GOING TOWARDS THE END

OF THE DISCUSSION ABOUT THE TWIN PARADOX

- - - - - **Section I - Percival's comments**

1 - Subject: Re: It gets better

Date: Tue, 30 May 2000 12:30:51 -0400

From: Nick Percival <NickP@SMNT.com>

Organization: Semantech

To: umberto bartocci <bartocci@dipmat.unipg.it>

References: 1 , 2

In your paper "Most Common Misunderstandings About Special Relativity", your section 5 on the Twin Paradox, you

"explain" the net time difference in PROPER TIMES in terms of the asymmetry of the travelling twin's turnaround

acceleration and that fact that one twin's path is geodesic and the other's is not. This, in itself, is probably the most

common misunderstanding about SRT.

On the one hand, both the travelling twin's turnaround acceleration and that fact that one twin's path is geodesic and the

other's is not geodesic are indeed relevant, but they play a relatively minor role in understanding the physics of how the net

time difference in proper times accumulates. This is suggested when we modify the classic Twin Paradox scenario to

produce the accelerationless scenario, which you said you were familiar with, which also has the three "triplets" all

confined 100% to geodesic paths AND YET we get the SAME NET TIME DIFFERNCE IN PROPER TIMES (aside

from the contribution to net time difference due to acceleration per se that can be made to be arbitrarily small with respect

tot the total net time difference by increasing the length of the trip). The exercise I propose in the section "It gets better .."

confirms the above referenced suggestion - I look forward to your participation.

The physics community's "consensus" explanations of the Twin Paradox scenario have asserted that 1/2 the net time

difference accumulates during the outbound leg and 1/2 accumulates during the inbound leg of the round trip (Note:

Explantions of the net time difference as due entirely to the turnaround acceleration by individuals such as Max Born have

never received "consensus" support). Hence, many have said that the well known muon experiments that you reference

give experimental confirmation of the expected results for a Twin Paradox scenario. Hence, it surprised me to see that you

discuss the muon experiments in quite different terms than you discuss the Twin Paradox.

My explanation of the Twin Paradox (and your other related paradoxes) and the muon experiments is the same and is

straightforward and without a hint of paradox. I may be misreading your latest email, but I am surprised at the lack of

consistency in your treatment of seem to be the same physical phenomenon. I tend to agree with your basic thinking vis a

vis the muons, but I say that counter-intuitively not only is your Aether-like construct compatible with SRT, but a

necessary consequence of SRT. (Note: In my explanation, there is neither an Aether nor a unique frame of space-time but

rather the unique frame is the rest frame of the gravitational field which plays a physics role not unlike the rest frame of an

electromagnetic field.)

When I said the Twin Paradox debate was plagued with "qualitative, vague, semantic, ambiguous, marshmellowy"

exchanges, I was being somewhat glib. Some of the best minds of physics have written on both sides of the debate. It

would be more accurate to say that the debate has been an exercise in "talking at cross purposes" with paradoxical issues

never being addressed - as Chang in his extensive review also concluded. The problem is that each side puts down what

they think is the right answer and since there are no mistakes in the math, each side remains convinced that they are

correct and are amazed at how dense the other side is in not being able to follow their logic. The problem is that neither

issues rised by Dingle are finessed rather than answered - the meat of the subject does not appear in the equations per se

- the important points do not get put on paper and there are problems of ambiguous physics and semantics.

I labeled one section of my letter "Qualitative, Vague, Semantic, Ambiguous, Marshmellowy Remarks", I was again being

glib - this time about my own comments. In that section, I raised some questions vis a vis your section 5 explanation of the

Twin Paradox that have proven impossible for others who share your views to answer. You have not addresssed those

points and I do NOT ask you to for the reasons discussed in the previous paragraph.

I decided to try a new technique for discussing the Twin Paradox. Rather than stating my views and expecting them to

convince others, I ask others for there views and then analyze them - not with words, but just with numbers - then the

logical contradictions become more evident, more compelling and more convincing. I look forward to your participation in

the exercise I propose in the section "It gets better .."

Thanks

2 - Subject: Re: It gets better

Date: Fri, 16 Jun 2000 00:34:35 -0400

From: Nick Percival <NickP@SMNT.com>

Organization: Semantech

To: umberto bartocci <bartocci@dipmat.unipg.it>

References: 1 , 2

Dear Prof. Bartocci:

Thanks very much for your reply and for putting so much effort into it and for participating in my little project - I greatly

appreciate it.

I usually respond very quickly to correspondence, but the past ten days have been unusually busy so I have only just

read your email of June 5. I hope to be able to spend tomorrow producing a quality response. A few quick thoughts for

now.

1) I have been pursuing two debates with professors other than yourself using "my new methodology". One of these, an

American who specializes in SRT and is spending a few years at Oxford, is in agreement with you - although there may be

some nuances of difference. (I have gotten 3 letters from him in the past couple of weeks which I have not even had time

to open yet.) WE are at about stage 5 whereas you and I are only at stage 1. His initial responses paralleled yours so I

will be able to just edit my letters to him to send to you. He, like yourself, is most gracious and a pleasure to debate. He

has said (I believe sincerely and not just politely) that my questions to him have been most thought provoking and

challenging - though, like you, he was quite confident that it was a simple matter. I hope that my future thoughts will be of

interest to you also.

2) The 3rd professor that I am debating with holds a mutually exclusive view of how to explain the net time difference than

you do (or than above referenced "American in Oxford" professor) and he initially held that view with what he himself

characterized as arrogant confidence. This leads me to wonder if you and your "Italian colleague" are in agreement on the

Twin Paradox. By "Italian colleague" (I'm sure you have many), I mean the one you refer to as "one of the Italian

physicists most competent in relativity" who thinks the Twin Paradox is "not worthwhile to discuss". I wonder if he would

agree with your statements:

"but in this "short" ACCELERATION PERIOD his twin A

goes from G1 to G3, spending 7500060 sec!! Then B goes from E3 to E4, spending once again 2500000 sec, and A goes from G3 to E4, spending 1250000 sec.

