Characteristic formulation of the Regge-Wheeler and Zerilli Green functions

TL;DR

This paper introduces a characteristic initial value approach for calculating the Green functions of the Regge-Wheeler and Zerilli equations, improving accuracy in scalar self-force computations and energy flux analysis in Schwarzschild spacetime.

Contribution

It develops a novel numerical scheme combining known methods with new initial data, achieving higher-order convergence and enhanced accuracy in Green function calculations.

Findings

Achieved up to sixth-order convergence in numerical schemes.

Improved accuracy in scalar self-force calculations in Schwarzschild spacetime.

Validated results against frequency-domain methods for gravitational Green functions.

Abstract

We present a characteristic initial value approach to calculating the Green function of the Regge-Wheeler and Zerilli equations. We combine well-known numerical methods with newly derived initial data to obtain a scheme which can in principle be generalised to any desired order of convergence. We demonstrate the approach with implementations up to sixth-order in the grid spacing. By combining the results of our numerical code with late-time tail expansions and methods of subtracting the direct part of the Green function, we show that the scalar self-force in Schwarzschild spacetime can be computed to better accuracy than previous Green-function based approaches. We also demonstrate agreement with frequency-domain methods for computing the Green function in the gravitational case. Finally, we apply the Regge-Wheeler and Zerilli Green functions to the computation of the gravitational energy flux.