Spectral method for computing gravitational quasinormal-mode frequencies of black hole

Since the first direct detection of gravitational waves, the black-hole ringdown phase has been subjected to extensive analyses because of its novel application to probe fundamental physics in the strong-field regime. To thoroughly understand the detected ringdown signals, we need to calculate the gravitational quasinormal-mode frequencies of a black hole under various conditions. However, most existing methods are adequate to study quasinormal modes at sufficient accuracy only for some limited black-hole models. In this talk, I will describe a novel technique to solve the linearized Einstein equations for the quasinormal-mode frequencies of a perturbed black hole using spectral decompositions. Our method works directly with the metric perturbations, and in particular, it does not require simplifying the linearized field equations into master equations through special master functions. Moreover, our method turns the resulting, coupled and partial differential equations into a linear algebra problem, without requiring the decoupling of the radial and angular coordinates. All of this makes this new method extremely powerful in the study of black hole ringdown because, as a consequence, it can be applied to a general black hole background. Applying the method to the Kerr black hole background, we can accurately and efficiently calculate the 022-mode frequency with a relative error in the real and imaginary parts smaller than 1e-3 and we can easily calculate the 033 and 132 modes with similar accuracy.