Beyond the geodesic approximation: Conservative effects of the gravitational self-force in eccentric orbits around a Schwarzschild black hole

We study conservative finite-mass corrections to the motion of a particle in a bound (eccentric) strong-field orbit around a Schwarzschild black hole. We assume the particle’s mass $\mu$ is much smaller than the black hole mass $M$, and explore post-geodesic corrections of $O(\mu /M)$. Our analysis uses numerical data from a recently developed code that outputs the Lorenz-gauge gravitational self-force (GSF) acting on the particle along the eccentric geodesic. First, we calculate the $O(\mu /M)$ conservative correction to the periastron advance of the orbit, as a function of the (gauge-dependent) semilatus rectum and eccentricity. A gauge-invariant description of the GSF precession effect is made possible in the circular-orbit limit, where we express the correction to the periastron advance as a function of the invariant azimuthal frequency. We compare this relation with results from fully nonlinear numerical-relativistic simulations. In order to obtain a gauge-invariant measure of the GSF effect for fully eccentric orbits, we introduce a suitable generalization of Detweiler’s circular-orbit “redshift” invariant. We compute the $O(\mu /M)$ conservative correction to this invariant, expressed as a function of the two invariant frequencies that parametrize the orbit. Our results are in good agreement with results from post-Newtonian calculations in the weak-field regime, as we shall report elsewhere. The results of our study can inform the development of analytical models for the dynamics of strongly gravitating binaries. They also provide an accurate benchmark for future numerical-relativistic simulations.

Figure 1

Plot of some of the numerical data presented in Table . The left and right panels show, respectively, the absolute and relative GSF corrections, $\Delta \delta$ and $\Delta \delta /{\delta }_0$ (divided by the mass ratio $q\equiv \mu /M$), as functions of $p_0$ for a variety of eccentricities $e_0$. The data are shown on a log-log scale, and we have in fact plotted $-\Delta \delta$ and $-\Delta \delta /{\delta }_0$ since $\Delta \delta$ itself is negative (while ${\delta }_0$ is positive). Note $\Delta \delta$ depends very weakly on $e_0$ at large $p_0$, where it appears to fall off with a power-law $\propto p_0^{-4}$. The background advance ${\delta }_0$ falls off as $\sim 3p_0^{-1}$ [recall Eq. (11)]. The dashed line in the left panel is a reference line $\Delta \delta =-2000q/p_0^4$.

Figure 2

Tentative comparison of GSF and NR data for the periastron advance of slightly eccentric orbits. Shown is the frequency ratio ${\Omega }_{\phi }/{\Omega }_r$ as a function of ${\widehat{\Omega }}_{\phi }$ (adimensionalized using the total mass $M+\mu$), for a variety of mass ratios $q=\mu /M$ between $1:2$ and $1:6$. The dashed line corresponds to a test particle ($q=0$). Single data points describe results from NR simulations, reproduced here from Fig. 8 of Mroué et al. [23]. Solid lines are interpolated $O(q)$ GSF predictions, calculated using Eqs. (92, 94) with the numerical values of the GSF coefficients tabulated in Appendix . The horizontal scale of this plot roughly coincides with that of Fig. 8 of 23 for easy reference. Despite the manifest low accuracy of the NR data, this preliminary comparison is already rather instructive (as described in the text), and motivates further study.

Figure 3

Singularity of the transformation $(p_0,e_0)\to ({\Omega }_{\phi 0},{\Omega }_{r0})$ for bound (eccentric) geodesics in Schwarzschild spacetime. The plot displays the level lines ${\Omega }_{\phi 0}=\mathrm{const}$ (dotted, red) and ${\Omega }_{r0}=\mathrm{const}$ (solid, blue) over a portion of the $p_0$, $e_0$ parameter space. The diagonal $p_0=6+2e_0$ (straight black line) is the separatrix, where ${\Omega }_{r0}=0$; stable geodesic orbits exist in the region $p_0>6+2e_0$. In the domain shown, ${\Omega }_{\phi 0}$ decreases monotonically with $p_0$ while ${\Omega }_{r0}$ increases monotonically with $p_0$ (the sign of $\partial {\Omega }_{r0}/\partial p_0$ reverses further out to the right of the region shown). The thick (black) curve, $p_0=p_s(e_0)$, is the locus of points where the ${\Omega }_{\phi 0}$ and ${\Omega }_{r0}$ level lines are tangent to one another; along this curve the Jacobian of the transformation $(p_0,e_0)\to ({\Omega }_{\phi 0},{\Omega }_{r0})$ vanishes. Each orbit left of the singular curve has a dual isofrequency orbit associated with it, which lies to the right of the singular curve. One such pair is indicated in the plot (black crosses), at $(p_0,e_0)=(6.3,0.05)$ and (0.59274, 0.27569).