Reconstruction of inhomogeneous metric perturbations and electromagnetic four-potential in Kerr spacetime

TL;DR

This paper develops a method to construct metric perturbations and electromagnetic potentials in Kerr spacetime for inhomogeneous sources, extending previous vacuum solutions to include sources like orbiting objects.

Contribution

It introduces a procedure to derive inhomogeneous perturbations in Kerr spacetime by constructing the potential from source mode functions, enabling the application of Chrzanowski's method to non-vacuum cases.

Findings

Constructed the potential from source mode functions.

Extended Chrzanowski's method to inhomogeneous perturbations.

Provided a framework for perturbations from orbiting objects.

Abstract

We present a procedure that allows the construction of the metric perturbations and electromagnetic four-potential, for gravitational and electromagnetic perturbations produced by sources in Kerr spacetime. This may include, for example, the perturbations produced by a point particle or an extended object moving in orbit around a Kerr black hole. The construction is carried out in the frequency domain. Previously, Chrzanowski derived the vacuum metric perturbations and electromagnetic four-potential by applying a differential operator to a certain potential $\Psi $. Here we construct $\Psi $ for inhomogeneous perturbations, thereby allowing the application of Chrzanowski's method. We address this problem in two stages: First, for vacuum perturbations (i.e. pure gravitational or electromagnetic waves), we construct the potential from the modes of the Weyl scalars $\psi_{0}$ or $\phi_{0}$. Second, for perturbations produced by sources, we express $\Psi $ in terms of the mode functions of the source, i.e. the energy-momentum tensor $T_{\alpha \beta}$ or the electromagnetic current vector $J_{\alpha}$.