Entropy-Area Relations in Field Theory

We consider the contribution to the entropy from fields in the background of a curved time-independent metric. To account for the curvature of space, we postulate a position-dependent UV cutoff. We argue that a UV cutoff on energy naturally implies an IR cutoff on distance. With this procedure, we calculate the scalar contribution in a background anti-de Sitter space, the exterior of a black hole, and de Sitter space. In all cases, we find results that can be simply interpreted in terms of local energy and proper volume, yielding insight into the apparent reduced dimensionality of systems with gravity.