Special Relativity and Twin Paradox

Subject: Proposal

From: Nick Percival <NickP@SMNT.com

As if you did not have enough to do already, I have another project to

suggest. I believe that it is compatible with your book and with your

objective of clarifying misunderstandings, etc about SRT. Your role

could be limited, if you choose, to "kicking things off".

Clearly, the Twin Paradox represents one of the key areas of SRT where

misunderstandings still flourish. I have focussed on this topic for 35

years and, in addition to the long paper that I sent you, I had

previously published another long paper examining the many alleged

physical causes for the net time difference that had been put foward in

the literature and described the logical problems associated with each.

I would suggest a formal debate of the Twin Paradox. The debate would be done by exchanging documents - primarily via the Internet. Sections 1 & 2 of the paper I sent you would give you a feel for how the exchanges might proceed.

Of course, Dingle and McCrea, et al have previously had a debate in the

published literature. However, as Chang of Harvard's extensive review

paper notes, basically both sides were talking at cross purposes and

Dingle's chief question was never addressed. In addition, such a

difficult topic, which is now made more diffcult by both sides having

firmly entrenched biases, requires many, many exchanges and relatively

rapid turnaround (e.g., a week and far faster turnarounds than a debate

where each round must be published in a journal).

I would very much like to be the proponent of "SRT per se can NOT

explain the net time difference in the Twin Paradox." and use the "new"

methodology as described in my paper. You or someone of your choosing or some volunteer can take the other side. (I think the debate needs to be one against one as trying to do it by committee versus committee would bog things down. Perhaps, a second round could be done by committee.)

I would also suggest that someone play the role of "moderator". Their

role would NOT be to adjudicate a winner. Their role would be to try and keep the debate on a sharp focus. Their task would be to say things like "Debater A, I don't think that your response addresses Debater B's

point, please address the point." Up to three people could play the role

of the moderator. Anyone playing the moderator role needs to be

"neutral" and open minded. They can not think that the Twin Paradox is a dead issue that was resolved long ago nor can they think that SRT should be junked and that the Twin Paradox is just one of its many problems.

Your role could be limited, if you choose, to being the organizer or the

catalyst. However, I think your role in creating your book puts you in a

unique position to be the catalyst for a succesful and fruitful debate.

Clearly, there is much to be learned; 1) either we gain a better

understanding of what people like myself are confused about or 2) we

gain a better understanding of the limitations of SRT in explaining the

net time difference and maybe much more. What's needed is more structure to keep the debate on track and to force true communication between both sides.

Even though you are very busy now, I would argue that we should NOT wait until your book is published, etc. to start the debate. If you can just

start the process and delegate tasks to others, it should not take much

of your time. I have volunteered to take one side of the debate; surely

there must be someone who will volunteer now to take the other side.

That's what we need to get started! I have been preparing for this for

35 years - I can NOT delay too many years more. Of course, there are

many possible variations on the structure I described above for the

debate. What do you think?

* * * * *

Dear Percival,

I am leaving today for a couple of weeks, and unfortunately I have yet a lot of things to do. Anyway, I wished to thank you immediately for your proposal, even if I am not sure about its success. As you know very well, the greatest majority of people do believe that there is nothing new to be said about the Twin paradox, so I do not know whether I will be able to find some people willing to spend his time for defending the thesis opposite to the one you claim.

I am not sure to have explained myself well, I shall try again. You are willing to defend the thesis: "SR per se cannot explain the net time difference etc." (I have still to study your paper!); I, and with me many other people, would say that SR "explains" this difference in a "geometrical" way, in the same way as for instance gravity is "inserted" (I cannot find a better term) in the space-time geometry in General Relativity. Was this an explanation of gravity? In the same way, the net time difference we are talking about is a function of the "curvature" of the trajectory of the twins; geodesics have null curvatures, so the twin which follows the "straight" trajectory in Minkowski space-time has a different time than the twin which does not follow a geodesic path. Would you say that this is an "explanation"? This is the point about which I think it would be difficult to obtain a general agreement, apart the problem of the twins in itself…

Anyway, I agree with the "rules" you propose for the debate, I shall contact some competent people asking what are they thinking about, and we shall hear again when I shall be come back,

best wishes, Umberto Bartocci

* * * * *

Dear Percival, [...]

