Hi everyone.

This is a modification to the train-embankment experiment. I am hoping someone can solve this problem.

Assume when M and M' are co-located, lightning strikes their common location. There is also an observer B' in the negative direction of M’.

After some time elapses in each frame, B' and M become co-located.

Considering only the negative and positive x directions, the goal is to determine all of the special relativity truths for the lightning light beams at the instant M and B' are co-located.

1) The light beams are equidistant from M in the M frame. So, label these locations as (d,0,0) and (-d,0,0).

2) The light beams are equidistant from M’ in the M’ frame. So, label these as (d’,0,0) and (-d’,0,0)

To make it simple, label the common frame event of B’ and M being co-located as P. So, P is true if and only if B’ and M are co-located. Below are additional truths.

3) If P is true in M, then P is true in M’.

4) If P is true in M’, then P is true in M.

5) Therefore, if P is false in M’ then P is false in M.

6) Also, if P is false in M then P is false in M’.

Next, use the lorentz transformations on (d,0,0). Note, for this case t=d/c. Then , so . Now assume . Then, P is false for M' since P is only true for the M’ frame if the light beam is at (d’,0,0). But, P is true for M iff P is true for M’. So, that means P is false for M. But, we used (d,0,0) for the lorentz transforms which means P is true for M. So, if then P is true for M and false for M, which is a contradiction.

The same argument holds for applying lorentz transformations to (-d,0,0) with . It ends up as above if then P is true for M and false for M.

So, there is only one possibility left, and . Then, from above, since and , then and . So, and . Combine the two and you get . Simplification leads to , which is a contradiction.

The same conditions result if you apply the lorentz transformations to (d’,0,0) and (-d’,0,0).

So, it looks like nothing the lorentz transformations output can be true.

Thus, it seems all the truths of special relativity can’t hold true for this experiment when P is true.

Can anyone make all the truths of special relativity hold true for this thought experiment?