Relativistic Astronomy. III. test of special relativity via Doppler effect

The "Breakthrough Starshot" program is planning to send transrelativistic probes to travel to nearby stellar systems within decades. Since the probe velocity is designed to be a good fraction of the light speed, \citet{zha18} recently proposed that these transrelativistic probes can be used to study astronomical objects and to test special relativity. In this work, we further propose some methods to test special relativity and constrain photon mass using the Doppler effect with the images and spectral features of astronomical objects as observed in the transrelativistic probes. We introduce more general theories to set up the framework of testing special relativity, including a parametric general Doppler effect and Doppler effect with massive photon. We find that by comparing the spectra of a certain astronomical object, one can test Lorentz invariance and constrain photon mass. Besides, using imaging and spectrograph capabilities of transrelativistic probes, one can test time dilation and constrain photon mass. For a transrelativistic probe with velocity $v\sim0.2c$, aperture $D\sim3.5~{\rm cm}$ and spectral resolution $R\sim100$ (or $1000$), we find that the probe velocity uncertainty can be constrained to $\sigma_v\sim0.01c$ (or $0.001c$), and the time dilation factor uncertainty can be constrained to $\Delta\gamma=|\hat\gamma-\gamma|\lesssim0.01$ (or $0.001$), where $\hat\gamma$ is the time dilation factor and $\gamma$ is the Lorentz factor. Meanwhile, the photon mass limit is set to $m_\gamma\lesssim10^{-33}~{\rm g}$, which is slightly lower than the energy of the optical photon.