Scalar Field Quantum Inequalities in Static Spacetimes

TL;DR

This paper derives quantum inequalities for scalar fields in static spacetimes, showing how negative energy densities are constrained and extending flat space results to curved backgrounds like black holes and de Sitter space.

Contribution

It provides a general expression for quantum inequalities in static spacetimes and applies it to various geometries, including black holes and de Sitter space, using Euclidean two-point functions.

Findings

Quantum inequalities limit negative energy magnitudes and durations.

In short sampling times, inequalities resemble flat space forms with geometric corrections.

A quantum averaged weak energy condition (QAWEC) bounds energy densities in black hole spacetimes.

Abstract

We discuss quantum inequalities for minimally coupled scalar fields in static spacetimes. These are inequalities which place limits on the magnitude and duration of negative energy densities. We derive a general expression for the quantum inequality for a static observer in terms of a Euclidean two-point function. In a short sampling time limit, the quantum inequality can be written as the flat space form plus subdominant correction terms dependent upon the geometric properties of the spacetime. This supports the use of flat space quantum inequalities to constrain negative energy effects in curved spacetime. Using the exact Euclidean two-point function method, we develop the quantum inequalities for perfectly reflecting planar mirrors in flat spacetime. We then look at the quantum inequalities in static de~Sitter spacetime, Rindler spacetime and two- and four-dimensional black holes. In the case of a four-dimensional Schwarzschild black hole, explicit forms of the inequality are found for static observers near the horizon and at large distances. It is show that there is a quantum averaged weak energy condition (QAWEC), which states that the energy density averaged over the entire worldline of a static observer is bounded below by the vacuum energy of the spacetime. In particular, for an observer at a fixed radial distance away from a black hole, the QAWEC says that the averaged energy density can never be less than the Boulware vacuum energy density.