# Liouville's Theorem and Black Holes

 P: 215 In Professor Susskind's lecture 7 on Classical Physics, he discusses Liouville's Theorem. He said that a consequence was that points in the phase space can not coalesce and lose their identity. In Professor Susskind's book, Black Hole War, he discusses why destruction of information at the event horizon of a black hole would violate conservation of information. My question is: The conservation of information in Liouville's theorem and the conservation of information in the black hole debates; are the the same concept? The reason I ask, is that in Black Hole War, Susskind says that violation of conservation of information would violate the most fundamental laws of physics. Yet is is fuzzy what kind of information he means, and which fundamental laws mandate conservation of information.
 Sci Advisor P: 8,785 Yes there is an analogue of the Liouville theorem in quantum mechanics, where the quantum state can be described using a density matrix. In quantum mechanics a pure density matrix always remains pure, if time evolution is unitary, and no information is lost. However, Hawking's calculation showed that at his level of approximation, a pure density matrix turned into a mixed density matrix, suggesting that time evolution is not unitary, and information is lost. Here are two links that mention the density matrix in the context of Hawking radiation. http://arxiv.org/abs/hep-th/0409024 http://arxiv.org/abs/0909.1038
 P: 19 Just in case, here's a short version of the book - http://www.youtube.com/watch?v=pf0D8A0jRiY "Leonard Susskind on The Black Hole Wars"
P: 215
Liouville's Theorem and Black Holes

 Quote by atyy Here are two links that mention the density matrix in the context of Hawking radiation. http://arxiv.org/abs/hep-th/0409024 http://arxiv.org/abs/0909.1038
I looked at ghost two abstracts. Thanks.

In your opinion, is the Hawking-Susskind debate about information at event horizons still open, or is it settled? Is there a consensus now among physicists? If yes, which way?
 P: 30 I feel abased by my lack of knowledge concerning Liouvilles theorem.
 P: 215 See wikipedia article here. Or better still, watch the video of Susskind's lecture linked in the first post of this thread.
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P: 7,279
 Quote by anorlunda My question is: The conservation of information in Liouville's theorem and the conservation of information in the black hole debates; are the the same concept?
I would answer categorically, NO. Liouville's theorem is mathematics. The behavior of black holes is physics.

Keeping a clear head about the difference between math and physics is essential IMO. Of course you can't do much physics at any level without mathematics, but "the map is not the territory".

As one of my early mentors used to say, talking about engineering rather than physics: "never forget that the thing you are testing hasn't read any textbooks to tell it how it ought to behave."
 P: 215 Alephzero, I beg to differ. See wikipedia Liouville's theorem (Hamiltonian). The article includes a section on the physical interpretation. See also Liouville's theorem disambiguation page. It lists many meanings for the theorem, one of which is physics. I quote: Liouville's theorem has various meanings, all mathematical results named after Joseph Liouville: In complex analysis, see Liouville's theorem (complex analysis); there is also a related theorem on harmonic functions. In conformal mappings, see Liouville's theorem (conformal mappings). In Hamiltonian mechanics, see Liouville's theorem (Hamiltonian). In linear differential equations, see Liouville's formula. In transcendence theory and diophantine approximations, the theorem that any Liouville number is transcendental. In differential algebra, see Liouville's theorem (differential algebra) In differential geometry, see Liouville's equation