Computing the gravitational self-force on a compact object plunging into a Schwarzschild black hole

Abstract

We compute the gravitational self-force (or ``radiation reaction'' force) acting on a particle falling radially into a Schwarzschild black hole. Our calculation is based on the ``mode-sum'' method, in which one first calculates the individual $\ell$-multipole contributions to the self-force (by numerically integrating the decoupled perturbation equations) and then regularizes the sum over modes by applying a certain analytic procedure. We demonstrate the equivalence of this method with the $ζ-$function scheme. The convergence rate of the mode-sum series is considerably improved here (thus notably reducing computational requirements) by employing an analytic approximation at large $\ell$.

Figures (3)

• Figure 1: Upper figure: The "full" modes $\bar{F}^{r\ell}(r_p)$ at $\ell=0,\ldots,8$, for a particle released from rest at $r_0=14M$. Note how a limiting curve [given analytically by $B^r(r)$] is approached at large $\ell$. The wavy feature near $r_p=14M$ is due to the radiation content of the initial data (chosen here as conformally flat). This "spureous" feature damps down by the time the particle reaches $r_p\sim10M$, exposing the inherent self force effect. The bottom figure demonstrates the anticipated $\propto\ell^{-2}$ convergence of the difference $F_{\rm reg}^{r\ell}\equiv\bar{F}^{r\ell}-B^r$.
• Figure 2: Analytic approximation vs. numerical results: The plot shows the numerically calculated (regularized) modes $F_{\rm reg}^{r\ell}$ along with their large-$\ell$ analytic approximation, $F^{r\ell}_{\rm analytic}$, as given by the $O(L^{-2})$ term in Eq. (\ref{['NOr']}). Shown are the modes $\ell=0,1,2,3,4$ for $r_0=14M$ [times $L^2$ for the sake of comparison]. The inset shows the reminder $F^{r\ell}_{\rm reg}-F^{r\ell}_{\rm analytic}$ at $r_p=6M$, demonstrating its anticipated $\propto\ell^{-4}$ behavior. The wavy feature at the onset of the plunge is associated with the "spurious" radiation content of the initial data; the inherent SF is exposed only after these waves are dissipated away.
• Figure 3: The upper and bottom panels show, respectively, the $r$ and $t$ components of the overall SF on a particle starting at rest at $r_0=14M$. The plots labeled as "numerical" are produced by summing up the numerically calculated (regularized) modes up to $\ell=8$, and then incorporating our analytic approximation at $\ell>8$ (these higher modes contribute up to $\sim20\%$ of the total force). Also given, for comparison, is a curve based entirely on the analytic approximation (\ref{['NO']}) (summed over $\ell=2,\ldots,\infty$ plus the exact solutions for $l=0,1$); and a curve showing the mere $\ell=2$ contribution. The latter serves to illustrate the importance of higher $\ell$ contributions. All curves reach a finite value at the horizon.