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## Gravitational entropy

In ordinary QM and QFT entropy is defined using a density operator for a generalized state:

$$S = -\text{tr}\left(\rho\,\ln\rho\right)$$
b/c for the gravitational field we do neither know the fundamental degrees of freedom nor the Hilbert space states, a definition like

$$\rho = \sum_np_n\,|n\rangle\langle n|$$
is not available.

Questions:
1) are there attempts to formulate entropy for a QFT on a gravitational background for a "finite volume"?
2) are there attempts to formulate entropy for the gravitational field "within this finite volume"?
3) has this been done in string theory and / or spin networks for several different spacetimes (black holes, some other finite volume, expanding spacetime with e.g. co-moving dust, ...)?
4) how does the holographic principle show up?

(I know some special cases like the state counting for black holes in LQG, but I have never seen a general construction)

 Quote by tom.stoer In ordinary QM and QFT entropy is defined using a density operator for a generalized state: $$S = -\text{tr}\left(\rho\,\ln\rho\right)$$ b/c for the gravitational field we do neither know the fundamental degrees of freedom nor the Hilbert space states, a definition like $$\rho = \sum_np_n\,|n\rangle\langle n|$$ is not available. (I know some special cases like the state counting for black holes in LQG, but I have never seen a general construction)
Carlo Rovelli gave a talk related to this today, maybe a new paper will come out soon.

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## Gravitational entropy

Is it even well understood how to define gravitational entropy in the classical case? I was under the impression that there were nontrivial issues. Do we have a general definition that applies to cases like inflationary models, spacetimes with CTCs, spacetimes that aren't time-orientable, ...? Is this the reason for Tom's "finite volume" with quotation marks? Is it hopeless to try to cover the classical case before moving on to quantum gravity because accretion of hot matter onto a black hole then violates the second law?

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