Simulating gravitational waves passing through the spacetime of a black hole

TL;DR

This paper uses time-domain numerical simulations to study how gravitational waves interact with Schwarzschild black hole spacetime, revealing complex wave behaviors and back-scattering effects that influence observed waveforms.

Contribution

The work introduces a covariant perturbation formalism that avoids horizon boundary conditions and regularity issues, enabling detailed simulations of gravitational wave propagation through black hole spacetime.

Findings

Wave zones are highly twisted by the black hole.

Continuous waves cannot be fully sheltered, unlike geometric optics predictions.

Back-scattering lengthens the waveform during merger and ringdown phases.

Abstract

We investigate how GWs pass through the spacetime of a Schwarzschild black hole using time-domain numerical simulations. Our work is based on the perturbed 3+1 Einstein's equations up to the linear order. We show explicitly that our perturbation equations are covariant under infinitesimal coordinate transformations. Then we solve a symmetric second-order hyperbolic wave equation with a spatially varying wave speed. As the wave speed in our wave equation vanishes at the horizon, our formalism can naturally avoid boundary conditions at the horizon. Our formalism also does not contain coordinate singularities and, therefore, does not need regularity conditions. Then, based on our code, we simulate both finite and continuous initially plane-fronted wave trains passing through the Schwarzschild black hole. We find that for the finite wave train, the wave zone of GWs is wildly twisted by the black hole. While for the continuous wave train, unlike geometric optics, GWs can not be sheltered by the back hole. A strong beam and an interference pattern appear behind the black hole along the optical axis. Moreover, we find that the back-scattering due to the interaction between GWs and the background curvature is strongly dependent on the direction of the propagation of the trailing wavefront relative to the black hole. Finally, for a realistic input waveform generated by binary black holes, we find that the lensed waveform in the merger and ringdown phases is much longer than that of the input waveform due to the effect of back-scattering.