RESTRICTIONS ON NEGATIVE-ENERGY DENSITY IN A CURVED SPACETIME

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