Entropy for quantum fields in curved space time

Abstract

We discuss that entropy can be assigned to local domains for quantum fields only if we relate it to two domains nested in one another such that the entropy includes a surface effect depending on the distance of the two domains. We give upper and lower limits for the corresponding expressions, based on assumptions on the nuclearity of the quantum field and on the existence of a scaling limit. We apply these estimates to local domains in flat space and in de Sitter space. We show that in both cases, the total system is in a pure state with vanishing entropy, but also that the entropy of domains with vanishing size tends to 0. For quantum fields on a black hole, we consider the Schwarzschild spacetime and its extension to the Kruskal spacetime. The quantum field on the Schwarzschild spacetime has infinite entropy, even if we regularize over the horizon. Nevertheless, for domains in the Schwarzschild spacetime, the entropy tends to 0 if the size tends to 0. If, however, we consider domains that include the total Schwarzschild domain, the entropy is always ∞.