Solving the Teukolsky Equation with spectral methods

General Relativity is to this date the best theory of gravity we have and probing the very nature of Black Holes is possible through events such as Gravitational Waves. In this work we describe our time domain solver of the 1+1D homogeneous Teukolsky equation, an equation that encodes the physics of linear perturbations in Kerr spacetime. We solve it in hyperboloidal slices to be able to extract the signals at $\mathcal{I^+}$ without the need for extrapolation.

To do so we developed a code that employs two different kinds of spectral methods: a known pseudo-spectral scheme with collocation and a novel fully spectral scheme without collocation. With this second approach we obtain more accurate late-time power-law tails, as well as their decay rates, and we have achieved results for negative spin-weights without the use of quad precision that many suggest is necessary.


Our results are in agreement with analytical and empirical generalizations of Price’s Law for Kerr Black Holes. Then we proceed with a study of the convergence properties of both spectral schemes and of the time-symmetric integrator we use. This integrator is implicit and it shows advantages relative to the more usual explicit integrators mainly in terms of efficiency, but possibly also in terms of accuracy whenever we introduce source terms into the equation. Thus, this work marks a first step to solve the full Teukolsky system and thus evolve Extreme-Mass Ratio Inspirals simulations accurately and efficiently.