Unpacking Entropy Bounds and Their Violations

[martinbn]

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?

 

[PeterDonis]

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.

 

[martinbn]

The surface area is better than the vague $R$, but it still depends on the space-like slice. And how is the mass defined?

 

[PeterDonis]

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.

 

[Demystifier]

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

I think this could be the answer:
https://arxiv.org/abs/hep-th/9905177