26th Dec, 2013

N/A

Question

Asked 14th Dec, 2013

Herbert Dingle's argument is as follows (1950):

According to the theory, if you have two exactly similar clocks, A and B, and one is moving with respect to the other, they must work at different rates,i.e. one works more slowly than the other. But the theory also requires that you cannot distinguish which clock is the 'moving' one; it is equally true to say that A rests while B moves and that B rests while A moves. The question therefore arises: how does one determine, 3 consistently with the theory, which clock works the more slowly? Unless the question is answerable, the theory unavoidably requires that A works more slowly than B and B more slowly than A - which it requires no super- intelligence to see is impossible. Now, clearly, a theory that requires an impossibility cannot be true, and scientific integrity requires, therefore, either that the question just posed shall be answered, or else that the theory shall be acknowledged to be false.

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Vladimir: Infinitesimal parts of every differentiable curve are straight lines, but from this I cannot move on and prove that the curves themselves are straight lines. You are trying to apply finite (not infinitesimal!) transformation laws that are valid for inertial systems in the case of systems that are non-inertial, claiming that infinitesimally, every system is inertial. This is the same logical fallacy. Now it is of course possible to use a local form of the Lorentz transformation to deal with accelerating systems, but in this case, the parameters of the transformation (the 4-velocity) will change from point to point and thus derivatives of these parameters (i.e., the 4-acceleration) will also appear in the final, non-infinitesimal form of the transformations. The math is clear and relatively elementary.

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In special relativity, acceleration is absolute. This is basic, and the fact that Dingle never accepted it is his problem, not special relativity's. The fact that each reference frame sees the other's clocks run slow is no more a paradox than the fact that if you put two rulers at an angle to each other, and look at each along a line perpendicular to the other, each one looks shorter than the other. Dingle's misunderstanding was explained in great detail in the infamous correspondence in the letters page of Nature, though he seemed incapable of understanding that.

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Different observers will disagree as to which clock moves (or moves faster). Unless you bring the two clocks together, there is no observer-independent way to synchronize them. And once you do bring the two clocks together, whichever clock spent more time accelerating* is the one that shows less time elapsed. This is not a counter-argument against relativity, just a trivial misunderstanding/incomplete understanding of the theory.

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*A clarification to my now almost decade-old post: I said "whichever clock spent more time accelerating" but that's an oversimplification. What is important to recognize is that at least one of the clocks has to change direction, i.e., its reference frame cannot be the same inertial frame throughout the journey. And indeed, if the two clocks follow, say, symmetrical mirror trajectories, they'll be in sync once they are brought back together; however, throughout the journey they will disagree on what events are simultaneous and what aren't.

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V. Toth, if clock A accelerates relative to clock B, the inverse is also true with the same acceleration duration, so how can one clock show less accelerating time elapsed? Dingle's argument uses the same formulation Einstein used in his well known paragraph about the “peculiar consequence” of time dilation (1905 paper), with constant velocity, with the only difference that Dingle considers the two oposite situations: A-B and B-A. So who had the “trivial misunderstanding” and which is it?

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In special relativity, acceleration is absolute. This is basic, and the fact that Dingle never accepted it is his problem, not special relativity's. The fact that each reference frame sees the other's clocks run slow is no more a paradox than the fact that if you put two rulers at an angle to each other, and look at each along a line perpendicular to the other, each one looks shorter than the other. Dingle's misunderstanding was explained in great detail in the infamous correspondence in the letters page of Nature, though he seemed incapable of understanding that.

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Arno, I don't understand your sentence “ I consider SRT as a valid observation.” and at the end iyho “...in SRT-interpretations isn't imho conclusive.” Should I conclude that iyho there is not yet a solid counter-argument to Dingle's?

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Charles, Dingleś argument is different than the twin paradox. He is comparing opposite situations when one clock moves at constant velocity relative to the other and vice versa. Neither in Einstein's note nor in Dingle's argument is implied acceleration.

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Robert, your affirmation: “In special relativity, acceleration is absolute” sounds to me as magical rather than mathematical because I don't see how the first derivative of the position vector (velocity) is relative while the second derivative (acceleration) becomes absolute. Any way, acceleration has nothing to do with Dingle's argument.

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Ivan: if the clocks move at constant velocity, then you can only synchronize them once, and they will never cross paths again so you can never compare them again in an observer-independent fashion. As the clocks move apart, different observers will disagree as to which clock appears to run slower or faster. If you want to synchronize them again, they need to be brought back together, so at least one of the clocks must accelerate to change its trajectory in order to encounter the other clock. So acceleration is necessarily involved.

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Movement occurs always on some gravitational background which can be used as a reference frame for the measurement of movement. You can do thought experiments and models without gravitational background but that's just absurd.

I think V. Toth in his last post has hit right on the nail saying: “As the clocks move apart, different observers will disagree as to which clock appears to run slower or faster.” It also means imho that if clocks appear to run differently then they really and truly run at an absolute pace given by real absolute time as Newton would say.

In this sense I would further reformulate above Charles's phrase the following way: "A appears to work more slowly than B in B's frame and B more slowly than A in A's frame and both really and truly run at the same absolute pace in their own frames."

I am aware this is a heretical proposal, but imho it is the solution. For more mathematical details please read the papers:

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Ivan, I agree this is very perplexing and many people have either remained confused, or refused to accept it. I have a very good physicist friend who describes himself a complete relativist, but when it gets down to specific points about time and length he goes bonkers on me. :D))

First realize that the problem is far larger than just clock rates. Each observer thinks the lengths along the direction of motion of the other are shorter, and if they could be seen they'd appear rotated even though they are not. All these effects have to do with trying to use light to "see" something which is moving nearly as fast as light, which is an absurd thing to try and do in the first place, but light or other electromagnetic effects are all we have. Even matter is just a bundle of electromagnetic energy, and the proof is that you can annihilate it with an anti-particle and off it all goes as electromagnetic energy. But the length problem is even worse, because sometimes not just objects but space itself appears contracted, as for example the distance from the upper atmosphere to earth as seen by mu-mesons. But sometimes it doesn't, as in the distance to the mu-mesons as seen by us!

Second, realize that none of this has an actual affect on the observed object or system. If a spaceship speeds past our planet and sends us a message saying all our clocks run slow, in fact our clocks haven't changed their speed at all. It is just an artifact of the space ship interacting with us inefficiently because from the point of view of the space ship, it is very hard to push us around. The space ship's electromagnetic energy doesn't work very well on us. Nor does ours on the spaceship. So each of us thinks the other is very massive, because it takes a lot of force to push it. By conservation of momentum then, all the clocks seem to be running slow. But no one can say really whose clock runs slow within Special Relativity, because there is no way to get the clocks together.