Summing up, from the point of view of B, the most of the time which A has spent in his life - the very reason for his (A) becoming asymmetrically older with respect to him (B) - it has been spent exactly in that short acceleration period."

(I know that the above quote is taken out of context so by itself it would mean little to your "Italian colleague", but I

would be most interested if he is in agreement with the essence of what you are saying above.)

It has been my experience that about only 1 in 10 explain the net time difference in terms of the acceleration period -

even though at the moment 2 in 3 of my debating "opponents" are of that school. I suspect that your "colleague" may well

not be in agreement with you. I would be most interested to know if you were in agreement and I would think that you

would be interested also.

3) My apologies - I was not clear. When I said "With Part II of the attached, you will see the issues beginning to become

sharper", I did NOT mean that anything in that note would make things sharper. Your response "I do not see anything

"sharper" ..." is 100% correct. What I meant to say is it BEGINS the phase in our correspondence where things will get

sharper - although I realize you may well remain skeptical on that point.

I will try to move the ball ahead and send you something substantial soon.

Thanks again, Prof. Bartocci!.

3 - Subject: Re: It gets better

Date: Fri, 16 Jun 2000 00:51:57 -0400

From: Nick Percival <NickP@SMNT.com>

Organization: Semantech

To: umberto bartocci <bartocci@dipmat.unipg.it>

References: 1 , 2

One quick post script I must add.

You quote Delbert Larson as saying:

"If we try to come up with theoretical arguments to show how special

relativity is wrong [or "incomplete", or whatever else you like], we will

lose. SR has been studied and celebrated for generations now. If there was a theoretical flaw it would have been found long ago ... from a mathematical (and therefore theoretical) sense, special relativity is completely consistent and correct. Arguing that point merely shows a misunderstanding of the theory ... it takes a long time, generally, to find out why the opponent is wrong. But after days of thought one generally (90% or more) finds that the opponent is wrong".

While I agree with Delbert Larson that SRT is 100% correct, I find his argument to be totally without substance. He

simply states that we who hold SRT is correct contend that those who disagee with us are wrong.

But that's not the point. I believe that you added the parenthetical phrase: [or "incomplete", or whatever else you like]

This was clearly a logical error. It is a non sequitor, It does not follow. Let's say that Larson made a more astute argument

and actually did logically prove that SRT is correct. It does NOT then follow that therefore SRT is complete.

There is major difference, a qualitative difference, between what I am saying and what Dingle and most other SRT

critics have been saying.

4 - Subject: Re: It gets better

Date: Sun, 18 Jun 2000 12:58:15 -0400

From: Nick Percival <NickP@SMNT.com>

Organization: Semantech

To: umberto bartocci <bartocci@dipmat.unipg.it>

References: 1 , 2

My reply is attached in MS WORD 95 format.

Please let me know if you have any problems with the file and I will resend.

J N Percival Round 2 - 6 18 00.doc

Name: J N Percival Round 2 - 6 18 00.doc

Type: Microsoft Word (application/msword)

Encoding: base64

Dear Prof. Bartocci:

Thanks again for your major effort in responding to my letter of 5/9/00. At last, I can spend time to respond to you. I have printed out your email and will refer to the page numbers on my printout as a way of giving you an approximation as to how far down your email I’m referring, but I will also give the first words of the paragraph I refer to.

I start my analysis of your response about 1/3 the way down what is for me page 2 of 7 with your paragraph beginning "Your computations in Point II are indeed correct …"

We are seemingly in agreement __until__ your paragraph beginning "The simple answer is .." (at the top of my page 3 of 7).

First, let me clear up some terminology.

1) When you referred to the events in my scenario, you were correct except that you forgot that E0 to E1 was the period of initial acceleration and not the constant velocity part of the outbound leg. Hence, all the segments go out of sync with my original definition. Let me know if I am incorrect on this. You can refer to my original document emailed to you on 5/11/00. But let me put, on the next page, a synopsis of the segments as defined by their boundary events so that we can be in sync in our terminology. (If you wish to change the scenario, I probably will have no objection as I can handle any modification that does not change the essence of the scenario. However, I want to be sure that we are in sync in our terminology).

2) As you point out, all the events are defined in terms of the travelling twin. (The initial and final event, E0 and E6, are also defined in terms of the stay-at-home twin.) You also correctly point out that E1-E5 are NOT defined in terms of the stay-at-home twin. Hence, you recommend that we not refer to the stay-at-home’s "E1-E5", but rather refer to the stay-at-home’s "F1-F5" that are seen by the stay-at-home’s as being simultaneous with E1-E5 respectively. (Please note that I said essentially the same thing in my 2^{nd} paragraph of my page 7: "*All the events will be defined "in terms of the travelling twin" (e.g., the travelling twin begins accelerating, the travelling twin ends accelerating). Hence, for the stay-at-home twin, we will really be giving the number of ticks that occurred on his clock between events that he says are simultaneous with our defined events.)*

I will, therefore, try rewriting and give a short hand version of that table below in which I will explicitly write in F1-F5. I am also sensitive that for each segment in the table where I have accumulated times for both the stay-at-home twin and the travelling twin that we are not comparing "apples to apples". However I hope that the explicit inclusion of F1-F5 will be sufficient reminder of that fact so as not to induce errors.

I will put that summary table here, on a page by itself, in case you would like to have it as a convenient summary reference page.

