Didn’t it occur to Mr. Bethell at some point in the writing/planning proccess, that he might be dead wrong…and that the overhwhelming likelihood was precisely that he was dead wrong? What was he going to do the minute anyone asked him to prove his ability to discuss the matters in question? Is it possible that Mr. Bethell really does believe that there is some kind of neccesary metaphysical relationship between relativity theory and cognitive relativism? Surely this is an incredibly simple thing to get mixed up about. I can’t believe that Bethell’s thought is that simple but…

It sounds like you may know how to calculate gravitational time dilation after all, but were making a kind of “a fortiori” argument that might be paraphrased thus “Even if one were to calculate the gravitational time dilation in this (wrong) way, so as to get a bigger effect, it would still not be big enough to save Einstein.”

The Styer article to which you refer sounds very interesting. Thanks for calling my attention to it. From your description of it, it sounds like he shows that in a simple case of the twin paradox, the way things look in the travelling twin’s frame CAN be computed using a gravitational time dilation, where the “gravitation” is due to the traveller’s acceleration. It sounds like he is talking about uniform acceleration in a straight line, and probably using the Rindler metric to describe physics in the traveller’s frame. My guess is that this way of analyzing the problem would be more complicated in the case of the twin moving around the stationary twin in a circular orbit. But maybe it is possible. So what you have been saying makes more sense than it appeared at first. Nevertheless, the way you are doing the calculation is certainly not correct. It is going to be more involved.

I think you are trying to construct a proof that the special relativistic time dilation effect leads to a logical or mathematical contradiction. In that you will surely fail. Whether or not Special Relativity is a correct description of nature, there can be no doubt that it is at least a mathematically self-consistent theory. There is nothing mathematically inconsistent about Minkowski space, and once Minkowski space is assumed, the twin “paradox” follows quite simply.

What this means is that if you manage to analyze the GPS situation using a method similar to (but surely more complicated than)
the method of Styer, you must find what he presumably found, namely that there is nothing inconsistent about the predictions of SR.

Anyway, I encourage you to do the calculation carefully and correctly. If you do, I am sure you will become convinced that SR involves no self-contradiction.

There is an easier way to look at all this, by the way — which, as you know, I am willing to discuss by phone, since that is much easier than tapping away on a keyboard.

I might add that it is refreshing to see that you are actually “putting your money where your mouth is” and attempting a real calculation. It remains, though, for you to do the calculation rigorously and correctly all the way through.

