I sent an email to Norman Levitt, a well-known and widely published mathematics professor at Rutgers University, on Friday, December 6, 2002, 5:40pm., concerning special relativity and the derivation of gamma. Specifically on the meaning of the equation x' = ct'. I showed my equations from my paper and stated that in my opinion the slowing of time implied a larger period, and therefore a larger x. If this is correct, then x' = ct' is not a working equation, and Einstein's derivation of gamma fails. Dr. Levitt placed this question before a "discussion group" he had, on which sat "several well-respected physicists" in the field. The two who responded were Brent Meeker and Peter Fimmel. What follows is my very interesting discussion with these three men. All text is directly quoted from emails, except for commentary added by me which I have put in brackets.

NL: I think you are wrong and the textbooks are right. For one thing, it strikes me that there's some confusion in your exposition between the change-of-co-ordinate transformations relating spatial parameters x and x', and the comparison of lengths, i.e., differences in value between the x (resp x') coordinates of two points. It also might pay to think about how the various parameters are operationalized. [Note this last sentence.]

PF: The email from Miles is the wrong interpretation. Time slowing increases the interval between ticks of the clock. When considering motion, or better still, change of locus in spacetime, x is distance and t is duration. Then as t dilates the duration reduces. Thus, at c, x (distance in the stationary frame observed from the photon's frame) is zero, as is duration in the same frame. The next tick of the clock never happens - it has stopped; no time passes between light's departure and arrival!

BM: Here's the problem. Time slows down in the moving system, so *less* time interval is measured - not more. Draw yourself a spacetime diagram to two "light-clocks", i.e. clocks that tic by reflecting a photon between two mirrors at the same separation in their rest frame. I recommend Lewis Carroll Epstein's, "Relativity Visualized". P.S.: I'm often mentioned in the same sentence as Einstein. You know, "That Brent Meeker is no Einstein."

NL: I see nothing wrong with the answers of Meeker and Fimmel who, I might add, are very sophisticated guys deeply knowledgeable about foundational issues. I suggest you think about whether you are merely hung up on semantics and whether you have really come to grips with the operational and predictive significance of your calculation. The time between ticks of the K' clock, as measured by K , is greater than on his own clock. His twin on K' appears to be aging slower than he, from his vantage. K' time is slowed down relative to his when K is doing the measuring. That's all there is to it.

MM: I think of it this way. We accept length contraction. The measuring rod in K' is accepted to be smaller. Smaller compared to what? Compared to our measuring rods (we being the observers in K). The rod in K' is measured by our rod, and found to be short. Likewise, the clock in K' is said to be slower. Slower compared to what? Our clock, the clock in K, of course. The clock in K' is measured by us, using our clock. If the clock in K' is slow, it has a longer period than us. Even Peter Fimmel agrees with that. So if we measure the clock in K' by our clock, we measure the time between two ticks in K' by the ticks on our clock. According to our clock, the period in K' will have more of our ticks than our own period. In other words, we will measure the period of K' to have more seconds in it. More seconds is a higher number than less seconds. So whether you think of time as the period,or as the number of seconds, t is larger when time dilates.

NL: Semantics again! K is measuring duration in K' time by watching ticks on the K' clock, and comparing it to his own time. That is, he compares t' to t . t , t' are, in essence, just the number of ticks the K-observer observes. Obviously, fewer ticks observed in K' relative to those observed in K , t' smaller than t. End of story.

PF: Miles, if my clock shows a 4 second duration when a passing clock shows a 3 second duration, then the passing clock is going slower than mine. That's all there is to it! Remember that from the passing clock's frame mine is going 3 second when its shows 4 seconds.

MM: Please be patient. Yes, I see how you are stating it. Ticks versus ticks. But may I point out one more thing? I can't agree that this is a question of semantics; it is substantive. For this reason. Einstein says that time is a local measurement. There is only our time. Therefore, you say that what we have as far as data is something like

our clock ----- 4 ticks observed clock -------- 3 ticks.

3 is smaller than 4, end of story.

But those 4 ticks on our clock are seconds, obviously. We have defined one tick as a second. What are the ticks on the observed clock? May we say they are seconds? The duration on the observed clock is 3 seconds? I don't see how, if time is a local measurement. Whose seconds are those? Not ours. It is clear that the "length" of those seconds is not equivalent to ours. To turn them into seconds, and therefore show how they relate to us, we must measure them by our clock. Einstein said that is the only operational definition of time. By our own clock. We want to find the "length" of their tick, their second. Just as we wanted to find the length of the rod. Obviously, to do this we divide 4 by 3. If 4 of our ticks equals three of their ticks, then one of their ticks equals 1.33 of ours. Their tick equals 1.33 seconds. The duration of a tick has increased.

We are disagreeing on how time is measured from t to t'. But if you will agree that this is an operational problem, I will argue strictly in that way. We must use the same operation for discovering that time is slow that we used for discovering that x was short. Our method for asserting time dilation must be equivalent or analogous to our method for asserting that the rods appear short.

Our raw data in one case is rods. Our raw data in the other is ticks. You are saying that we simply add up the ticks. But we do not simply add up the rods. If their rods are shorter, then for a given distance, it will take more of their rods. But more of their rods is a greater number, and contradicts length contraction. The truth is we do not add up their rods. We measure the length of their rods relative to ours, just as we must measure the length of their seconds relative to ours. The "second" is analogous to the "meter". As the meter gets shorter, the second gets longer. Inverse proportion.

PF: No. Their rods and the given distance, being in the same co-moving reference frame, are equally short. So no extra rods are required.

MM: It doesn't matter if the given distance is in their reference frame or ours. In either case, it will take more of their rods to measure it or fill it than it will ours. If we take a given distance in K', it takes more prime rods to measure it than non-prime rods. Likewise, if we take a given distance in K, it takes more prime rods to measure it than non-prime rods. But this doesn't mean that length has expanded. Length contracting is not measured operationally by counting rods, it is measured by the length of the rod.

NL: Well, that's the whole point, isn't it? Duration, length, mass, etc., are not absolute properties of events and matter but, to coin a phrase, "relative" to the framework of observation. Your question is precisely what an understanding of relativity dispels as ill-posed

PF: Quite apart from the technical difficulty of making the measurement, if two one metre rods are congruent when they and the observer and all are at rest; when the rods are in relative motion (in the direction of their length) one will be seen to be longer than the other. The longer one will be the one at rest with respect to the observer. This is what follows from special relativity.

NL: I think the best approach for you right now is to sit somewhere quiet for an hour or so and "t'ink a little t'ink," as Uncle Albert used to say.