# Entropy bounds

• A
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
Mentor
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.

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
Mentor
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