Two approaches for the gravitational self force in black hole spacetime: Comparison of numerical results

TL;DR

The paper reconciles two independent gravitational self-force calculations for a particle on a circular Schwarzschild orbit—one in the Lorenz gauge (BS) and one in the Regge-Wheeler gauge (SD)—by establishing a formal mapping between the two trajectory descriptions and applying a gauge adjustment. It derives a gauge-invariant measure, $\Delta U(R)$, to compare conservative finite-$\mu$ effects across gauges, and shows that the resulting $O(\mu)$ shifts in $u^t$ agree to fractional discrepancies on the order of $10^{-5}$ to $10^{-7}$, within numerical errors. The work confirms the equivalence of the two descriptions and demonstrates a constructive approach for cross-validating self-force calculations, with potential extension to more general orbits and Kerr spacetime. This establishes a robust framework for validating future GSF computations and for extracting physically meaningful, gauge-invariant information from multiple formalisms.

Abstract

Recently, two independent calculations have been presented of finite-mass ("self-force") effects on the orbit of a point mass around a Schwarzschild black hole. While both computations are based on the standard mode-sum method, they differ in several technical aspects, which makes comparison between their results difficult--but also interesting. Barack and Sago [Phys. Rev. D {\bf 75}, 064021 (2007)] invoke the notion of a self-accelerated motion in a background spacetime, and perform a direct calculation of the local self force in the Lorenz gauge (using numerical evolution of the perturbation equations in the time domain); Detweiler [Phys. Rev. D {\bf 77}, 124026 (2008)] describes the motion in terms a geodesic orbit of a (smooth) perturbed spacetime, and calculates the metric perturbation in the Regge--Wheeler gauge (using frequency-domain numerical analysis). Here we establish a formal correspondence between the two analyses, and demonstrate the consistency of their numerical results. Specifically, we compare the value of the conservative $O(μ)$ shift in $u^t$ (where $μ$ is the particle's mass and $u^t$ is the Schwarzschild $t$ component of the particle's four-velocity), suitably mapped between the two orbital descriptions and adjusted for gauge. We find that the two analyses yield the same value for this shift within mere fractional differences of $\sim 10^{-5}$--$10^{-7}$ (depending on the orbital radius)--comparable with the estimated numerical error.