Perturbations of rotating black holes in Lorenz gauge: method and applications
- Sam Dolan (The University of Sheffield)
Abstract
Roy Kerr's solution to the vacuum Einstein field equations is believed to provide an accurate description of the spacetime geometry of all the rotating black holes in our Universe. The metric perturbations of Kerr's geometry are governed by a set of 10 coupled PDEs that arise from linearizing Einstein's equations. In the 1970s, Teukolsky showed that two "Weyl scalars" (certain projections of the perturbed Weyl tensor) satisfy decoupled equations which are also separable. At a stroke, important physical questions could now be answered by analysing ODEs rather than PDEs. The challenge of reconstructing a metric from these Weyl scalars was also addressed in the 1970s, with some success; but it becomes a non-trivial problem in the presence of sources, and there is substantial freedom in the choice of gauge, which can help or hinder.
I will describe new work (in collaboration with Wardell and Kavanagh) in which we calculate the 10 components of the metric perturbation in Lorenz gauge from (differential operators acting on) six Teukolsky-like scalars, each of which satisfies a decoupled, separable equation in the frequency domain. I will present numerical results for the metric perturbation generated by a compact body in a circular orbit about the black hole, and I will discuss the extension of the method to eccentric and non-equatorial orbits. I will briefly outline three applications for this line of work: (i) generating source terms for second-order self-force calculations in order to accurately model Extreme Mass-Ratio Inpirals (EMRIs) for space-based gravitational wave detectors (LISA); (ii) assessing the effect of black hole environments on the evolution of EMRIs, and vice versa; (iii) non-linear perturbation theory for e.g. modelling quasinormal mode ringdowns of black hole coalescences.