__ E0 to E1:__ (Initial acceleration)

__ E0 to F1:__ (Viewed by stay-at-home twin as simultaneous with Initial acceleration)

Ticks Accumulated Between the Two Named Events

Stay-At-Home Clock: 30

Travelling Clock: Approximately 24

__ E1/F1 to E2/F2:__ (Constant Velocity part of outbound leg)

Ticks Accumulated Between the Two Named Events

Stay-At-Home Clock: 5 Million

Travelling Clock: 2.5 Million

__ E2/F2 to E3/F3:__ (Deceleration wrt the stay-at-home frame – 1

Ticks Accumulated Between the Two Named Events

Stay-At-Home Clock: 30

Travelling Clock: Approximately 24

__ E3/F3 to E4/F4:__ (Re-acceleration wrt the stay-at-home frame – 2nd part of turn around )

Ticks Accumulated Between the Two Named Events

Stay-At-Home Clock: 30

Travelling Clock: Approximately 24

__ E4/F4 to E5/F5:__ (Constant Velocity part of outbound leg)

Ticks Accumulated Between the Two Named Events

Stay-At-Home Clock: 5 Million

Travelling Clock: 2.5 Million

__ E5/F5 to E6:__ (Final deceleration to come to rest in the stay-at-home frame)

Ticks Accumulated Between the Two Named Events

Stay-At-Home Clock: 30

Travelling Clock: Approximately 24

Summary

Total Ticks Accumulated Between the Start and End Events E0-E6

Stay-At-Home Clock: Approximately 10 Million

Travelling Clock: Approximately 5 Million

Observed Time Vs Proper Time

These two very different constructs are continually confused in SRT - especially when trying to reconcile the net time difference with SRT per se. In the context of SRT, one is often forced into making this tacit switch in order to come up with the right answer.

Let me repeat my breakdown of the Twin Paradox here:

The Twin Paradox can be viewed as containing the following elements.

- How many ticks (proper time) does A’s (and B’s) clock accumulate for each segment of the round trip.
- What are "A’s observations of B’s clock during the trip" and what are "B’s observations of A’s clock during the trip".
- How does one reconcile the symmetric observations of A and B (i.e., each observes the other’s clock to be running slow during the constant velocity parts of the trip) with the asymmetric result of a net time difference in proper time.

I’m trying to focus just one part 1), namely, how does the net time difference accumulate.

Clearly, in general, when we study special relativistic phenomena, observed times are NOT accurate descriptions of how proper times are really accumulating. For example, when A and B are travelling at constant relative velocity, both *observe* the other’s clock to be running slow. Clearly, both sets of observations can not both be accurate descriptions of how proper time is accumulating on A’s clock versus B’s clock and vice versa.

Hence, where I have a problem with your description of "how and where and when" the net time difference accumulates is when you say (top of my page 3 of 7 in your paragraph beginning "The simple answer .."), "Then B goes from E1 to E3 (*really E2 to E4 in terms of my, Nick’s, notation*), spending only 48 sec of his time, but in this "short" ACCELERATION PERIOD his twin A goes from G1 to G3, spending 7500060sec!!"

It seems clear to me that you are NOT directly describing how the net time difference accumulates. Instead, it seems to me, that you are describing what B *observes *to be happening on A’s clock during B’s turn around acceleration. In other words, it seems, to me, your sentence is addressing parts 2) and 3) of the Twin Paradox (see above) when the topic is to understand part 1).

ON the one hand you seem to be explicitly saying that the net difference in proper times is due to the turnaround acceleration and yet you explanation is given in terms of what the traveling twin observes about the stay-at-home twin.

Let’s say that you are correctly describing what B observes about A’s clock during B’s turnaround. You also seem to be explicitly saying that that __also__ accurately describes how the net time difference accumulates (which is the question I’m asking). That implies, in effect, that when B is doing his turnaround acceleration, his clock his going at (48)*/*(7500060ths) the rate of A’s clock or something equivalent.

**Please confirm that that is what you are saying or alternatively please restate what it is that you are saying.** If that is NOT what you are saying, then I do not know what you mean by the net time difference accumulates during the turnaround acceleration. If it is what you are saying, I have a number of very strong physics arguments against that view (I will not burden you with them unless/until you confirm that the above is indeed your position.)

Appendix - The Physics of the Net Time Difference

It is clear that the net time difference represents a real physical phenomenon, namely, a difference in proper times accumulated for two identical clocks that have traversed different paths through space-time. I want to see if the physics of this net time difference of proper times can be consistently and logically explained with currently accepted physics.

When I talk about being clear and explicit about the physics of how the net time difference accumulates, I mean not only that one should use equations and/or numbers, but also that one should be clear as to the *physical meaning* of both. For example, be clear as to when those numbers refer to *proper* time and when they refer to *observed* time. But I also mean something beyond that which I discuss below.

To better illustrate this additional point, let me discuss an analogous scenario which *is* correctly understood in terms of current theory.

Let’s switch topics and discuss General Relativity. Let’s have two twins, a stay-at-home twin and a travelling twin, who start together in a region with a strong gravitational field that has a sharp gradient.

The travelling twin moves rapidly (but not at relativistic speeds) to a new position where the difference in gravitational potential requires that his clock’s rate slows to one half the rate of the stay-at-home twin’s, and let’s say that he stays there, according to his clock, for one hour. Then, we have the travelling twin move rapidly to a new position where the difference in gravitational potential requires that his clock’s rate slows to one quarter the rate of the stay-at-home twin’s, and let’s say that he stays there, according to his clock, for one hour. Finally, the travelling twin returns rapidly to the starting position to reunite with the stay-at-home twin.

We can not only accurately say how much the total net time difference will be, but we can accurately describe how much of the net time difference accumulated for each segment of the round trip. We also have at least a first level description of the physics of the cause of the net time difference, namely, the difference in gravitational potential.

We can provide that accurate, unambiguous and detailed description because we *do* know the physics of that scenario. We can also provide that accurate, unambiguous and detailed description from the point of view of the stay-at-home twin or the travelling twin or some third party.

I’m looking for the same level of physics explanation of how the net time difference accumulates in SRT’s Twin Paradox scenario.

