Post-Newtonian and Numerical Calculations of the Gravitational Self-Force for Circular Orbits in the Schwarzschild Geometry

TL;DR

The paper performs a rigorous cross-check between post-Newtonian (PN) and gravitational self-force (SF) approaches for circular orbits in Schwarzschild geometry, advancing PN to 3PN order using dimensional regularization (DR) and comparing with high-precision SF results. It derives a gauge-invariant observable, $u^T$, and demonstrates that DR poles and regulator scales cancel in the final observable, providing a robust consistency test between two very different regularization schemes. The SF calculation yields a precise 3PN coefficient, which, when matched against the PN expansion, confirms agreement to within a small error and validates the conservative dynamics in both methods. The work strengthens confidence in PN and SF as complementary tools for modeling compact binaries across regimes relevant to LIGO/Virgo and LISA, and sets the stage for future higher-order analyses and potential inclusion of logarithmic terms at 4PN/5PN orders.

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<<1, and is most appropriate for small orbital velocities v. The perturbative self-force (SF) analysis requires an extreme mass ratio m1/m2<<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 proportional to 1/(d-3) 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 SF 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.

Figures (3)

• Figure 1: Different analytical approximation schemes and numerical techniques are used to study black hole binaries, depending on the mass ratio $m_1/m_2$ and the orbital velocity $v^2 \sim G m/r_{12}$, where $m = m_1 + m_2$. The post-Newtonian theory and black hole perturbation theory can be compared in the slow motion regime ($v\ll c$ equivalent to $r_{12}\gg G m/c^2$ for circular orbits) of an extreme mass ratio ($m_1 \ll m_2$) binary.
• Figure 2: The self-force contribution $u^T_\mathrm{SF}$ to $u^T$ plotted as a function of the gauge invariant variable $y^{-1}$. Note that $y^{-1}$ is equal to $R_\Omega/{m_2}$, an invariant measure of the orbital radius, scaled by the black hole mass $m_2$ [cf. Eq. \ref{['yPN']}]. The "exact" numerical points are taken from Ref. Det08.
• Figure 3: Numerically derived residuals, i.e., after removal of the 2PN and 3PN self-force contributions to $u^T_\mathrm{SF}$, plotted as a function of the gauge invariant variable $y^{-1}$. Compare with scales in Fig. \ref{['ut_SF']}. Note that $y^{-1}$ is equal to $R_\Omega/{m_2}$, an invariant measure of the orbital radius, scaled by the black hole mass $m_2$ [cf. \ref{['yPN']}].