Rapid Evaluation of Radiation Boundary Kernels for Time–domain Wave Propagation on Blackholes†

For scalar, electromagnetic, or gravitational wave propagation on a background Schwarzschild blackhole, we describe the exact nonlocal radiation outer boundary conditions (robc) appropriate for a spherical outer boundary of finite radius enclosing the blackhole. Derivation of the robc is based on Laplace and spherical– harmonic transformation of the Regge–Wheeler equation, the pde governing the wave propagation, with the resulting radial ode an incarnation of the confluent Heun equation. For a given angular index l the robc feature integral convolution between a time–domain radiation boundary kernel (tdrk) and each of the corresponding 2l+ 1 spherical–harmonic modes of the radiating wave. The tdrk is the inverse Laplace transform of a frequency–domain radiation kernel (fdrk) which is essentially the logarithmic derivative of the asymptotically outgoing solution to the radial ode. We numerically implement the robc via a rapid algorithm involving approximation of the fdrk by a rational function. Such an approximation is tailored to have relative error ε uniformly along the axis of imaginary Laplace frequency. Theoretically, ε is also a long–time bound on the relative convolution error. Via study of one–dimensional radial evolutions, we demonstrate that the robc capture the phenomena of quasinormal ringing and decay tails. Moreover, carrying out a numerical experiment in which a wave packet strikes the boundary at an angle, we find that the robc yield accurate results in a three–dimensional setting. Our work is a partial generalization to Schwarzschild wave propagation and Heun functions of the methods developed for flatspace wave propagation and Bessel functions by Alpert, Greengard, and Hagstrom (agh), save for one key difference. Whereas agh had the usual armamentarium of analytical results (asymptotics, order recursion relations, bispectrality) for Bessel functions at their disposal, what we need to know about Heun functions must be gathered numerically as relatively less is known about them. Therefore, unlike agh, we are unable to offer an asymptotic analysis of our rapid implementation. † Based on Reference [1].