For example, if someone tells me that their view is that the net time difference (primarily) accumulates during the constant velocity parts and that it is due to the traveling twin’s relative velocity wrt the stay-at-home twin, then I know what physics he has in mind when he’s discussing how the net time difference accumulates. I do not agree with that explanation, but I know what he’s talking about and that’s what I’m asking to be given and that’s what I want to analyze.

If one says that the net time difference is (primarily) due to the turnaround acceleration, then I assume 1) that he does NOT believe in the effect described in the previous paragraph and 2) he has in mind some physical effect such as the acceleration induces a powerful gravitational field which causes the accelerated clock to slow down versus the stay-at-home twin’s. I don’t care what if any specific physical effect (e.g., induce gravitational field) you may have in mind (we can call it the X factor), but are you of this school of thought? Your words seem to me to be saying so.

I’m looking for a description of how the net time difference accumulates like the one’s described in this section. I am NOT interested (in terms of our discussion) in discussing how to reconcile "how the net time difference actually accumulates" with "A’s and B’s observations of each other’s clocks during the various segments".

5 - Subject: PS

Date: Sun, 18 Jun 2000 21:50:01 -0400

From: Nick Percival <NickP@SMNT.com>

Organization: Semantech

To: umberto bartocci <bartocci@dipmat.unipg.it>

As a post script to my last email and attachment, I want to remind

you that the number of clock ticks that I assign to the segments of the

round trip, for each twin separately, are what I have observed to be the

majority opinion among physicists (about 90%). Those numbers do NOT represent my view and you should change them to be consistent with your view - as I said originally, I just assigned the consensus numbers as I thought an example would help explain what I was looking for. Most physicists hold that the net time difference accumulates

(primarily) during the constant velocity legs of the round trip and

that, for the classic scenario, the outbound leg contributes the same

amount to the net time difference as the inbound leg and that the net

time difference accumulates at a constant rate during those constant

velocity segments.

Obviously, the number of clock ticks between two events on a twin's

worldline represent his proper time between those events.

- - - - - **Section II - Interlude**

Dear Percival,

just a quick answer for saying that I received everything OK, and that I shall be more precise, as you seem to wish, as soon as I shall be able to find some free time. But le me even repeat frankly that I do not believe that this would be a very usefully spent time! For instance, I see that you insist on arguments like "the majority opinion among physicists", but scientific truth is no matter of democracy! I could agree a priori with you, with no need of any "sociological" proof (for this same reason, I shall not bother my colleague asking him whether he agrees or not with my analysis!), that most people do not understand the geometry of SR, since it is highly counterintuitive with respect to the ordinary mind models of space and time. For instance, I see that you introduce now a "theoretical" difference between Observed Time and Proper Time, saying that "These two very different constructs are continually confused in SRT". As a matter of fact, are just these confusions which are at the origin of all "misunderstandings" (including my own misunderstandings!), and I cannot do too much in order to eliminate them, even from my mind or from the mind of other people (you do not imagine with how many people I am always discussing similar things). There is not an "observed time", but just a COORDINATE TIME, and in order to introduce a coordinate time one must first rigorously introduce a coordinate system. But which is the "natural" coordinate system associated to the accelerated twin?? The plain mathematical truth is that it does not exist one, since only inertial observers have a natural coordinate system, and this FACT (a mathematical fact) implies that the word "observed time" has not univocal meaning (about this argument you should study O'Neill's "Semiriemannian Gemetry"). If you change systems, event this time would change, only proper time remains unchanged; furthermore one should not even forget that some systems could be defined only "locally", which means that they would not possibly contain in their coordinate domain ALL events. This is the heart of the dilemma which is troubling you since many years…

I hope not to be disappointing, but I really do believe that what I have written before is definitive, and that I do not believe that this effort of exchanging long mails would produce (perhaps) more than a better reciprocal understanding of something which is already very clear IN ITSELF, for people who really understand relativity (I am sure to know some of them, whose opinion would be able to put a final end to this discussion, but I do not dare bothering them…)

Best wishes from yours most sincerely UB

P.S. Perhaps you will be interested in this quotation from a mail I received very recently:

"I'm very much interested in the ''equivalence principle'', and I'm planning since a long time ago to write something on the subject. I already have some material. According to my point of view such a principle does not exist. Gravity is characterized by the Riemann tensor and this is zero or it is not. Accelerated frames in a gravity free region are not equivalent to a real gravity field. Physicists always confuse local (which means in an open set) with punctual…"

I add that for saying to you that actually I do not understand myself completely the (mathematical and physical) meaning of the "principle of equivalence" (I would say today that I rather agree with the opinion of the previously quoted colleague), and that I am asking to friends in all the world in order to get help in this effort of further understanding.

Subject: Re: PS

Date: Mon, 19 Jun 2000 23:48:54 -0400

From: Nick Percival <NickP@SMNT.com>

Organization: Semantech

To: umberto bartocci <bartocci@dipmat.unipg.it>

References: 1 , 2

See comments below in BLUE.

umberto bartocci wrote:

Dear Percival,

just a quick answer for saying that I received everything OK, and that I

shall be more precise, as you seem to wish, as soon as I shall be able to

find some free time. But le me even repeat frankly that I do not believe

that this would be a very usefully spent time! For instance, I see that you insist on arguments like "the majority opinion among physicists", but scientific truth is no matter of democracy!

No. Instead, I was saying that virtually all physicists, including yourself and your "Italian colleague", think that the Twin

Paradox is a "dead issue". Yet, this consensus group holds at least two mutually exclusive views on the topic. Hence, at

least one part of the consensus group is wrong! I thought it might make you more receptive to discussing the Twin

Paradox if it was pointed out that others who share your view that it is a "dead issue", and whom you respect, disagree

with you as to how its resolved.

I could agree a priori with you,

with no need of any "sociological" proof (for this same reason, I shall not bother my colleague asking him whether he agrees or not with my analysis!), that most people do not understand the geometry of SR, since it is highly counterintuitive with respect to the ordinary mind models of space and time.