Steve Barr

Steve,
Sorry about the delay. I was playing poker until 4am. You are one of the few physicists who have taken the time to understand my argument so that your criticisms can be specific and meaningful. Thank you. It’s much better than what I usually get. Anyway, I finally have my wits about me so here goes:
You said:
To be frank, my “fellow professors” — not just at my university but anywhere — would not give such nonsense a moment’s notice — unless you were in their class, in which case you’d find all sorts of red ink written all over it. I, however, take a certain clinical interest in it.
I say:
Really? Since many in the physics community have been spending their entire careers embracing such nonsense as string theory and its nonexistent extra dimensions, I would have thought they would give my “nonsense” at least five minutes? Oh, well.
You also said:
Besides that — not that it really matters here — you don’t appear to know how to calculate gravitational time dilation. You seem to be under the impression that it is simply proportional to acceleration. (That is evidently how you get the figure of 2,800 ns/d for the “correction”: 2,800 ns/d is to 0.62 m/s^2 as 45,000 ns/d is to 9.8 m/s.) But gravitational time dilation is NOT just proportional to acceleration. For example, at a distance r from a spherical body of mass M, a clock runs slower (relative to a clock at infinity) by a fraction that is (for weak gravitational fields) GM/r, whereas the gravitational acceleration there is GM/r^2 (i.e. r squared, not r, in the denominator.
I say:
I do know how to calculate gravitational time dilation and I actually did it in the most generous way possible for Einstein and his theory still doesn’t hold up. That was my point. I used the gh/c2 formula which is the same formula that Neil Ashby, Robert Nelson and every other GPS expert has used to calculate time dilation. As you surely know, it was originated by Einstein in 1911 and is used to calculate dilation in gravitational fields that are induced by acceleration as well as real gravitational fields. Because this is rotation, you could try using other derivations, but since the distance (h) is in the numerator – I figured I’d give Einstein more bang for his buck, so to speak, rather than use a formula that will calculate a difference less than 2,800 ns/d.
And finally, you said:
This correction factor you conceive of as a gravitational time dilation factor that arises from the “induced gravity” experienced in an accelerating frame. Unfortunately for you, one CANNOT calculate how things will appear in an accelerating frame in such a way. You are trying to use simple analogies to guess an answer which can only be arrived at through a careful mathematical derivation. It is not simply a matter of applying an extra “induced gravity” time dilation. Your calculation has no resemblance whatever to the correct way of calculating how things look in an accelerating frame.
I say:
Again, I am giving Einstein the benefit of the doubt and he still comes up 11,200 ns/d short.
I’m not sure what your exact argument is on this point. Is it that: since there is induced gravity, any time dilation from the relative velocity no longer contributes to the overall clock rate? If that is your argument – then you are not correct. In Einstein’s world of relativity, there are plenty of examples of determining net dilations from multiple factors. The GPS is one example. They don’t suddenly pretend that relative velocity makes no contribution since there are dilation differences from gravity. Both are factored in to get a net total. (See Ashby in particular.) Another example is the turnaround portion of the twin paradox trip. At that point the proper calculation (for an Einsteinian) includes the relative velocity AND the gravitational potential difference from acceleration. If you like – check out Dan Styer’s article in the Am J phys 75 (9) Sept 2007. Although I don’t agree with his premise, at least he represents consistency in the Einsteinian framework on that point.
So to sum up the GPS system, the satellite is whizzing past all of us on the ground and we have a relative velocity from the perspective of the satellite of nearly 4000 m/s. We also (possibly) have an additional minor contribution from a difference in gravitational potential due to a centripetal acceleration.
I will say it one more time: Einstein’s relativity says the ground should not be in sync with the satellites – yet they are.

Bethell in the American Spectator article: “Petr Beckmann, a Czech immigrant who taught electrical engineering at the University of Colorado”. Not only did I look up Beckmann’s record, I read a few of his papers. I also am aware of what he published early in his life, before he went to Colorado.

Mr. Bethell, I would never say to you “go away silly little amateur”. You don’t rise to the level of a decent amateur. If you were an amateur — which implies someone who really loves the field — you might actually have taken the trouble to learn physics at the required level to talk about its technicalities accurately, instead of taking a purely journalistic approach to it. How often must I say it? It is not a question of credentials but of knowledge and understanding. Every year all over this country tens of thousands of students, some of them probably no more intelligent than you, take real courses in college-level physics and learn quite a lot — and they don’t all go on to become professional physicists. Some become teachers, some even become journalists.
These courses are open to “amateurs” also. There is no excuse — even for a journalist — to talk about things he doesn’t understand.

I am not sure what substantive point Bethell is making about the term “gravitational time dilation”. There is a well-known effect that is normally called “gravitational time dilation” (see Steven Weinberg’s classic textbook, for example, or many other standard texts, or if you’re in a hurry just look at the wikipedia article on “gravitational time dilation”). The effect results in what is called the “gravitational red shift of light” (or the opposite “gravitational blue shift”), it also results in identical clocks at different altitudes registering different times. The effect has been observed and measured. The standard (GR) explanation is that this is a gravitational effect. But even if one has some other explanation of the origin of the effect than the GR one, the magnitude of the effect is not in question, because it has been measured. Moreover, very general theoretical principles that fall well short of Einstein’s theory of gravity imply that such an effect must exist and must have a certain magnitude. You can call the effect something else, if you want. But this effect — the altitude dependence of clock readings — is not enough to account for the bending of light. It does (with the fermat principle) give a bending, but the wrong amount.

