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1977ub said:2) Actually I take it back. Since the hurling process takes time, and they are both moving wrt rest frame by the end of the boulder hurling process, I expect the forward person to finish hurling later that the back individual as seen from the rest frame.
Well, you can actually calculate it. For simplicity, let F = the rest frame before hurling the boulder. F' = the rest frame afterward. Let v = the relative velocity between the frames. Let T = the time required to hurl the boulder, as measured in frame F. Let D = the distance traveled while hurling the boulder, as measured in frame F. Let L = the distance between people, as measured in frame F.
Then identify a number of events:
- e_1: the rear person starts to hurl the boulder.
- e_2: the rear person finishes.
- e_3: the front person starts to hurl the boulder.
- e_4: the front person finishes.
Our assumptions are that the corresponding actions are synchronized in frame F. So we have the coordinates for these events in frame F:
- x_1 = 0,\ \ t_1 = 0
- x_2 = D,\ \ t_2 = T
- x_3 = L, \ \ t_3 = 0
- x_4 = L+D, \ \ t_4 = T
Now use the Lorentz transforms to see the coordinates in frame F':
- x_1' = 0, \ \ t_1' = 0
- x_2' = \gamma (D - vT), \ \ t_2' = \gamma (T - \dfrac{vD}{c^2})
- x_3' = \gamma L,\ \ t_3' = -\gamma \dfrac{vL}{c^2}
- x_4' = \gamma (L+D - vT), \ \ t_4' = \gamma (T - \dfrac{v(L+D)}{c^2})
While in frame F, t_1 = t_3 and t_2 = t_4, in frame F', the order of events is: t_3', then t_1' or t_4' and then finally t_2'. (The order of t_1' and t_4' depends on the size of L, D, and T.)
So what things look like in frame F' is this:
- Initially, both people are traveling at speed v in the -x direction.
- t' = t_3': the front person throws a boulder. His speed in the -x direction starts slowing down. But the rear person continues to travel at speed v in the -x direction.
- t' = t_1': the rear person starts to throw a boulder, as well.
- t' = t_4': the front person comes to rest.
- t' = t_2': the rear person comes to rest.
So between times t_3' and t_2', the front person is traveling slower than the rear person, so experiences less time dilation. So when they come to rest, the front clock will have gained more time than the rear clock, and also, the distance between the rear and the front will have increased.