Bekenstein bound

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Jacob D. Bekenstein (2008), Scholarpedia, 3(10):7374. doi:10.4249/scholarpedia.7374 revision #121148 [link to/cite this article]
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Figure 1: The Bekenstein bound is a limit on the entropy S that can be contained in a system with given total energy E and which can be enclosed in a sphere of given radius R [see equation(1)].


The Bekenstein bound or universal entropy bound is a limit on the entropy that can be contained in a physical system or object with given size and total energy (Bekenstein J D, 1981). Other entropy bounds have also been proposed, e.g. the holographic entropy bound by ('t Hooft G, 1993 and Susskind L, 1995), the covariant entropy bound by (Bousso R, 1999), its extended form by (Flanagan E E, Marolf D et al, 2003) and the causal entropy bound by (Brustein R and Veneziano G, 2000), among others.

Contents

Statement of the bound

If \(R\) is the radius of a sphere that can enclose a given system, while \(E\) is its total energy including any rest masses, then its entropy \(S\ ,\) expressed in dimensionless form as customary in physics today, is bounded according to (Bekenstein, J D, 1981, 2004) \[\tag{1} S\leq {2\pi R E\over \hbar c}. \]

Here \( \hbar\) is the reduced Planck constant and \(c\) is light's speed. This bound is primarily intended for a system whose gravitational potential is everywhere small compared to \(c^2\) (weak self-gravitation).

What use is a bound on entropy?

The second law of thermodynamics assures us that the entropy of a closed physical system will grow until it reaches its maximum value. But how high a maximum ? While an appeal to statistical mechanics can yield the maximum value for a particular system, such approach cannot provide a generic rule for the maximum entropy of all, or a class, of systems. The various entropy bounds, of which bound (1) was the first proposed, attempt to fill this gap in our understanding of thermodynamics. An entropy bound can be viewed as an extension of the second law of thermodynamics: it limits the process of entropy growth.

Now in thermodynamics entropy is an extensive quantity. For example, in a homogeneous system, such as a quantity of liquid in a bottle, the entropy should be proportional to the mass of the system. With the discovery of black hole entropy (Bekenstein-Hawking entropy) by (Bekenstein J D, 1973 and Hawking S W, 1975) it became clear that extensivity of entropy is not a universal property: for a Schwarzschild black hole the entropy scales as the mass squared. The various entropy bounds strive to encompass both of these types of entropy behavior, as well as any others. Thus both the universal entropy bound and the holographic entropy bound are saturated by the Schwarzschild black hole's entropy, but are a bit above the entropy of other black holes, and greatly exceed the entropies of most mundane systems.

Because the peak entropy of a system is a direct measure of its capacity to hold information. e.g. (Shannon C E and Weaver W, 1949 or Bekenstein J D, 2004), all entropy bounds, and bound (1) in particular, are simultaneously bounds on information capacity, the amount of information that can be encoded in the system by exploiting all of its degrees of freedom. In this context the tighter the bound, the more useful constraints it sets in practical terms. But even the notoriously loose holographic bound has been useful in formulating crucial questions regarding, for instance, the information capacity of the universe. This activity is reviewed by (Bekenstein J D, 2003 and Bousso, 2002).

How is the universal bound inferred?

Figure 2: Lowering the system to the horizon and dropping it provides a way to determine the coefficient in bound (1).

A poor man's way to see that a bound like (1) must exist is to imagine dropping a definite system of mass \(m\) (including all mass-energy) and bearing entropy \(S\) into a Schwarzschild black hole of mass \(M\gg m\) from far away. Neglecting energy losses to gravitational radiation and Hawking radiation (both of which are suppressed by making \(M\) very large), we see that the black hole's mass will grow to \(M+m\ .\) Since initially the hole's entropy was \(S_{BH}=4\pi(G/\hbar c)M^2\ ,\) it will have grown by \(8\pi(G/\hbar c)Mm \) plus a negligible term of order \(m^2 \ .\) Meanwhile the entropy \(S \) has gone out of sight of observers outside the hole. But the generalized second law introduced by (Bekenstein J D, 1974 Bekenstein-Hawking entropy) demands that the sum of ordinary entropy outside black holes and the total black hole entropy shall never decrease. In our gedanken experiment this means that \(-S+ 8\pi(GM/\hbar c) m\geq 0 \ .\)

The initial radius of the hole's horizon (gravitational or Schwarzschild radius) is \(r_h =2GM/c^2\ ,\) so the above is the same as the inequality \[\tag{2} S\leq {4\pi r_h mc^2\over \hbar c}. \]

We now imagine making the black hole radius smaller and smaller while keeping it at least a few times larger than the largest "radius" \(R\) of the falling system, so that the last can be swallowed by the hole as stipulated. This is compatible with our condition \(m\ll M\) because the system is presumed to be much larger than its own gravitational radius \(2Gm/c^2\) (weak self-gravitation). It now follows from inequality (2) that \[\tag{3} S\leq {\xi \pi R E\over \hbar c}, \]

where \(E=mc^2\) and \(\xi\) is a positive number which could be as small as a few times unity. A similar result is obtained by (Bekenstein J D, 2004). It uses only the ordinary second law, but arranges for the dropping point to be just so far that the black hole's Hawking radiation energy emitted during the fall equals the system's energy at drop. Both routes make it clear that a bound of style (1) must exist, though the arguments do not provide the precise numerical coefficient.

