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An Interesting Exchange
by Miles Mathis
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 pretended to be a graduate student in physics, knowing that otherwise I could not hope to engage him in any conversation at all. What was required was to be an insider—and a lower-ranking insider. Only then might I be able to ask a question. 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 t. 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.
MM: I am a graduate student in the physics department at UMASS, Amherst, and I
have come to you because my professors are not able to answer a question I
have about the derivation of gamma. It is a mathematical question, one
I thought would be easy to answer, but I have been told to basically accept
what I am reading without trying to understand it. This seems to me to be
unscientific, to say the least. My professors get impatient, treat my
questions as somehow insubordinate, and we get nowhere. I have read some
your books and many of your articles, and I admire your stance against
pseudoscience and deconstruction and the like. That is why I have come to
you. You are one of the few people I have even heard of who seems to know
what he is talking about.
We were doing a derivation of gamma using tensor calculus, and I felt that
some of the suppositions were fuzzy. So I decided to look into the problem
little further. The way the problem was stated made me think that the math
we were using was a bit overcomplex. It seemed to me to be a problem that
was amenable to a simple algebraic solution. I always prefer to use the simplest math that will solve the problem, since this often makes a problem
of applied mathematics more transparent conceptually. Special Relavitity
already dense enough conceptually. Tensor calculus made it even moreso, it
seemed to me. I discovered that Einstein had come to the same conclusion,
since he threw out the tensor calculus derivation he had used at Princeton in
favor of the algebra derivation he had originally used in 1905 (updated slightly in form). He used this albegraic derivation in the book Relativity. Then I discovered that many
physics textbooks had used, and still were using, a simple algebraic
derivation of gamma, using Einstein's initial equations and his other
givens.... [Here I copied the derivation, and my critique of x' = ct', etc.]
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 clock1. 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! [Notice that Fimmel admits that the interval between ticks increases—which is the period by definition]
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." [Meeker contradicts Fimmel, saying that the interval is less]
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: Thank for your comments and for your consideration of my question. All three of you are telling me that time dilation implies a smaller t. But I am not convinced. 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.
[The most important thing to glean from this exchange is that, operationally, time must be thought of as measurement of length. Just as we measure the rod to be shorter, we measure the period to be longer. We are measuring the length of the period, in seconds. A "second" is the length of the period in K. That is the definition of a second. If the period is longer in K', it cannot be a second. A second is a given length. In comparing the operations, the length of the period is analogous to the length of the rod. Therefore x and t are in inverse proportion.]
[As you might imagine, if you know me at all, I have one final email to send to NL. But I am holding it for a couple of weeks, to see if anyone in this "discussion group" has some sort of revelation. I don't want to burn my bridge until I am sure it will hold no weight at all. Meeker and Fimmel, at least, seem to be friendly. Their statements here do not lend me much hope for their ability to analyze this problem, but you never know.
Fimmel contradicts himself in his first email. He says, "Time slowing increases the interval between ticks of the clock." Once sentence later, he says this: "Then as t dilates the duration reduces." For this to make any sense at all, he must show that the interval and the duration are opposites, but he never attempts to do this. I think it is clear that they are the same thing. At the end of the same email he says, "The next tick of the clock never happens—it has stopped; no time passes between light's departure and arrival!" I have no idea what this means. It is false in any possible context. If light has a finite speed, light cannot depart and arrive without time passing, in any co-ordinate system. In his last email, Fimmel is even more confused and confusing. He says, "If two one meter 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." Here he has the longer rod both at rest and moving relative to the observer. It would be funny if it weren't pandemic. The whole point is that the longer rod is not moving relative to the observer. If both rods are in relative motion, as he says, then both of them will appear shorter.
Levitt attempts to save face simply by not arguing past a certain point. Once he begins to lose, he reverts to posturing and browbeating. His method of discourse left much to be desired from the beginning, with his condescending "that's all there is to it," and "End of story." And apparently every word but his own is a word that falls under the heading of semantics. Notice that he asks for an operational analysis of the problem, then not only fails to provide one, but fails to follow the straightforward one of mine (or to show any fault in it). This might be passed over as only a symptom of the final stages of smugness if it weren't for the fact that we are talking about the failure to follow a simple algebraic exposition by one of the self-professed masters of math in the world. It seems to me that this conversation can be taken as proof that ones knowledge of higher mathematics says nothing about ones ability to apply it. If Levitt cannot do an operational analysis of high-school algebra, what does that say of his analysis of his own esoteric equations?
And this leads to another, more widely drawn question: is the ever more arcane mathematics of contemporary physics a true advance, or is it in fact an obscuring of the concepts of physics, purposefully, in order hide them from view? Billions are being spent on research that depends upon the equations of relativity and quantum mechanics. It is just a coincidence that no theoretical work is being done in either field? Is it coincidence that the equations of relativity have not been questioned in a hundred years? It is a coincidence that gamma has survived untouched in all that time despite being a mathematical mirage, a false ghost? And that its uncovering had to come from someone so far outside of academia and science as to be a ghost himself? It will make a great comedy someday, but for the present, it is a tragedy beyond the imagination of Sophocles or Shakespeare.]