Third, realize that gravity is an altogether different matter. Clocks deep in a gravitational field really do run slow. They are not moving away. They sit right there, perhaps a thousand feet below, where you can study them and count the ticks they give off for a hundred years if you like, and indeed not enough ticks will be given off. The reason I would give for this would be controversial in the minds of other physicists so I won't confuse this thread with it, but all would agree that many experiments have been carried out just as I described and the result always is that the clock deeper in the field really does run slow. The first such experiment was Pound-Rebka, but in fact GPS satellite clocks have to be adjusted for this effect. (and for relative motion as well)

So, that's the fourth thing. All this impossible to understand stuff is the result of experiment. Maybe some of it wasn't when it was first thought of, but it is now and there is no question about any of it. Except what it means. At this point, our theories are just attempts to understand what we know to be true. How could an ancient man understand the sun going down in the west and coming up in the east? Well, obviously it had to journey through the underworld overnight. In a strange sort of way, he is right. Geometrically the sun is under his feet during the night. Perhaps this analogy is not useful. I'm just trying to illustrate that there are facts of nature we can observe, but we are not always equipped to make sense of them.

Special Relativity is particularly nonsensical because it just "assumes" that the speed of light is the same in all frames of reference, or inertial frames anyway. Nobody originally suspected that. In the late 1800s Michelson and Morley did a little experiment to measure the speed of the earth relative to a hypothetical medium in which light was propagating, and they came up with the little mystery that there appeared to be no medium, or at least the speed of light was the same in all directions.

When one mathematically works out what this means, it means that the speed of light is an upper limit for momentum or energy transactions, and that all kinds of energy and momentum transactions must be in some way the same. Otherwise, it would be possible to violate conservation of momentum and energy by crossing reference frame boundaries with one kind, then converting to a more efficient kind (propagating at a different speed) and crossing back to the original reference frame.

The derivation of the Lorentz transform (the culprit in time and length changes) from Special Relativity is a little different than the ether derivations that Fitzgerald and Lorentz originally worked out prior to Einstein. It is a big assumption to start from the postulate that light travels at the same speed in all directions in every inertial frame, as in SR. It is sort of assuming the answer, I have always felt. I have worked 40 years to find a more primitive basis. I have just uploaded the answer. It came actually after I worked a few years in Finance and got an appreciation for a very simple thing from which it turns out the Lorentz transform is composed. I don't know if this alternate explanation and derivation of the SR effects would just annoy or confuse you or if it might possibly provide a glimmer of sanity to the wonderland-like contradictions of SR, but I offer it below in case you'd like to give it a try. - Robert

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Ivan ""A appears to work more slowly than B in B's frame and B more slowly than A in A's frame and both really and truly run at the same absolute pace in their own frames." As I read this, this isn't heretical, it's just what we mean by proper time. (Though whether what I read is what you had in mind when you wrote it is a different question...)

When clocks are moving inertially, we can say without any contradiction that each appears (to the other) to be running slowly. This is explained perfectly clearly in any number of introductions to special relativity, and Dingle's argument simply doesn't make any sense. It is a direct analogy to the case of the two rulers that I described earlier.

In order for the two clocks to be compared directly, they must be in the same place at some initial time, and then brought together again later, which means that at least one must accelerate. When you calculate the elapsed proper time of two such clocks, if one accelerates and the other doesn't then the accelerated clock shows less elapsed time. The proper time of a trajectory in Minkowski space is the analogue of arc length in Euclidean space, with the somewhat mind-expanding difference that in Minkowski space the straight lines are of maximal 'length', not minimal.

This is not saying that acceleration affects the clock---indeed it is an assumption that acceleration does not affect the clock. This is also in direct analogy to the fact that a curved joining two points has a different length from a straight line joining them, even though the integral for length does not depend on the acceleration of the curve. In each case, the total length/proper time depends only on the speed at each point along the trajectory.

But it does not follow that one can deduce the existence of some absolute time which you can compare all clock rates with. There just isn't a one of them.

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Charles: OK, but this takes us away from Minkowski space and special relativity to the somewhat different context of a space-time geometry with a preferred timelike direction. I don't mind thinking of it as a genuine parameter, because I don't share your instrumentalist philosophical standpoint . But I'm sure we both agree that there is no local experiment that can be done which identified the clocks measuring that time out of a collection of clocks moving at all possible velocities.

Einstein's text (paragraph from seccion 4 of 1905 paper)

From this it results following peculiar consequence: if at the points A and B of K there are clocks at rest which, considered from the system at rest, are running synchronously, and if the clock at A is moved with the velocity v along the line connecting B, then upon arrival of this clock at B the two clocks no longer synchronize, but the clock that moved from A to B lags behind the other which has remained at B by ½tv2/c2 sec. (up to quantities of the fourth and higher order), where t is the time required by the clock to travel from A to B.

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Quite so. If two clocks are initially at rest in the same frame, and then one is brought to the other, they cannot both be inertial throughout the procedure. One is accelerated, and it is that one which shows the lesser elapsed time, just as Einstein noted.

All this confusion goes away if one will only work in the geometry of Minkowski space, and regard proper time as an integral along a world line, independent of choice of inertial frame, while coordinate time is frame dependent. You may disagree profoundly with whether this is a good model of reality, but there is no problem with internal consistency. Indeed, John Schutz (in Independent Axioms for Minkowski Space) has proven in great detail that the geometry of Minkowski space is exactly as consistent as that of Euclidean space.

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it was done in the spirit of the Hilbert approach to Euclidean geometry, trying to find a set of axioms that describe the geometry, without building it up as R^4 with a particular inner product. Consistency relative to Euclidean geometry was an incidental consequence. It's an interesting exercise on the foundations of the mathematics of Minkowski space, rather than a useful approach to carrying out calculations in relativity. To see what he was up to, you could take a look at

which gives an overview.

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Perhaps I'm perverse: it makes me realise how much is happening 'under the bonnet' of things I think I already understand, and happy that somebody else is worrying about it so that I can get on with thinking in the terms I'm comfortable with. (I, for one, find that the harder I think about the real number system, the less I think I understand. There may be a moral there...)

@Robert & @Charles, as long as one does not perform rotations on systems of objects Minkowski space is fine. But rotations on systems of objects do not always give a physical result. Physical clocks do not automatically re-synchronize. See Swann, 1960, link below.

Yes that's all I was saying also. I have seen write-ups of accelerating spaceships that make incorrect predictions about clocks on the spaceships, for example, not realizing that the new coordinate frame, once the spaceships stop accelerating, is not automatically in place. Interesting philosophical implications, I think. Opinions will certainly differ.

