No, no, no, and no again.
Jack, look at the premises we agreed on:
- For M at rest, two points A and B are each a distance d from M.
- For M at rest, M' moves past M at velocity v.
- For M at rest, when M' and M are co-located, lightning strikes at A and at B.
- For M' at rest, two points A' and B' are each a distance d/γ from M'.
- For M' at rest, when M and M' are co-located, A is co-located with A' and B is co-located with B'.
Your logic hasn't progress past this point, because you keep introducing unproven premises. You need to use the agreed premises to prove at least one of the following:
- For M' at rest, the observers B' and A' are co-located at the lightning strikes at A and B.
- For M' at rest, when M' and M are co-located, A and B are struck simultaneously.
- For M' at rest, when M' and M are co-located, A' and B' are struck simultaneously.
In each subsequent argument you have introduced at least one of these premises with either no proof, or with a proof relying one one of the other two unproven premises.
Please, Jack, show me that you can construct a rigorous proof. For each new statement, list the premises from which it stems. Don't introduce any new statements without that support.
Here it is again.
Now, the reductio ad absurdum argument is applied by assuming Einstein’s conclusion that M' sees the light from B prior to A from the view of M' as stationary.
To support Einstein’s conclusion that M' sees the light from B prior to A one of the following two possibilities must be true:
1. M' moves toward the lightning strike at B closing the distance for light to travel relative to the strike at A.
2. The strike at B occurs prior to the strike at A in the time coordinates of M'.
Possibility 1
Since M' is stationary, it is not moving. A, B and M are moving relative to M'. Sure, B closes the distance to M' as the light travels toward M' but this has nothing to do with the distance light traveled in the frame of M'. The distance light traveled would be measured from B' to M' which is the distance from the light emission point in the frame to the strike point in the coordinates of the frame. That distance is d/γ. The same logic applies to A' and M' where that distance is also d/γ. This logic is supported by length contraction under SR. In particular, if a moving rod is of length d and a stationary rod is of length d/γ, then from the view of the stationary frame, the two ends of the rods can be simultaneously co-located. Hence, possibility one is not viable when taking M' as stationary since B' and A' are equidistant from M and M' when they are co-located and the lightning strikes.
Possibility 2
Since, there are the observers B' and A' co-located at the lightning strikes at A and B, it is impossible there is any disagreement between the frames as to whether light is moving along the x-axis or not. Hence, for example, if B' claims lightning just struck, B will make the same claim as well. So, it cannot be claimed the lightning appears for one frame at some location while a co-located observer claims light is not at that location. Therefore, perhaps the time on the clock of B' will show an earlier time than the clock of A' for the light strike and this explains it. In other words, the light emitted from B' before it emitted from A'.
So, let tB' be the time of the lightning strike at B' and tA' be the time of the strike at A'. Therefore, tB' < tA'. By the experiment, B' is located a distance d/γ from M' at the time of the strike and is co-located with B. At that strike at B', as required by the experiment, M and M' are co-located. But, that also implies M moves a distance (tA' - tB')v between the two lightning strikes in the time of M' if they indeed occur at different times in the M' frame. Thus, when the strike at A/A' occurs, M has moved a distance (tA' - tB')v, hence, M and M' are no longer co-located at the time of the strike at A/A'. So, possibility 2 is not viable.
Therefore, using this simple argument, the stationary frame of M' will be struck by two simultaneous equidistant lightning strikes and M will be struck by the lightning from A prior to B.