On the Green functions of gravitational radiation theory

TL;DR

This paper analyzes the Green functions for gravitational radiation in black-hole collisions, deriving integral representations and solutions for the associated hyperbolic operators, advancing understanding of perturbative gravitational fields.

Contribution

It provides a detailed analysis of the Green function for second-order hyperbolic operators in gravitational radiation theory, using canonical form and Riemann functions.

Findings

Derived integral representation of the Green function.

Solved characteristic initial-value problem analytically.

Provided numerical methods for solution approximation.

Abstract

Previous work in the literature has studied gravitational radiation in black-hole collisions at the speed of light. In particular, it had been proved that the perturbative field equations may all be reduced to equations in only two independent variables, by virtue of a conformal symmetry at each order in perturbation theory. The Green function for the perturbative field equations is here analyzed by studying the corresponding second-order hyperbolic operator with variable coefficients, instead of using the reduction method from the retarded flat-space Green function in four dimensions. After reduction to canonical form of this hyperbolic operator, the integral representation of the solution in terms of the Riemann function is obtained. The Riemann function solves a characteristic initial-value problem for which analytic formulae leading to the numerical solution are derived.