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.
We present a novel analytic extraction of high-order post-Newtonian (pN) parameters that govern quasi-circular binary systems. Coefficients in the pN expansion of the energy of a binary system can be found from corresponding coefficients in an extreme-mass-ratio inspiral (EMRI) computation of the change ∆U in the redshift factor of a circular orbit at fixed angular velocity. Remarkably, by computing this essentially gauge-invariant quantity to accuracy greater than one part in 10 225 , and by assuming that a subset of pN coefficients are rational numbers or products of π and a rational, we obtain the exact analytic coefficients. We find the previously unexpected result that the postNewtonian expansion of ∆U (and of the change ∆Ω in the angular velocity at fixed redshift factor) have conservative terms at half-integral pN order beginning with a 5.5 pN term. This implies the existence of a corresponding 5.5 pN term in the expansion of the energy of a binary system.Coefficients in the pN series that do not belong to the subset just described are obtained to accuracy better than 1 part in 10 265−23n at nth pN order. We work in a radiation gauge, finding the radiative part of the metric perturbation from the gauge-invariant Weyl scalar ψ0 via a Hertz potential. We use mode-sum renormalization, and find high-order renormalization coefficients by matching a series in L = ℓ+1/2 to the large-L behavior of the expression for ∆U . The non-radiative parts of the perturbed metric associated with changes in mass and angular momentum are calculated in the Schwarzschild gauge. * a.g
We present the first numerical calculation of the (local) metric perturbation produced by a small compact object moving on an eccentric equatorial geodesic around a Kerr black hole, accurate to first order in the mass ratio. The procedure starts by first solving the Teukolsky equation to obtain the Weyl scalar ψ4 using semi-analytical methods. The metric perturbation is then reconstructed from ψ4 in an (outgoing) radiation gauge, adding the appropriate non-radiative contributions arising from the shifts in mass and angular momentum of the spacetime.As a demonstration we calculate the generalized redshift U as a function of the orbital frequencies Ωr and Ω φ to linear order in the mass ratio, a gauge invariant measure of the conservative corrections to the orbit due to self-interactions. In Schwarzschild, the results surpass the existing result in the literature in accuracy, and we find new estimates for some of the unknown 4PN and 5PN terms in the post-Newtonian expansion of U . In Kerr, we provide completely novel values of U for eccentric equatorial orbits. Calculation of the full self-force will appear in a forthcoming paper. * M.vandeMeent@soton.ac.uk † a.g.shah@soton.ac.uk 1. Homogeneous solutions (generic case ωmn = 0 and ma = 0)
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