Bekenstein bound
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In physics, the Bekenstein bound imposes a limit on the entropy S or information that can be contained within a three-dimensional volume. Additionally, in Computer Science, the Bekenstein bound implies that there is a maximum information processing rate and that Turing machines, with their (by definition) infinite memory tape, are physically impossible if they are to have a finite size and bounded energy. The bound was originally found by Jacob Bekenstein in the form
,
where R is the radius of the region, and E is the energy of the contained matter as measured when the matter is moved to an infinite distance, i.e., accounting for binding force potential energies.
Applying this to the area A of the event horizon of a black hole, we find that since a black hole's radius and energy are both proportional to its mass, the limit to the entropy (and information) contained within a spherical volume is proportional to the area:
where A is the two-dimensional area of the black hole's event horizon in units of the Planck area, 
.
Since a black hole has maximal entropy, this then imposes an upper limit for the amount of entropy that can be contained within any region. Gerard t' Hooft later generalized it to the form involving A/4. This is the holographic principle. The argument, in more detail, is that since the bound is known to hold for black holes on relatively firm model-independent grounds, it must hold more generally based on the second law of thermodynamics. If the bound was violated by a region of space that didn't contain a black hole, then mass could be brought into the region in order to form a black hole, resulting in a net decrease of entropy. If this argument is correct, then it is often taken to imply that spacetime is discrete, since continuous spacetime should be able hold an infinite amount of information within a finite volume.
Intuitively, the motivation for the Bekenstein bound is that, regardless of the existence of black holes in our universe, event horizons exist. An event horizon will be perceived by any observer who is in an accelerated frame of reference (which, by the equivalence principle, is equivalent to being in a gravitational field). When pairs of particles are produced at the horizon due to vacuum fluctuations, one comes to the observer, while the other moves out of the region of the universe observable to her. As seen by the observer, the horizon therefore radiates. All event horizons must look the same locally (except for differences due to the observer's own motion), since otherwise it would be possible for the observer to use the differences between them to obtain information about what was behind them. A horizon thus looks like a generic black-body radiator with a temperature 
. (In the special case where the horizon is that of a black hole, the radiation being described is that of black hole evaporation.) The entropy of the region behind the event horizon was taken there by the members of the particle pairs that went away from the observer rather than toward her.
[edit] See also
[edit] References
- J. D. Bekenstein, "Generalized second law of thermodynamics in black hole physics", Phys. Rev. D 9, 3292 (1974).
- J. D. Bekenstein, "A universal upper bound on the entropy to energy ratio for bounded systems", Phys. Rev. D 23, 287 (1981).
