As already mentioned in the previous Section 4.1, these considerations led Bekenstein  to propose a universal bound on the entropy-to-energy ratio of bounded matter, given by – although in such cases in may still be possible to rescue Eq. (16) by postulating a suitable minimum energy of the box walls .
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 .) In particular it is easily seen that for a system consisting of non-interacting species of particles with identical properties, Eq. (16) must fail when becomes sufficiently large. However, for a system of species of free, massless bosons or fermions, one must take to be enormously large  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, , of the system via the canonical ensemble, i.e., , where denotes the canonical ensemble density matrix, indicates that (16) holds at low temperatures when is defined via the microcanonical ensemble, i.e., where is the density of quantum states with energy . 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 “” and “” are defined in Minkowski spacetime , 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” .) With regard to “”, 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 “”: Given any spacelike 2-surface, , in a 4-dimensional spacetime and given any open neighborhood, , of , there exists a spacelike 2-surface, (composed of nearly null portions) contained within 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  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  have been criticized by Bekenstein on the grounds that:
Responses to criticism (i) were given in  and ; a response to criticism (ii) was given in ; and a response to (iii) was given in . As far as I am a aware, no response to (iv) has yet been given in the literature except to note  that the arguments of  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  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 must satisfy [89, 18, 86]
Unlike Eq. (16), the bound (18) explicitly involves the gravitational constant (although we have set 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, , contained within the region, so the difficulty in defining in curved spacetime does not affect the formulation of (18). However, the above difficulty in defining the “bounding area”, , 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 : Suppose we had a spherically symmetric system that was not a black hole (so ) and which violated the bound (18), so that . Now collapse a spherical shell of mass onto the system. A Schwarzschild black hole of radius should result. But the entropy of such a black hole is , 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 species of (boson or fermion) particles confined by a spherical box of radius . Then (neglecting self-gravitational effects and any corrections due to discreteness of modes) we have
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 be a null hypersurface such that the expansion, , of is everywhere non-positive, (or, alternatively, is everywhere non-negative, ). In particular, is not allowed to contain caustics, where changes sign from to . Let be a spacelike cross-section of . Bousso’s reformulation conjectures that
In  it was argued that the bound (21) should be valid in certain “classical regimes” (see ) 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 be a null hypersurface which starts at a cross-section, , and terminates at a cross-section . Suppose further that is such that its expansion, , 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 . 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 , not merely to an event horizon of a black hole – which would reduce to (21) in a suitable classical limit.
© Max Planck Society and the author(s)