Self-force effects on the marginally bound zoom-whirl orbit in Schwarzschild spacetime

TL;DR

This paper calculates the conservative gravitational self-force effects on a particle in a marginally bound orbit around a Schwarzschild black hole, providing numerical results that agree with theoretical predictions and paving the way for more complex orbit analyses.

Contribution

First direct numerical calculation of self-force effects on unbound orbits in Schwarzschild spacetime, confirming theoretical estimates and demonstrating a method for future hyperbolic orbit studies.

Findings

Self-force causes measurable shifts in orbital parameters.

Numerical results agree with Effective One Body estimates.

Method applicable to hyperbolic scattering orbits.

Abstract

For a Schwarzchild black hole of mass $M$, we consider a test particle falling from rest at infinity and becoming trapped, at late time, on the unstable circular orbit of radius $r=4GM/c^2$. When the particle is endowed with a small mass, $\mu\ll M$, it experiences an effective gravitational self-force, whose conservative piece shifts the critical value of the angular momentum and the frequency of the asymptotic circular orbit away from their geodesic values. By directly integrating the self-force along the orbit (ignoring radiative dissipation), we numerically calculate these shifts to $O(\mu/M)$. Our numerical values are found to be in agreement with estimates first made within the Effective One Body formalism, and with predictions of the first law of black-hole-binary mechanics (as applied to the asymptotic circular orbit). Our calculation is based on a time-domain integration of the Lorenz-gauge perturbation equations, and it is a first such calculation for an unbound orbit. We tackle several technical difficulties specific to unbound orbits, illustrating how these may be handled in more general cases of unbound motion. Our method paves the way to calculations of the self-force along hyperbolic-type scattering orbits. Such orbits can probe the two-body potential down to the "light ring", and could thus supply strong-field calibration data for eccentricity-dependent terms in the Effective One Body model of merging binaries.