Restrictions on negative energy density in a curved spacetime

TL;DR

This paper investigates quantum inequalities restricting negative energy densities in curved spacetimes, deriving bounds for compact hypersurfaces and analyzing their behavior in short and long sampling time limits.

Contribution

It extends quantum inequality bounds to compact curved spacetimes and explicitly evaluates dominant terms in the short sampling time limit.

Findings

Derived asymptotic expansion of the lower bound in the short sampling time limit.

Explicitly evaluated dominant terms in terms of invariant quantities.

Provided estimates for the bounds in the long sampling time limit.

Abstract

Recently a restriction ("quantum inequality-type relation") on the (renormalized) energy density measured by a static observer in a "globally static" (ultrastatic) spacetime has been formulated by Pfenning and Ford for the minimally coupled scalar field, in the extension of quantum inequality-type relation on flat spacetime of Ford and Roman. They found negative lower bounds for the line integrals of energy density multiplied by a sampling (weighting) function, and explicitly evaluate them for some specific spacetimes. In this paper, we study the lower bound on spacetimes whose spacelike hypersurfaces are compact and without boundary. In the short "sampling time" limit, the bound has asymptotic expansion. Although the expansion can not be represented by locally invariant quantities in general due to the nonlocal nature of the integral, we explicitly evaluate the dominant terms in the limit in terms of the invariant quantities. We also make an estimate for the bound in the long sampling time limit.