Mukkamala-Pere\~niguez master function for even-parity perturbations of the Schwarzschild spacetime

TL;DR

This paper explores a new master function for even-parity perturbations of Schwarzschild spacetime that satisfies the Regge-Wheeler equation, simplifying some aspects but complicating others like radiation field extraction and metric reconstruction.

Contribution

It extends previous work by identifying a source term, relating the master function to radiation fields, and reconstructing the metric perturbation from this new function.

Findings

The new master function satisfies the Regge-Wheeler equation for even-parity perturbations.

Relating the master function to radiation fields reveals increased complexity in practical applications.

Reconstruction of metric perturbations from the master function is more complex despite the simpler governing equation.

Abstract

Mukkamala and Pere\~niguez recently discovered a new master function for even-parity metric perturbations of the Schwarzschild spacetime. Remarkably, this function satisfies the Regge-Wheeler equation (instead of the Zerilli equation), which was previously understood to govern the odd-parity sector of the perturbation only. In this paper I follow up on their work. First, I identify a source term for their Regge-Wheeler equation, constructed from the perturbing energy-momentum tensor. Second, I relate the new master function to the radiation fields at future null infinity and the event horizon. Third, I reconstruct the metric perturbation from the new master function, in the Regge-Wheeler gauge. The main conclusion of this work is that the greater simplicity of the Regge-Wheeler equation (relative to the Zerilli equation) is offset by a greater complexity of obtaining the radiation fields and reconstructing the metric.