Green's function in general relativity

TL;DR

This paper derives Green's functions for gravitational fields in general relativity, introducing a novel momentum space and linearisation method that applies to highly curved space-times without weak-field assumptions.

Contribution

It presents a new approach to defining Green's functions in curved space-time and linearising Einstein's equations without relying on weak-field approximations.

Findings

Green's functions exist in curved space-time with indefinite metrics.

A new momentum space definition in curved space-time is proposed.

Green's functions are explicitly derived for plane wave and Schwarzschild solutions.

Abstract

This report provides Green's functions (classical propagators) of gravitational fields appearing in general relativity. The existence of Green's function of the wave equation in curved space with an indefinite metric is ensured owing to the Hodge harmonic analysis. The analyticity of Green's function is determined intrinsically to keep a causality. This report proposed a novel definition of momentum space in curved space-time and the linearisation of the Einstein equation as a free field consistent with that of the Yang-Mills gauge field. The proposed linearisation does not utilize the weak-field approximation; thus, the method applies to highly caved space-time. We gave two examples of Green's function of gravitational fields: the plane wave solution and the Schwarzschild solution.