Echoes and quasi-normal modes of perturbations around Schwarzchild traversable wormholes

We investigate the waveforms and quasi-normal modes around Schwarzschild traversable wormholes under different field perturbations, including the scalar field, the electromagnetic (vector) field, and the axial gravitational (tensor) field perturbations. Our results indicate that under the influence of the matter at the throat of the wormhole, a Dirac $\delta$-function distribution of matter appears in the effective potentials of the scalar and axial gravitational perturbations and it affects the propagation of these two types of perturbations in spacetime. However, the matter at the throat has no influence on the propagation of electromagnetic perturbations. Furthermore, we quantify the impact of throat matter on both the perturbation waveforms and quasi-normal modes for all three field types. Through comparative studies between Schwarzschild traversable wormholes and Schwarzschild black holes, we identify two distinct features. Firstly, the perturbation waveforms exhibit echoes and damping oscillations around wormholes, whereas they solely display damping oscillations around black holes. Secondly, the difference between the adjacent peaks varies with the mass parameter and the throat radial coordinate in the waveform around Schwarzschild traversable wormholes, while a constant peak spacing occurs, which is determined solely by mass, in the waveform around Schwarzschild black holes. Based on these findings, we propose a framework to estimate the mass parameter and throat radial coordinate of Schwarzschild traversable wormholes through waveforms and quasi-normal modes. Our analyses provide a more profound comprehension of the inherent characteristics of Schwarzschild traversable wormholes.