Revisiting the quasinormal modes of the Schwarzschild black hole: Numerical analysis

We revisit the problem of calculating the quasinormal modes of spin 0, 1/2, 1, 3/2, 2, and spin 5/2 fields in the asymptotically flat Schwarzschild black hole spacetime. Our aim is to investigate the problem from the numerical point of view, by comparing some numerical methods available in the literature and still not applied for solving the eigenvalue problems arising from the perturbation equations in the Schwarzschild black hole spacetime. We focus on the pseudo-spectral and the asymptotic iteration methods. These numerical methods are tested against the available results in the literature, and confronting the precision between each other. Besides testing the different numerical methods, we calculate higher overtones quasinormal frequencies for all the investigated perturbation fields in comparison with the known results. In particular, we obtain purely imaginary frequencies for spin 1/2 and 3/2 fields that are in agreement with analytic results reported previously in the literature. The purely imaginary frequencies for the spin 1/2 field are exactly the same as the frequencies obtained for the spin 3/2 field. In turn, the quasinormal frequencies for the spin 5/2 perturbation field are calculated for the very first time, and purely imaginary frequencies are found also in this case. We conclude that both methods provide accurate results and they complement each other.