Gravitational vacuum polarization II: Energy conditions in the Boulware vacuum

TL;DR

This paper demonstrates that in the Boulware vacuum around a Schwarzschild black hole, all standard energy conditions are violated throughout the exterior region, indicating potential issues for semiclassical quantum gravity solutions.

Contribution

It provides a detailed analysis of energy condition violations in the Boulware vacuum, contrasting with the Hartle--Hawking vacuum, using analytical and numerical methods.

Findings

All energy conditions are violated outside the event horizon.

Violations occur at all points and for all null/timelike vectors.

Violations may extend inside the event horizon, though less clearly.

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.