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

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 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 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 () shift in u^t (where 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 10^-5--10^-7 (depending on the orbital radius) ---comparable with the estimated numerical error.