Completion of metric reconstruction for a particle orbiting a Kerr black hole

Abstract

Vacuum perturbations of the Kerr metric can be reconstructed from the corresponding perturbation in either of the two Weyl scalars ψ0 or ψ4, using a procedure described by Chrzanowski and others in the 1970s. More recent work, motivated within the context of self-force physics, extends the procedure to metric perturbations sourced by a particle in a bound geodesic orbit. However, the existing procedure leaves undetermined a certain stationary, axially-symmetric piece of the metric perturbation. In the vacuum region away from the particle, this "completion" piece corresponds simply to mass and angular-momentum perturbations of the Kerr background, with amplitudes that are, however, a priori unknown. Here we present and implement a rigorous method for finding the completion piece. The key idea is to impose continuity, off the particle, of certain gauge-invariant fields constructed from the full (completed) perturbation, in order to determine the unknown amplitude parameters of the completion piece. We implement this method in full for bound (eccentric) geodesic orbits in the equatorial plane of the Kerr black hole. Our results provide a rigorous underpinning of recent results by Friedman et al. for circular orbits, and extend them to non-circular orbits.

Self-force effects on the marginally bound zoom-whirl orbit in Schwarzschild spacetime

“…In the past few years there has been progress in the development of time-domain methods based on the Teukolsky formalism, with the idea of computing the self-force from a radiation-gauge metric perturbation constructed from numerical, time-domain solutions of the spin-±2 Teukolsky equation [86][87][88][89][90]. This offers improved computational efficiency (since one has to solve a single scalar-like equation instead of 10 coupled equations in the Lorenz-gauge method), and also entirely circumvents the complications involved in computing the Lorenz-gauge monopole and dipole modes [91,92]. The implementation of this method in 1+1-dimensions appears to be numerically efficient even in the Kerr case, where mode-coupling has to be accounted for [93,94].…”

“…Specifically, these self-force terms were shown to be precisely proportional to a( 1 4 ) and a ( 1 4 ), respectively, where a(u) denotes the self-force piece of the basic radial potential A(u; ν) = 1 − 2u + νa(u) + O(ν 2 ) of EOB dynamics; see Eqs. (91) below. [Here, u := (M + µ)/r EOB , while ν := µM/(M + µ) 2 = η/(1 + η) 2 denotes the symmetric mass ratio.]…”

Numerical computation of the effective-one-body potentialqusing self-force results

“…Since this operator is not injective, this inversion is ambiguous up to an element of its kernel. The gauge-invariant content of this kernel is simply given by a shift in mass and angular momentum of the system [52], and can be extracted unambiguously [53]. As shown in Ref.…”

Fundamental frequencies and resonances from eccentric and precessing binary black hole inspirals

“…In this case the ratio of the particle mass µ to that of the Kerr black hole M Kerr serves as a small expansion parameter. While SF results for Kerr are still in development [67][68][69][70], mounting evidence suggests that SF results may be extended much closer to the nearly equal mass cases than expected [71][72][73], which is the regime appropriate for our our high (but not extreme) mass ratio binaries. Meanwhile, the lowest order approximation to the SF-expansion, namely the test particle limit, incorporates the effects of strongly curved spacetime, highly relativistic motion, and is fully understood.…”