ (unless $B$ is on a boundary of $M$): $F_1,\cdots,F_4$.

3. As shown in the previous section, we can assume without loss of generality that the expansion of $F_1$ has the same sign everwhere on $B$. If the expansion is positive (in the direction away form $B$), i.e., if the corss-sectional area is increasing , $F_1$ must not be used for an entropy/area comparison, If the expansion is zero or negative, $F_1$ will be allowed. Repreating this test for each family, one will be left with at least two allowed families. If the expansion is zero in some directions, there may be as many as three or four allowed families.

4. Pick one of the allowed families, $F_i$. Construct a null hypersurface $L_i$, by following each light-ray until one of the following happens:

(a) The light-ray reaches a boundary or singularity  of the space-time.

(b) The expansion becomes positive, i.e., the cross-sectional area spanned by the family begins to increase in a neighborhood of the light-way.

The hypersurface $L_i$ obtained by this procedure will be called a lightsheet of the surface $B$. For every allowed family, there will be a different light-sheet.

5. The conjecture states that the entropy $S_i$ on the light-sheet $L_i$ will not exceed a quarter of the area of $B$: $$\begin{align}S_i\le \frac{A}{4}.\end{align}$$ Note that the bound applies to each light-sheet individually. Since $B$ may posses up to four light-sheets, the total entropy on all light-sheets could add up to as much as $A$.

3.2 Caustices as light-sheet endpoints

We must understand perfectly well what it is that can cause the expansion to become positive. Our conclusion will be that the expansion becomes positive only at caustics. The simplest example of such a point is the center of a sphere in Minkowski space, at which all ingoing light-rays intersect. 

Raychauduri’s equation for a congruence of null geodesics with tangent vector field $k^a$ a and affine parameter $\lambda$ is given by $$\begin{align}\frac{d\theta}{d\lambda}=-\frac{1}{2}\theta^2-\hat{\sigma}_{ab}\hat{\sigma}^{ab}-8\pi T_{ab}k^ak^b+\hat{\omega}_{ab}\hat{\omega}^{ab},\end{align}$$ where $T_{ab}$ is the stress-energy tensor of matter. The expansion $\theta$ measures the local rate of change of an element of cross-sectional area $\mathcal{A}$ spanned by nearby geodesics: $$\begin{align}\theta=\frac{1}{\mathcal{A}}\frac{d\mathcal{A}}{d\lambda}.\end{align}$$ The vorticity $\hat{\omega}_{ab}$ and shear $\hat{\sigma}_{ab}$ are defined in Refs. [18, 19]. The vorticity vanishes for surface-orthogonal null congruences. The first and second term on the right hand side are manifestly non-positive. The third term will be non-positive if the null convergence condition holds: $$\begin{align}T_{ab}k^ak^b\ge 0\mbox{ for all null }k^a.\end{align}$$ The dominant energy condition, which we are assuming, implies that the null convergence condition will hold. (It is also implied by the weak energy condition, or by the strong energy condition.)

Therefore the right hand side of Eq. (2) is non-positive. It follows that $\theta$ cannot increase along any geodesic. (This statement is self-consistent, since the sign of $\theta$ changes if we follow the geodesic in the opposite direction.) Then how can the expansion ever become positive? By dropping two of the non-positive terms in Eq. (2), one obtains the inequality $$\begin{align}\frac{d\theta}{d\lambda}\le -\frac{1}{2}\theta^2.\end{align}$$ If the expansion takes the negative value $\theta_0$ at any point on a geodesic in the congruence, Eq. (5) implies that the expansion will become negative infinite, $\theta\rightarrow -\infty$, along that geodesic within affine time $\Delta\lambda\le 2/|\theta_0|$ [19]. This can be interpreted as a caustic. Nearby geodesics are converging to a single focal point, where the cross-sectional area $\mathcal{A}$ vanishes. When they re-emerge, the cross-sectional area starts to increase. Thus, the expansion $\theta$ jumps from $-\infty$ to $\infty$ at a caustic. Then the expansion is positive, and we must stop following the light-ray. This is why caustics form the endpoints of the light-sheet.