For instance, I see that you introduce now a "theoretical" difference

between Observed Time and Proper Time, saying that "These two very different constructs are continually confused in SRT". As a matter of fact, are just these confusions which are at the origin of all "misunderstandings" (including my own misunderstandings!), and I cannot do too much in order to eliminate them, even from my mind or from the mind of other people (you do not imagine with how many people I am always discussing similar things).

There is not an "observed time", but just a COORDINATE TIME, and in order to introduce a coordinate time one must first rigorously introduce a coordinate system. But which is the "natural" coordinate system associated to the accelerated twin?? The plain mathematical truth is that it does not exist one, since only inertial observers have a natural coordinate system, and this FACT (a mathematical fact) implies that the word "observed time" has not univocal meaning (about this argument you should study O'Neill's "Semiriemannian Gemetry"). If you change systems, event this time would change, only proper time remains unchanged; furthermore one should not even forget that some systems could be defined only "locally", which means that they would not possibly contain in their coordinate domain ALL events. This

is the heart of the dilemma which is troubling you since many years…

I only want to discuss proper time. (It seemed to me that you were not giving the proper time for the stay-at-home twin,

but rather were giving what the travelling twin observed to be the elapsed time for stay-at-home twin during the travelling

twin's acceleration. My papers refer to coordinate time for inertial observers where that term is appropriate.)

I hope not to be disappointing, but I really do believe that what I have written before is definitive, and that I do not believe that this effort of exchanging long mails would produce (perhaps) more than a better reciprocal understanding of something which is

already very clear IN ITSELF,

Sorry, I was so lengthy. Just give me the proper times for the two twins for the segments as defined. I will send my

analysis - it will require a little time on your part to analyze - but it's focussed - it will be to the point - not an endless or

prolonged exchange. Thanks

- - - - - **Section III - (Final?!) answer**

Dear Percival,

here it is in attachment my final and complete answer to all your questions (I hope not to have made too many computational mistakes!).

Best regards, UB

Dear Percival,

first of all let me say that I am sorry that I have to answer to you in a foreign language - moreover, which I do not even know well enough - because first I am very slow in aswering, second I cannot answer in the "light" way which I would wish to, and I am forced to be rather cool, "academic" (but don't forget that I write even for other friends, or students, interested in such arguments)...

Anyway, I am happy because I see that our discussion is starting to "converge". As a matter of fact, I have seen that you accepted my suggestion to be more precise specifying the events which define the "segments" which are object of your attention, both for twin A (the stay-at-home twin, the inertial twin) and for twin B (the accelerated twin, the one which comes back), since the greatest problem in relativity is exactly to compare the "same" time interval for different observers, or different coordinate systems - even if you did not accept until now my suggestion to consider this issue a DEAD ISSUE, as it really is!

For this reason, I did a major effort in order to answer to ALL questions you asked until now, adding something more to the considerations I have previously made. I shall divide my reply in sections, I hope that everything will be understandable, and that I have not made any computational mistake!

* * * * *

SECTION 1 - INTRODUCTION

Let me start from your own "program", according to your three points, giving a general answer point by point:

> 1 - How many ticks (proper time) does A’s (and B’s) clock accumulate for each segment of the round trip.

Answer: this is very easy, once you specify exactly of which "segments" you are talking about. Every question in special relativity (SR), which is quite a simple mahematical theory - and so it is COMPLETE, at least no less than other mathematical theories - has an answer, perhaps more counterintuitive than it is usual when one has left apart ordinary space and time categories...

> 2 - What are "A's observations of B's clock during the trip" and what are "B's observations of A's clock during the trip".

Answer: this is very easy too, even if one must be more careful with the DEFINITIONS, and accurately distinguish between proper time and "true observed time", as I shall show you soon.

> 3 - How does one reconcile the symmetric observations of A and B (i.e., each observes the other's clock to be running slow during the constant velocity parts of the trip) with the asymmetric result of a net time difference in proper time.

Answer: I have already shown to you in my last mail how this "reconciliation" is well possible, and easy, using only the PROPER TIME, and I am sorry that you apparently have not been persuaded. Anyway, in this reply I shall give you TWO different reconciliations, using different TIMES...

In order to prove this last assertion, I shall first repeat my general argument about PROPER TIMES, going immediately to the point at which you seem more interested, namely the "table" of the CORRESPONDING proper times. As a matter of fact, you asked:

> Just give me the proper times for the two twins for the segments as defined...

and then:

> I will put that summary table here, on a page by itself, in case you would like to have it as a convenient summary reference page.

__ E0 to E1:__ (Initial acceleration)

__ E0 to F1:__ (Viewed by stay-at-home twin as simultaneous with Initial acceleration)

Ticks Accumulated Between the Two Named Events

Stay-At-Home Clock: 30

Travelling Clock: Approximately 24

__ E1/F1 to E2/F2:__ (Constant Velocity part of outbound leg)

Ticks Accumulated Between the Two Named Events

Stay-At-Home Clock: 5 Million

Travelling Clock: 2.5 Million ... etc.

I shall give a complete answer to your question, but first let me repeat that there is no need to introduce SO MANY events. I already suggested to you to introduce just ONE "smoothing", but then you replied:

> When you referred to the events in my scenario, you were correct except that you forgot that E0 to E1 was the period of initial acceleration and not the constant velocity part of the outbound leg. Hence, all the segments go out of sync with my original definition.

My comment is that I did not "forget" anything, I was just eliminating useless prolixities, and trying to go STRAIGHT TO THE POINT, studying only the ESSENTIAL acceleration period, the one in which the travelling twin, B, is starting to come back towards A. The first two acceleration periods you introduce are inessential, you do not really need to introduce THREE of them. Perhaps, your approach would seem, but just seem, more "physical", but the "substance" of matters would not change. You can well suppose that A and B just meet TWICE, B having a constant uniform speed with respect to A both at the beginning (first meeting), both at the end (second meeting), and that is enough.