[I will add my response to NL when it is delivered to him.]
Brent Meeker replied on December 11 to my initial comments:
BM: Contemplate the attached graphic. It shows the world lines of three identical clocks. Each clock consists of two mirrors exactly one light-year apart with a photon that bounces back and forth between them (thus ticking). One (ours) is stationary relative to us. The others are moving to the right at speeds of 0.5c and 0.9c respectively. Note the Lorentz contraction of the distance between the mirrors. Note that the photons all travel at the same speed (slope=1). Note the number of ticks on the other clocks during the twenty years I've graphed. The tick marks along the mirror world lines are 1/10 of year apart. Note how the time dilation is measured by the photon of the clock. Is it clear now?
MM: Thank you for the diagram. I am aware that there will be fewer ticks with increased velocity. But my discussion with Drs. Fimmel and Levitt progressed past the number of ticks. I am trying to get someone to see that time is not measured by number of ticks. Not in our coordinate system or in any other. If you will read my brief comments again more closely, you may see my point. Let me be very clear: I am not arguing against time dilation. I believe in it. I am not arguing against relativity. I find it unassailable, as a theory. I am arguing against the interpretation of t as number of ticks. I still maintain that t must be understood as the period, not the number of ticks. Notice that in your diagram, both t and x may be thought of as distance. At velocity, there are fewer ticks, as you say. AND there is more distance between ticks. It is this "distance" that is t, not the number of ticks. If you will follow my operational analysis in my last email of the way we measure x and t in real life, in problems of relativity, you will see that the length of the rod is analogous to the length of period. Not to the number of ticks. Time is dependent on the length of the second, not on the number of seconds. I have thought through all of the things you and Dr. Levitt have asked me to. I have offered operational analyses and cogent argments. I am not convinced that any of you have thought through my comments, since I have received no feedback on my specific claims, just a repeat of the time is ticks claim. My thoughts have been dismissed out of hand, I feel. I understand that you all think that you are busy men, and that I am some errant knight, lost in the fields; but I assure you that I would not bother you with stupid claims and empty bluster. I have been working on this specific problem for several years, on and off. I didn't just chime in from some high-school science class or gathering of cranks on the web. Believe me, it would be so much easier to accept all the equations as is. I would love to be convinced that you are right. Unfortunately, I do not find the time-as-ticks argument compelling. Operationally and as a definition of time, it makes no sense. Einstein did not think of time as ticks of the clock. He thought of time as a measurement of the "time" between ticks of the clock. It is my contention that he simply borrowed the wrong math of Lorentz and never realized it. Correcting this mistake, even if I am right, does not contradict Relativity, but it is an important finding nonetheless, and I wish I could get someone to consider it seriously. I promise it is worth your time.
BM: But how would you measure time between ticks of the clock—it would require another, faster ticking clock. In fact that's why I pointed out in the diagram that the tick marks on the mirror world lines were 1/10 of a year apart. They would be created by a local, smaller light-clock at the mirror. Light-clocks keep physical time because we already know from independent experiments that the speed of light is the same in all reference frames—something you used in deriving the Lorentz transform. You apparently imagine some metaphysical time that is not tied to operational measurement. You refer to the t parameter along the vertical axis in my diagram when you say the the moving clocks have more 'time' between ticks. But the only operational significance of the vertical axis is that it is the time marked by the stationary clock. If you can't grasp the distinction between mathematical parameters and operationally measured values, you'd better drop physics and go into mathematics, philosophy, or theology. [And in response to my claim that it was worth his time, he said this:] No it's not. Nor yours either.
MM: You say,"But how would you measure time between ticks of the clock—it would require another, faster ticking clock?" You measure the time of the moving clock by the stationary clock. The stationary clock is the faster moving clock in that instance. You do not have to measure the time between ticks on the stationary clock, since that measurement is DEFINED as a second. But one second is the time between two ticks. One second is not a tick. Then you say, "But the only operational significance of the vertical axis is that it is the time marked by the stationary clock." That is precisely my point, which you continue to miss in favor of ad hominem remarks. The length of the period of the moving clock must be measured by the stationary clock, else it remains metaphysical. It remains uninterpreted data.
I did study philosophy, as a matter of fact, and still do. Maybe that is why I am able to see the logical significance of these equations and charts, and not just their intended use as a tool of indoctrination. I don't understand the pleasure you and Levitt seem to get from being condescending. Nor do I understand the point of telling another man what is worth his while. If you cannot fathom my argument, say so, and we will be done with it. Or I will. We are done with it. You were right about one thing, you are no Einstein. The sad part is you are not even useful to an Einstein. History will say, "That Brent Meeker, he was just another stone in the road." P.S. I am not really a student at Umass. This was all a test.
