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

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.