Entropy bounds

4.2 Entropy bounds

As discussed in the previous Section 4.1, for a classical black hole the GSL would be violated if one could lower a box containing matter sufficiently close to the black hole before dropping it in. Indeed, for a Schwarzschild black hole, a simple calculation reveals that if the size of the box can be neglected, then the GSL would be violated if one lowered a box containing energy $E$ and entropy $S$ to within a proper distance $D$ of the bifurcation surface of the event horizon before dropping it in, where

(This formula holds independently of the mass, $M$ , of the black hole.) However, it is far from clear that the finite size of the box can be neglected if one lowers a box containing physically reasonable matter this close to the black hole. If it cannot be neglected, then this proposed counterexample to the GSL would be invalidated.

As already mentioned in the previous Section 4.1, these considerations led Bekenstein [15 ] to propose a universal bound on the entropy-to-energy ratio of bounded matter, given by

where $R$ denotes the “circumscribing radius” of the body. Here “ $E$ ” is normally interpreted as the energy above the ground state; otherwise, Eq. (16

) would be trivially violated in cases where the Casimir energy is negative [72

] – although in such cases in may still be possible to rescue Eq. (16

) by postulating a suitable minimum energy of the box walls [21

Two key questions one can ask about this bound are: (1) Does it hold in nature? (2) Is it needed for the validity of the GSL? With regard to question (1), even in Minkowski spacetime, there exist many model systems that are physically reasonable (in the sense of positive energies, causal equations of state, etc.) for which Eq. (16 ) fails. (For a recent discussion of such counterexamples to Eq. (16 ), see [70 , 71 , 72 ]; for counter-arguments to these references, see [21 ].) In particular it is easily seen that for a system consisting of $N$ non-interacting species of particles with identical properties, Eq. (16 ) must fail when $N$ becomes sufficiently large. However, for a system of $N$ species of free, massless bosons or fermions, one must take $N$ to be enormously large [17 ] to violate Eq. (16 ), so it does not appear that nature has chosen to take advantage of this possible means of violating (16 ). Equation (16 ) also is violated at sufficiently low temperatures if one defines the entropy, $S$ , of the system via the canonical ensemble, i.e., $S(T ) = − tr[ρln ρ]$ , where $ρ$ denotes the canonical ensemble density matrix,

where $H$ is the Hamiltonian. However, a study of a variety of model systems [17

] indicates that (16

) holds at low temperatures when $S$ is defined via the microcanonical ensemble, i.e., $S(E ) = ln n$ where $n$ is the density of quantum states with energy $E$ . More generally, Eq. (16

) has been shown to hold for a wide variety of systems in flat spacetime [17 , 22 ].

The status of Eq. (16 ) in curved spacetime is unclear; indeed, while there is some ambiguity in how “ $E$ ” and “ $R$ ” are defined in Minkowski spacetime [72 ], it is very unclear what these quantities would mean in a general, non-spherically-symmetric spacetime. (These same difficulties also plague attempts to give a mathematically rigorous formulation of the “hoop conjecture” [68 ].) With regard to “ $E$ ”, it has long been recognized that there is no meaningful local notion of gravitational energy density in general relativity. Although numerous proposals have been made to define a notion of “quasi-local mass” associated with a closed 2-surface (see, e.g., [77 , 30 ]), none appear to have fully satisfactory properties. Although the difficulties with defining a localized notion of energy are well known, it does not seem to be as widely recognized that there also are serious difficulties in defining “ $R$ ”: Given any spacelike 2-surface, $𝒞$ , in a 4-dimensional spacetime and given any open neighborhood, $𝒪$ , of $𝒞$ , there exists a spacelike 2-surface, $𝒞′$ (composed of nearly null portions) contained within $𝒪$ with arbitrarily small area and circumscribing radius. Thus, if one is given a system confined to a world tube in spacetime, it is far from clear how to define any notion of the “externally measured size” of the region unless, say, one is given a preferred slicing by spacelike hypersurfaces. Nevertheless, the fact that Eq. (16 ) holds for the known black hole solutions (and, indeed, is saturated by the Schwarzschild black hole) and also plausibly holds for a self-gravitating spherically symmetric body [83 ] provides an indication that some version of (16 ) may hold in curved spacetime.

