Teukolsky-like equations in a non-vacuum axisymmetric type D spacetime

TL;DR

This paper derives a generalized Teukolsky-like equation for gravitational waves in a non-vacuum, axisymmetric, Petrov type D spacetime, extending the classical vacuum Kerr solution to include matter or other non-vacuum effects.

Contribution

It introduces a new class of axisymmetric metrics conformal to Kerr with a deformation function, and derives a Teukolsky-like wave equation in this non-vacuum setting.

Findings

Separation of variables in the wave equations is possible under certain gauge conditions.

The resulting radial equation generalizes the classical Teukolsky equation to non-vacuum backgrounds.

The metric deformation allows for analysis of gravitational waves beyond vacuum Kerr spacetime.

Abstract

We study an axisymmetric metric satisfying the Petrov type D property with some additional ansatze, but without assuming the vacuum condition. We find that our metric in turn becomes conformal to the Kerr metric deformed by one function of the radial coordinate. We then study the gravitational-wave equations on this background metric in the case that the conformal factor is unity. We find that under an appropriate gauge condition, the homogeneous wave equations admit the separation of the variables, which is also helpful for solving the nonhomogeneous equations. The resultant ordinary differential equation for the radial coordinate gives a natural extension of the Teukolsky equation.