Gravitational self force on a particle in circular orbit around a Schwarzschild black hole

TL;DR

This paper presents the first fully self-consistent calculation of the gravitational self-force for a particle in a circular orbit around a Schwarzschild black hole, carried out entirely in the Lorenz gauge using time-domain evolution. It combines direct computation of the Lorenz-gauge metric perturbation, mode-sum regularization, and a detailed validation program, revealing both the dissipative temporal SF and the conservative radial SF, which together produce the $O(\mu)$ shifts in energy, angular momentum, and orbital frequency. The temporal SF satisfies energy balance with the radiated flux, while the radial SF provides gauge-dependent conservative corrections, including an explicit expression for the frequency shift and analytic large-$r$ and near-ISCO fits. The results demonstrate the feasibility of an all-Lorenz-gauge, 1+1D time-domain approach to gravitational self-force problems and pave the way for extending to eccentric orbits and Kerr spacetime, with important implications for EMRI waveform modeling and gravitational-wave data analysis.

Abstract

We calculate the gravitational self force acting on a pointlike particle of mass $μ$, set in a circular geodesic orbit around a Schwarzschild black hole. Our calculation is done in the Lorenz gauge: For given orbital radius, we first solve directly for the Lorenz-gauge metric perturbation using numerical evolution in the time domain; We then compute the (finite) back-reaction force from each of the multipole modes of the perturbation; Finally, we apply the ``mode sum'' method to obtain the total, physical self force. The {\em temporal} component of the self force (which is gauge invariant) describes the dissipation of orbital energy through gravitational radiation. Our results for this component are consistent, to within the computational accuracy, with the total flux of gravitational-wave energy radiated to infinity and through the event horizon. The {\em radial} component of the self force (which is gauge dependent) is calculated here for the first time. It describes a conservative shift in the orbital parameters away from their geodesic values. We thus obtain the $O(μ)$ correction to the specific energy and angular momentum parameters (in the Lorenz gauge), as well as the $O(μ)$ shift in the orbital frequency (which is gauge invariant).

Figures (10)

• Figure 1: A numerical grid cell, of dimensions $h\times h$ (see description in the text). Our 2-D grid is based on characteristic (Eddington--Finkelstein) coordinates $v$ and $u$. These are related to the Schwarzschild coordinates through $v=t+r_*$ and $u=t-r_*$, where $r_*=r+2M\ln[r/(2M)-1]$.
• Figure 2: Finite-difference scheme for terms in the field equations involving single $v$ derivatives (diagrams to go with the description in the text). The dashed line represents the worldline, with two cases shown.
• Figure 3: Diagram to explain how $r$ derivatives are taken at the worldline (see description in the text). The dashed line represents the particle's trajectory on the numerical grid. The SF is calculated at the point labeled '0'.
• Figure 4: Analytic fitting for the large-$l$ tail of the SF, exemplified here for $r_0=6M$. We used the fitting formula (\ref{['eq:highL-formula']}) with $N=2$, and based on the modes $10\le l\le 15$. Circles ('$\odot$') represent actual data obtained for $F_{\rm reg}^{r l}$ (calculated from $r_0^-$), for the various modes $3\leq l\leq 25$. The dashed line is the analytic fit. Left/right panels show the same data on linear/log scales. The large-$l$ tail of the mode-sum series shows the $l^{-2}$ fall-off expected from theory (cf. Fig. \ref{['fig:large l']}).
• Figure 5: Numerical convergence of the calculated SF, demonstrated here for the $r$ component, for $r_0=6M$. The left and right panels show $l=2$ and $l=15$, respectively. Plotted is the difference $F^{rl}_{\rm reg}[h_i]-F^{rl}_{\rm reg}$ between the value of the regularized mode computed with step size $h_i$, and the value extrapolated to $h\to 0$. Each panel displays both one-sided values of the force: "left" and "right" stand for $r_0^-$ and $r_0^+$ values, respectively. The reference lines (dotted) have slops $\propto h^2$. This demonstrates the quadratic convergence of the numerical calculation.