I think Mr. Bethell is accusing me of “conflating” special and general relativity, because the word “time dilation” is used for an effect in special relativity, and I am using it for an effect in general relativity (the altitude dependence of clock readings). There is no conflating going on, however. There are two distinct effects: the SR effect is usually called simply “time dilation”, while the GR effect is usually called “gravitational time dilation”. For example, if one looks at Steve Weinberg’s classic book “Gravitation and Cosmology”, one sees in the Index on page 656: “Time dilation (gravitational): 79-85″. If one turns to the pages 79-85, one finds that this is Section 5 of Chapter 3, and is entirely devoted to the purely GR effect of gravitational time dilation. The title of that section is, by the way, just “Time Dilation”.

Mr. Bethell tells me that science is not “authoritarian”. I completely agree. Now let me just make a practical suggestion. Why doesn’t Mr. Bethell (perhaps with the aid of a “credentialed” person to facilitate the effort, or even with a credentialed person submitting it under his own name) just submit a paper explaining Beckmann’s/Bethell’s ideas on relativity, bending of light, or gravity to Physical Review, or Physics Letters, or some other standard refereed journal? I think I know why. If he did it is virtually certain that the paper would be rejected by the referees as unsuitable for publication. In fact, even more likely the editor would not bother to send it to a referee. Certainly Petr Beckmann never published any paper in Phys. Rev., Phys. Lett., etc. containing his gravitation theories.
I guess this refusal of the “white coated priesthood” (Mr. Bethell’s term, I believe) to take Bethell & Beckmann seriously would, in Mr. Bethell’s eyes just show that the whole scientific world is “authoritarian”. So, at least I am not alone in this horrible vice.

I guess I am part of Mr. Bethell’s white-coated priesthood. I think I will end my part of the conversation here. Maybe if Mr. Bethell answers in QUANTITATIVE terms — including the all-important and historically famous factor of 2 — the question I posed to him about Beckmann’s theory, I will return to the discussion. We white-coated types care about numbers. In physics, if you cannot get the numbers to work out, the discussion is over.

i would also be very happy to return to this discussion, if mirabile dictu, Mr. Bethell decides to accept my challenge to show the work that he can calculate and get the right answers even in very simple problems (undergraduate level) in SR, GR —- or for that matter Newtonian physics. The terms of the challenge are described above. (I’ll add a few more here: If he takes such a timed test, and it is proctored to prevent cheating, and he passes it, I’ll pay him $1,000.)

I think the fog of angry denunciations of me as an authoritarian, a bad arguer, a mean-y who calls people nasty names, a guy who doesn’t know my own business, and all the rest are to distract readers from the fact that he won’t accept my challenge, or answer the key question I posed.

WE BOTH LOVE ENGINEERS

First let me deal with the “engineer” issue. Barr wrote at the outset: “Bethel tells us that Beckmann was an engineer.” But, he adds, “knowledge of engineering in itself no more qualifies a person to talk about fundamental physics than does knowing about baseball or butterfly collecting.”

Mr. Barr here strongly implies that Beckmann was not qualified to “talk about” fundamental physics, including relativity theory. He refers, arrogantly, to “Beckmann’s crackpot ideas about relativity theory,” when he hasn’t even opened a book by Beckmann. He really shouldn’t have done that. It was not the hallmark of a scientist.

He now says that I would like readers of his blog to think that he has “contempt for engineers.”

The man cannot think or argue straight. The impression he gave was not that he had contempt for engineers but that he had deliberately misrepresented Beckmann’s credentials in order to make it seem that he was unqualified to criticize relativity theory. Beckmann, in short, was out of his depth and had crackpot ideas. The knowledge that engineers possess no more qualifies them to “talk about fundamental physics” than does the knowledge that butterfly collectors possess.

I never in my life called Beckmann an engineer, which was Mr. Barr’s first mistake. Later, he reveals that he had looked up Beckmann’s qualifications so he actually knew perfectly well that Beckmann taught electrical engineering at the University of Colorado. Further, Barr pretended that I had called him an engineer in order to justify his own mislabeling. I call that dishonest.

Later, I asked him in what way Beckmann’s training in Prague fell short of what is needed to understand “fundamental physics.” He has not done so. Maybe in the interim he took a look at the list of PB’s publications.

