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I watched Leonard Susskind's great layman lecture "The World As Hologram" the other day (https://www.youtube.com/watch?v=2DIl3Hfh9tY) and it kind of makes sense to me. If Information is a conserved quantity I would expect that some kind of Gauss divergence theorem holds. So if we enclose some space with a sphere it should be possible to somehow encode incoming/outgoing information just on the sphere. I'm not saying though that this analogy is used in the actual mathematics of the holographic world model, it just helps me to get a more intuitive grasp on it. (But I welcome comments if this is a valid view.)

But what got me thinking is his statement: "every finite region of the universe can be seen as a hologram". And the "holographic principle" says: "The maximum amount of information in a region of space is proportional to the area of the region". So lets take a sphere of radius $r$. Then we know the maximum information contents of a sphere of radius $r/10$ has to be $100$ times smaller. But since its volume is a $1000$ times smaller we could pack about $1000$ of them into the original sphere in a non-overlapping way. So in total they could encode about $10$ times more information than the enclosing original sphere. This discrepancy could be made as big as we wish, wenn we use even smaller spheres.

So how is this discrepancy resolved? Or did I just misunderstand something? The only resolution I currently can think of is that maybe the state of nearby regions is correlated in such a way, that it could be compressed so that it only increases quadratic by radius in total. (Clearly there have to be some correlations, e.g. I couldn't have a neutron star in my wardrobe and life a normal life in my living room.) But Susskind doesn't mention if that is the case and why it should resolve in such a way.

I hope someone can give a clarification?

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