the usual notion of entropy for classical systems is then predicated on theassumption that generic dynamical orbits “sample” the entire energy shell,spending “equal times in equal volumes”; a similar assumption underlies thenotion of entropy for quantum systems (see  for further discussion). Now,an appropriate notion of “time translations” is present when one considersdynamics on a background spacetime whose metric possesses a suitable one-parameter group of isometries, and when the Hamiltonian of the system isinvariant under these isometries. However, such a structure is absent in gen-eral relativity, where no background metric is present.4The 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 ingeneral relativity. Notions like the “entropy density of the gravitational field”are not likely to fare any better. It may still be possible to use structureslike asymptotic time translations to define the notion of the total entropy ofan (asymptotically flat) isolated system. (As is well known, total energy canbe defined for such systems.) However, for a closed universe, it seems highlyquestionable that any meaningful notion will exist for the “total entropy ofthe universe” (including gravitational entropy).The comments in the previous paragraph refer to serious difficulties indefining the notions of gravitational entropy and total entropy in general rel-ativity. However, as I now shall explain, even in the context of quantum fieldtheory on a background spacetime possessing a time translation symmetry—so that the “rigid” structure needed to define the usual notion of entropy ofmatter is present—there are strong hints from black hole thermodynamicsthat even our present understanding of the meaning of the “ordinary entropy”4Furthermore, it is clear that gross violations of any sort of “ergodic behavior” occur inclassical general relativity on account of the irreversible tendency for gravitational collapseto produce singularities, from which one cannot then evolve back to uncollapsed states—although the semiclassical process of black hole evaporation suggests the possibility thatergodic behavior could be restored in quantum gravity.19
of matter is inadequate.Consider the “thermal atmosphere” of a black hole.As discussed inSection 3 above, since the locally measured temperature is given by eq.(12), ifwe 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 shortwavelength modes.5As already noted in Section 3, if we insert a cutoff ofthe order of the Planck scale, then the thermal atmosphere contributes anentropy of order the area,A, of the horizon (in Planck units). Note that thebulk 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.