Entropy bounds

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martinbn
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There is something that is unclear to me, and because entropy bounds and their violations were discussed in the other thread, I thought it is a good opportunity to learn something. The problem is essentially a matter of impression. The statements go roughly in the following way: for a system with entropy ##S## and energy ##E##, which is contain in space of radius ##R## a certain inequality involving the above must hold. The problem for me is that the ##E## and the ##R## are never defined (well, I haven't seen it, it might very well be explained somewhere). And in a general relativistic setting they are meaningless.

So the question is how does one make the statements precise?
 

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  • #2
PeterDonis
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The precise definition of the Bekenstein bound is that the entropy of an object with externally measured mass ##M## and enclosed within a surface with surface area ##A## must be less than or equal to the entropy of a black hole with mass ##M## and horizon area ##A##. The latter has a precise formula first derived by Hawking, which amounts to the entropy being the log of the horizon area divided by the Planck area.
 
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martinbn
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The surface area is better than the vague ##R##, but it still depends on the space-like slice. And how is the mass defined?
 
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PeterDonis
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The surface area is better than the vague ##R##, but it still depends on the space-like slice.
Technically, yes, but for an asymptotically flat (i.e., isolated) system, one can define what amounts to a center of mass frame and use that to define the spacelike slices. Until we get a proper theory of quantum gravity, that's probably the best we're going to be able to do, since without one we simply don't know the precise microscopic degrees of freedom of a system including the spacetime geometry.

how is the mass defined?
The ADM mass or the Bondi mass would be the simplest definitions, since they apply to any asymptotically flat system. I would lean towards the latter since it takes into account radiation emitted out to infinity.
 

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