# Help at dismissing a time dilation "paradox" [duplicate]

If Observer A moves at speed $$v$$ respect to observer B, then there is a time dilation given by

$$t_B=t_A\gamma$$

as is known from special relativity. Here, $$t_A$$ is the time interval measured by A and $$t_B$$ is the time interval measured by B. However, a friend of mine asked me about the inverse case, what if we take the frame of reference in which A is stationary and B is moving? in that case, $$t_A=t_B\gamma$$

In this case, A's time should be dilated with respect to B's, since B is moving away, so we get conflicting results. I know that this case is somehow erroneous, but I am now sure how to justify it. I think that it has something to do with the fact that in the first equation, $$t_A$$ is the proper time, and that is the one that is dilated, but I have been unsuccesful at developing a convincing enough argument. How can I solve this "paradox"?

• I wrote an answer a while back about essentially the same misconception, but dealing with length contraction instead of time dilation, you should be able to figure out the answer using the arguments there. Commented May 29, 2020 at 23:12
• Does this answer your question? How can time dilation be symmetric?
– user87745
Commented May 29, 2020 at 23:16