Tuesday, June 28, 2011

Rubik's cubes inside black holes

If you want to get a fair intuition how a black hole emits the Hawking radiation so that it stays unitary and the exterior and the interior remain separated in spite of that, you should sort Rubik's cube many times:

Full screen... (click).

Turn the sound off in the upper right corner of the Flash applet. I wonder whether a TRF reader can solve the cube in this applet edition, at least the 3x3 case.

At least, this is the conclusion of a very creative new preprint by Bartłomiej Czech, Klaus Larjo, and Moshe Rozali:
Black Holes as Rubik's Cubes
String theory - including its manifestly unitary descriptions in terms of Matrix theory and AdS/CFT - shows that the Hawking radiation can ultimately carry the entropy away.

However, attempts to understand the "mechanism" how the information is encoded and how much of it is being carried away by the radiation - and how much the entanglement entropy changes as the black hole evaporate - tended to disagree with the broad conclusion of string theory. Simply speaking, it seemed that the entanglement between the black hole interior and the black hole exterior was growing too large and the Hawking radiation therefore couldn't contain enough information.

The three authors - by the way, Czech is not a Czech name, it's a Polish name, much like Bartoloměj Čech, Robert Polák, and Miloš Němec (Czech, Pole, and German) are Czech names - propose a creative fix. The interior of the black holes acts as a Rubik's cube. This Hungarian gadget, if you allow me to add another Central European nation :-), is a proof of a concept demonstrating that all the conditions we expect or demand can actually be reconciled with each other.

If a black hole emits a new Hawking particle, it may be found in one of the states that are identified with the moves of a Rubik cube - or its generalization. For example, there is a state for the upper layer of the cube rotated by 90 degrees in the clockwise direction, and it corresponds to the photon emitted in a certain state.

Once a Hawking particle is emitted, the Rubik's cube inside the black hole is rotated by the corresponding move. ;-) Only if the interior black hole's Rubik's cube is "solved", we say that the interior is in the "vacuum state" and it means that the black hole has already evaporated and there will be no more nontrivial "moves" or other nontrivial evolution in time.

This is the rough idea but this "model I" doesn't manage to decrease the entanglement entropy - ever. Also, the total time needed for a black hole to evaporate - the number of moves to solve the Rubik's cube - would be too variable.

So they offer a refinement, their model II. The size of the black hole is identified with the minimum number of moves needed to solve the Rubik's cube inside it. Also, when we apply a Rubik move connected with the evaporation of a pair, we manually erase the parts of the wave function in which the Rubik's cube didn't get closer to being solved. The rest, which is easily seen to be nonzero, is renormalized.

This model II restored a problem - the horizon is not "information free" because we could directly learn about some features of the internal state by looking at the absence of certain particles etc.: the particles emitted outside depend on the internal state of the black hole which violates causality - too much.

In model III, they don't merge the advantages of models I, II yet. Instead, they solve another problem - an apparent immediate influence of the Hawking radiation on the internal state which is "bad". To eliminate it, they introduce a buffer for the internal moves. So the Rubik's cube is really modified by the move linked not to the most recent emitted Hawking particle but the 3rd (in their example) before it - the cache has 3 slots in it. Eventually, the cache should be much longer but it shouldn't be infinitely large because we would return back to those models where the entanglement entropy grows forever.

The final model, in which the cache of the model III is ignored once again :-) so that the previous paragraph was just a distraction, unifies the virtues of model I and model II. The interior of the black hole is composed of many Rubik's cubes and some of them may already be solved. In that case, the total black hole energy is reduced. The black hole states with a solved Rubik's cube emit a new "q-particle". This model is unitary, "nice", and has an information-free horizon. After some time, the entanglement entropy begins to drop. The information is not lost.

It is a good toy model although I don't expect that the Rubik cube's group is exactly the right set of operations for the emitted particles from any black hole in string theory.

Instead, I think it could be damn realistic to consider Witten's monstrous AdS3 black hole with the monster group M, replacing the Rubik's cube's group. However, in the simplest model, there's no massless particles this black hole may emit. Still, I think that the monster group is a better approximation of the kind of "moves" that are occurring to the "puzzles" living inside the black holes.

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