As far as my personal opinion is concerning (but I still could not study your paper!), I do not find ANY circumstance which can put SR in trouble with the "twin" argument. You claim that there are "many alleged physical causes for the net time difference ... in the literature", which is true, but just shows that SR has not been generally well understood. Then your main thesis is that: "SRT per se can NOT explain the net time difference in the Twin Paradox", which, as I alreday told you, does not seem true to me. The fact that people is not able to find the correct one, or that people disagree on the "solution", is just a "sociological" remark: it does not mean that the solution does not exist.

There are many things which I would like to communicate to you about this question, and I shall try to write at least some of them, hoping that you would at least appreciate my effort. Excuse me if I shall be very "didactic", saying things that you know very well, but this can be the only way in which I accept your "challenge", and start taking "the other side" of the discussion you propose. Then we shall see if there are other people willing to join me in this kind of "defense" of SR - which is, please believe me, a rather unpleasant role for me! - or you, in the defense of your point of view.

First of all, let me say that I am sending to you in attachment, as a jpg file, a diagram which tries to establish symbols and definitions, that we should use in the future.

[See Diagram]

You have a first inertial reference frame R in the 2-dimensional Mnkowski space M (I shall assume throughout geometrical unities, namely c = 1), in which there are TWO still "observers" A and B, say A in the origin, and B in the point L 0. The clocks of A and B are synchronized in such a way that the one-way speed of light in R is always c, and that is OK.

Then you can see in the diagram the world-line of a THIRD observer A*, which is NOT an inertial one, and does NOT "belong" to any inertial reference frame of M. A* can be thought of as a TWIN of A: they meet, and their clocks are synchronized, in the event:

e1 = (0,t0 = L/k - (sqr(L^2-g^-2))/k)

(all coordinates are meant with respect to R; g is any real parameter such that g 0 and 1/g < L; k is any real parameter 0 < k < 1).

After the event e1 in which they are together, A* moves away from A, of course with respect to R, with speed:

v = -k(kt-L)/sqr(g^-2 + (kt-L)^2)

from the instant t0 to the instant t1 = L/k, in which A* reaches the greatest distance from A, that is to say L-1/g. After this time t1, A* comes back towards A, and the two twins meet again in the instant:

t2 = L/k + (sqr(L^2-g^-2))/k [namely, in the event e2 = (0,t2)].

All the trip of A* lasted, from the point of view of R, a time:

Dt(A*) = [2*sqr(L^2-g^-2)]/k .

Let us remark that:

1 - The speed of A* is always "almost equal" to k, in the first part of the trip, and to -k in the second, the acceleration appearing only in a "small" neighborhood of the instant t = L/k.

2 - g is a measure" for the acceleration of A*. As a matter of fact, the ordinary acceleration a of A* in R is given by the expression:

a = -k^2/(g^2*[g^-2 + (kt-L)^2]^3/2) =

= -g*k^2/([1 + g^2*(kt-L)^2]^3/2) ,

which is always "almost equal" to zero, except that in a neighborhood of t = L/k, in which instant its value is -g*k^2 . Furthermore, the ds^2 of the 2-acceleration of A* is given by:

g^2*k^4/([1 + g^2*(1-k^2)*(kt-L)^2]^3) ,

and this function is once again always "almost equal" to zero, except that in a neighborhood of t = L/k, in which its value is of course g^2*k^4.

Other Remark: In my paper on "Misunderstandings..." [see point 2 in the page Foundations of Physics, or even the more extensive paper in the point 4 of the same page, but this is written in Italian] I assumed k = 1, which makes the whole matter - particularly in the next development - much simpler. In this case, while te ordinary acceleration in R is again almost always equal to zero, except that in a neighborhood of t = L, in which its value is -g, the ds^2 of the 2-acceleration (a space-kind vector) is a CONSTANT: g^2; namely, the module of this vector is the constant g, and A* could be said "uniformously accelerated"...

