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I have a question about the Bekenstein bound or the holographic principle.

I make the following assumptions:

  1. The universe is infinite
  2. The universe is homogeneous

These assumptions are at least considered plausible (on large scale). The Bekenstein bound (or holographic principle) say that the information contained in a ball of radius $R$ is at most $q A$ where $q$ is a proportionality constant and $A$ the area of the sphere. But if the universe is homogeneous, its density is constant and so the information contained in a sphere should be proportional to $R^3$; while the bound is proportional to $R^2$. That seems impossible, a big enough sphere will eventually exceed the bound.

A similar argument seems to imply that some sufficiently large portion of the universe should necessarily be a black hole.

What I am doing wrong here?

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Bekenstein's bound does indeed fail when applied to an infinite universe, but then Bekenstein specified conditions for the validity of his bounds:

The system must be of constant, finite size and must have limited self-gravity, i.e., gravity must not be the dominant force in the system. This excludes, for example, gravitationally collapsing objects, and sufficiently large regions of cosmological space-times. Another important condition is that no matter components with negative energy density are available.

Various attempts have been made to extend the Bekenstein bound. The current front runner appears to be Bousso's covarient entropy bound - see also the summary on Wikipedia. This works for an infinite universe as well as for systems conforming to Bekenstein's original restrictions.

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  • $\begingroup$ I have looked at the wikipedia article and indeed there are conditions that are apparently not satisfied. Maybe you have a link explaining these conditions in a not too technical condition? This wikipedia article seems to tell that the bound apply to a black hole (but maybe not inside it) does it have limited self-gravity? $\endgroup$ – olivier Dec 10 '15 at 14:02
  • $\begingroup$ Also what about the argument that the Schwarzschild radius of a sufficiently large sphere will eventually exceed the radius of the sphere itself; which would imply that a sufficiently large sphere should necessarily be a black hole? Does it fail in the same way? $\endgroup$ – olivier Dec 10 '15 at 14:11
  • $\begingroup$ @Olivier: Re your first comment: I suspect that stating Bousso's argument in a non-technical form would be hard. It is not something I feel any great urge to attempt. Re your second comment: the universe is expanding so the bigger you make your sphere the more rapidly the matter at the boundary is moving away from the centre. No matter how big you make your sphere the matter within it will never form a black hole. $\endgroup$ – John Rennie Dec 10 '15 at 17:32

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