Green functions of the Regge-Wheeler and Teukolsky equations in Schwarzschild spacetime

TL;DR

This paper calculates the retarded Green functions for gravitational perturbations in Schwarzschild spacetime, revealing their singularity structures and oscillatory behaviors near caustics, using advanced numerical and analytical techniques.

Contribution

It provides the first detailed computation of the full retarded Green functions for the Regge-Wheeler and Teukolsky equations in Schwarzschild spacetime, including their singularity structures and physical oscillations.

Findings

Green functions exhibit 4-fold singularity structure away from caustics

At caustics, Green functions show 2-fold singularity structure

Gravitational perturbations display physical oscillations near singularities

Abstract

We present a calculation of the full retarded Green functions of the Regge-Wheeler and Teukolsky equations obeyed by gravitational field perturbations of Schwarzschild spacetime. We perform the calculations for spacetime points along: (i) a timelike circular geodesic (where null-separated points are not at caustics); and (ii) a static worldline (where null-separated points are at caustics). These Green functions show a 4-fold singularity structure away from caustics, and 2-fold at caustics (similarly to the case of scalar field perturbations, which we also reproduce). Physical oscillations near the singularities appear in the gravitational case, which were not present in the scalar case. We obtain our results by developing various numerical and analytical methods.