3 - When g approaches infinity, the motion of A* approaches the two line segments, corresponding to the motions of TWO different inertial observers, A' from the event (0,0) to the event (L,L/k), and A" from the event (L,L/k) to the event (0,2L/k); 2L/k represents indeed the maximum possible value for the duration Dt(A*) of the trip of A*, as measured in R.

The so-called "twin paradox" relies on the ascertainment that the PROPER TIME DT(A*) of A* during this trip, namely the duration of the trip from HIS point of view (the measure given by HIS clock), is LESS than the previous one:

DT(A*) < Dt(A*) .

Remark - The exact value of DT(A*) is given by a rather complicated integral, which is equal to 1/k times the integral from -sqr(L^2 - g^-2) to sqr(L^2 - g^-2) of the function:

sqr[(1 + g^2*(1-k^2)*x^2)/(1 + g^2*x^2)] ,

but it is not so important to find the exact expression of DT(A*).

We can call the difference Dt(A*) - DT(A*) 0 the TWIN EFFECT, and this value is maximum for g approaching infinity, namely:

2L/k - [2L*sqr(1-k^2)]/k .

It is perhaps instructive to remark that the curvature of the hyperbolic branch in its vertex [the event (L-1/g,L/k)], when g increases, approaches infinity, independently from k (even if k is almost zero, this curvature goes to infinity!); this shows in some sense that the "curvature" of the broken line made up of the two line segments (0,0)(L,L/k) and (L,L/k)(0,2L), in the common event (L,L/k) should be considered as equal to infinity, and the curvature has physically the meaning of an acceleration (a mathematician would rather say that the curvature is NOT defined on that point).

The so called "Dingle paradox" is easily RULED OUT by SR, just by observing that A* is not inertial (nor it is the PAIR A', A"!), and that there is NO symmetry at all in the Minkowski space between A and A*. Namely, from the point of view of A*, it is true that A is moving away, and that then, after reaching a maximum distance from A*, he is coming back towards A*. But it is even true that:

- A* DOES NOT SEE ANY "INVERSE" TWIN EFFECT,

in the sense that the clock of A does NOT appear to A* slowed down, due to the speed of A with respect to him, as conversely the clock of A* appeared slowed down to A, due to the speed of A* with respect to him. This computation, when k is different from 1, is more complicated than the one I offered in the recalled paper, but the substance does not change.

In conclusion, the twin effect, is ASYMMETRIC, as it MUST be. The reason for the net time difference you suggested to to discuss, does not properly rely on "speed" (which, as a matter of fact, could be just relative, and then "symmetric"), but only on the "acceleration parameter" g, or if you prefere in the curvature of the world-line of the accelerated twin; namely, at last, on the "geometry" of the Minkowski space-time. It is an ABSOLUTE fact that A is inertial, his world-line is a geodesic, and that A* is NOT.

Remark - Some people could be misleaded by the fact that the QUANTITATIVE size of the "twin effect" does depend from BOTH the parameters: speed AND acceleration. This is not at all "strange", on the contrary it is quite ununderstandable. As Silvio Bergia correctly remarks [S. Bergia, M. Valleriani, "Relativita' ristretta: convenzione o nuova concezione del mondo?", Giornale di Fisica, XXXIX, 4, 1998, p. 205], when you have a car driving straight away, then you have some kinetic energy depending only on its speed with respect to the road. When you turn suddenly and crash against a tree, then the quantitative effect of the crash does depend from that energy, namely from that speed, but the very "physical reason" for the crash was that sudden (infinitesimal) turning of the wheel.

This is for me ALL one can say about the twin effect, and I understand that, for the moment being at least, I cannot be but disappointing for you, since I have not yet studied your paper, and I do not know which are your objections. Perhaps I will change idea after that reading, but I frankly doubt it, and I admit that this is a feeling "a priori" - which I always reproach in different circumstances to my interlocutors! - but when a question is settled down, it is settled down.

