Gauge-independent metric reconstruction of perturbations of vacuum spherically-symmetric spacetimes

Perturbation theory of vacuum spherically-symmetric spacetimes (including the cosmological constant) has greatly contributed to the understanding of black holes, relativistic compact stars, and even inhomogeneous cosmological models. The perturbative equations can be decoupled in terms of (gauge-invariant) master functions satisfying $1+1$ wave equations. In this work, building on previous work on the structure of the space of master functions and equations, we study the reconstruction of the metric perturbations in terms of the master functions. To that end, we consider the general situation in which the perturbations are driven by an arbitrary energy-momentum tensor. Then, we perform the metric reconstruction in a completely general perturbative gauge. In doing so, we investigate the role of Darboux transformations and Darboux covariance, responsible for the isospectrality between odd and even parity in the absence of matter sources, and also of the physical equivalence between the descriptions based on all the possible master equations. We also show that the metric reconstruction can be carried out in terms of any of the possible master functions and that the expressions admit an explicitly covariant form.