The surface area to volume ratio of a sphere and the Bekenstein bound

Question

I am trying to relate the surface-area-to-volume-ratio of a sphere to the Bekenstein bound. Since the surface-area-to-volume-ratio decreases with increasing volume, one would surmise that, per unit of volume, a small space is richer in information than a large one. How can this be and how can this bound work for black holes of various sizes?

Thank you very much Mr. Rennie. I appreciate and have investigated your answer. It turns out that I am familiar with the AdS/CFT correspondence and have sufficient understanding of the math (just barely) to be intrigued with the conjecture and, of course, the holographic theory. If the correspondence only works for a certain diameter black hole, the conjecture seems, to me, weak because of the changing surface-area-to-volume-ratio of a sphere. For myself, it would appear to be, likely, a mathematical curiosity or fluke. However, if, through some aspect that I do not understand, the correspondence holds for varying diameters, in fact, all diameters of black holes, then it seems quite astonishing, indeed. After searching for some time, I have once seen the amount described as trivial and possibly in another instance, that it may have something to do with informational redundancy. I’m afraid I cannot site these references as they were far too brief to be of any help.

This post imported from StackExchange Physics at 2014-03-07 13:38 (UCT), posted by SE-user Jim McKenzie

Comments

You ask "How can this be?". If I had a really good answer to that I would be writing off to Stockholm for my Nobel prize. As far as I know (which is not much) the AdS/CFT conjecture (en.wikipedia.org/wiki/AdS/CFT_correspondence) is the best currently known approach to the issue.

This post imported from StackExchange Physics at 2014-03-07 13:38 (UCT), posted by SE-user John Rennie

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blogs.discovermagazine.com/outthere/2013/08/20/… I think this answers my question. Apparently I understand the concept and can follow the math but I fail to believe it. As a black hole gets bigger and bigger, it is less and less dense, until eventually it isn't particularly dense at all. Faith in mathematics would seem to be my issue. Thank you for your help. Jim McKenzie.

This post imported from StackExchange Physics at 2014-03-07 13:38 (UCT), posted by SE-user Jim McKenzie