5 Calculations of Black Hole 4 The Generalized Second Law 4.1 Arguments for the validity

4.2 Entropy bounds 

As discussed in the previous subsection, 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

  equation273

(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 subsection, these considerations led Bekenstein [16] to propose a universal bound on the entropy-to-energy ratio of bounded matter, given by

  equation279

where R denotes the ``circumscribing radius'' of the body. Here `` E '' is normally interpreted as the energy above the ground state; otherwise, Eq. (16Popup Equation) would be trivially violated in cases where the Casimir energy is negative [70Jump To The Next Citation Point In The Article] - although in such cases in may still be possible to rescue Eq. (16Popup Equation) by postulating a suitable minimum energy of the box walls [13Jump To The Next Citation Point In The Article].

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. (16Popup Equation) fails. (For a recent discussion of such counterexamples to Eq. (16Popup Equation), see [71, 72, 70Jump To The Next Citation Point In The Article]; for counter-arguments to these references, see [13].) In particular it is easily seen that for a system consisting of N non-interacting species of particles with identical properties, Eq. (16Popup Equation) 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 [18Jump To The Next Citation Point In The Article] to violate Eq. (16Popup Equation), so it does not appear that nature has chosen to take advantage of this possible means of violating (16Popup Equation). Equation (16Popup Equation) also is violated at sufficiently low temperatures if one defines the entropy, S, of the system via the canonical ensemble, i.e., tex2html_wrap_inline1454, where tex2html_wrap_inline1456 denotes the canonical ensemble density matrix,

equation296

where H is the Hamiltonian. However, a study of a variety of model systems [18Jump To The Next Citation Point In The Article] indicates that (16Popup Equation) holds at low temperatures when S is defined via the microcanonical ensemble, i.e., tex2html_wrap_inline1462 where n is the density of quantum states with energy E . More generally, Eq. (16Popup Equation) has been shown to hold for a wide variety of systems in flat spacetime [18, 22].

The status of Eq. (16Popup Equation) in curved spacetime is unclear; indeed, while there is some ambiguity in how `` E '' and `` R '' are defined in Minkowski spacetime [70], 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, tex2html_wrap_inline1232, in a 4-dimensional spacetime and given any open neighborhood, tex2html_wrap_inline1484, of tex2html_wrap_inline1232, there exists a spacelike 2-surface, tex2html_wrap_inline1490 (composed of nearly null portions) contained within tex2html_wrap_inline1484 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. (16Popup Equation) 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 (16Popup Equation) may hold in curved spacetime.

With regard to question (2), in the previous section 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:

  1. A ``thin box'' approximation was made [17Jump To The Next Citation Point In The Article].
  2. It is possible to have a box whose contents have a greater entropy than unconfined thermal radiation of the same energy and volume [17].
  3. 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].
  4. 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 [21Jump To The Next Citation Point In The Article].

Responses to criticism (i) were given in [95Jump To The Next Citation Point In The Article] and [75Jump To The Next Citation Point In The Article]; 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 [43Jump To The Next Citation Point In The Article] that the arguments of [21] should pose similar difficulties for the ordinary second law for gedankenexperiments involving a self-gravitating body (see the end of subsection 4.1 above). Thus, my own view is that Eq. (16Popup Equation) is not necessary for the validity of the GSL Popup Footnote . 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, 20, 86Jump To The Next Citation Point In The Article]

  equation336

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 (18Popup Equation) would follow from (16Popup Equation) under the additional assumption of small self-gravitation (so that tex2html_wrap_inline1498). Thus, many of the arguments in favor of (16Popup Equation) are also applicable to (18Popup Equation). Similarly, the counterexample to (16Popup Equation) obtained by taking the number, N, of particle species sufficiently large also provides a counterexample to (18Popup Equation), so it appears that (18Popup Equation) can, in principle, be violated by physically reasonable systems (although not necessarily by any systems actually occurring in nature).

Unlike Eq. (16Popup Equation), the bound (18Popup Equation) 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 (18Popup Equation) applicable when gravity is ``turned off''. Also unlike (16Popup Equation), the bound (18Popup Equation) 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 (18Popup Equation). 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 (18Popup Equation) is necessary for the validity of the GSL [86]: Suppose we had a spherically symmetric system that was not a black hole (so R > 2 E) and which violated the bound (18Popup Equation), so that tex2html_wrap_inline1514 . 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 (18Popup Equation), 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 (18Popup Equation) 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

  equation357

We wish to consider a configuration that is not already a black hole, so we need E < R /2. To violate (18Popup Equation) - and thereby threaten to violate the GSL by collapsing a shell upon the system - we need to have tex2html_wrap_inline1528 . This means that we need to consider a model with tex2html_wrap_inline1530 . For such a model, start with a region R containing matter with tex2html_wrap_inline1528 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 tex2html_wrap_inline1540 . But the Hawking evaporation timescale in this model is tex2html_wrap_inline1542, since the flux of Hawking radiation is proportional to N . Since tex2html_wrap_inline1530, we have tex2html_wrap_inline1548, 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 (18Popup Equation) is ill defined in a general (non-spherically-symmetric) curved spacetime. There also are other difficulties with (18Popup Equation): In a closed universe, it is not obvious what constitutes the ``inside'' versus the ``outside'' of the bounding area. In addition, (18Popup Equation) 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 (18Popup Equation) 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 tex2html_wrap_inline1550 be a null hypersurface such that the expansion, tex2html_wrap_inline1096, of tex2html_wrap_inline1550 is everywhere non-positive, tex2html_wrap_inline1556 (or, alternatively, is everywhere non-negative, tex2html_wrap_inline1112). In particular, tex2html_wrap_inline1550 is not allowed to contain caustics, where tex2html_wrap_inline1096 changes sign from tex2html_wrap_inline1564 to tex2html_wrap_inline1566 . Let B be a spacelike cross-section of tex2html_wrap_inline1550 . Bousso's reformulation conjectures that

  equation374

where tex2html_wrap_inline1572 denotes the area of B and tex2html_wrap_inline1576 denotes the entropy flux through tex2html_wrap_inline1550 to the future (or, respectively, the past) of B .

In [43Jump To The Next Citation Point In The Article] it was argued that the bound (21Popup Equation) 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 tex2html_wrap_inline1550 be a null hypersurface which starts at a cross-section, B, and terminates at a cross-section B '. Suppose further that tex2html_wrap_inline1550 is such that its expansion, tex2html_wrap_inline1096, is either everywhere non-negative or everywhere non-positive. Then

  equation384

Although we have argued above that the validity of the GSL should not depend upon the validity of the entropy bounds (16Popup Equation) or (18Popup Equation), there is a close relationship between the GSL and the generalized Bousso bound (21Popup Equation). Namely, as discussed in section 2 above, classically, the event horizon of a black hole is a null hypersurface satisfying tex2html_wrap_inline1112 . Thus, in a classical regime, the GSL itself would correspond to a special case of the generalized Bousso bound (21Popup Equation). 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 [101Jump To The Next Citation Point In The Article], not merely to an event horizon of a black hole - which would reduce to (21Popup Equation) in a suitable classical limit.



5 Calculations of Black Hole 4 The Generalized Second Law 4.1 Arguments for the validity

image The Thermodynamics of Black Holes
Robert M. Wald
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