That coefficient can be determined as follows. Imagine the system in question lowered slowly (adiabatically) to very near the horizon, as in Figure 2. Accounting for the energy so extracted by the lowering agent shows that the system's energy as gauged by distant observers has been reduced from \(mc^2\) to \(\epsilon=(mc^4 R/8GM)\) (the calculation being easiest done by taking the gravitational redshift into account as described by (Misner, C W et al 1973). One then drops the system into the hole. The new \(S_{BH}\) is calculated using the mass \(M+\epsilon/c^2\ .\) It becomes evident that in order for the consequent growth in black hole entropy to at least compensate for the disappearance of \(S\ ,\) the latter must comply with bound (1) with coefficient \(2\pi\ .\) Doubts were raised by (Gibbons G W, 1973) about whether a cord strong enough to support mass \(m\) near the horizon is physically permitted; these have been laid to rest by (Fouxon, I et al, 2008). Likewise, the effects of quantum buoyancy of the system near the horizon, which were claimed by (Unruh W G and Wald R M 1982,1983) to undermine the above argument, have been incorporated into a more elaborate derivation of the bound by (Bekenstein J D 1994, 1999). An easy way to sidestep the buoyancy issue is to imagine, as done in (Bekenstein J D 1999), dropping the system not at the horizon, but from a small distance off. The prize is a slight increase of the coefficient in the bound.

Scope of application of the universal entropy bound

With the coefficient \(2\pi\) bound (1) also encompasses any Kerr-Newman black holes of mass \(M\ ,\) charge \(Q\) and angular momentum \(J\ .\) For these one can imagine the horizon radius expressed in Boyer-Linquist coordinates (see Bekenstein-Hawking entropy), \[\tag{4} r_h\equiv GM/c^2+\sqrt{(GM/c^2)^2-(G^{1/2}Q/c^2)^2-(J/Mc)^2}, \]

Figure 3: The universal bound for systems with average densities \(10^{-20}\, {\rm g cm}^{-3}\) (green), \(1\, {\rm g cm}^{-3}\) (blue) and \(10^{15}\,{\rm g cm}^{-3}\) (purple), as well as the holographic bound (grey). Overplotted in red and black are the entropy vs scale of various objects in the universe.

to play the role of \(R\) in the bound, and \(Mc^2\) to play that of \(E\ .\) Then \(S_{BH}\) indeed complies with (1). For the Schwarzschild black hole (\(Q=J=0\)) bound (1) is saturated, e.g. becomes an equality. For other systems, including all sorts of mundane ones, the entropies fall well below bound (1). This is clear from Figure 3 which displays the universal entropy bounds for systems with average densities \(10^{-20}\, {\rm g\, cm}^{-3}\) (green), \(1\, {\rm g\, cm}^{-3}\) (blue) and \(10^{15}\, {\rm g\, cm}^{-3}\) (purple) together with the holographic bound (black). Note that the neutron star's entropy should be judged against the purple line bound, the globular cluster's against the green line one, and the rest against the blue line bound. Schwarzschild black holes with densities \(1\, {\rm g\, cm}^{-3}\) and \(10^{15}\, {\rm g\, cm}^{-3}\) are also plotted; as mentioned, each saturates the relevant universal bound as well as the holographic one.

Problems with bound (1) or the holographic bound are known to occur in extreme situations. Both fail when the gravitational potential is large (strong self-gravity), e.g. system already collapsed inside a black hole. In an infinite universe the holographic bound fails when applied to a sufficiently large region. In a closed (finite) universe the specification of \(R\) or bounding area becomes ambiguous (Bousso 1999, 2002). The covariant entropy bound corrects these and other shortcomings. As shown by (Bousso R, 2003) this bound can be used to rederive bound (1) for situations when it is viable. Over the history of the subject a number of examples purporting to violate bound bound (1) in mundane situations have been exhibited; many involve baroque set ups. Some of these are rebutted in (Bekenstein J D, 2005). The principal lesson from such examples is that, if the bound is to be respected, \(E\) in it must include the mass-energy of any container or walls required to maintain the system's integrity, as could be guessed from the derivations of it sketched earlier. Although in some instances one may drop such "wall contribution" with impunity, this cannot be done uncritically.

References

  • Bekenstein J D (1973) Phys. Rev. D 7:2333
  • Bekenstein J D (1974) Phys. Rev. D 9:3292
  • Bekenstein J D (1981) Phys. Rev. D 23:287
  • Flanagan E E, Marolf D and Wald R M (2000) Phys. Rev. D 62:084035
  • Fouxon I, Betschart G and Bekenstein J D (2008) Phys. Rev. D 77:024016 (http//xxx.lanl.gov/abs/0710.1429)
  • Gibbons G W (1972) Nature Phys. Sci. 240: 77
  • Hawking S W (1975) Commu. Math. Phys. 43:199
  • Misner C W Thorne K S and Wheeler J A (1973) Gravitation Freeman (San Francisco)
  • Unruh W G and Wald R M (1982) Phys. Rev. D 25: 942
  • Unruh W G and Wald R M (1983) Phys. Rev. D 27: 2271
  • Shannon C E and Weaver W (1949) The mathematical theory of communication University of Illinois Press (Urbana)

Internal references

  • Teviet Creighton and Richard H. Price (2008) Black holes. Scholarpedia, 3(1):4277.
  • Tomasz Downarowicz (2007) Entropy. Scholarpedia, 2(11):3901.


Further Reading

  • Bekenstein J D (2003) Information in the holographic universe Scientific American 289: no. 2 58-65
  • Susskind L and Lindesay J (2005) An introduction to black holes, information and the string theory revolution: The holographic principle World Scientific (Singapore)
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