The second-order gravitational self-force

Abstract

This project makes progress towards a first calculation of the second-order gravitational self-force in extreme-mass-ratio binaries. This is an important component in the modeling of these key astrophysical sources of gravitational waves. Computing the secondorder self-force requires the second-order metric perturbation, which can be calculated by solving the Einstein field equations through second order in the mass ratio. Here we have developed, for the first time, a practical scheme for solving the second-order equations. The main ingredient is a certain “puncture” field, which describes the local metric perturbation near the small member of the binary, and for which we obtain a useful covariant-form expression. We apply this method to the case of a quasicircular binary of nonrotating black holes. As a first test we numerically solve the first-order field equations and compute the first-order self-force, finding good agreement with previous results obtained using a different method. The calculation of the second-order metric perturbation brings about two additional technical difficulties: the need for a certain regularization at infinity and on the event horizon of the large black hole, and the strong divergence of the second-order source of the field equations near the small object. We show how these issues can be resolved, first in a simple scalar-field toy model, and then in the second-order gravitational problem. We finally apply our method in full in order to numerically solve the second-order perturbation equations in the quasicircular case, focusing on the monopole piece of the perturbation as a first example.

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Identifiers

ORCID for Adam Pound: orcid.org/0000-0001-9446-0638

ORCID for Leor Barack: orcid.org/0000-0003-4742-9413

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