# The Holographic Principle

Hi there..

If was wondering about the relation between information and entropy, in the following context.
The way I understand Gerard 't Hoofd's holographic principle is this. We know that the entropy of a black hole is A/4. Now suppose that a volume V was found to have an excess in entropy of a black hole just big enough to fit inside V. By throwing in additional matter we could form such a black hole. But... this gives problems with the second law. The entropy decreases by this proces. Conclusion: the maximum entropy of a volume V is given by the area of it's boundary (in certain units).
In an article in the Scientific American (aug. 2003) Bekenstein says that the maximum information in a given volume is bounded by the area.
I don't see this relation between entropy and information clear.

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I am trying to understand the holographic principle in string theory. More specific, in de AdS/CFT correspondence. The Anti-de Sitter spacetime is an empty space with a negative cosmologial constant, a contracting universe, right.
Then on the other side of this correspondence we have the conformal field theories on the boundary of this five dimensional AdS space. Maldacena showed these two to be indistinguishable. Or maybe what he did can be summarized better by saying that he made a 'dictionary' for translating words in the AdS 5D-world to the CFT 4D-world.
What then has this to do with string theory? Are these spaces curved or something? I heard people saying that this AdS/CFT correspondence is one of the most important results of string theory.. but what is so stringy about it..

Something else: What is the use of this correspondence? We don't live in an empty space with negative cosmological constant, do we.

skowalcz said:
Something else: What is the use of this correspondence? We don't live in an empty space with negative cosmological constant, do we.
As far as I can tell, the "success" of the AdS/CFT correspondence is simply that it shows a clearly-defined relationship between two opposite (or distinct) physical formulations, one which is quantum and one which is cosmological.
While it isn't "realistic", the fact is that it can be done.

For me, the fascination of the HP is the intrinsic connection between geometry (area) and information.

In the mean time I found out that since the invention of AdS/CFT it has been generalized, so that it works also in other spaces. Not only in empty space with negative lambda.
This was done by Witten and others, but what they showed exactly I can not tell.

skowalcz said:
Conclusion: the maximum entropy of a volume V is given by the area of it's boundary (in certain units).
In an article in the Scientific American (aug. 2003) Bekenstein says that the maximum information in a given volume is bounded by the area.
I don't see this relation between entropy and information clear.
From http://arxiv.org/abs/hep-th/0203101,
the number of degrees of freedom = ln of the dimension of the Hilbert space = number of bits of information,
and if entropy = S, then $e^S$ = number of independent quantum states compatible with macroscopic parameters => entropy is a measure of our ignorance of the detailed microscopic state of a system.

Then, the number of degrees of freedom connected with information should be equal to the number of independent states compatible with mIcroscopic parameters, since information is negative entropy?

I wrote an article on this, but ... :yuck: it's not in english..

http://gene.science.uva.nl/~skowalcz/scoop_juni2004-18-20-holografie.pdf [Broken] (Scoop, 06-2004): voor als je nederlands verstaat

And if you don't speak dutch: check out my artist's impression of a black hole!

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