Gravitational vacuum polarization. II. Energy conditions in the Boulware vacuum.

Abstract

Building on techniques developed in the preceding paper, I investigate the various pointwise and averaged energy conditions for the quantum stress-energy tensor corresponding to a conformally coupled massless scalar field in the Boulware vacuum. I work in the test-field limit, restrict attention to the Schwarzschild geometry, and invoke a mixture of analytical and numerical techniques. In contradistinction to the case of the Hartle-Hawking vacuum, wherein violations of the energy conditions were confined to the region between the event horizon and the unstable photon orbit, I show that in the Boulware vacuum (1) all standard (pointwise and averaged) energy conditions are violated throughout the exterior region, all the way from spatial infinity down to the event horizon, and (2) outside the event horizon the standard pointwise energy conditions are violated in a maximal manner: They are violated at all points and for all null or timelike vectors. (The region inside the event horizon is considerably messier and of dubious physical relevance. Nevertheless, the standard pointwise energy conditions seem to be violated even inside the event horizon.) I argue that this is highly suggestive evidence, pointing to the fact that general self-consistent solutions of semiclassical quantum gravity might not satisfy the energy conditions and may in fact for certain quantum fields and certain quantum states violate all the energy conditions.