The usual notion of entropy for classical systems is

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the usual notion of entropy for classical systems is then predicated on the assumption that generic dynamical orbits “sample” the entire energy shell, spending “equal times in equal volumes”; a similar assumption underlies the notion of entropy for quantum systems (see [14] for further discussion). Now, an appropriate notion of “time translations” is present when one considers dynamics on a background spacetime whose metric possesses a suitable one- parameter group of isometries, and when the Hamiltonian of the system is invariant under these isometries. However, such a structure is absent in gen- eral relativity, where no background metric is present. 4 The absence of any “rigid” time translation structure can be viewed as being responsible for mak- ing notions like the “energy density of the gravitational field” ill defined in general relativity. Notions like the “entropy density of the gravitational field” are not likely to fare any better. It may still be possible to use structures like asymptotic time translations to define the notion of the total entropy of an (asymptotically flat) isolated system. (As is well known, total energy can be defined for such systems.) However, for a closed universe, it seems highly questionable that any meaningful notion will exist for the “total entropy of the universe” (including gravitational entropy). The comments in the previous paragraph refer to serious difficulties in defining the notions of gravitational entropy and total entropy in general rel- ativity. However, as I now shall explain, even in the context of quantum field theory on a background spacetime possessing a time translation symmetry— so that the “rigid” structure needed to define the usual notion of entropy of matter is present—there are strong hints from black hole thermodynamics that even our present understanding of the meaning of the “ordinary entropy” 4 Furthermore, it is clear that gross violations of any sort of “ergodic behavior” occur in classical general relativity on account of the irreversible tendency for gravitational collapse to produce singularities, from which one cannot then evolve back to uncollapsed states— although the semiclassical process of black hole evaporation suggests the possibility that ergodic behavior could be restored in quantum gravity. 19
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of matter is inadequate. Consider the “thermal atmosphere” of a black hole. As discussed in Section 3 above, since the locally measured temperature is given by eq.(12), if we try to compute its ordinary entropy, a new ultraviolet catastrophe occurs: The entropy is infinite unless we put in a cutoff on the contribution from short wavelength modes. 5 As already noted in Section 3, if we insert a cutoff of the order of the Planck scale, then the thermal atmosphere contributes an entropy of order the area, A , of the horizon (in Planck units). Note that the bulk of the entropy of the thermal atmosphere is highly localized in a “skin” surrounding the horizon, whose thickness is of order of the Planck length.
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