It's possible because t and t' mean different things in the two equalities. Consider two inertial coordinate systems S and S' with a common origin. Denote the velocity of S' in S by v. We assume that v>0. Denote the event at the origin by O. Let E be an event on the time axis of S, and denote the time coordinate of E in S by t. Now the time dilation formula tells us that t'=γt. Here t' is the time coordinate of E in S'. Note that E is
not on the time axis of S'.
But when you use the time dilation formula to go from S' to S instead, t' is the time coordinate of an event F on the time axis of S', and t is the time coordinate of F in S. So we're not just doing the calculation we did before "in reverse". It's a calculation that involves a different event.
This is what I said about the apparent contradiction in another thread:
Quote by Fredrik
"B's clock is slow relative to A" appears to contradict "A's clock is slow relative to B". To understand the problem here, it's essential that you understand that these statements are actually defined to mean something different from what they appear to be saying. What they actually mean is this:
"The coordinate system associated with A's motion assigns time coordinates to B's world line that increase faster along B's world line than the numbers displayed by B's clock"
"The coordinate system associated with B's motion assigns time coordinates to A's world line that increase faster along A's world line than the numbers displayed by A's clock"
