Does the GSL imply an entropy bound?

Warren G. Anderson, University of Wisconsin-Milwaukee

Recently, a number of papers have appeared on the gr-qc preprint archive concerning entropy bounds imposed on matter by the generalized second law of thermodynamics (GSL) for spinning and charged black holes[1]. However, the question of whether the GSL implies the existence of such bounds, even for the simple case of a Schwarzschild black hole, is still being debated in the literature. This question, first raised by Bekenstein[2] almost 30 years ago, arises from a gedankenexperiment. The goal of this article is to review this experiment and the key results which comprise our current understanding of whether or not the GSL implies an entropy bound for matter.

Let us begin by recalling the GSL: any process involving matter with entropy Smatter exterior to a black hole of area ${\cal A}$ should satisfy $\delta S_{matter} + \delta S_{BH} \ge 0$, where $S_{BH}={\cal A}/4$ is the Bekenstein-Hawking black hole entropy. The central premise of all the papers reviewed here is that this law should be valid.

Now, consider a gedankenexperiment involving a black hole (area ${\cal A}$) and a box of proper height b and cross-sectional area A. Far from the black hole, the box is filled with matter, so that the total energy of the box and contents is E and its total entropy S. The box is then lowered adiabatically toward the black hole, so that its entropy S remains constant. Its energy as measured by a distant static observer, $E_\infty$, however, does not; the box is doing work on the agent that is controlling the lowering process. This work is subtracted from the box in the form of a redshift. When the box is a proper height $\ell$ above the horizon, the energy in the box as measured at infinity is therefore $E_\infty(\ell)=\chi(\ell) E$, where $\chi(\ell)$ is the redshift factor.

Suppose that the box is lowered until it nearly touches the black hole. This is, of course, physically impossible, since the box and/or rope will fail mechanically before this can happen. However, for simplicity we consider this limiting case. In this limit the box is an average proper distance $\sim b$ from the horizon, and the energy of the box is $E_\infty(b) \approx
E~b~\sqrt{\pi/{\cal A}}$. If the box is then allowed to fall freely into the black hole, the entropy of exterior matter will be reduced by $\delta
S_{matter} = - S$, but the entropy of the black hole will increase by $\delta
S_{BH} = 2~E_\infty(b)~\sqrt{\pi {\cal A}}$. Putting these entropy changes into the GSL, one gets

S/E \le 2 \pi b. \eqno(1)

This inequality, originally derived by Bekenstein[2], seems to imply a new law of nature: that any matter of energy E confined to a volume whose smallest dimension is b has a fundamental upper bound on its entropy.

The derivation of this bound, however, depends on the box continuing to do work on the lowering agent all the way to the horizon. If this is not the case, the bound might be modified, or even removed. About a year after Bekenstein proposed bound (1), Unruh and Wald[3] pointed out that quantum effects could indeed alter the work done by the box, and hence the entropy bound.

The Unruh-Wald argument goes as follows: an adiabatic lowering process is quasi-static, and can therefore be treated as a sequence of static (i.e., accelerating) boxes. Accelerating observers see the quantum vacuum as a bath of thermal radiation, whose temperature is proportional to the acceleration. Since the bottom of the box is closer to the black hole than the top, it must have a greater acceleration to remain static. Thus, the bottom of the box is exposed to hotter radiation than the top, creating a net upward pressure. This pressure gradient buoys the box against gravity, reducing the net work done by the box on the lowering agent. In fact, since the acceleration (and hence temperature) diverges at the horizon, near the horizon the lowering agent would have to do work on the box (push the box) in order to lower it further. Thus, at some distance above the horizon, where the box stops doing work and the lowering agent starts doing work, the box must float freely above the black hole. Just as for Archimedes' buoyancy, the box floats when its energy is equal to the energy of the displaced fluid (i.e., acceleration radiation).

For a buoyant box, Unruh and Wald[3] showed that the box contributes the minimum entropy to the black hole when dropped from the floating point. For macroscopic boxes, this point is very close to the black hole[4], so we do not avoid the issue of mechanical instability, but again let us consider the limiting case (floating box) since this will minimize the entropy gain for the system. In this limit, the minimum change in black hole entropy is[3]

\delta S_{BH} = \frac{A}{T_{bh}}\int_0^b [\rho(l_0,y)-\rho_{ar}(l_0,y)]
\chi(l_0+y) dy + S_{ar}, \eqno(2)

where Tbh is the Hawking temperature of the black hole, $\rho$ and $\rho_{ar}$ are the energy densities of the box and the acceleration radiation respectively, l0 is the proper height of the bottom of the box above the black hole at the floating point, and Sar is the entropy of the acceleration radiation displace by the box.

