Quasinormal Modes of Black Holes: Efficient and Highly Accurate Calculations with Recurrence-Based Methods

Abstract: We discuss new recurrence-based methods for calculating the complex frequencies of the quasinormal modes of black holes. These methods are based on the Frobenius series solutions of the differential equation describing the linearized radial perturbations. Within the general method, we propose two approaches: the first involves calculating the series coefficients, while the second employs generalized continued fractions. Moreover, as a consequence of this analysis, we present a computationally efficient and convenient method that uses double convergence acceleration, consisting of the application of the Wynn algorithm to the approximants obtained from the Hill determinants, with the Leaver-Nollert-Zhidenko-like tail approximations taken into account. The latter is particularly important for stabilizing and enabling the calculations of modes with small real parts as well as higher overtones. The method demonstrates exceptionally high accuracy. We emphasize that Gaussian elimination is unnecessary in all of these calculations. We consider $D$-dimensional ($3<D<10$) Schwarzschild-Tangherlini black holes as concrete examples. Specifically, we calculate the quasinormal modes of the $(2+1)$-dimensional acoustic black hole (which is closely related to the five-dimensional Schwarzschild-Tangherlini black holes), the electromagnetic-vector modes of the six-dimensional black holes and the scalar (gravitational tensor) modes in the seven-dimensional case. We believe that the methods presented here are applicable beyond the examples shown, also outside the domain of the black hole physics.