Robert L. and Charles, excuse me that I don't follow your metaphysical disquisitions, which make me think that for you the question is closed. So let me also close it with my own answer: I read all over again both texts, Dingle's argument and Einstein's pertinent paragraph. I find Dingle's formulation a clean valid argument that touches directly into Eintein's belief that relative time is the only time to be considered in physics, therefore it is not apparent, it is real and true, so that clocks really run at different rhythm when moving one relative to the other. When they meet “the two clocks no longer synchronize”. From here, considering the opposite situation, Dingle shows by simple logic that there is a contradiction. He is not questioning time dilation nor Lorentz transfomation, it is Enistein's interpretation of relative time as clock time which leads to contradiction.

Einstein, very consistent with his belief in a chronological frozen universe, at the end of his days wrote: “A true physik-believer knows that the difference made between past and future is a pure capricious illusion.” Even though this is the belief of a genius I admire, I am sorry, I don't share it, I look at a clock to know what time is it now inorder to take desisions for future situations. I think RT is incomplete without absolute time (not to be confused with Minkowski's Eigenzeit), the time shown by all clocks.

Feliz Navidad amigos! Y próspero año 2014. I enjoyed our discussion, thanks!

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I've now gone entirely off topic. Sorry.

Charles: " I am very comfortable with thinking about the real number system because I know exactly how to to construct it, starting from the null set and without ever referring to anything real" Oh, that's no big deal: von Neumann integers, complete to an additive group, define multiplication, pass to the field of quotients, then take the Cauchy completion. I don't have a problem with any of that. My own background is as much maths as physics.

But my head hurts when I try to consider how the rationals and irrationals are interleaved (both dense, but one set of measure zero), non-measurable sets, and in omore than two dimensions the Banach-Tarski paradox, the relationship between the reals as a second-order axiomatic system and as a first-order one which has non-standard models, the continuum hypothesis, and so on.

There's plenty to worry about if you think too hard, so it's easiest just to mutter "the reals are the unique complete ordered field" under one's breath, keep calm, and carry on.

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Ivan, "Dingle shows by simple logic that there is a contradiction". But this is exactly what we disagree about. I see no contradiction whatever, and I also claim that it isn't just that I'm looking away or closing my eyes to it. But I don't suppose I'll convince you any more than you can convince me. Merry Christmas to you, too.

Ivan, you need to provide Dingle's text. Send it to me privately if you prefer. Nature articles are not freely available on the web and I am not going to spend the $32 to buy a 1967 article that should by now be in the public domain. Please do not try to summarize it. Arguments regarding Special Relativity are overly sensitive to minor nuances of expression.

It might also be useful to provide McCrea's reply and Dingle's reply to McCrea. I have found web pages on these but they appear to be a summary by Paul Ballard, and as I have stated I don't trust summaries.

Apparently there were several other answers as well, also locked down by copyright. I need them all, thanks.

It is very annoying to see that these guys who cannot handle Special Relativity can all be published in Nature. It makes me very angry.

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I don't think you're missing much by not reading the correspondence. As far as I can recall, it's mostly yet another example of a perfectly lucid explanation being completely ignored. It may be that the editor of Nature at the time (nearly half a century ago!) thought that the letters page of Nature was a good place to explain why the "clock paradox" argument is invalid.

Of course, no reputable journal will now publish papers "proving" that special relativity is inconsistent, for the same reason that they won't publish papers giving angle trisections, or proving that pi is exactly 22/7.

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Robert L., To disagree is healthy for science...

Robert S., Both texts I have provided here, please read my first post.

Saludos, igr

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Charles: I didn't realise you were so hard line about finitism. I can see the advantages, especially if you're willing to use (say) real analysis as a tool for your purposes, but without imputing to it anything beyond book-keeping convenience. But I'm kind of attached to the continuum model, so I'll continue to live in and explore a universe with the axioms of infinity and choice, real numbers, and all those monsters that I have to let in along with the bits I want.

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Charles, thanks for the rebuke, I am lucky that at the present the Holly Inquisition is not so powerful any more...

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Ivan: "To disagree is healthy for science..." There is some truth in that since disagreement is necessary for change and therefore progress. But while disagreement is necessary, it is not sufficient. Disagreement about the best explanation of experimental evidence may give rise to improved theories. Disagreeing about whether there is any validity to Dingle's claim that special relativity is inconsistent will not produce any progress, because Dingle was just plain wrong (at best, he was disproving only his idiopathic misinterpretation of special relativity). Alas, I can't see any way of convincing you of that, if the arguments and resources that have already been provided won't do it.

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Charles: by your standards I am surely a mathematician playing with ideas inspired by physics, rather than a physicist. I find the geometrical structures arising in general relativity sufficiently rich and fascinating that I'm willing to invest my time in understanding them better, even knowing that at some level they don't represent reality. If I were more fundamental physics minded, I'd probably be thinking more about some version of the causal sets program and how physics works there (and how the continuum arises as an approximation).

But then, I'm not as sanguine about physical theory as some people: I'm content to regard it as a model building exercise, where some models are better than others in some circumstances, without believing that we're somehow converging on the 'truth' about the universe. It's clear, for example, the special and general relativity both provide splendid ways of thinking about a whole bunch of phenomena, but it's also clear that neither is 'the truth'. That doesn't bother me, any more than accepting the axioms of infinity and choice to make mathematical reasoning more convenient bother me.

@Ivan, you are sure, I do not see them. I clicked "show all answers" and looked through the whole list.

Meanwhile I have proceeded with what I have and will post it shortly. The problem Dingle raises appears to be pretty much identical with the so-called Twin Paradox (Robert Low doesn't like that name, noted), except that it is just half of it ... starting from out, not the full out and back. But it is analyzed the same way.

Further, despite the huge number of papers on this, most of which I have read recently in connection with another research project, there is no one paper that settles the matter firmly to the satisfaction of common sense doubters who don't want to take any hand waving on faith. In other words, I have judged it important at least in terms of education about physics, and frankly, I learned some new things even myself when I worked out the full details. I will post shortly after proofreading it.

@Robert Low, regarding: "by your standards I am surely a mathematician playing with ideas" and also "I'm content to regard it as a model building exercise, where some models are better than others in some circumstances, without believing that we're somehow converging on the 'truth' about the universe..."

I figured you out when we first met. ; ) It is pleasing to meet someone who also understands themselves so well. I always enjoy your comments.

The two approaches - of mathematical fascination and obsession with how things work - should not be at odds, but have drifted that way in the field of General Relativity, I think, because of the difficulty of testing hypotheses with astronomical data. Indeed, data can never be obtained from black holes, perhaps even theoretically, and the rich space the mathematics allows for exploration. So I believe exploring this mathematical space has become a substitute for exploring the real universe, and I'm on a real tear to hit the reset button. :o!

@Iván, I am inclined to agree that no truly satisfactory answer to your question, or to Dingle, has been provided. I find the question to be essentially an abridged version of the famous Twins Paradox which received discussion in the scientific literature for many decades. The very length of time over which it was discussed seems to indicate the difficulty authors had in being crisp about their explanations. I believe the matter has dropped from the literature not because one good explanation was given, but just from the weight of many almost-but-not-quite good enough explanations, and a general fatigue of new approaches.

