Gravitational vacuum polarization IV: Energy conditions in the Unruh vacuum

TL;DR

This paper analyzes the energy conditions in the Unruh vacuum for a quantum scalar field around a Schwarzschild black hole, showing all point-wise energy conditions are violated and providing a semi-analytic model consistent with numerical results.

Contribution

It introduces a semi-analytic model for the stress-energy tensor in the Unruh vacuum that matches numerical data and explores violations of energy conditions in this context.

Findings

All point-wise energy conditions are violated in the Unruh vacuum exterior.

The averaged null energy condition is violated along all outgoing null geodesics.

The semi-analytic model accurately reproduces numerical results within 0.8%.

Abstract

Building on a series of earlier papers [gr-qc/9604007, gr-qc/9604008, gr-qc/9604009], I investigate the various point-wise and averaged energy conditions in the Unruh vacuum. I consider the quantum stress-energy tensor corresponding to a conformally coupled massless scalar field, work in the test-field limit, restrict attention to the Schwarzschild geometry, and invoke a mixture of analytical and numerical techniques. I construct a semi-analytic model for the stress-energy tensor that globally reproduces all known numerical results to within 0.8%, and satisfies all known analytic features of the stress-energy tensor. I show that in the Unruh vacuum (1) all standard point-wise energy conditions are violated throughout the exterior region--all the way from spatial infinity down to the event horizon, and (2) the averaged null energy condition is violated on all outgoing radial null geodesics. In a pair of appendices I indicate general strategy for constructing semi-analytic models for the stress-energy tensor in the Hartle-Hawking and Boulware states, and show that the Page approximation is in a certain sense the minimal ansatz compatible with general properties of the stress-energy in the Hartle-Hawking state.