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The "Breakthrough Starshot" program is planning to send transrelativisticprobes to travel to nearby stellar systems within decades. Since the probevelocity is designed to be a good fraction of the light speed, \citet{zha18}recently proposed that these transrelativistic probes can be used to studyastronomical objects and to test special relativity. In this work, we furtherpropose some methods to test special relativity and constrain photon mass usingthe Doppler effect with the images and spectral features of astronomicalobjects as observed in the transrelativistic probes. We introduce more generaltheories to set up the framework of testing special relativity, including aparametric general Doppler effect and Doppler effect with massive photon. Wefind that by comparing the spectra of a certain astronomical object, one cantest Lorentz invariance and constrain photon mass. Besides, using imaging andspectrograph capabilities of transrelativistic probes, one can test timedilation 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 canbe constrained to $\Delta\gamma=|\hat\gamma-\gamma|\lesssim0.01$ (or $0.001$),where $\hat\gamma$ is the time dilation factor and $\gamma$ is the Lorentzfactor. Meanwhile, the photon mass limit is set to$m_\gamma\lesssim10^{-33}~{\rm g}$, which is slightly lower than the energy ofthe optical photon.

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