Many of the approaches based on how clocks behave in acceleration that I have read are actually wrong. It took me many years to discover this and created much confusion.

The best explanations confine themselves to Special Relativity. Though Einstein declined to do this, it is not difficult and I believe history has shown he made a tactical error in omitting a full proof of his choice. Of course, he did not necessarily know at the time what confusion he was creating.

I realize you have a firmly held belief and that nothing may persuade you, but because of my own belief that in fact the best possible explanation in a short paper has not been given, and because I can't point to any of the answers to Dingle and say as other respondents here have said that it is perfectly clear to me, I have spent quite a bit of my time formulating my best shot at it. And made my wife somewhat angry in the process. So I hope you will take a few days to read the article I'm linking below and give it some thought and let me know your reaction. If you don't like it, I'd appreciate a careful point by point review, not a general dismissal, so that if possible I may repair any unclear or mis-formulated points.

Thanks for an interesting challenge!

Robert S.: you'll find it right below the question. You can also download (free) his book:"1972, Herbert DingleI, Science at the crossroads” (quote from page 27)

I agree, it is a topic important in terms of education about physics and it is a good exercise in logic. I'll be waiting for your new post on it.

Charles, well come back to subject! You ask: “How can there be a "time shown by all clocks" when we already know that all clocks can not show...” Here again the question is if what we meassure or adjust are apparent relative variables or absolute variables. I believe that RT would gain very fruitful results, like those in classical mechanics, if we learn how to deal with both time variables x4=ict (a relative coordinate) and T an independent scalar variable so that we study trajectories in M4 x=x(T) with formalisms like Hamilton-Jacobi's.

@Charles, I would greatly appreciate if you would read couple of my papers (ToEbi & Atom Model and Relativity). There I have outlined how QM and relativity are combined quite elegantly. Naturally, some old must be discharged in order to gain some new. BTW. I'm also mathematician trying to understand physics.

Article Atom Model and Relativity

@Charles, What are you talking about? Could you give me a reference where the mechanism behind relativity is explained? E.g. what mechanism causes clocks tick at the different rate?

On the other hand, if you think that the current RTs and QM are the right ones then how come we don't have the TOE already? And you are wrong about your claim that physicist who are bothered by relativity theory are those who don't understand it.

@Charles, I'm looking for a deeper mechanism, e.g. the phenomenon which alters atomic level events hence for example atomic clocks event rate. Your reference don't cover that.

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Robert Shuler: " I am inclined to agree that no truly satisfactory answer to your question, or to Dingle, has been provided." And yet I see what seem to me to be entirely satisfactory explanations of the situation in standard introductory textbooks, Rindler's "Introduction to Special Relativity", Taylor and Wheeler's "Spacetime Physics", Ellis and Williams' "Flat and curved space-times", to name the first three I dragged off my shelves. I admit that others don't necessarily spell things out in complete detail, but give enough information that a competent student should be able to fill in the gaps (and, indeed, filling in the gaps is a pretty standard exercise).

I don't think anybody has bothered to write papers on this recently (apart from those claiming that there is a paradox) because, to quote Huxley from another context, "Life is too short to occupy oneself with the slaying of the slain more than once."

All this is not to say that there isn't a great mystery about why there is a privileged class of frames of reference, namely the inertial frames. This is, indeed, a mystery. But fortunately we aren't dealing with that; we are examining the situation within the context of special relativity, in which this class of inertial frames is part of the framework. (Though it seems likely to me that it is the implicit and unacknowledged rejection of this part of the framework that underlies Dingle's confusion.)

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@Charles, The implication you suggest seems to be the case. I'm honestly amazed how misled contemporary physics really is. My theory is easily falsifiable and it predicts unique phenomena (which are also testable).

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Charles: de gustibus semper dubitandum est, I suppose. My reading of Rindler is that from a few basic assumptions you construct a mathematically elegant framework (which of, of course, provides predictions one can treat experimentally). And his goal is to provide the mathematical framework, rather than to provide an experimental physics book. However, putting all that aside, I wasn't really trying to argue that these particular texts are good (though they're all to my taste :-) but only that there are entirely adequate descriptions of how simultaneity works in SR and its consequences for relative clock speeds, etc.

But I do agree that Bondi's 'Relativity and Common Sense' is an outstanding paedagogical treatise. Rather wonderfully, it is also available for free from

He even provides a brief (verbal) explanation of the twin "paradox" on pp150-152, which is, depending on one's tastes, either mercifully or disappointingly free of mathematical formalism.

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@Robert Low, I agree completely with Charles Francis about Rindler, and Wheeler as well. No one wondering about this question is going to be reading any of those texts. With due respect, as I appreciate that you are an excellent mathematician, many if not most people do not see or believe the implications of subtle proofs as readily as you. : ) I either of us must accept the verdict of the confused or the disbelieving as to whether the explanation is clear. It so happened I was reading back through old literature on the Twin Paradox recently for some obscure reason. Also, Petkov published on the Bell Spaceship paradox within the last few years. The Bell Spaceship paradox is kind of a more sophisticated version of the problem focusing on length instead of time. So the topic is still alive. But we understand not for you. ; )

@Charles Francis, re "Believing in a substantive space time is taking them too far in my view."

Kudos for saying that out loud! But you had better duck, all the fantasy seeking believers will lock you up with Galileo. :D)) Pardon my sarcasm, but it was kind of a release to read that statement.

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Robert S: " I agree completely with Charles Francis about Rindler, and Wheeler as well. No one wondering about this question is going to be reading any of those texts." Some kind of selection effect, I suspect. Those of us unlucky enough to be struggling with the idea in the first place also struggle to cope with these explanations. I guess that my advice to those people (not that anybody really asked for it) is to get mathematically literate enough that these books make sense: Taylor and Wheeler is at the freshman level, though Rindler is rather more avanced. And, as always, there's no royal road to enlightenment.

Charles's suggestion of Bondi's book is a good place to start, and it gets easier with practice... And once a person gets used to Minkowski's space-time diagrams, these kinematic effects really do become straighforward.

But maybe I shouldn't be quite to blase about it all: Bell's spaceship caused some confusion amongst the professionals, or so I read. (And heaven knows, I've made enough blunders of my own :-()

@RL, I think we owe them a short succinct explanation using high school level algebra without any textbook. Take a look at my paper I wrote for Ivan and see what you think. I believe I gave one element of the computation that is not easy to find in the literature. : )

Sorry if I mistakenly maligned "Taylor and Wheeler," I have not seen it. I just have an automatic reaction to the mention of "Wheeler" after reading his book with Misner and Thorne many years ago, which is not very accessible. I have a GR text by Rindler, and a bunch of others besides. Rindler has a few useful points but is not easy to follow either. None of the ones I have is suitable to give to someone asking a question about SR derivations. in fact I've never seen a textbook that gave a comprehensible derivation. Most are concerned only with being mathematically correct and I guess the student is supposed to accept it on faith. Ages ago when I was an undergraduate, I remained confused about basic calculus concepts through three semesters of making mostly A's in the subject. Finally after it was all over I suddenly saw it one day, but textbooks and profs did not help.

