Analytical approximation of the stress-energy tensor of a quantized scalar field in static spherically symmetric spacetimes

TL;DR

This paper develops analytical approximations for the vacuum expectation values of the scalar field squared and stress-energy tensor in static spherically symmetric spacetimes, applicable to both massive and massless fields with arbitrary curvature coupling.

Contribution

It introduces a method to approximate these quantities by dividing into low- and high-frequency parts, including high-frequency mode contributions for any quantum state.

Findings

Derived expressions for <φ^2> and <T^μ_ν> in static spacetimes

Calculated low-frequency contributions in Minkowski vacuum

Discussed the applicability limits of the approximations

Abstract

Analytical approximations for ${< \phi^2 >}$ and ${< T^{\mu}_{\nu} >}$ of a quantized scalar field in static spherically symmetric spacetimes are obtained. The field is assumed to be both massive and massless, with an arbitrary coupling $\xi$ to the scalar curvature, and in a zero temperature vacuum state. The expressions for ${< \phi^2 >}$ and ${< T^{\mu}_{\nu} >}$ are divided into low- and high-frequency parts. The contributions of the high-frequency modes to these quantities are calculated for an arbitrary quantum state. As an example, the low-frequency contributions to ${< \phi^2 >}$ and ${< T^{\mu}_{\nu} >}$ are calculated in asymptotically flat spacetimes in a quantum state corresponding to the Minkowski vacuum (Boulware quantum state). The limits of the applicability of these approximations are discussed.