What motivated Einstein to start thinking about a General Theory of Relativity?

[PeterDonis]

This diagram is the scenario where the twins T1 (blue) and T2 (green) comove, then part company.

This is what we thought the scenario was, but now we're not sure. You have T1 and T2 separated in the x direction and moving in the x direction; but we think arindamsinha meant to have them separated in the y direction (no initial separation in the x direction) and moving in the x direction.

 

[PeterDonis]

Is that a correct description of the scenario as seen from Frame A?

I'm going to assume that it is and go ahead and post the analysis, since it's pretty simple.

We have the following events (coordinates t, x, y are given relative to Frame A):

#1: $(0, 0, 0)$ T1 starts the experiment, moving in the x direction at velocity v.

#2: $(0, 0, 1)$ T2 starts the experiment, moving in the x direction at velocity v.

#3: $(t_1, v t_1, 0)$ T1 stops moving.

#4: $(t_2, v t_2, 1)$ T2 stops moving and ends the experiment.

#5: $(t_2, v t_1, 0)$ T1 ends the experiment.

We have, by hypothesis, $t_2 > t_1$, and for convenience I will define $\delta t = t_2 - t_1$.

The proper times in Frame A are then:

$$\tau_1 = \frac{t_1}{\gamma} + \left( t_2 - t_1 \right) = \frac{t_1}{\gamma} + \delta t$$

$$\tau_2 = \frac{t_2}{\gamma} = \frac{t_1 + \delta t}{\gamma}$$

This makes it obvious that $\tau_1 > \tau_2$.

Now let's look at things in Frame B. Here are the event coordinates t', x', y' in that frame, obtained by Lorentz transforming the coordinates given above (note that we have assumed the origins of both frames are the same, at event #1):

#1: $(0, 0, 0)$ T1 starts moving in the x direction at velocity v.

#2: $(0, 0, 1)$ T2 starts moving in the x direction at velocity v.

#3: $(t_1 / \gamma, 0, 0)$ T1 stops moving.

#4: $(t_2 / \gamma, 0, 1)$ T2 stops moving.

#5: $(t_1 / \gamma + \gamma \delta t, - \gamma v \delta t, 0)$ T1 ends the experiment.

The proper times in this frame are then:

$$\tau_1 = \frac{t_1}{\gamma} + \frac{t_1 / \gamma + \gamma \delta t - t_1 / \gamma}{\gamma} = \frac{t_1}{\gamma} + \delta t$$

$$\tau_2 = \frac{t_2}{\gamma} = \frac{t_1 + \delta t}{\gamma}$$

In other words, the proper times are the same in both frames, as they should be. The key thing to note, of course, is that in Frame B, event #5 happens *later* than event #4, and the additional coordinate time that this adds to T1's "moving" segment in that frame more than compensates for the fact that T1 is moving while T2 is at rest. This is basically the same resolution as the previous scenario; the y coordinate drops out of the analysis since all the motion is in the x direction, but there is still separation in the x direction at the end of the experiment (even though there isn't at the start), so relativity of simultaneity still comes into play in making event #5 later than event #4 in Frame B.

 

[D H]

Thread locked pending cleanup.

Metz114 et al: Please remember to use the report button to let the mentors know about nonsense such as that which you highlighted.
OK. Cleanup complete. I deleted 50 posts. That's a bit much, perhaps too much. Those posts are still here; I soft-deleted them. Let me know if there's anything that you members strongly feel needs to be restored.