I am glad Mr. Barr has a high opinion of engineers. So do I (my grandfather was one). What I particularly admire about them is that they pay their own way and do not depend on government funding. The buildings they build have to stand up without support, which is not always the case with the structures of theoretical physics. (A man I know who teaches astronomy at Caltech was asked by his mother what he thought about when he gazed through a telescope at the heavens. “Funding,” was his one-word reply.)

I am glad that Barr now admits the possibility that Beckmann is saying “reasonable things.” If he is prepared to abandon his (ad hominem) argument against Beckmann being qualified to write a book critical of relativity, then I am prepared to abandon the engineer issue and say no more about it.

Furthermore, Mr. Barr can call me all the names he wants and I won’t mind. Ignorant-denialist-uttering-utter-nonsensical-balderdash. Keep ‘em coming. Water off a duck’s back. I am a journalist.

SCIENCE NOT AN AUTHORITARIAN FIELD

Mr. Barr he has to get over his obsession with qualifications. It is a serious mistake he is making. He seems to think that science is analogous to religion, or more specifically the Catholic Church. The church is a structure based on authority but science is not. You really can’t call yourself a Catholic and go around saying that the Pope is talking nonsense. But there is no analogy to that in science. The whole glory of science is that what someone says about a scientific theory or law is unaffected by his credentials. The denialist’s denial may after all be correct. Mr. Barr probably knows that with his head but his heart clearly rebels against it. (“I am the expert here! Go away you silly little amateurs!” is the attitude that he conveys.)

Secondly, he has to demonstrate some error in what Petr Beckmann is saying. I am sorry that for the time being he will have to depend on my own version of Beckmann’s ideas. But that is a problem that he has imposed on himself.

Now, to the substantive issue that he raises. I wrote above:

“Can someone please remind me where I said anything about Beckmann claiming he could derive the bending of starlight by use of time dilation and Fermat’s Principle? I don’t think I could have said any such thing because, for one thing, Petr didn’t believe that time dilation was a reality.”

He replied:

I’d be happy to tell you where you said this Mr. Bethell. It was in your previous post here. Your exact words:

“[I]t did bring the theory into line with Petr Beckmann’s theory, in which the local gravitational field is equivalent to the luminiferous medium. Motion through this field causes clocks (not time) to slow down. When the clocks are moved to a higher altitude, the gravitational field becomes more attenuated, and there the clocks speed up (much as planes encounter less air resistance and therefore save fuel when they fly higher). Thus does Beckmann’s theory give the same results as general relativity, but in a much simpler manner; bending of starlight is explained by Fermat’s Principle, no need for “curvature of spacetime,” etc.”

“So according to your own words,” Barr continues, “Beckmann explains the bending by (a) the faster clocks at higher altitude and (b) Fermat’s principle.”

Then Mr. Barr writes:

“Now the slowing of clocks at low altitude and faster clocks at high altitude is what is called by everyone “gravitational time dilation”. You and Beckmann don’t call it that, but it really doesn’t matter what you call it or how you explain it.”

CONFLATES SPECIAL AND GENERAL

It is in the last paragraph that Mr. Barr goes off the rails. He conflates the special and the general theories when normally they are kept separate. He even raises the question in my mind that he may be the one who is not competent to discuss “fundamental physics.”

Anyway, could he give me a reference to his claim that “everyone” conflates gravitational (altitude) effects and time dilation (special relativity) effects by referring to “gravitational time dilation”? Does “everyone” include Einstein I wonder? No doubt it is possible to mix them together – e.g. by imagining a clock moving both horizontally and vertically (diagonally, as it were) but it certainly makes things more complicated.

The “slowing of clocks” (as Petr Beckmann explained what is really going on) does not belong to special relativity, and “time dilation” is no part of Beckmann’s theory. That is the point that I hope to convey to Mr. Barr.

In special relativity, time dilates in the reference frame that is moving with respect to an observer. And here we are talking about inertial reference frames in which gravity plays no role.

