Wiki wrote:∆t is the time interval between two co-local events (i.e. happening at the same place) for an observer in some inertial frame (e.g. ticks on his clock)
That's not what's happening here, so that's not the equation we should be using. In fact, we are looking for the coordinates of the flashers in the stations frame of reference (call it (x',t')) given their coordinates in the train's frame (x,t). Specifically, we want the time coordinate.
James wrote:... due to the timing devices being all set to read the same at the point of interest with respect to BOTH frames, "Flash time".
Isn't this just what we're trying to decide? You can't stipulate the conclusion. My point is that it is impossible to set them to flash at the same time at when the clocks pass in both frames: the phrase "the same time" is frame-relative.
James S Saint wrote:If you want to use that diagram AFTER the "Flash point"...
The diagram plots the entire exchange, both before and after, equally well. Could you expand on this point?
James S Saint wrote:Note that at T0, P0, the time READINGS are set to be different intentionally between the station and the train items so that at flash time, there will be no dt, difference in time readings.
This is one of two scenarios, which I called "run #2" a few posts ago. It's possible to set to the flashers beforehand so that they will be simultaneous for the station clock, but in that case they are not simultaneous for the train clock.
James S Saint wrote:The relative positioning of the flashers to the train clock; x, x', do not vary thus there is no time dilation or delta time related to those items. Make a diagram showing the relative motion between the two flashers and you should see them on the same x and y angle along with the train clock, merely displaced fore and aft in train clock's x positioning but identical timeline (y).
First, I'd like to suggest that we use the standard notation that the variables in one frame of reference be plain (x,y,z,t) and those in the other be primed (x',y',z',t'). Accepting a convention will help us avoid confusion. In that case, the locations of the flashers x1 and x2 do not vary in the frame of the train, but t1 and t1', and t2 and t2' do vary.
The clocks will have parallel timelines for both frames of reference, however their location in time, their 't' coordinate, will vary between the frames of reference. In one, frame, they will hit y=0 at the same time, i.e. they will both be on the x axis at the same time. But in the 'squished' frame, they will be before and after 0. Here are a couple new diagrams to illustrate this:
This is the train's frame of reference. The clocks are set to be simultaneous with respect to the train in this scenario (again, this decision is arbitrary, there are two scenarios, and each will have a lack of simultaneity in one frame). At time zero (the x axis), the flashers flash, the light travels to the clock at the origin (x=0) and the clock stops.
This is the frame of reference of the station. I left it skewed to illustrate that the skewing doesn't change the facts within it when it is skewed, it is still a perfectly good Cartesian plane. Here, the flashers flash early and late; the red lines indicate the approximate value of t when each flashes (that the red line hits the green circle is due solely to the selection of values for v and x).
These two diagrams are useful just for separating out the two frames of reference. They more clearly convey the information contained in one of my earlier graphs, but in two separate graphs. They also make it easier to see that the skewed coordinate plane is still a full coordinate plane. It is the plane of the stations frame of reference, and must be used to evaluate t relative to the station.