With regard to question (2), in the previous Section 4.1 we reviewed arguments for the validity of the GSL that did not require the invocation of any entropy bounds. Thus, the answer to question (2) is “no” unless there are deficiencies in the arguments of the previous section that invalidate their conclusions. A number of such potential deficiencies have been pointed out by Bekenstein. Specifically, the analysis and conclusions of [94 ] have been criticized by Bekenstein on the grounds that:

A “thin box” approximation was made [16 ].
It is possible to have a box whose contents have a greater entropy than unconfined thermal radiation of the same energy and volume [16 ].
Under certain assumptions concerning the size/shape of the box, the nature of the thermal atmosphere, and the location of the floating point, the buoyancy force of the thermal atmosphere can be shown to be negligible and thus cannot play a role in enforcing the GSL [19 ].
Under certain other assumptions, the box size at the floating point will be smaller than the typical wavelengths in the ambient thermal atmosphere, thus likely decreasing the magnitude of the buoyancy force [20 ].

Responses to criticism (i) were given in [95 ] and [75 ]; a response to criticism (ii) was given in [95 ]; and a response to (iii) was given in [75 ]. As far as I am a aware, no response to (iv) has yet been given in the literature except to note [43 ] that the arguments of [20 ] should pose similar difficulties for the ordinary second law for gedankenexperiments involving a self-gravitating body (see the end of Section 4.1 above). Thus, my own view is that Eq. (16 ) is not necessary for the validity of the GSL³ . However, this conclusion remains controversial; see [2 ] for a recent discussion.

More recently, an alternative entropy bound has been proposed: It has been suggested that the entropy contained within a region whose boundary has area $A$ must satisfy [89 , 18 , 86 ]

This proposal is closely related to the “holographic principle”, which, roughly speaking, states that the physics in any spatial region can be fully described in terms of the degrees of freedom associated with the boundary of that region. (The literature on the holographic principle is far too extensive and rapidly developing to attempt to give any review of it here.) The bound (18

) would follow from (16

) under the additional assumption of small self-gravitation (so that $E ≲ R$ ). Thus, many of the arguments in favor of (16

) are also applicable to (18

). Similarly, the counterexample to (16

) obtained by taking the number, $N$ , of particle species sufficiently large also provides a counterexample to (18

), so it appears that (18

) can, in principle, be violated by physically reasonable systems (although not necessarily by any systems actually occurring in nature).

Unlike Eq. (16 ), the bound (18 ) explicitly involves the gravitational constant $G$ (although we have set $G = 1$ in all of our formulas), so there is no flat spacetime version of (18 ) applicable when gravity is “turned off”. Also unlike (16 ), the bound (18 ) does not make reference to the energy, $E$ , contained within the region, so the difficulty in defining $E$ in curved spacetime does not affect the formulation of (18 ). However, the above difficulty in defining the “bounding area”, $A$ , of a world tube in a general, curved spacetime remains present (but see below).

The following argument has been given that the bound (18 ) is necessary for the validity of the GSL [86 ]: Suppose we had a spherically symmetric system that was not a black hole (so $R > 2E$ ) and which violated the bound (18 ), so that $2 S > A ∕4 = πR$ . Now collapse a spherical shell of mass $M = R ∕2 − E$ onto the system. A Schwarzschild black hole of radius $R$ should result. But the entropy of such a black hole is $A ∕4$ , so the generalized entropy will decrease in this process.