I would rather suggest to discuss another feature of the previous set-up, namely to discuss only THE FIRST HALF OF THE TRIP, in which we can substitute to A* the observer A' (for g going to infinity). In this case, A' is NOT a "twin" of A, but it seems to me that the essence of the "paradox" remains quite UNCHANGED.

We could call a "twin effect" the simple DILATION OF TIME which is now registered by B - and NOT by A! - which is given by the difference of the times:

L/k - [L*sqr(1-k^2)]/k ,

exactly the half of the maximum twin effect we were previously investigating.

Let me explain more clearly the situation (once again, excuse me for my being didactic, but I write even for other people - or still better, I shall put this discussion in my web site; perhaps, some other unknown people would be willing to join us in this discussion).

B "knows" - he belongs to the reference frame R, which gives a ccordinate system for the whole space-time - that A' is going with speed k; he knows that the clock of A' was marking zero exactly as the clock of his "companion" A, which is at a distance L from him, when A' was in front of A; B should then expect that te clock of A' would mark L/k when A' will be in front of him, since k is the speed of A', and L the distance that he travels; surprisingly enough, according to Einstein, B sees instead that the clock of A' is marking less than L/k, namely [L*sqr(1-k^2)]/k !

Would you agree that this is an EFFECT exactly of the same kind as before?! This is the phenomenon one must explain in SR; this is, with your words, "the key area of SRT where misunderstandings still flourish"; and NOT, in my opinion, the twin effect PER SE.

Why do not then discuss the "physical reason" for THIS net time difference? Why choose instead the effective twin case, which does necessarily introduce acceleration? I understand of course that in this case the ascertainment that the two clocks are marking a different time after reunion can be made by the SAME observers which at the beginning remarked instead that the two (identical) clocks were marking the same time, bu the question is the same, and it is the very heart of SR conception of time - quite against the ordinary intuition of it. Furthermore, in the case I suggest to study, there is indeed COMPLETE SYMMETRY between the inertial reference frame R, in which we suppose that A and B are still, and the inertial reference frame R' uniquely associated to A' (if we suppose that the clocks of A and A' are synchronized in the event (0,0)). Here it lies the essence of "relativity of smultaneity", the effect clearly does depend now just on speed, which is relative, and then the effect MUST BE symmetric, even if it does not possibly SEEM symmetric. I always wondered why people is so struck by the twin effect, and not by this "simple" dilation of time; or even by LENGTH CONTRACTION, which is in some sense, as we shall see, exactly the same thing as the dilation of time. After all, space and time are now "the same thing", as one can well realize when he is assuming, as we did, that c = 1 (and c is dimensionless).

Remark - And, after all, this case is exactly the famous muon experiment: a muon is "alive" in front of A, and it is still alive in front of B, while the "life" of a muon which is still in the reference frame R is less than the time L/k which the travelling muon takes going from A to B!

Let me come back to the case I am proposing to discuss instead than the twin effect, and to analyze some of its symmetric features.

As we have said, A and A' are synchronized in (0,0), but when A' is in front of B his clock marks [L*sqr(1-k^2)]/k , while the clock of B is marking L/k. So B thinks that the clock of A' is slowing down by the effect of the speed k. Well, first of all let us see how things appear from the point of view of A'. A' sees A passing in front of him with speed -k, and when B passes next in front of him, with the same speed -k, A' must of course ascertain as before that while his clock is marking the value [L*sqr(1-k^2)]/k, the clock of B is marking instead L/k, which is greater, and not smaller, than the value marked by the clock of A'.

Why that? Is there any difference with the previous case, when B was looking at a smaller value of the clock of the travelling observer with respect to the value marked by his own clock?

The explanation is of course very simple, and quite SYMMETRIC, and does involve of course length contraction - showing that as we have said time dilation and length contraction are in some sense the same thing. The distance between A and B (the "train's length", if you wish) is, from the point of view of R', L*sqr(1-k^2) instead than L, and then A' must obviously expect that the clock of B measures

[L*sqr(1-k^2)=length]/[k=speed],

which is exactly the time that HIS clock is marking when B is in front of him. As far as that, everything is OK. But as we have said, the clock of B marks instead L/k, which is a bigger value than the time measured by the clock of A'. Some people sees a possible asymmetry in that: while B was concluding that the moving clock is slowing down, A' should conclude at the contrary that the moving clock is going faster. But this is not correct, since from the point of view of the "expectations" of A', he must foresee the value that the clock of B is measuring, when he is in front of him, taking into account TWO informations:

- first, the value that he sees when B is in front of him;

- second, the value that the clock of B WAS measuring in the same instant when A is in front of A'!