I read that about Bell's paradox and the professionals too, but I am not quite sure what was confusing to them. The problem is more adequately explained by Swann 1960 who was writing about the Twins, not about Bell's Spaceships. He observed that when accelerating a reference frame, it does not remain a synchronized reference frame.

BTW, I have seen a book, not a textbook, that gave a comprehensible derivation of SR. I suppose there may be many of them, but the one I learned from was B. H. Goode 1968 which I bought in that year, just before I graduated from high school. It is very clear on SR. It makes some mistakes in trying to over simplify GR which threw me off the track for many years.

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RS: I've taken a look at your note, and it seems clear enough to me: there are, of course, lots of ways of analysing the situation and counting how often each person sees the other clock tick (taking into account separation and light speed) is as good as any.

For the 'round trip' clock paradox, with linear motion apart followed by a short period of acceleration, then linear motion together I like to draw the space-time diagram showing how the accelerating clock's version of simultaneity changes so that lots of the inertial clock's time passes while only a short period of proper time elapses for the accelerating one, to give a geometric understanding of what the formulae are saying.

One minor point. You don't need to worry about aberration if you're just considering one dimensional motion---there isn't any, just the (relativistic) Doppler effect to worry about.

On texts: I think that in both cases you might be thinking about different books from me. MTW is authoritative, magisterial, and very hard to read unless you already understand it (and if you find MTW after already studying the subject you wish you'd read MTW first, but you're wrong). Likewise, the Rindler book I'm referring to is his introduction to SR, with no GR in it at all, though it is at the advanced rather than the beginning undergraduate level.

I understand your comments about calculus. My experience, and I think that of many of my colleagues, is that you often only start to understand stuff when you start to teach it. Up until then, you're perhaps very good at 'doing' it, which isn't the same thing at all. And once you've been teaching it for a couple of decades you forget that there was a time you didn't really understand it yourself, and get all snotty about how you students don't understand it properly in spite of the perspicuity of your classes. Oh well.

@RL, concur that the first year or two of teaching is a time of increasing understanding. I've only had limited teaching experience, but it seemed that way, and I often benefitted from trying to explain to my friends.

I think we are losing Ivan until January, but I have a question about your space-time diagrams. Do you draw these only from the point of view of the inertial clock? Or do you use some trick to some up with some diagrams from the point of view of the traveling clock?

I used to use the S-T diagrams, but since I read Swann 1960 a couple of years ago, and truthfully I was already having misgivings, I am not convinced the time axis automatically comes into existence. I'm basically certain it does not. That would imply automatic adjustment of clocks.

I think the thing people want, at least the thing that makes it a "non-paradox" for me, is to have a way of calculating from each point of view. The person who calculates from the moving clocks point of view usually makes a great many mistakes. The first one is to forget about length contraction. That gets the clocks back to even. What I have seen in some articles is to add an accounting of the "speed up" of remote clocks during the acceleration phase, using GR. But often it is done incorrectly and it is not necessary.

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RS: space-time diagrams are a way of showing the situation as it is represented in any inertial frame. Since the accelerating clock is not stationary in any inertial frame throughout, they don't show things as if the accelerating clock were inertial. But you can see directly that all inertial observers come to the same conclusion about the elapsed time for both clocks. You can also use the diagram to analyse the situation from each clock's point of view while it is in inertial motion, and see what is going on when the accelerating clock goes from moving in one direction to the other. (And a picture helps to understand the tick-counting argument, too.)

I'm not sure what you mean by 'the the time axis automatically comes into existence'. Any timelike straight line can be taken as the time axis, and then the space axis is just a line perpendicular to the time one, in just that same way that in the Euclidean plane any straight line can be taken to be the x-axis, and any line perpendicular to it can then be taken to be the y-axis (though perpendicular means something different in this geometry).

After all, the whole point of a mathematical formalism is that you can do all your thinking about the situation in the mathematical picture, then translate back to physical ideas at the end as appropriate, rather than having to argue directly about the physical model all the time. And Minkowski space gives such a good formalism.

Hi Robert, yes I understand well how to use the diagrams for inertial observers. As earlier postulated, I believe those who remain puzzled want an explanation from the point of view of both clocks. I thought at first you might be implying use of an S-T diagram for this accelerated observer and I was going to pounce. :D)) But of course you are too smart to be caught on that.

There is an animation linked from several sites around the web that particularly annoys me. You can see it here: http://en.wikipedia.org/wiki/Lorentz_transformation It is carefully labeled "The momentarily co-moving inertial frames along the world line of a rapidly accelerating observer (center)." About half way down on the right margin.

While the author is careful not to make a technically incorrect statement, the implication that those instantaneously co-moving frames are meaningful is very misleading. It is in particular the time axis which cannot be verified by during a brief moment of being co-moving with an inertial frame, and there is no time for signals from remote clocks. On the other hand, the accelerated observer can see exactly the same things that the co-moving inertial observer sees for that instant. The same photons are accessible. It is easy to correct for Doppler shift and aberration. The two will "see" exactly the same past time value of all visible clocks.

Einstein set up his reference frames in a peculiar way. They are not naturally occurring. I would say the grids of meter sticks are reasonable. But the clocks require synchronization. Einstein insisted on making certain measurements dependent on these synchronized clocks. Minkowski took them and proposed they exist naturally in a sort of 4D S-T, and for whatever reason Einstein didn't correct him. Possibly there were personal reasons and Einstein was glad of favorable mention from Minkowski, who had been very critical of Einstein as an undergraduate physics student. Probably Einstein couldn't think clearly where Minkowski was concerned.

But as Swann pointed out, the clocks don't automatically assume the Minkowski times (i.e. the times given by the Lorentz transform), if you take a reference frame (made of some suitably semi-rigid material) and accelerate it to a different inertial state. The lengths of the meter sticks will indeed come to be automatically. But not the states of clocks.

If Minkowski space were a "physical" thing, I believe the clocks would not need re-synchronizing. In fact for 35 years I "assumed" this to be the case without checking. A couple of years ago I was doing the analysis for my paper on the isotropy of inertia (Dec. 2011) using an equivalence set up. I was reading through a bunch of related references and came across Swann. I was appalled and shocked. Both at my own naiveté and the scam I felt had been perpetrated on me by all the coordinate-based treatments I'd been fed in countless books and papers.

