Gravitational vacuum polarization III: Energy conditions in the (1+1) Schwarzschild spacetime

TL;DR

This paper analyzes energy conditions of quantum stress-energy tensors in a (1+1)-dimensional Schwarzschild spacetime, providing exact results for different vacua and serving as a benchmark for higher-dimensional studies.

Contribution

It offers exact analytical results for energy conditions in (1+1) dimensions, clarifying their behavior across various quantum states in Schwarzschild spacetime.

Findings

NEC is satisfied outside the horizon for Hartle--Hawking vacuum.

NEC is violated for Boulware and Unruh vacua.

DEC is violated everywhere in the spacetime.

Abstract

Building on a pair of earlier papers, I investigate the various point-wise and averaged energy conditions for the quantum stress-energy tensor corresponding to a conformally-coupled massless scalar field in the in the (1+1)-dimensional Schwarzschild spacetime. Because the stress-energy tensors are analytically known, I can get exact results for the Hartle--Hawking, Boulware, and Unruh vacua. This exactly solvable model serves as a useful sanity check on my (3+1)-dimensional investigations wherein I had to resort to a mixture of analytic approximations and numerical techniques. Key results in (1+1) dimensions are: (1) NEC is satisfied outside the event horizon for the Hartle--Hawking vacuum, and violated for the Boulware and Unruh vacua. (2) DEC is violated everywhere in the spacetime (for any quantum state, not just the standard vacuum states).