Gravitational Self-force in a Radiation Gauge

Details Gravitational Self-force in a Radiation Gauge Gravitational Self-force in a Radiation Gauge

2010-04-13

arxiv.org/abs/1004.2276v3

Phys.Rev.D82:124012,2010

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

33 pages, 3 Tables. Changes for publication. Accepted to PRD

Scientific paper

10.1103/PhysRevD.82.124012

In this, the first of two companion papers, we present a method for finding the gravitational self-force in a radiation gauge for a particle moving on a geodesic in a Schwarzschild or Kerr spacetime. The method involves a mode-sum renormalization of a spin-weight $\pm 2$ perturbed Weyl scalar and the subsequent reconstruction from a Hertz potential of the renormalized perturbed metric. We show that the Hertz potential is uniquely specified by the requirement that it have no angular harmonics with $\ell\leq 2$. The resulting perturbed metric is singular only at the position of the particle: It is smooth on the axis of symmetry. An extension of an earlier result by Wald is needed to show that the perturbed metric is determined up to a gauge transformation and an infinitesimal change in the black hole mass and spin. We show that the singular behavior of the metric and self-force has the same power-law behavior in $L=\ell+1/2$ as in a Lorenz gauge (with different coefficients). We 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 compute the singular field by matching the retarded field to a power series in $L$ and subtracting off the leading and subleading terms in this series. We do so, and compare the accuracy of the two methods. Details of the numerical computation of the perturbed metric, the self-force, and the quantity $h_{\alpha\beta}u^\alpha u^\beta$ (gauge invariant under helically symmetric gauge transformations) are presented for this test case in the companion paper.

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