We investigate the waveforms and quasi-normal modes around Schwarzschild traversable wormholes under different field perturbations, including the scalar field perturbation, the electromagnetic (vector) field perturbation and the gravitational (tensor) field perturbation. By comparing Schwarzschild traversable wormholes with Schwarzschild black holes, we find some unique properties for the former. At first, the perturbation waveform includes echoes and damping oscillations around Schwarzschild traversable wormholes, while it has only the damping waveform around Schwarzschild black holes. Secondly, the difference between adjacent peaks of echoes varies with the mass parameter and the defect parameter in the waveform around Schwarzschild traversable wormholes, while it always keeps constant around Schwarzschild black holes. Thirdly, the ordinary isospectrality between the odd and even parities no longer exists in the quasi-normal modes of gravitational perturbations around Schwarzschild traversable wormholes, but an alternative isospectrality appears. According to these properties, we summarize a scenario for estimating the mass parameter and the defect parameter of Schwarzschild traversable wormholes through the waveforms and quasi-normal modes. Our analyses provide a more profound comprehension of the inherent characteristics of Schwarzschild traversable wormholes.
Comment: v1: 28 pages, 12 figures, 9 tables