Significance of black hole quasinormal modes: A closer look

It is known that approximating the Regge-Wheeler potential with step functions significantly modifies the Schwarzschild black hole quasinormal mode spectrum. Surprisingly, this change in the spectrum has little impact on the ringdown waveform. We examine whether this issue is caused by the jump discontinuities and/or the piecewise constant nature of step functions. We show that replacing the step functions with a continuous piecewise linear function does not qualitatively change the results. However, in contrast to previously published results, we discover that the ringdown waveform can be approximated to arbitrary precision using either step functions or a piecewise linear function. Thus, this approximation process provides a new mathematical tool to calculate the ringdown waveform. In addition, similar to normal modes, the quasinormal modes of the approximate potentials seem to form a complete set that describes the entire time evolution of the ringdown waveform. We also examine smoother approximations to the Regge-Wheeler potential, where the quasinormal modes can be computed exactly, to better understand how different portions of the potential impact various regions of the quasinormal mode spectrum.