So, let me precisely define the "scenario" which I suggest to discuss together from now on. Introduce an inertial observer A in 2-dimensional Minkowski space time M, and introduce a Lorentzian coordinate system (x,t) associated with A, call this coordinate system R. Suppose that A is defined by the worldline x = 0. Call E0 the event which has R-space-time coordinates (0,0), and introduce an observer B defined by the following worldline: given any "length" L > 0, any speed v > 0, any quantity e less than L/v (this e stands for the more common epsilon, a quantity which will be thought of as "infinitesimal"), B does coincide with the inertial observer B' given by x = vt, for any t until L/v - e. In the event E1, given by R-coordinates (L-ev, L/v - e) B starts decelerating, going from speed v to speed 0 at the instant L/v, and at the distance

L - ev/2 from A (I am arranging things in a quite symmetric way). I shall call this event E2. From this event on, B starts accelerating again, towards A, until it reaches the speed -v in the event E3, which is given by R-coordinates (L-ev, L/v + e). From now on, B (the worldline of B) does coincide with the inertial observer B'', given by x = -vt + 2L, and so it meets again the worldline of A in the event E4, given by R-coordinates (0,2L/v) (see Figure I in attachment).

I suggest to you this scenario, first because it is simpler than yours, second because I would rather prefer to find general formulae, and not just simple numbers, whose meaning would not be so clear at first sight. If you wish, just choose values for the aforesaid parameters L, v and e, and you will get of course any number you wish. This observation leads me to the following:

DIGRESSION - The starting point is something that you asserted before, namely:

> You give a lot of equations and very little explicit physics

but I hope that you are not meaning that the difference between Physics and Mathematics, for instance in the case we are discussing, would be just to introduce more acceleration periods, or to give determinate numbers in place of parameters! As a matter of fact, the only difference which is really meaningful, is not between Physics and Mathematics, but between rigorous Physics and imprecise, poor, approximated, Physics, which becomes sometimes even a plainly wrong Physics, as many of my anti-relativistic friends usually do. I do not say that because I am a mathematician, and I consider Mathematics, or even worst mathematicians, AT THE TOP. Quite on the contrary, mathematics is not at the top, but AT THE GROUND. Mathematics belong to everybody, as the unique common thought-language of Mankind, so even physicists MUST BE mathematicians, or, in order to be more accurate: MUST BE MORE than mathematicians!

Mathematics is just a pure "shape", and the only one difference between Mathematics and Physics could be found in the "contents", in the "purposes" of the research. The mathematician studies mathematical phenomena "per se", the physicist makes use of Mathematics in order to study natural phenomena: the difference is quite great, but the amount of mathematics, of RIGOUR in going on with the study, is the same.

Let me give immediately an example of this difference coming from our present discussion. There is one point in which you are of course right, but in which there is no possibility to give an univocal answer, because the question is not a mathematical one, and only mathematical questions can have a definitive answer. It is when you ask "HOW" (would it be the same as to ask "WHY"?!) the time difference accumulates. Of course this is very different to ask "where", or "when", or even "if" a difference does accumulate or not, or if this difference is in LOGICAL contradiction or not with other consequences of the theory (this is of course IMPOSSIBLE, I am no afraid to claim it A PRIORI). I mean now in which "parts" of the trip, or in which "segments", according to your approach, even if, when introducing these concepts, one should be very careful. As a matter of fact, one is now making use of terms which belong to the ORDINARY LANGUAGE, and so they cannot be meaningful but with respect to the ORDINARY space and time categories. This is the origin of all misinterpretations of SR!

If you wish to investigate instead which are the possible "physical reasons" to which the time dilation phenomenon is due, then an answer in my opinion CANNOT BE FOUND. Nobody in the world could tell you why clocks "go slower" or "sticks" contract in the "mathematical model" of space-time built by people like Einstein, Minkowski, Hilbert, Weyl, Pauli, and many others… Moreover, this is a phenomenon which in some sense is not even "true", it is not even a correct description of the relativistic conception of "space" and "time" (which we must not forget are relative, and not absolute). Clocks just SEEM to go slower, sticks just SEEM to contract, with respect to TWO different observers in the SAME inertial coordinate system - so one needs in all THREE observers, one moving with respect to the other two, which are instead standing still one with respect to the other. Furthermore, as I already said, the twin case is not the more interesting case to study with the purpose to examine possible "physical causes" for the relativistic time dilation: more interesting would be in my opinion the "muon set-up"...

Anyway, I always would avoid arguments as:

> he has in mind some physical effect such as the acceleration induces a powerful gravitational field...

As a matter of fact, I have not in mind such arguments. Moreover, in my opinion, they would not mean anything. What is a "field"? What is a "gravitational field" in absence of matter? Ancients philosophers clearly indicated the epistemological risk to explain unknown things using even more unknown concepts: IGNOTUM PER IGNOTIUS! Do not forget that before (or even above, again "human" time and space!) I started doing just simple GEOMETRY of Minkowski space-time (and more "Geometry" I shall do very soon!), or if you prefer KINEMATICS, which is meant just as a DESCRIPTION of the motion, and not as an analysis of its CAUSES, which not even DYNAMICS is in my opinion able to do ENTIRELY! Is the concept of "Newtonian force" a really satisfactory explanation for the causes of some motion? After all, we must not forget that there are in Newtonian mechanics "motions", with respect to some coordinate system of the Newtonian space-time, which have not CAUSE at all! Moreover, for instance, as you know very well, in GR the concept of gravitational force disappears, and even in SR all "forces", namely 3-forces as magnetic or electric forces, are just relative to the observers.

The only possible way to do "more Physics", as you apparently ask, is to put ourselves in the etheric conception of the "space", which I fully accept (a space which is not a simple geometrical space, but a real physical space, endowed with physical characteristics). Then things are "physically" much simpler: there is only one possible force, which is due to a perturbation of the ether, which could be possibly due to many different causes. A force cannot be but a VIS A TERGO, a "push" from the back, a push that the space itself exerts on bodies embedded on it, etc. etc., one could go very far with this "fluid-dynamic" analogy...