The worst one was an article I'll have to try to find - or maybe it was in a book - which tried to say that during the acceleration phase where the traveling twin turns around, distant clocks "run very fast" due to a sort of gravitational potential caused by the acceleration. This is rubbish and not part of the Twins analysis. Swann made me realize that the "time" required for the turnaround can be assumed as small as one likes, very close to zero. The photons in the vicinity are the same photons accessible to an inertial observer at the turnaround point. So remote clocks cannot visually appear to jump ahead. In fact, if they do not jump ahead for any nearby observer, they do not jump ahead for the Twin, because the Twin could simply enter that observer's frame.

In fact, and I'll call this "Shuler's Clock Principle," any actual clocks at remote locations have the same time values for all observers at a particular location simultaneously, regardless of their states of motion. Said actual clocks can be synchronized for at most one inertial reference frame. An observer's transition from one frame to another has no effect on the clocks. The reading of any of these clocks when an observer arrives, moreover, has nothing to do with coordinate time transformations, and can be computed from the following three things:

1. its current reading in any inertial frame

2. the length coordinate transformation from the frame of reading to the traveling observer's frame

3. the observer's velocity relative to the clock (assuming the observer is making straight for it and undergoes no further acceleration)

Some of this I didn't learn until I wrote the answer for Ivan. I had originally intended to make use of the coordinate time transformation. So this simple problem that in principle we who have faith in mathematically reasoning didn't need to look into, in fact taught me something I find very interesting. ; )

And thank you for drawing out the discussion from me. I continued to learn things even as I wrote this response!

Iván, I have become uncertain that my paper really and fairly answers your question. The use of photons calculated from the time before the acceleration of A in predicting the observations of A after acceleration, though logical, may violate the idea of calculating the rate of B's clock entirely with SR from A's new viewpoint. In fact it avoids any real calculation of this rate. So for now I withdraw it.

@Robert Low, I have waded through some algebra. One MUST use the discontinuity in shifting viewpoints through Minkowski space in order for A to attribute the expected relativistically slow clock rate to B and find agreement in elapsed time upon arrival. This adds suddenly the time value vL/c^2 to B's clock. No one observes this. No photons sent from B ever record such a discontinuity. No co-moving observer sees such a thing. It is just a mathematical artifact. When one adds this to A's computation of B's clock increment slowed by the lower distance L/gamma and the slower tick rate by gamma, giving an elapsed transit time on B's clock of L/v*gamma^2, and does the algebraic simplification, A would conclude B's time should be L/v just as B in fact measures.

However, if I were Ivan, I would not find this acceptable. In fact, being me, I am not at all sure I buy it. This is exactly what Swann says is not automatic. At this time of night I won't jump to any far reaching conclusions, but I'm not overly optimistic you'll have an explanation for me in the morning. If you can, that would be great, I won't have to end my short career as a physicist. :D)

Just out of curiosity, I searched for experimental verification of relativistic doppler, which is the only way of getting at a moving object's clock rate. It's verified for moving particles, but not for a moving detector. I wouldn't expect any surprises, necessarily, but someone ought to do the experiment. It would rule out the pre-Einstein interpretation of the Lorentz formula. When I did my derivation of the relativity principle from geometric interaction impact, I did get a strange gamma squared factor divided into everything at one point. I just assumed it was normalized away, but maybe I should have paid attention?

@Robert Low ... here's a thought. Just before launch, if A sends B a message querying "what time is it when you receive this message?" B will respond L/c, where L is the distance separating them.

Just after launch if A sends the same message, the answer is still L/c, even though later, after traversing the length, A will conclude that it had been shortened to L/γ.

It appears there is more than one possible transform that satisfies SR. I've never heard of that, and I've even seen proofs to the opposite I believe. But if A takes a realist approach to sensor data, then not finding any gap in clock signals from B, A will conclude something different about remote events. It is impossible to ever prove A wrong in this conclusion I believe. It does not change any predictions of local fact, and SR retains its locally predictive power. But A can use a different mathematics which is more compatible with a realist interpretation of remote events, and also explain local facts. A will not, for example, ever be able to find an ether nor even any reason to look for one.

1. I see why Einstein abandoned the Lorentz transform over any appreciable distance in GR. Been wondering about that for years. It does not give reasonable predictions about physically distant events from accelerated frames.

2. I now also understand, at last, Swann's second paper on the Twins: Am. J. Phys. 28, 319 (1960); doi: 10.1119/1.1935795. He analyzes the problem from the point of view of an inertial frame which is initially moving with the traveling twin. But when the traveling twin turns around, that frame, being inertial, keeps going. And in that frame the traveling twin is going now at a much higher velocity with a very slow clock, and winds up younger.

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RS: too much stuff to do, too little time to do it in, so I'm going to try to cherry-pick out what I think is a central part of the issue. the consequence is that I'm about to write a minor essay which will probably fail to address your actual concern.

First, there is a terrible confusion in SR about the word 'observe'. In this context, an 'observation' by A means the assignment of space and time cordinates in his rest frame, and is very different from what A 'sees', which means which photons are arriving at A. So I'll be using 'observes' in this sense, not to mean a measurement of the photons just arriving, and will try to distinguish carefully, though I've probably been guilty of using 'sees' when I mean 'observes' in previous contributions.

So, we have the question of what B observes (as opposed to sees) during the period of acceleration. I think that you are thinking of B as being attached to a physical scaffolding of rods and clocks which have to be re-synchronised when the acceleration is over: I agree that this is problematic, and probably not well-defined at all.

However, there's another picture. Rather than thinking of a single physical framework, we think of space as full of all possible ones (magically above to inter-penetrate without interfering---but that should be ok, because we have exactly the same issue if there's only one). In mathematical terms, we have a whole bunch of coordinate systems that can be used, each of which is associate with one choice of system of rods and clocks. Now, at any given point in B's journey, there is one of these frames in which B is at rest just at that moment. At each time, B's observations are those which are made in the instantaneous rest frame. There is no problem of elasticity or re-synchronization, B is just using the most appropriate set of coordinates at each point of his journey. (And doing this does give a bad coordinate system far enough from B in the opposite direction from A, but it's fine in the vicinity of A.)

In this picture we see that B's notion of simultaneity is constant during the inertial phases, and swinging around during the acceleration: B's 'now' just before the acceleration is different fro that after, and (because of the spatial dependence) a small amount of B's proper time elapses while his observation of elapsed time at A increases enormously. (An accelerating observer can observe inertial clocks to run fast, and this is why. It's only for inertial clocks that you have the symmetry of each observing the other to run slow.) This smooths out the instantaneous jump you get if you imagine B's velocity just suddenly changing sign, in such a way that is the period of acceleration tends to zero, the change tends to the one of the sudden value, so it's not unreasonable to take the instantaneous change in velocity as a good model of the situation. (And in this latter case B is instantaneously going from one notion of simultaneity to a different one, and there is a whole chunk of A's path that never gets observed, this length being the mismatch: but this non-observation is a mathematical fiction, caused by making a discrete jump in the choice of inertial frame.)

