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Where and how is the entropy of a black hole stored?

Is it around the horizon? Most of the entanglement entropy across the event horizon lies within Planck distances of it and are short lived.

Is it stored near the singularity? How can you pack so much information in such a small region?

Can you please help me?

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One specific idea is that the information is stored in the degrees of freedom of fuzzballs. Fuzzballs can be visualized as a messy tangle of strings and higher dimensional objects. – Dilaton Jan 8 '13 at 12:20

You can find most of entropy of black hole on the surface of Event Horizon associated with strings. The rest is associated with Hawking Radiation etc. As region inside event horizon is fully disconnected from our universe, this must be true to prevent "entropy decrease" of system (our universe).

The same duplicate entropy can be found inside event horizon, too. But, the most of entropy is found at the singularity.

How can you pack so much information in such a small region?

Actually, its nothing in front of singularity seed of big bang. There's no limit to the process of packing.

How does the packing works?

We don't know yet. Remember, when a theory gives you monstrously high or low number like infinity etc, it means it has failed to describe the situation in our working domain. When General Relativity predicts singularity, it means describing singularity is beyond its level. General Relativity is for big bodies and singularity is very very small where probability rules. So, to understand the process, we need a theory of gravity for quantum world. And, we are working on it.

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This is a very profound question in physics. Given that a black hole has an entropy which scales as $$S_{BH} \sim \frac{A}{4}, $$ the question is how does this relate to $S_{Boltzmann} = K_B \ln W$. As in, what are the microstates of the theory which hold the information in the black hole. This was answered in part by a series of papers by Vafa, Strominger, Callan, Maldacena in the 1990's and the answer is that the microstates of the black hole are actually realised by compactifying type $IIB$ string theory on $T_4 \times S_1$. On the resulting geometry, $n_5$ number $D_5$ branes are wrapped around the torus and $n_1$ number $D_1$ branes around the circle. In the limit the string coupling is small, the effective geometry correctly counts the microstates of the black hole. Since then, there has been a large number of work in constructing the microstates and precisely counting them for a large family of black holes. However, these solutions work only at or near the extremal limit where the entire physics of the black hole can be written in terms of the charges of the theory. There is a further restriction that it works best in supersymmetric cases and when the coupling is small.

A more robust technique was put forward by Mathur, Lunin in the so called fuzzball approach which over comes the coupling restriction by considering the black hole to be an effective geometry of $e^S$ number of bound string states, with $S$ here being the entropy.

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