* * * * *

SECTION 2 - FIRST "RECONCILIATION" USING PROPER TIMES

Coming back to the point I was discussing before this long digression, I cannot but repeat what I told you last time:

- "Your computations [at the time I wrote these words, it was your point II, now it is the aforesaid list] are indeed correct - there is no doubt about that, after all they are very simple - but the point is: what do they teach to us? What can we learn from them?".

As a matter of fact, even now your computations are correct, but if you hope to find by means of them something "interesting" concerning the fundamental point 3 in your argument, you will be deluded. I restate again what you have written: " How does one reconcile the symmetric observations of A and B ... with the asymmetric result of a net time difference in proper time". I shall show once again how this reconciliation is very easy, hoping that you will realize at last that there is no hope to find anything "strange", or "unexpected" (for instance, that SR is in some sense "incomplete") doing all the computations you are doing.

The fundamental point is that you have now correctly examined TWO corresponding decompositions of the travels of A and B, but, in order to do a COMPLETE analysis of the situation, you must introduce and study THREE of them! As I already tried to explain, GIVEN at the beginning the events E0, E1, etc. in the life of B (now I am accepting your decomposition), you have the corresponding events F0 (= E0), F1, etc., in the life of A, the ones which, as you say, are "viewed by stay-at-home twin as simultaneous" with E0, E1, and so on. But then you MUST even introduce the events G0, G1, etc., which again belong to the life of A, and are defined as the events in the life of A which are simultaneous with the events E0, E1, etc. FROM THE POINT OF VIEW OF B, the travelling twin.

Remark - As far as this makes sense, since in order to define this simultaneity one would need in fact a coordinate system associated with B, which is globally an accelerated observer, and so things are not so simple (see Section 4 of my paper on "Most common misunderstandings..."). In our case we are essentially using the possibility to "split" B in the two inertial observers B' and B''…

Once you do that, you fill find that all problems disappear, and that:

i) from the point of view of the inertial twin A, the definitively ASYMMETRIC net time difference accumulates during the "constant velocity parts of the trip" of its twin B:

ii) from the point of view of B, which is "symmetric" with respect to A, but ONLY during the constant velocity parts of the trip (so in some sense it does SPLIT in two different inertial observers, the aforesaid B' and B''), the asymmetric net time difference does accumulate during the "short" acceleration part of its travel, whatever short this part you can think of (although "infinitesimal" you take the parameter e, the time elapsed for A in the corresponding "segment", from B's point of view, will be always greater than a fixed determinate quantity)!!

Remark - One could of course ask: why THREE decompositions, for TWO twins? And why TWO for A, and only ONE for B? The answer is very simple, and it does depend from the ASYMMETRY of the situation, poor Prof. Dingle notwithstanding. Since A is inertial, there is no possibility to introduce ONE canonical decomposition of its travel, since the choice of the events in its life (A's life) would be purely arbitrary. But since B is accelerated, one can correctly introduce, as you do, the canonical decomposition of its travel separating the constant velocity parts from the accelerated ones, and then this decomposition is the only one possible we can start with. Going on, corresponding to this CANONICAL decomposition of B's travel, one can introduce TWO other decompositions, because there are TWO twins, and then you have a first decomposition in the life of A taking the events in its life which are simultaneous to E0, E1, etc. ACCORDING TO A, or the events in this same life which are simultaneous to the previous ones ACCORDING TO B. [In what follows I shall introduce even two more decompositions for A's travel (corresponding to the "true observed time"), which will bring the total to FIVE...].

Let me now find the explicit formulae which give a complete answer to your question in the scenario I have previously specified.

We start from the events E0, E1, E2, E3, E4 in the life of B, which are respectively given by R-space-time-coordinates:

(0,0), (L-ev, L/v - e), (L - ev/2, L/v), (L-ev, L/v + e), (0,2L/v).

To these events Ei, i=0,...,4, do correspond events Fi in A's life (the ones which are seen by A, or better by the coordinate system R, as simultaneous with Ei), namely:

(0,0), (0, L/v - e), (0, L/v), (0, L/v + e), (0,2L/v)

(obviously, F0 = E0, F4 = E4).

TABLE 1 - Proper time elapsed for A

from F0 to F1: L/v - e

from F1 to F3: 2e

from F3 to F4: L/v - e

Total time: L/v - e + 2e + L/v - e = 2L/v, OK.

TABLE 2 - "Corresponding" proper time elapsed for B

from E0 to E1: (L/v - e)*sqr(1-v^2)

from E1 to E3: I shall call this time T(e), which is the result of the integral, from (L/v - e) to (L/v + e), of the function sqr(1-w^2), where w, which is actually a function of the R-COORDINATE TIME t, is the speed of B, with respect to A, from E1 to E3. In all cases, we obviously have: T(e) < 2e*sqr(1-v^2) .

from E3 to E4: (L/v - e)*sqr(1-v^2), again.

Total time: DT(B) = (L/v - e)*sqr(1-v^2) + T(e) + (L/v - e)*sqr(1-v^2) = 2(L/v - e)*sqr(1-v^2) + T(e).

This value is greater than 2(L/v - e)*sqr(1-v^2), and less than

2(L/v - e)*sqr(1-v^2) + 2e*sqr(1-v^2) = 2(L/v)*sqr(1-v^2), OK.

Consequence ("twin paradox"): at reunion in E4, the clock of B (which we suppose is marking 0 in E0) marks DT(B), which is less than 2(L/v)*sqr(1-v^2), and so it is less than the value marked by the clock of A (once again, we suppose that this clock is marking 0 in E0 = F0), which is 2L/v.

Remark (proof of previous assertion i):

the B-proper time from E0 to E1 is less than the A-proper time from F0 = E0 to F1;

the B-proper time from E1 to E3, which is T(e), is less than the A-proper time from F1 to F3, which is 2e;

the B-proper time from E3 to E4 is less than the A-proper time from F3 to F4 = E4.