Interpretation of Eq. (2) is not difficult. It simply states that the minimum change in black hole energy (SBH Tbh) due to an adiabatically lowered box is the energy of the box at the floating point ($\int \rho dV$) minus the work done by the box against the buoyancy force of the acceleration radiation during lowering ( Tbh Sar - Ear). This interpretation seems reasonable, and until recently Eq. (2) has been accepted as correct in the literature (I will deal with a recent exception a bit later in this article).

However, the conclusions that can be drawn from Eq. (2) regarding entropy bounds seem to depend heavily on the nature of the acceleration radiation (i.e., on $\rho_{ar}$ and Sar). In particular, the crucial assumption seems to be the following: since acceleration radiation is thermal, it should have the maximum entropy density at any given energy density. If this is the case, it has been shown that Eq. (2) implies the GSL without assuming a bound such as Eq. (1)[3,5,6] (although Eq. (1) itself may follow from this assumption together with some other plausible assumptions about the acceleration radiation[6]). Conversely, in those papers[2,4,7] where this maximal entropy density assumption is not made, the GSL is shown to be violated unless an entropy bound exists.

That the resolution of the entropy bound question rests on properties of acceleration radiation is slightly troubling, because in some senses this radiation is not physical. For instance, even though the vacuum expectation value of the stress-energy tensor for an electromagnetic field in Minkowski space vanishes, accelerating observers see that vacuum as a thermal bath of photons. However, these photons clearly carry no energy or momentum on average. One must therefore be careful as to the properties one requires of acceleration radiation.

Recently, Bekenstein[8] has placed even more emphasis on the nature of acceleration radiation. He has pointed out that since the temperature far from the black hole is low, the average wavelength of the radiation can be longer than the height of the box. Therefore, he argues, the acceleration radiation will not behave like a fluid there; rather, one should treat the interaction of the radiation with the box as a scattering process. This can significantly reduce the buoyancy, causing Eq. (2) to be modified. He then finds that an entropy bound of the form (1) is necessary to preserve the GSL, even if the acceleration radiation is assumed to be maximally entropic.

If it is troubling to invoke the properties of acceleration radiation in resolving questions about this gedankenexperiment, one might ask how such questions can be resolved in an observer independent way. Such a resolution was originally provided by Unruh and Wald[3], and somewhat elucidated later[9], although it has not played a part in the recent literature. The resolution lies in the fact that accelerating surfaces emit quantum fluxes. These fluxes can have negative energy, and it is just such a flux that is created inside the box due to its accelerating walls. As the box is lowered the energy in the box decreases, since the negative energy deposited in the box by the quantum flux is added to the energy density of the box itself. This negative energy is just what one would have to add to the acceleration radiation in order that an initially empty box, lowered adiabatically toward the black hole, should continue to look empty to an observer accelerating with the box. In other words, it preserves the adiabatic vacuum (Boulware) state inside the box. In this picture, the box floats because the negative energy of the accelerating vacuum inside the box cancels the positive energy of the box itself.

Most interestingly, Eq. (2) can be derived by the quantum flux analysis as well. One might suspect, therefore, that this analysis needs to be modified so as not to be at odds with Bekenstein's calculation of the the scattering of acceleration radiation at low temperatures[8]. A natural modification which might be analogous to the scattering process would be the exclusion of long wavelength components of the flux due to the scale set by the size of the box. However, at least in two-dimensional examples, it can been shown that Eq. (2) follows from an exact analysis, regardless of any exclusion of long wavelength components[3]. The question of understanding Bekenstein's modification to (2) without invoking acceleration radiation therefore seems to be open. It is open questions such as this that will need to be resolved before we can truly understand whether or not the generalized second law of thermodynamics implies an entropy bound.


[1] T. Shimomura and S. Mukohyama, gr-qc/9906047; A. E. Mayo, gr-qc/9905007; B. Linet, gr-qc/9905007; S. Hod, gr-qc/9903010, gr-qc/9903011; J. D. Bekenstein and A. E. Mayo, gr-qc/9903002; S. Hod, gr-qc/9901035.

[2] J. D. Bekenstein, Phys. Rev. D23, 287-297, (1981).

[3] W. G. Unruh and R. M. Wald, Phys. Rev. D25, 942-958, (1982).

[4] J. D. Bekenstein, Phys. Rev. D49, 1912-1921, (1994).

[5] W. G. Unruh and R. M Wald, Phys. Rev. D27, 2271-2276, (1983).

[6] M. A. Pelath and R. M. Wald, gr-qc/9901032.

[7] J. D. Bekenstein, Phys. Rev. D27, 2262-2270, (1983).

[8] J. D. Bekenstein, gr-qc/9906058.

[9] W. G. Anderson, Phys. Rev. D50, 4786-4790, (1994).

Jorge Pullin