This is the first of two papers on computing the self-force in a radiation gauge for a particle of mass m moving in circular, equatorial orbit about a Kerr black hole. In the EMRI (extreme-massratio inspiral) framework, with mode-sum renormalization, we compute the renormalized value of the quantity H := 1 2 h αβ u α u β , gauge-invariant under gauge transformations generated by a helically symmetric gauge vector; here h αβ is the metric perturbation, u α the particle's 4-velocity. We find the related order m correction to the particle's angular velocity at fixed renormalized redshift (and to its redshift at fixed angular velocity), each of which can be written in terms of H. The radiative part of the metric perturbation is constructed from a Hertz potential that is extracted from the Weyl scalar by an algebraic inversion [1]. We then write the spin-weighted spheroidal harmonics as a sum over spin-weighted spherical harmonics sYℓm and use mode-sum renormalization to find the renormalization coefficients by matching a series in L = ℓ + 1/2 to the large-L behavior of the expression for H. The non-radiative parts of the perturbed metric associated with changes in mass and angular momentum are calculated in the Kerr gauge.
Gravitational waves from the inspiral of a stellar-size black hole to a supermassive black hole can be accurately approximated by a point particle moving in a Kerr background. This paper presents progress on finding the electromagnetic and gravitational field of a point particle in a black-hole spacetime and on computing the self-force in a "radiation gauge." The gauge is chosen to allow one to compute the perturbed metric from a gauge-invariant component ψ 0 (or ψ 4 ) of the Weyl tensor and follows earlier work by Chrzanowski and Cohen and Kegeles (we correct a minor, but propagating, error in the Cohen-Kegeles formalism). The electromagnetic field tensor and vector potential of a static point charge and the perturbed gravitational field of a static point mass in a Schwarzschild geometry are found, surprisingly, to have closed-form expressions. The gravitational field of a static point charge in the Schwarzschild background must have a strut, but ψ 0 and ψ 4 are smooth except at the particle, and one can find local radiation gauges for which the corresponding spin ±2 parts of the perturbed metric are smooth. Finally a method for finding the renormalized self-force from the Teukolsky equation is presented. The method is related to the Mino, Sasaki, Tanaka and Quinn and Wald (MiSaTaQuWa) renormalization and to the DetweilerWhiting construction of the singular field. It relies on the fact that the renormalized ψ 0 (or ψ 4 ) is a sourcefree solution to the Teukolsky equation; and one can therefore reconstruct a nonsingular renormalized metric in a radiation gauge.
In this, the first of two companion papers, we present a method for finding the gravitational self-force in a modified radiation gauge for a particle moving on a geodesic in a Schwarzschild or Kerr spacetime. An extension of an earlier result by Wald is used to show the spin-weight ±2 perturbed Weyl scalar (ψ0 or ψ4) determines the metric perturbation outside the particle up to a gauge transformation and an infinitesimal change in mass and angular momentum. A Hertz potential is used to construct the part of the retarded metric perturbation that involves no change in mass or angular momentum from ψ0 in a radiation gauge. The metric perturbation is completed by adding changes in the mass and angular momentum of the background spacetime outside the radial coordinate r0 of the particle in any convenient gauge. The resulting metric perturbation is singular on the trajectory of the particle and discontinuous across the sphere r = r0. A mode-sum method can be used to renormalize the self-force, but the justification given in the published version of this paper [1] referred to work by Sam Gralla [2] to justify the use of the renormalized self-force, and the radiation gauge we use does not satisfy the regularity conditions required by Gralla. Instead we show that the renormalized self-force, computed either from the retarded field for r > r0 or for r < r0 gives the correct equations of motion in a gauge smoothly related to a Lorenz gauge; and Pound et al. [3] argue that the average of the self-force obtained in the way described in our paper for r > r0 and for r < r0 gives the correct equation of motion for our gauge (what Pound et al. call the no-string gauge).In a Lorenz gauge one can renormalize the self-force by subtracting the leading and subleading terms in a mode-sum expansion of the retarded field, and the self-force in the two gauges just described is again obtained by subtracting only these leading and subleading terms. A companion paper [4] finds that these two terms coincide with the leading and subleading Lorenz-gauge terms.We explicitly compute the singular Weyl scalar and its mode-sum decomposition to subleading order in L for a particle in circular orbit in a Schwarzschild geometry and obtain the renormalized field. Because the singular field can be defined as this mode sum, the coefficients of each angular harmonic in the sum must agree with the large L limit of the corresponding coefficients of the retarded field. One may therefore compute the singular field by numerically matching the retarded field to a power series in L. To check the accuracy of the numerical method, we analytically compute leading and subleading terms in the singular expansion of ψ0 and compare the numerical and analytic values of the renormalization constants, finding agreement to high precision. Details of the numerical computation of the perturbed metric, the self-force, and the quantity h αβ u α u β (gauge invariant under helically symmetric gauge transformations) are presented for this test case in the companion paper.
This is the second of two companion papers on computing the self-force in a radiation gauge; more precisely, the method uses a radiation gauge for the radiative part of the metric perturbation, together with an arbitrarily chosen gauge for the parts of the perturbation associated with changes in black-hole mass and spin and with a shift in the center of mass. In a test of the method delineated in the first paper, we compute the conservative part of the self-force for a particle in circular orbit around a Schwarzschild black hole. The gauge vector relating our radiation gauge to a Lorenz gauge is helically symmetric, implying that the quantity h αβ u α u β must have the same value for our radiation gauge as for a Lorenz gauge; and we confirm this numerically to one part in 10 14 . As outlined in the first paper, the perturbed metric is constructed from a Hertz potential that is in term obtained algebraically from the the retarded perturbed spin-2 Weyl scalar, ψ ret 0 . We use a mode-sum renormalization and find the renormalization coefficients by matching a series in L = ℓ + 1/2 to the large-L behavior of the expression for the self-force in terms of the retarded field h ret αβ ; we similarly find the leading renormalization coefficients of h αβ u α u β and the related change in the angular velocity of the particle due to its self-force. We show numerically that the singular part of the self-force has the form fα ∝ ∇αρ −1 , the part of ∇αρ −1 that is axisymmetric about a radial line through the particle. This differs only by a constant from its form for a Lorenz gauge. It is because we do not use a radiation gauge to describe the change in black-hole mass that the singular part of the self-force has no singularity along a radial line through the particle and, at least in this example, is spherically symmetric to subleading order in ρ.
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