3.3 Light-sheet examples and first evidence

For a spherical surface surrounding a spherically symmetric body of matter, the ingoing light-rays will end on a caustic in the center, as for an empty sphere. If the interior mass distribution is not spherically symmetric, however, some light-rays will be deflected into angular directions, and will form “angular caustics” (see Fig. 2). This does not mean that the interior will not be completely swept out by the light-sheet. Between two light-rays that get deflected into different overdense regions, there are infinitely many light-rays that proceed further inward. It does mean, however, that we have to follow some of the light-rays for a much longer affine time than we would in the spherically symmetric case.

This does not make a difference for static systems: they will be completely penetrated by the light-sheet in any case. In a system undergoing gravitational collapse, however, light-rays will hit the future singularity after a finite affine time. Consider a collapsing ball which is exactly spherically symmetric, and a future-ingoing light-sheet starting at the outer surface of the system, when it is already within its own Schwarzschild horizon. We can arrange things so that the light-sheet reaches the caustic at $r = 0$ exactly when it also meets the singularity (see Fig. 3). Now consider a different collapsing ball, of identical mass, and identical radius when the light-rays commence. While this ball may be spherically symmetric on the large scale, let us assume that it is highly disordered internally. The light-rays will thus be deflected into angular directions. As Fig. 2 illustrates, this means that they take intricate, long-winded paths through the interior: they “percolate.” This consumes more affine time than the direct path to the center taken in the first system. Therefore the second system will not be swept out completely before the singularity is reached (see Fig. 3).

The first ball is a system with low entropy, while the second ball has high entropy. One might think that no kind of entropy bound can apply when a highly enthropic system collapses: the surface area goes to zero, but the entropy cannot decrease. The above considerations have shown, however, that light-sheets percolate rather slowly through highly enthropic systems, because the geodesics follow a kind of random walk. Since they end on the black hole singularity within finite affine time, they sample a smaller portion of a highly enthropic system than they would for a more regular system. (In Fig. 2, the few lightrays that go straight to the center of the inhomogeneous system also take more affine time to do so than in a homogeneous system, because they pass through an underdense region. In a homogeneous system, they would encounter more mass; by Raychauduri’s equation, this would accelerate their collapse.) Therefore it is in fact quite plausible, if counter-intuitive, that the covariant entropy bound holds even during the gravitational collapse of a system initially saturating Bekenstein’s bound.

3.4 Recovering Bekenstein's bound

The covariant entropy conjecture can only be sensible if we can recover Bekenstein’s bound from it in an appropriate limit. 

Let A be the area of a closed surface B possessing at least one futuredirected light-sheet L. Suppose that L has no boundary other than B. Then we shall call the direction of this light-sheet the “inside” of B. Let the spatial region V be the interior of B on some spacelike hypersurface through B. If the region V is contained in the causal past of the light-sheet L, the dominant energy condition implies that all matter in the region V must eventually pass through the light-sheet L. Then the second law of thermodynamics implies that the entropy on V , SV , cannot exceed the entropy on L, SL. By the covariant entropy bound, SL ≤ A/4. It follows that the entropy of the spatial region V cannot exceed a quarter of its boundary area: SV ≤ A/4.

The condition that the future-directed light-sheet L contain no boundaries makes sure that none of the entropy of the spatial region V escapes through holes in L. Neither can any of the entropy escape into a black hole singularity, because we have required that the spatial region must lie in the causal past of L. Since we are always assuming that the space-time is inextendible and that no naked singularities are present, all entropy on V must go through L. We summarize this argument in the following theorem.

Spacelike Projection Thoerem. Let $A$ be the area of a closed surface $B$ possessing a future-directed light-sheet L with no boundary other than $B$. Let the spatial region $V$ be contained in the intersection of the causal past of $L$ with any spacelike hypersurface containing $B$. Let S be the entropy on $V$. Then $S \le A/4$.

Now consider, in asymptotically flat space, a Bekenstein system in a spatial region V bounded by a closed surface B of area A. The future-directed ingoing light-sheet L of B exists (otherwise B would not have “limited selfgravity”), and can be taken to end whenever two (not necessarily neighbouring) light-rays meet. Thus it will have no other boundary than B. Since the gravitational binding of a Bekenstein system is not strong enough to form a black hole, V will be contained in the causal past of L. Therefore the conditions of the theorem are satisfied, and the entropy of the system must be less than A/4. We have recovered Bekenstein’s bound.

