When an unstable particle is accelerated to relativistic speeds, will its decay width increase or decrease? According to the uncertainty relation ${\Delta E\Delta t}\sim{\hbar/2}$, there is an uncertainty in the rest masses of unstable particles which is inversely proportional to their half-lives. When these particles are accelerated to relativistic speeds, their half-lives will be prolonged by a Lorentz factor $\gamma$. As a result, the uncertainty $\Delta E$ should be reduced by $\gamma$. However, according to relativity, the energy of relativistic particles is their rest mass boosted by $\gamma$, which means the $\Delta E$ should be expanded, not shrunk by $\gamma$. So what’s wrong with my understanding?
 A: As you correctly say, the $\Delta E$ is about the rest mass of the particles (or, more accurately, the decay width as explained in this question and its answers), and so the relation $\Delta E\Delta t \geq \frac{\hbar}{2}$ is in the rest frame of a particle. It's not supposed to hold in other frames. You get the life time of a moving particle simply by applying this relation in the rest frame and then applying time dilation to the life time - there is no need to think about any changes in $\Delta E$.
A: Muons reaching the ground from cosmic rays
Is a nice example of this. The lifetime is very short in the reference frame, and even though from a cosmic ray shower they are moving very fast, they would not have time to reach the ground. Since the time is dilated the muon can still be detected at the ground. Or you can also look at it from the perspective the length of the atmosphere is contracted.
If you do the calculation of how many half-lives of the particle you can do it in both the muon reference frame or the ground reference frame and find that the number of half-life of the particle the particle experiences is the same. So you will get the same number of counts in your experiment looking at it from either frame.
This connects to your question since the number of half-lives each reference frame is the same, the number of particles reaching the detector is the same. Since the reference frame that you observing gives you the half life and you can use that used to find the lifetime, and from the lifetime the line width, I think you then end up being consistent.
