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The decay of a moving system is studied in case the system is initiallyprepared in a two-mass unstable quantum state. The survival probability$\mathcal{P}_p(t)$ is evaluated over short and long times in the referenceframe where the unstable system moves with constant linear momentum $p$. Themass distribution densities of the two mass states are tailored as power lawswith powers $\alpha_1$ and $\alpha_2$ near the non-vanishing lower bounds$\mu_{0,1}$ and $\mu_{0,2}$ of the mass spectra, respectively. If the powers$\alpha_1$ and $\alpha_2$ differ, the long-time survival probability$\mathcal{P}_p(t)$ exhibits a dominant inverse-power-law decay and isapproximately related to the survival probability at rest $\mathcal{P}_0(t)$ bya time dilation. The corresponding scaling factor $\chi_{p,k}$ reads$\sqrt{1+p^2/\mu_{0,k}^2}$, the power $\alpha_k$ being the lower of the powers$\alpha_1$ and $\alpha_2$. If the two powers coincide and the lower bounds$\mu_{0,1}$ and $\mu_{0,2}$ differ, the scaling relation is lost and dampedoscillations of the survival probability $\mathcal{P}_p(t)$ appear over longtimes. By changing reference frame, the period $T_0$ of the oscillations atrest transforms in the longer period $T_p$ according to a factor which is theweighted mean of the scaling factors of each mass, with non-normalized weights$\mu_{0,1}$ and $\mu_{0,2}$.

Time dilation in the oscillating decay laws of moving two-mass unstable quantum states