There are many other interesting applications of the theorem. In particular, it can be used to show that area bounds entropy on spacelike sections of anti-de Sitter space. This can be seen by taking B to be any sphere, at any given time. The future-ingoing light-sheet of B exists, and unless the space contains a black hole, has no other boundary. Its causal past includes the entire space enclosed by B. This remains true for arbitrarily large spheres, and in the limit as B approaches the boundary at spatial infinity.

The theorem is immensely useful, because it essentially tells us under which conditions we can treat a region of space as a Bekenstein system. In general, however, the light-sheets prescribed by the covariant entropy bound provide the only consistent way of comparing entropy and area.

We pointed out in Sec. 3.1 that the covariant entropy bound is T-invariant. The spacelike projection theorem is not T-invariant; it refers to past and future explicitly. This is because the second law of thermodynamics enters its derivation. The asymmetry is not surprising, since Bekenstein’s bound, which we recovered by the theorem, rests on the second law. We should be surprised only when an entropy law is T-invariant. It is this property which forces us to conclude that the origin of the covariant bound is not thermodynamic, but statistical (Sec. 7).

6. Testing the conjecture in gravitational collapse

6.1 Light-sheet penetration into collapsing systems

Consider the Oppenheimer-Snyder collapse of a dust ball [20], commencing from a momentarily static state with $R = 2M$, as shown in Fig. 6. 

A future-directed ingoing light-sheet, starting at the surface B at a sufficiently early time but inside the event horizon, can easily traverse the ball before the singularity (or the Planck density, at R ∼ M1/3 ) is reached. But this light-sheet would endanger the bound only if the system collapsed from a state in which it nearly saturated Bekenstein’s bound. So how much entropy does the dust star actually contain? Strictly, the Oppenheimer-Snyder solution describes a dust ball at zero temperature. Since it also must be exactly homogeneous, it contains not even the usual positional entropy equal to the particle number. Thus the entropy is zero. In order to introduce a sizable amount of entropy, we have to violate the conditions under which the solution is valid: homogeneity and zero temperature. This collapse will be described by a different solution, for which a detailed calculation would have to be done to determine the penetration depth of light-sheets.

By definition, highly enthropic systems undergoing gravitational collapse are very irregular internally and contain strong small scale density perturbations. This will make the collapse inhomogeneous, with some regions reaching the singularity after a shorter proper time than other regions. One might call this effect Local Gravitational Collapse. A saturated Bekenstein system is globally just on the verge of gravitational collapse: R = 2M. But as it contracts, individual parts of the system, of size ∆R < R will become gravitationally unstable: ∆R ≤ 2∆M. Particles in an overdense region will reach a singularity after a proper time of order ∆R2/∆M, which is shorter than the remaining lifetime of average regions, ∼ R2/M. This makes it more difficult for a light-sheet to penetrate the system completely, unless it originates near the beginning of the collapse, when the area is still large.

The internal irregularity of highly enthropic systems also enhances the effect of Percolation, discussed in Sec. 3.3. Inhomogeneities that break spherical symmetry will deflect the rays in the light-sheet and cause dents in their spherical cross-sections. Such dents will develop into “angular” caustics. At any caustic, the light-sheet ends, because the expansion becomes positive (Sec. 3.2). Any light-rays that do not end on caustics will follow an irregular path through the object similar to random walk. Since they waste affine time on covering angular directions, they may not proceed far into the object before the singularity is reached. Thus it may well be impossible for a lightsheet to penetrate through a collapsing, highly entropic system far enough to sample excessive entropy.

The quantitative investigation of the formation of angular caustics on light-sheets penetrating collapsing, highly enthropic systems lies beyond the scope of this paper. A strong, quantitative case for the validity of the bound may still be made by eliminating the percolation effect. We will consider a system containing only radial modes. This system is spherically symmetric even microscopically, and cannot deflect light-rays into angular directions. With no constraints on mass and size, it can contain arbitrary amounts of entropy, but cannot lead to angular caustics on the light-sheet. 

6.2 A quantitative test

Consider a Schwarzschild black hole of horizon size r0 (see Fig. 7). Let B be a sphere on the apparent horizon at some given time; thus B has area $$A=4\pi r_0^2$$