I am not aware of any counter-argument in the literature to the argument given in the previous paragraph, so I will take the opportunity to give one here. If there were a system which violated the bound (18 ), then the above argument shows that it would be (generalized) entropically unfavorable to collapse that system to a black hole. I believe that the conclusion one should draw from this is that, in this circumstance, it should not be possible to form a black hole. In other words, the bound (18 ) should be necessary in order for black holes to be stable or metastable states, but should not be needed for the validity of the GSL.

This viewpoint is supported by a simple model calculation. Consider a massless gas composed of $N$ species of (boson or fermion) particles confined by a spherical box of radius $R$ . Then (neglecting self-gravitational effects and any corrections due to discreteness of modes) we have

We wish to consider a configuration that is not already a black hole, so we need $E < R∕2$ . To violate (18

) – and thereby threaten to violate the GSL by collapsing a shell upon the system – we need to have $S > πR2$ . This means that we need to consider a model with $N ≳ R2$ . For such a model, start with a region $R$ containing matter with $S > πR2$ but with $E < R∕2$ . If we try to collapse a shell upon the system to form a black hole of radius $R$ , the collapse time will be $≳ R$ . But the Hawking evaporation timescale in this model is $3 tH ∼ R ∕N$ , since the flux of Hawking radiation is proportional to $N$ . Since $N ≳ R2$ , we have $tH ≲ R$ , so the Hawking evaporation time is shorter than the collapse time! Consequently, the black hole will never actually form. Rather, at best it will merely act as a catalyst for converting the original high entropy confined state into an even higher entropy state of unconfined Hawking radiation.

As mentioned above, the proposed bound (18 ) is ill defined in a general (non-spherically-symmetric) curved spacetime. There also are other difficulties with (18 ): In a closed universe, it is not obvious what constitutes the “inside” versus the “outside” of the bounding area. In addition, (18 ) can be violated near cosmological and other singularities, where the entropy of suitably chosen comoving volumes remains bounded away from zero but the area of the boundary of the region goes to zero. However, a reformulation of (18 ) which is well defined in a general curved spacetime and which avoids these difficulties has been given by Bousso [25 , 26 , 27 ]. Bousso’s reformulation can be stated as follows: Let $ℒ$ be a null hypersurface such that the expansion, $𝜃$ , of $ℒ$ is everywhere non-positive, $𝜃 ≤ 0$ (or, alternatively, is everywhere non-negative, $𝜃 ≥ 0$ ). In particular, $ℒ$ is not allowed to contain caustics, where $𝜃$ changes sign from $− ∞$ to $+ ∞$ . Let $B$ be a spacelike cross-section of $ℒ$ . Bousso’s reformulation conjectures that

where $AB$ denotes the area of $B$ and $S ℒ$ denotes the entropy flux through $ℒ$ to the future (or, respectively, the past) of $B$ .

In [43 ] it was argued that the bound (21 ) should be valid in certain “classical regimes” (see [43 ]) wherein the local entropy density of matter is bounded in a suitable manner by the energy density of matter. Furthermore, the following generalization of Bousso’s bound was proposed: Let $ℒ$ be a null hypersurface which starts at a cross-section, $B$ , and terminates at a cross-section $B ′$ . Suppose further that $ℒ$ is such that its expansion, $𝜃$ , is either everywhere non-negative or everywhere non-positive. Then

Although we have argued above that the validity of the GSL should not depend upon the validity of the entropy bounds (16 ) or (18 ), there is a close relationship between the GSL and the generalized Bousso bound (21 ). Namely, as discussed in Section 2 above, classically, the event horizon of a black hole is a null hypersurface satisfying $𝜃 ≥ 0$ . Thus, in a classical regime, the GSL itself would correspond to a special case of the generalized Bousso bound (21 ). This suggests the intriguing possibility that, in quantum gravity, there might be a more general formulation of the GSL – perhaps applicable to an arbitrary horizon as defined on p. 134 of [101 ], not merely to an event horizon of a black hole – which would reduce to (21 ) in a suitable classical limit.