Second Order Perturbations of Kerr Black Holes: Reconstruction of the Metric

Abstract

Motivated by gravitational wave observations of binary black hole mergers, we present a procedure to compute the leading order nonlinear gravitational wave interactions around a Kerr black hole. We describe the formalism used to derive the equations for second order perturbations. We develop a procedure that allows us to reconstruct the first order metric perturbation solely from knowledge of the solution to the first order Teukolsky equation, without the need of Hertz potentials. Finally, we illustrate this metric reconstruction procedure in the asymptotic limit for the first order quasinormal modes of Kerr. In a companion paper [1] we present a numerical implementation of these ideas.

New metric reconstruction scheme for gravitational self-force calculations

“…We conclude by noting that our puncture scheme represents only one possible path to obtaining the metric perturbation in a regular gauge. Other reconstruction methods are also under development [57][58][59][60]. Although these are not yet able to reconstruct a nonvacuum metric perturbation, they may soon offer a viable alternative to our method.…”

Numerical computation of second order vacuum perturbations of Kerr black holes

“…This can partly be motivated by the presence of a family of slowly damped modes, whose damping timescale grows without bound as the black hole spin approaches its extremal value [11,12] 2 . The slower damping of perturbations implies nonlinear effects could be more pronounced; most intriguing in this regard is the suggestion that mode coupling induces 1 We note though that comparable mass mergers cannot produce near-extremal remnants, see e.g. [7][8][9][10], and it is unknown how rapidly the typical supermassive black holes in the universe, of relevance to EMRIs, rotate.…”

“…For some applications it may be necessary to go beyond linear perturbations. Here for brevity we only mention a couple of key incentives (a more thorough discussion that motivates this study can be found in our companion paper [1]). In order to extract subleading modes of the ringdown following comparable mass mergers, it may be necessary to consider nonlinear effects.…”

“…Nonlinear physics may also play an important role in the ringdown of near-extremal Kerr black holes 1 . This can partly be motivated by the presence of a family of slowly damped modes, whose damping timescale grows without bound as the black hole spin approaches its extremal value [11,12] 2 .…”

“…Motivated by the desire to understand the leading order nonlinear gravitational wave interactions around arbitrarily rapidly rotating Kerr black holes, we describe a numerical code designed to compute second order vacuum perturbations on such spacetimes. A general discussion of the formalism we use is presented in [1]; here we show how we numerically implement that formalism with a particular choice of coordinates and tetrad conditions, and give example results for black holes with dimensionless spin parameters a = 0.7 and a = 0.998. We first solve the Teukolsky equation for the linearly perturbed Weyl scalar Ψ…”