I have a problem understanding what proper time really means, well actually, it's the symmetry that confuses me... Suppose for convenience that particle A has a mean life-time of 10 seconds in its own rest frame, as in, when it was created in a laboratory, it decayed after 10 seconds (without it moving in a relativistic speed).
The scenario is: Let S be the laboratory's frame of reference, and S' be the frame of reference stuck to particle A. Suppose at t=0seconds, according to the lab's stationary clock (the clock in the lab), particle A is created and moves at a relativistic speed relative to the lab frame, which has a Gama factor equal to 5 (for convenience). (Particle A moves in X axis direction relative to the lab's frame)
As I understand, when 50 seconds pass according to the lab's clock, particle A will decay, cause relative to the lab's frame, particle A's clock has been moving slowly, and when 50 seconds pass according to the lab's clock, particle A's clock will look like that it had only ticked for 10 seconds relative to an observer (Bob) in the lab's frame. Thus according to the equation (Delta T = Gama * proper time---> 50 = gama* proper time---> proper time for particle A = 50/5 = 10seconds).
So now, we basically know that 10 seconds exactly passed in Particle A's frame of reference. As in, if an observer (Joe) was in S' frame of reference all along, he will say that according to the watch on his wrist, 10 seconds exactly passed before particle A decayed.
However, relative to observer Joe (in the particle's frame of reference), during these 10 seconds, the lab's frame of reference S has been moving with a gama factor of also 5, thus during these 10 seconds, he will see the lab's frame clock moving slowly, and he will conclude that the lab's frame clock (which is moving slowly relative to him) has ticked for 2 seconds only, cause delta T = gama * lab proper time ---> 10 = gama *proper time of lab ---> Proper time of lab = 2seconds.
How is this possible? We know that 50 seconds passed in the lab's frame and not 2 seconds... the situation is supposed to be symmetric, none of the observers could tell whether they were moving or not. I would be glad if someone could explain where I had gone wrong in my explanation. Thanks!