Of course, this second value must be measured from the point of view of R', namely from a "companion" B' of A', belonging to his same reference frame, which can see the required value of the clock of B when B is in front of him at the time t'=0, the time of R'.

Well, if we do the easy computations, when A and A' are one in front of the other, this companion B' of A', which sees B in front of him at the instant t'=0, must be of course at a distance L*sqr(1-k^2) from A', and at the time t'=0 we have the event (L*sqr(1-k^2),0) in R', which does correspond in R to the event which has L as space coordinate (as it must be!), and t° = kL/sqr(1-k^2) as time coordinate.

[we are of course making use of the Lorentz transformations:

x = (x'+kt')/sqr(1-k^2), t = (t'+kx')/sqr(1-k^2)]

So, A' "knows" that the clock of B was measuring that value t° when A is in front of him, and would then EXPECT that the clock of B is measuring:

t° [initial time] + [L*sqr(1-k^2)]/k [time required by the trip] =

kL/sqr(1-k^2) + [L*sqr(1-k^2)]/k = L/[k*sqr(1-k^2)]

when B is in front of him.

But, as we know, this is not true: contrarily to A' expectations, the clock of B is measuring instead L/k, which is, AS IN THE PREVIOUS CASE, a value LESS than the previous one. So even A' thinks that the clock of B is indeed slowing down, and exactly in the same proportion as before, since:

L/k = L/[k*sqr(1-k^2)] (expected time) * sqr(1-k^2) ,

which is a PERFECT SYMMETRY.

Of course, I could say much more about this case*, trying for instance to explain why, according to SR, it is enough the relative speed k in order to produce the effect we are discussing; or to discuss, which is even more important, if we are in front of something which could be said just coventional (namely, depending from the "appearance", from the conventions we use in our measurements), or not. That is to say, if we are in front of an objective FACT of Nature, or not. This is a difficult question indeed, in which there is no complete agreement even by people handling correctly relativity. "Today" I would rather say that, even if experiments of the kind of the extendend life of the speedy muons seem undoubtedly in favour of relativity (apart the fact that there exist other possible explanations for their outcome, from the point of view of an aether theory), and do not SEEM to depend on any sinchronization, we are here in front of a "conventional" feature, since in all case we must measure lengths, speeds, and so on. But this does not mean that SR is just a "convention"; I agree with Bergia and others that, if SR is true, it teaches us something new, it gives us a new image of the world. The non-conventional essence of relativity relies, in my opinion, in its first postulate, the Principle of relativity [see papers at the points 8, 10 in this same page, or point 5, which is written in Italian], but here the discussion would become harder, and I stop it, waiting for your reply to these first comments...

Thanks for the meditation hint you have offered to me, and best wishes

yours most sincerely

Umberto Bartocci

* For instance, I find interesting to discuss even another TRUE paradox: suppose to have a natural phenomenon, for instance the lighting of a candle, which lasts a time D less than L/k , but more than

[L*sqr(1-k^2)]/k . Well, when A' is in front of A, A' lights the candle, and when A' is in front of B the candle is still lighting on, while the elapsed time, from the point of view of B (of R) is L/k, and then greater than the candle's life. This is a "paradox" we have solved before. But we can think of another possibility: B lights an his own candle, not when his clock is measuring zero (we would then fall in the previously discussed case), but when he directly sees the candle of A' lighting on. That is to say, from SR point of view, when his clock (the clock of B) measures L (=L/c). At the time L+D (of the clock of B), the candle of B is extinguished; is it possible that the candle of A' is still lighting on when A' passes in front of B, at the time L/k? This would be truly paradoxical, and this in fact IMPOSSIBLE, as one could easily check...