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

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 $\mathrm{\ell }$-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 $\zeta -$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 $\mathrm{\ell }$.