And of course, if you just chase through what each person actually sees (i.e. how many ticks of the distant clock arrive during each interval of proper time), it all matches up. B sees A's ticks arrive at one (slow) rate for the first half of the journey, and a faster rate in the second half. A sees B's ticks arrive at the same slow rate until he sees (i.e. really sees, no observes) B turn round (which happens more than half way between the time of B's departure and re-arrival) and at the same faster rate after he sees B turn round, so he sees fewer ticks in total than A does. And although I didn't read it in detail, I read enough to be convinced that you have calculated these two totals and found that it does all match up as advertised above.

I don't think I have anything to add that isn't just going to be repetition, so I'll stop now.

@RL thanks for long note. When I use a co-moving observer, it is the same as when you select one of the infinitely many available pre-existing reference frames. The distinction between observing and seeing is critical, I agree, but strangely is rarely made in literature on Twins. Offhand I do not remember an article on the Twins specifically discussing the difference. The two I know of based on "seeing" are very different. Darwin 1957 in Nature uses relativistic Doppler and is probably correct, I've only seen a summary. Unruh 1981 uses parallax, and "sees" the stay at home twin rapidly recede during turnaround. If I think about parallax and aberration too hard in the context of a constant c I get confused.

I have convinced myself that Einstein avoid using changing Lorentz frames in his original exposition which Ivan cites because he viewed the time jumps of remote objects as unphysical. He uses SR to analyze a rotating disk, but this is "local" and doesn't require the time jumps. For many years I was confused as to why GR astrophysicists do not use Lorentz, and I now suspect Einstein limited it so that the new theory of GR would give more physical predictions. I'm roughly satisfied but cannot think of anything that I believe would convince Ivan.

Arno, can you explain what you mean, or possibly offer an alternative? On the broad stroke I'm inclined to agree, but I wouldn't know with what I was agreeing exactly. : )

Hi Arno, yes I do agree and that was the point of my question. I think you would enjoy the speculative paper I linked with the question description above. In it I factor the Lorentz gamma factor into (1+v/c) and (1-v/c) and observe that the actual gamma factor is the inverse of the geometric mean of those two quantities.

The arithmetic average is just "1". Geometric averages of quantities that deviate equally about a central point (1 in this case) are always less than arithmetic means. They are usually physically meaningful if some sort of serially dependent effect is present, like compound growth rates in biology, or investment return rates in economics. In the paper above, I do not speculate about what this serial dependency might be. Initially I did have an idea but it did not produce a fully relativistic result. So what I have is just an assumption. But it may be that someone better able to handle Maxwell's Equations could easily figure it out.

Arno, I'm sorry, I didn't realize what QA blog I was on for a minute. I have a similar one one, here:

Dingle is right. Each observer sees nothing change in his own frame of reference, but claims to see changes in the other observer’s frame of reference. Let m1 be the mass of a clock in the hands of observer 1 to which is attached a coordinate system S1. Let m2 be the mass of an identical clock in the hands of observer 2 to which is attached a coordinate system S2. Let S1 and S2 be oriented in the same fashion. Let these clocks measure time t1 and t2 and have lengths L1 and L2 respectively. Now according to Einstein, if there is a constant relative motion in the direction of the lengths of the clocks, with speed v, 0 < v < c, where c is assumed a universal constant, then S1 watches S2 and claims that m2 > m1 and that t2 < t1 and that L2 < L1. However, since Einstein’s theory is symmetric, Einstein says S2 watches S1 and claims that m1 > m2 and that t1 < t2 and that L1 < L2. The between-observer effects are therefore mutual.

Now observer S1 sees no change in m1 or L1 and his t1 marks time as usual for him. S2 sees no change in m2 or L2 and his t2 marks out time as usual for him. Thus, the changes Einstein alleges occur between observers, not at the observers themselves. No observer sees anything in his hands change in any strange way.

Einstein detaches time and mass and all else from reality with mathematical hokum and irrational arguments. What is a year? It is the execution of an Earth-Sun orbit. Primitive Man knew about the year by its seasons and the azimuth of the Sun, but had no clocks that ticked. Let us therefore let the aforementioned clocks t1 and t2 tick Earth-Sun years instead of seconds. In other words, the ticks of the clocks count Earth-Sun orbits instead of seconds. This is a one-to-one correspondence: 1 clock tick = 1 Earth-Sun orbit (i.e. 1 year). Now Einstein claims that even though his theory is symmetric, asymmetric phenomena occur therein. This is a contradiction. Einstein and his followers rely for this asymmetry upon his famous Twins Paradox. Einstein and his followers invent all sorts of nonsense to justify this contradiction, which they always call a ‘paradox’. Now according to Einstein, if one of two twins stays on Earth whilst his twin brother travels on a high speed relativistic journey to a distant star and returns, then the twin on Earth will have aged more than he who travelled to the distant star. This violates the symmetry built into Special Relativity. No matter, Einstein and his followers invent things like ‘accelerations’, ad hoc. They say, for instance, that the travelling twin accelerated up to and down from his cruising constant velocity when outward bound for the star, and then accelerated up to and down from his cruising constant velocity during the return leg of his journey. These accelerations they say cause the asymmetry in time.

Consider our two chronometers t1 and t2. They are ticking Earth-Sun orbits, not seconds. So if the twin on Earth aged say 50 orbits (years) by the Earth-Sun orbit counting clock in his hands and the travelling twin aged 10 orbits (years) by the Earth-Sun orbit counting clock in his hands, as Einstein and his followers would have us believe, then the twin on Earth counted 50 Earth-Sun orbits but the travelling twin counted only 10 Earth-Sun orbits when he returned to Earth to meet his brother. Really? Einstein dictates whether or not the Earth-Sun executes an orbit! The Earth-Sun either executes an orbit or it does not. Both twins must count the same number of Earth-Sun orbits. Since according to Einstein and his followers they do not do so, Einstein and his followers are talking rubbish. The twins must age the very same. Thus, when the twins are reunited, t1 = t2.

Since both clocks are identical, m2 = m1 and L1 = L2 before the journey to the distant star begins. After the fantastic voyage to a distant star, when twin 2 gets back to Earth is m1 = m2 and L1 = L2? Of course they are the same. So there is no absolute change in the masses or lengths of the clocks. The alleged change in mass and length occurs only during the journey when the clocks are in constant relative motion, and this occurs only between observers, not at each observer. S1 claims m2 > m1 and L2 < L1 and S2 claims that m1 > m2 and L1 < L2 during the constant relative motion. But S1 sees no change in the mass and length of clock t1 and S2 sees no change in the mass and length of clock t2. Also S1 sees no dilatory change in his clock time t1 and S2 sees no dilatory change in his clock time t2. Nonetheless, Einstein and his followers claim that time actually changes so that after the journey t1 > t2. Earth-Sun therefore executes different numbers of orbits, according to Einstein and his followers, according to the motion of observers. That’s impossible!