[Amazingly, the emails continued after that.]
BM: But then one second is also the time between two ticks of the moving clocks. Remember they are identical physical mechanisms. There is no absolute motion so any one of them can be take as the 'stationary' clock.
MM: But once one of them has been chosen as the stationary clock, all measurements are made from there. That becomes your "at rest" system, and all other systems are relative systems. You must choose a point of view. You cannot argue from two systems at once, which is what you are trying to do. Only the period in the "at rest" system has a period defined as one second. All other periods are measured, not defined.]
BM: This assumes that 'stationary' is an objective property—but it isn't. If you wish to assume that it is, you can, like Lorentz assume that the contraction of rulers and slowing of clocks moving relative you is a real physical effect that is mysteriously related to your person. It means that all the clocks on Earth slow down as you fly from Los Angeles to New York.
MM: No, "stationary" is an operational fact. The measurer must measure himself as stationary, and all measurement must be made from a system thought of or defined as stationary. Clocks on earth slow down as you fly from Los Angeles to New York only if they are measured from Los Angeles. If "you" measure them, they do not slow down at all, obviously. [Meeker continues his sloppy thinking and sloppy word choice.]
BM: [In response to my Postscript:] How'd the test come out? [signed] Brent "Rocky" Meeker.
MM: You failed, Sylvester.
MM: [to Norman Levitt, December 12] I did not reply to your last email immediately, since I decided to let a few days pass, to see if anyone in your "discussion group" might have a revelation. And I had a few more emails to trade with Brent Meeker. But now I am free to burn this bridge thoroughly. I can only guess at the pathology behind your being amused at yourself for being smug and condescending to avowed students. Superiority as a pose to scare your enemies is understandable and forgiveable, perhaps. Treating students with such contempt is a sign of something else entirely. You may have thought you were coming off as one of the mighty, but you only succeeded in appearing vile.
Nor did your arguments come off any better than your character. In fact, past a certain point you made no argments. You asked for an operational analysis, but did not give one, or respond to mine (except with unctuous ad hominem remarks). We need not agree. My arguments, though cogent, may be false. But at least they had content. Your emails had no content. I showed several contradictions in your bald statements of "fact", and you showed nothing. Nothing but blind acceptance of a dogma you admit to not having thought through.
Science is not built on reputations, it is built on theory. And no theory is so fundamental that it may not be questioned. Otherwise Einstein never would have questioned Newton.
Now is the time to tell you that this was all a test. I am not a student at UMass. I have not read much of what you have written. And although I agree with your contempt for much of the sloppy thinking in academia, especially in "Science Studies" and other fake fields, I cannot say that you appear to fare much better. You brag about your knowledge of the tools and discourse of math and science, and yet you can't penetrate a simple application of high-school algebra to a real problem. A tool and the application of that tool are two different things. As you have made clear with language, another tool. In your toolbag, "semantics" is word that applies to all statements that do not come out of your mouth. And for you, "to coin a phrase" means the same as to state a cliche. Don't bother replying with more of your unamusing wind. Because the truth is, I no longer give a damn what you think.
[If my final reactions to Levitt and Meeker seem a bit highhanded, reconsider the sequence of events. I treated them with politeness and deference in the beginning. I assume a person is a friend until he proves to be an enemy. Once the argument devolves to browbeating, I can give as good as I get. Levitt, on the otherhand, assumes a person is an enemy until he capitulates completely—at which point he is no longer an enemy. He is a slave. Likewise Meeker mistakes personal attacks for argument. Snide remarks about philosophy from a man who has no understanding of the concepts underlying his own field does not sit well with me at all. Physicists are now taught tensor calculus and Feynman diagrams and Minkowski space, but they are not taught a simple definition of time. They believe that having a firm conceptual knowledge of your terms is "semantics", and therefore beneath them as scientists. They juggle complex equations while having no precise idea what the terms mean. To them, the terms are just variables, and may be applied willynilly, to achieve any desired result. It is equivalent to giving yourself the moniker "Rocky", despite the fact that you are clearly a self-satisfied wonk.
Addendum: It occurred to me later that maybe Meeker was talking about Rocky the flying squirrel.
1These physicists and mathematicians are not arguing from the "Standard Model of Special Relativity", since even today there is still much confusion regarding the subject of time dilation. Although the physicists here argue that time dilation implies a smaller t variable, other mainstream physicists say the opposite when explaining time dilation. For instance, James Walker, the author of a widely used physics textbook, says this, "Since the time between ticks is greater, the clock runs slow. We refer to this phenomenon as time dilation, because the time interval for one tick has been increased, that is, dilated." [p. 949] Walker appears to understand my point, but he has also accepted the current math, as expressed by the light clock visualization. He sees that time dilation implies a greater time, but he does not see that the math of SR is faulty.
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