Unpacking Entropy Bounds and Their Violations

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Discussion Overview

The discussion revolves around the concept of entropy bounds and their potential violations, particularly in the context of general relativity. Participants explore the definitions and implications of entropy, energy, and spatial parameters in relation to these bounds, seeking clarity on how to formulate precise statements regarding these concepts.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about the definitions of energy (##E##) and radius (##R##) in the context of entropy bounds, noting their lack of clarity in general relativity.
  • Another participant defines the Bekenstein bound, stating that the entropy of an object with mass ##M## enclosed within a surface area ##A## must not exceed the entropy of a black hole with the same mass and horizon area, referencing a formula derived by Hawking.
  • Concerns are raised about the dependence of surface area on the choice of space-like slice, questioning how mass is defined in this context.
  • A later reply suggests that for asymptotically flat systems, one can define a center of mass frame to establish spacelike slices, but acknowledges the limitations due to the absence of a complete theory of quantum gravity.
  • Participants mention the ADM mass and Bondi mass as potential definitions for mass in asymptotically flat systems, with a preference for Bondi mass due to its consideration of emitted radiation.
  • A participant reiterates the need for precision in making statements about entropy bounds.
  • A link to a research paper is provided as a potential resource for further clarification.

Areas of Agreement / Disagreement

Participants express uncertainty regarding the definitions and implications of the parameters involved in entropy bounds, indicating that multiple competing views remain on how to approach these concepts. The discussion does not reach a consensus on the precise definitions or the implications of the entropy bounds.

Contextual Notes

Participants highlight limitations in the definitions of energy and radius, as well as the dependence on the choice of space-like slices. There is an acknowledgment of the unresolved nature of these concepts in the absence of a complete theory of quantum gravity.

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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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?
 
martinbn said:
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.

martinbn said:
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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