This result means that time dilation is false and so Special Relativity collapses entirely.

Stephen J. Crothers

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@Stephen, I do NOT agree that time dilation is false, but I DO agree that time is simply the motion of objects. See link below.

1 Recommendation

No, there is no solid counter-argument to Dingle's objection. The simple reason is that one is forced to invent an asymmetry in a situation which could be made totally symmetric, with or without an acceleration assumed.

V. Toth: You say "whichever clock spent more time accelerating is the one that shows less time elapsed".

1.) How do you measure "accelerating time"? By which clock do you measure this time?

2.) We can make, or at least imagine, symmetric situation, when all observers have

identical "accelerating time". What does the theory say what will the relevant clocks show when they meet again?

Vladimir: You measure the time spent accelerating and the magnitude of acceleration in the proper time of the accelerating clock. This is unambiguous and the result is relativistically invariant. Regarding your second question, if the two observers' world lines are symmetrical, when they meet, their clocks will be in synch, and the result will be consistent with all the measurements they made during their respective journeys, including times when they observe their own clocks running faster or slower than the other's.

A proper formulation of the question is whether or not it is possible to connect two events (the initial and final time when the two observers meet) with two world lines that are symmetrical, yet the proper time measured along them is different. Apart from the signature of the metric (pseudo-Euclidean vs. Euclidean) this is the same as drawing two symmetrical curves between two points on a sheet of paper and measuring different path lengths along the two: clearly impossible.

V. Toth:

In the analogy with two symmetrical curves between the two points, I, being on the one curve and you on the other, can always say that the infinitesimal parts (straight lines) of your curve are shorter than the parts of my curve. However, when we meet at the other point, I am forced to conclude that the total sum of the parts of your curves equals the total sum of the parts of my curve.

Vladimir: I'm attaching a simple spacetime diagram that shows two observers/clocks along symmetric worldlines that do meet again, and how equal proper time intervals for the right-hand-side ("red") observer map into unequal proper time for the left ("blue") observer (dark red dots on the left). Dashed lines represent constant-time hypersurfaces in the instantaneous frame of the "blue" observer, corresponding to the equal proper time intervals measured by the "red" observer. (I.e., ticks of the "red" observer's clock.)

When they meet again, the two observers will have measured the same amount of proper time with their respective clocks. (The same also works of course if we interchange the "red" and "blue" observers.)

While the drawing is "freehand", so to speak, I think it is reasonably accurate. Needless to say, I am not suggesting that a diagram should be taken as a substitute for rigorous proof, but rigorous proofs can be found in any decent relativity textbook... here, I think, the issue is how one can appreciate the seemingly counterintuitive nature of these apparent paradoxes, and in that sense, I hope my diagram helps.

- 10.84 KBworldlines.jpg

V. Toth:

Thank you for the drawing. Yes, the parts are UNEQUAL, but you cannot say that the red observer line sees every part of the blue observer line as being shorter than the corresponding part of the red observer line. See what happens between the last point and next to the last point---there the red observer sees LONGER blue lines, while the other blue lines he does see shorter that his red lines. Of course, that must be so in order to obey unmerciful math, but to obey special relativity the red observer has the right to say that EVERY blue part is shorter than the corresponding red part. And that is the impossibility. The possible way out of this is to say that the right to say that every blue part is shorter than the corresponding red part is a mere illusion. It's is not real. However, if we adopt such a view, then we have problems with the experimental results which say that the time dilations and length contractions are real, not just an illusion.

Vladimir: Your argument would be valid if the red observer and the blue observer were moving inertially. However, in that case they would never meet again. So some non-inertial motion (curved worldlines) is needed to steer them to meet again, and that negates your reasoning: some of the "red" intervals along the "blue" worldline will in fact be shorter than the corresponding "blue" intervals. I assure you, the math is self-consistent, fairly trivial to those who practice it regularly, and works very well (and indeed, it is confirmed by many experiments that involve, e.g., spacecraft and precision clocks, such as GRACE, GRAIL or the GPS/Glonass constellations), even though it may appear counterintuitive at first.

V. Toth:

Infinitesimal parts of the curves are straight lines, i.e. the two observers are LOCALLY inertial. There are no such things as the "curved infinitesimal elements". Correct me if I'm wrong. Dividing two perfectly symmetrical curves into equal and arbitrary big number of (almost) infinitesimal parts does the job--we have the correspondence between two inertial frames (or two equally imperfect but almost inertial frames). The observers can agree at the beginning of their journeys that they will , by their own initially synchronized clocks, each divide their journeys into equal number of paths. They can also agree that they will at every, or at least at some (numbered) parts of their journeys measure the relevant quantities belonging to the other observer, as seen in their own frames. And what will they see? One will see that the lengths of the other observer is shorter and clocks slower. At each and every part of the journey. At no points of the two symmetrical journeys there is any asymmetry.

So, are the length contractions and time slowing at each and every part of the journeys real, or not?

Discussing the consequences of the existence of a rest frame in the universe

Discussion

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- Asked 6th Jun, 2022

- Sydney Ernest Grimm

The detection of the existence of the Cosmic Microwave Background Radiation (CMBR) from everywhere around in the universe has puzzled theorists. Not least because of the discovery of a Doppler effect in the data that can only be interpreted as direct related to the velocity and the direction of the motion of the solar system. But if it is correct we have to accept that there exist a rest frame in the universe. Actually we can determine the existence of absolute space and that is not in line with the “belief” of most of the theorists.

There is another method to verify the results: counting the numbers and measuring the brightness of galaxies from everywhere around. The first results – using visible light – were not convincing. But a couple of days ago The Astrophysical Journal Letters published a paper from Jeremy Darling with results that were obtained with the help of radio waves: “*The Universe is Brighter in the Direction of Our Motion: Galaxy Counts and Fluxes are Consistent with the CMB Dipole”* (https://iopscience.iop.org/article/10.3847/2041-8213/ac6f08).

In other words, it is real. We can determine the existence of "absolute space". Moreover, we know from set theory (mathematics) that absolute space and phenomenological reality must share the same underlying properties otherwise we cannot detect the existence of absolute space. The consequence is that absolute space has a structure too, because phenomenological reality shows structure.

None of the grand theories in physics is founded on the structure of absolute space. Therefore we are facing a serious problem in respect to the foundations of theoretical physics (the conceptual framework of physics).

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Experimental findings and developments in theoretical Physics sug-gest the need to modify special relativity. Doubly special relativity is one of the most widely studied approaches. However, on closer in-spection, it appears that the theory is just a non-linear disguise of the standard Lorentz transformation[1] if the classical picture of space-tim...

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