Divergent part of the stress-energy tensor for Maxwell's theory in curved space-time: a systematic derivation

TL;DR

This paper systematically derives the divergent part of the stress-energy tensor for Maxwell's theory in curved space-time using Green functions and asymptotic expansion, providing explicit formulas for various space-times.

Contribution

It presents a new, concise formula for the divergent part of the stress-energy tensor in curved space-time, applicable to various metrics including Kerr, Schwarzschild, and de Sitter.

Findings

Derived explicit formulas for the divergent stress-energy tensor components.

Applied the formulas to physically relevant space-times like Kerr and Schwarzschild.

Provided a general framework for regularizing stress-energy tensor divergences in curved space-time.

Abstract

In this paper the Feynman Green function for Maxwell's theory in curved space-time is studied by using the Fock-Schwinger-DeWitt asymptotic expansion; the point-splitting method is then applied, since it is a valuable tool for regularizing divergent observables. Among these, the stress-energy tensor is expressed in terms of second covariant derivatives of the Hadamard Green function, which is also closely linked to the effective action; therefore one obtains a series expansion for the stress-energy tensor. Its divergent part can be isolated, and a concise formula is here obtained: by dimensional analysis and combinatorics, there are two kinds of terms: quadratic in curvature tensors (Riemann, Ricci tensors and scalar curvature) and linear in their second covariant derivatives. This formula holds for every space-time metric; it is made even more explicit in the physically relevant particular cases of Ricci-flat and maximally symmetric spaces, and fully evaluated for some examples of physical interest: Kerr and Schwarzschild metrics and de Sitter space-time.