Mr. Barr may not have grasped the big difference between the claim that a clock is running slow and that time is running slow. Here is the entirely orthodox relativist A.P. French whose book on special relativity is or was used in college courses. After quoting French I will quote Beckmann so that the difference between the two theories is made clear. Beckmann does not agree with French, but French correctly states the Einstein theory:

The claim that “moving clocks run slow” is “misleading,” French says. It

“suggests that some essential change occurs in the operation of the clock itself, that the physical basis of its operation has somehow been modified, whereas it is a central feature of relativity theory that just the opposite is true—that the operation of the clock described in its own frame of reference is completely unaffected.” [French, Special Relativity, p 105]

Beckmann says, on the other hand, that the inner working of an atomic clock really is affected by its motion. To illustrate the point, he asks us to imagine two identical pendulum clocks that are transported around the globe, one to the east, one to the west. If flown eastward the transported clock will read fast, if westward, it will read slow. Time dilation? No, says Beckmann. “The period of a pendulum varies inversely as the square root of the downward force on it, and that force is the vector sum of gravitational attraction and the centrifugal force due to the earth’s rotation.” That force increases when the clock moves east, because its angular velocity about the earth’s center increases, etc. “This is an inherent change, one that an observer traveling with the clock (i.e. at rest with respect to it), could measure by comparison with an equally accurate wrist watch—if it is unaffected by centrifugal force.”

Beckmann cites this as a change in a clock “which might easily be mistaken for a change in the flow of time.” [Beckmann, Einstein Plus Two, p. 78]

Mr. Barr mixes special and general relativity together. Time dilation is explained purely by special relativity, and, as I hope to have shown, it forms no part of Beckmann’s theory. In his theory clocks do inherently slow down as they move horizontally through the luminiferous medium (the local gravitational field of the Earth, in his theory.)

At a higher altitude what Clifford Will calls “gravitational blue shift” takes effect and the “flying clock will tick faster than the ground clock.” This is where the explanation according to general relativity enters the picture. “The two effects, gravitational blue shift and time dilation, tend to offset one another.” [Will, Was Einstein Right? p 56] Which effect predominates depends on whether the clock motion is to the east or the west.

Beckmann argues that at higher altitude the gravitational field is thinner, more rarified, and here clocks really do run faster (than clocks lower down). Clocks moving horizontally really do run more slowly (than stationary clocks) because the field offers resistance to its motion as a clock moves through it.

The ether that Michelson was looking for was thought to be uniform throughout the entirety of space. The most important change introduced by Beckmann is that for him the ether or luminiferous medium is real but NON-uniform. It is denser closer to massive objects. Therefore, if a light beam from a distant star passes by the Sun, it is bent simply by Fermat’s Principle. As the ray approaches the sun it “slews around” slightly (as Eddington put it). It was one of the key things that Einstein’s GR predicted. As the light beam enters the denser field, it also slows down slightly in addition to changing direction. Here everything is explained by classical physics plus a non-uniform luminiferous medium. No need for general relativity or the “curvature of four dimensional space-time.”

(Whether it’s needed for Mercury’s orbit is a complex and interesting question to which I devote a chapter but I won’t go into it here.)

In 1968, Frank Tangerlini published an article in Am J. Physics arguing that a correct value for the gravitational deflection of light could be calculated without general relativity by using Snell’s Law, formulated in the 17th century. (Closely related to Fermat’s Principle.) When a light ray enters a denser medium it is refracted at a certain angle and Snell’s Law demonstrates how the change in angle is related to the change in velocity. (This route was not open to Einstein, who had postulated a constant speed for light as a cornerstone of his theory.) Tangerlini showed that these calculations could be used to give exactly the formula that Einstein used for the refraction of starlight. [Tangerlini, On Snell’s Law and the Gravitational Deflection of Light,” Am J. Physics, Nov 1968, pp 1001-04.] Something similar was done by Tian and Li, also in Am J. Physics, in 1990.

If I have still not given a clear explanation of what Petr Beckmann is about, then that is my fault. I was warned before I wrote this book that for those who have accepted SRT and GRT for decades, and teach it, the whole subject gets cemented into their heads and they are most unlikely to accept anything new. But one has to start somewhere, and there is no harm in trying. For younger people, there is at least a hope. What is that quote (Max Planck?) about scientific theories not changing until department heads die off? There’s a lot of truth to it.

–Tom Bethell