Gravitational vacuum polarization II: Energy conditions in the Boulware vacuum

TL;DR

This work analyzes the impact of gravitational vacuum polarization on energy conditions for a conformally coupled massless scalar field in the Boulware vacuum on Schwarzschild spacetime. Using a mix of analytic approximations (Page–Hartle–Hawking plus Brown–Ottewill) and numerical data, it shows that all standard energy conditions are violated throughout the exterior region, with outside-horizon violations being maximal for all null and timelike vectors. Inside the horizon the situation is more subtle, with some directions allowing partial null energy condition satisfaction, but overall violations persist. The results suggest that in semiclassical gravity, energy conditions may not hold universally, having profound implications for singularity theorems and energy-positivity proofs, while also highlighting caveats related to the unphysical aspects of the Boulware state near the horizon.

Abstract

I show that in the Boulware vacuum (1) all standard (point-wise 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 point-wise energy conditions are violated in a maximal manner: they are violated at all points and for all null/timelike vectors. (The region inside the event horizon is considerably messier, and of dubious physical relevance. Nevertheless the standard point-wise energy conditions also seem to be violated even inside the event horizon.) This is rather different from the case of the Hartle--Hawking vacuum, wherein violations of the energy conditions were confined to the region inside the unstable photon orbit. These calculations are 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. This *suggests* 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.