Post-Newtonian and numerical calculations of the gravitational self-force for circular orbits in the Schwarzschild geometry

Abstract

The problem of a compact binary system whose components move on circular orbits is addressed using two different approximation techniques in general relativity. The post-Newtonian (PN) approximation involves an expansion in powers of $v/c\ensuremath{\ll}1$, and is most appropriate for small orbital velocities $v$. The perturbative self-force analysis requires an extreme mass ratio ${m}_{1}/{m}_{2}\ensuremath{\ll}1$ for the components of the binary. A particular coordinate-invariant observable is determined as a function of the orbital frequency of the system using these two different approximations. The post-Newtonian calculation is pushed up to the third post-Newtonian (3PN) order. It involves the metric generated by two point particles and evaluated at the location of one of the particles. We regularize the divergent self-field of the particle by means of dimensional regularization. We show that the poles $\ensuremath{\propto}(d\ensuremath{-}3{)}^{\ensuremath{-}1}$ appearing in dimensional regularization at the 3PN order cancel out from the final gauge invariant observable. The 3PN analytical result, through first order in the mass ratio, and the numerical self-force calculation are found to agree well. The consistency of this cross cultural comparison confirms the soundness of both approximations in describing compact binary systems. In particular, it provides an independent test of the very different regularization procedures invoked in the two approximation schemes.