which of _my_ states corresponds to the state of the finnish line (who apparently has the most to do with this assertion) observing the first runner crossing it ; which of _your_ states corresponds to that state of the finnish line, etc. Having obtained this calibration, we might go on to derive pairwise distance from that (e.g. through Einstein's associated distance definition); we might find that one or the other observer pair moved wrt. each other in the course of the run (most of all each runner wrt. the finnish line , perhaps). Further, having obtained those coordinate relations and using them as constraints in a problem of variational calculus, we could calculate the most probable potential in the region of this experiment, as it is sampled by each observer individually (e.g., given enough measured constraints, we might find that the track let's the runners bounce off throughout the race, etc.). At least that's the procedure being conducted in QFT. The Lorentz equations ... a.k.a. Lorentz transformations , IIUC ... are an approximation. They do not give an absolute depiction of reality. An approximation wrt. which other measured quantities? How, if not through the Einstein procedures, do you suggest to obtain a depiction of reality which (being reality) should be reproducible/understandable by all others? Frank,     I have no idea what you are doing with four dimensions.  I have no way of even visualizing it.  I use three dimensions.  One thing that is obvious using three dimensions is that  four dimensions have you overlooking some very obvious facts.    First of all, all clocks in a _frame_ of reference will agree with one another.  Now you say that from another _frame_ of reference they do not using the Lorentz equations.  What difference does that make?    The value of t is measured from the _frame_ of reference in which it exists , not from another _frame_ of reference.  Any clock in a _frame_ of reference can be put next to an event and will show the time of that event, and, secondly, will agree with all other clocks in that _frame_ of reference, assuming all clocks are operating correctly. Robert B. Winn

What do they read from each _frame_ of reference when the astronaut reaches the second mark? Since t' denotes a state/time of the second mark'/clock', the corresponding t' = the astronaut reaches the second mark' . Which (if any) state/time/reading t of the first mark/clock corresponds to that, I don't know a priori. Again, which calibration procedure would you suggest in order to determine that experimentally? Regards,    Frank  W ~@) R

and t' = 0 , i.e. one of the readings/states at the _other_ clock'/mark'. Since t denotes a state/time of the first mark/clock, apparently the states/times t = 0 and t = the astronaut reaches the first mark are the same states/times; we might as well write t = 0_the astronaut reaches the first mark . The point being: it is not abvious how to do arithmetic with readings/states/times. Consequently, secondly: What do you mean by t' when the astronaut reaches the first mark ?, IOW, which procedure do you suggest to determine whether or not the state/time t = 0_the astronaut reaches the first mark of the first clock/mark corresponds to the state/time t' = 0 of the second clock'/mark' ? (In order to discard the trials in which this were not found, or in order to adjust otherwise.) Hint: Recall Einstein's train and lightning example to which you referred earlier. What do they read from each _frame_ of reference when the astronaut reaches the second mark? Since t' denotes a state/time of the second mark'/clock', the corresponding t' = the astronaut reaches the second mark' . Which (if any) state/time/reading t of the first mark/clock corresponds to that, I don't know a priori. Again, which calibration procedure would you suggest in order to determine that experimentally? Frank,  You are making this too complicated.   Event 1  ,the astronaut reaches the first mark Event 2 the astronaut reaches the second mark   Now you are saying that no other events can be proven to happen at the same time these events happen.  I went all through this with some other scientists last summer with Einstein's train and lightning problem.  I understand what you are saying, but it is wrong.    Let me show you what I showed them, and maybe it will clarify this for you. Here is what scientists are saying happens in the train and lightning problem.    We run a train by an observer by the track in the manner in which Einstein described.  When an observer on the train at the middle is opposite an observer on the ground, theoretically, the observers could reach out and touch hands, although, at the speeds described, if they did so, they would both lose the hands in question.  So at the exact instant that the observers touch hands, lightning strikes the front and rear of the train simultaneously in both _frame_s of reference.  Now, according to the Lorentz equations, this will take four bolts of lightning, so let us consider where the four bolts of lightning hit.   We stipulate that all four bolts of lightning hit the front and rear of the train, each bolt of lightning leaving a mark on the railroad track.  Now we look at where the Lorentz equations say the marks on the track are. FROM THE _frame_ OF REFERENCE OF THE TRACK _A________B_____O_____C__________D 1. The rear of the train reaches A and lightning strikes the rear of the train. 2.  The front of the train reaches C and the rear of the train reaches B, and lightning strikes the front and rear of the train simultaneously. 3. The front of the train reaches D and lightning strikes the front of the train. If you need the distances for these points, I can give them to you. _frame_ OF REFERENCE OF THE TRAIN _A_______B_________C_________D___ 1.  The front of the train reaches C and lightning strikes the front of the train. 2.  The front of the train reaches D and the rear of the train reaches A, and lightning strikes the front and rear of the train. 3.  The rear of the train reaches B, and lightning strikes the rear of the train.    Now I do not believe that this is the way it happens, but this is what scientists say happens.  This is what the Lorentz equations show.    What I believe happens is that there are only two bolts of lightning which strike in the following manner: __A_______B____o_____C_______D____ The  observer on the train reaches the position of observer o on the ground, the front of the train is at C, and the rear of the train is a B, lightning strikes the front and rear of the train simultaneously in both _frame_s of reference.  The distance from B to C is the rest length of the train.    This is what I believe happens.  If you can find an error in what I say, go ahead and show it. Robert B. Winn

lightning strikes the front and rear of the train simultaneously in both _frame_s of reference. cannot be realized at all (unless front and rear are the same and the trainlength is zero; or the _frame_s are same throughout the signal exchange and therefore cannot move wrt. each other). Best regards,  Frank  W ~@) R

cannot be realized at all (unless front and rear are the same and the trainlength is zero; or the _frame_s are same throughout the signal exchange and therefore cannot move wrt. each other). Now, you can consider the problem any way you want to as long as you concede one thing:  If we put a row of clocks along the ground at intervals of one foot for a mile, in the _frame_ of reference of the track, the clocks all agree with one another.  If we put a row of clocks down the middle of the train from front to back at intervals of one foot, in the _frame_ of reference of the train, all of these clocks agree with one another.    That is all I need to prove my equations. Robert B. Winn

As I have said, I am a welder, not a mathematician. But I'll assume that you can count nevertheless.