Giving only TWO Tables, as you did, and I did until now, one does NOT answer to the SYMMETRIC requirement of your point 2. As a matter of fact, you intended to study not only B from the point of view of A, but even A from the point of view of B: where is the Table which describes A's travel in terms of its twin B? This is surely NOT Table 1. Anyway, one can compute this "missing table" very easily indeed, and after having achieved this task, one will also obtain the "reconciliation" you were looking for in point 3.

So, we must introduce the following Table 3, introducing the events G0 = E0, G1, G3, G4 = E4, in the life of A, the ones which are seen respectively as simultaneous by B' (which is a "part" of B) with E0 and E1 (these are indeed events in the life of B'), and then by B'' with E3 and E4 (these are events in the life of B'').

G1 is defined as the intersection between the worldine of A, x = 0, and the (space-kind) line which is Lorentz-orthogonal to x = vt in E1, namely: v^2*x - v*t + (1-v^2)(L-ev) = 0 , that is to say:

G1 = (0, (1-v^2)(L/v - e)) .

In a similar way, we find:

G3 = (0,2L/v - (1-v^2)(L/v -e)).

From these expressions we get at last:

TABLE 3 - "New" proper time elapsed for A (description of A's travel from the point of view of B, better of B' and B'')

from G0 to G1: (1-v^2)(L/v - e)

from G1 to G3: 2L/v - (1-v^2)(L/v - e) - (1-v^2)(L/v - e) =

= 2Lv + 2e*(1-v^2)

from G3 to G4: (1-v^2)(L/v - e)

Total time: (1-v^2)(L/v - e) + 2Lv + 2e*(1-v^2) + (1-v^2)(L/v - e) =

= 2L/v , OK!!

Remark 1 (proof of ii)- The proper time spent by A from G1 to G3, the "acceleration period" of B, is not at all infinitesimal, since it is always greater than 2Lv!

Remark 2 - The first and last value in the previous Table 3 coincide exactly with the relativistic time dilation values obtained from the first and the last values in Table 2:

(1-v^2)(L/v - e) = [(L/v - e)*sqr(1-v^2)]*sqr(1-v^2) .

This identity is the mathematical expression of the PARTIAL SYMMETRY of the argument we are discussing.

This is the real HEART of the problem, either in your approach, or in Dingle's approach; this is the real origin of all troubles which people usually show in wrong discussions of the twin issue. One has indeed an ASYMMETRIC result coming back from a situation which APPEARS, if not completely symmetric, at least LARGELY SYMMETRIC, from a quantitative point of view. The "originality" of your approach, in my opinion, consists in the proposal to examine more deeply this quantitative side, trying to show that it was so "small" that it could not account for the rather great final asymmetric difference, but this is not true…

* * * * *

SECTION 3 - A GENERAL INTRODUCTION TO THE SECOND "RECONCILIATION", USING OBSERVED TIMES

I have done all previous computations explicitely, even because you have said that:

> It seemed to me that you were not giving the proper time for the stay-at-home twin, but rather were giving what the travelling twin observed to be the elapsed time for stay-at-home twin during the travelling twin's acceleration.

This is of course NOT CORRECT, and for TWO reasons. The proper time for the stay-at-home twin is the most important thing to evaluate in all this EXERCISE, and I have done it TWICE in my previous mail. Let me give you just an example of that, I wrote for instance:

- "Then B goes from E3 to E4, spending once again 2500000 sec, and A goes from G3 to E4, spending 1250000 sec."

In this sentence, the 2500000 sec are B's proper time, and the 1250000 sec are A's proper time. I did already give to you ALL proper times of A in the "new" scenario I suggested, which is exactly your scenario, but for a simplification (one smoothing of the corners, instead of three).

The second, and most important, reason for which your previous assertion is not correct, is that your are speaking now of the time which "the travelling twin observed...", but I was not speaking of any really OBSERVED TIME, both in my previous mail and in the present one - at least until now! I was just speaking of PROPER TIMES, computed in events which are considered simultaneous with the events E0, E1 etc. either from A or from B, NO TRUE OBSERVED TIME AT ALL.

As a matter of fact, this is in agreement with one of your recent assertions:

> I only want to discuss proper time...

and with your own "tables". When you write:

>** E1/F1 to E2/F2:** (Constant Velocity part of outbound leg)

Ticks Accumulated Between the Two Named Events

Stay-At-Home Clock: 5 Million

Travelling Clock: 2.5 Million

you are giving proper times of A and B respectively. You are not giving neither the time that B observes in the clock of A when B goes from E1 to E2, nor the time that A observes in the clock of B when A goes from F1 to F2.

The point is that we must be VERY CLEAR about the meaning of the words that you used in your previous point 2, of expressions such as: "A's observations of B's clock during the trip" etc..

We have only two possibilities. Either we are in front of a pure "linguistic ambiguity", and the word "observed time" means only, as we did until now, that we compare proper times in corresponding events, namely events in the life of some observer which are considered simultaneous with other events in the life of another observer. Or we mean something else, and we intend to speak literally of the time which some observer DIRECTLY sees, in some event of its life, as marked in the clock of another observer (in this case, the first observer just sees the "past", as we see the past of the stars we are watching in the sky during the night).

The first case would correspond to all times we (I) have computed until now, and if you intended only to do that, then we can put an end to our discussion.

Anyway, this possible "ambiguity" suggests to me to go on with the analysis, giving other Tables for the TRUE observed time - Tables which will be in any case rather instructive. After all, you did even dedicate a whole section to the problem of the DIFFERENCE between "Observed Time" and "Proper Time", asserting that:

> These two very different constructs are continually confused in SRT...

Of course, I shall defend the opinion that it is not so, and that perhaps, as usual, you could be right only on a "sociological" ground (the "sociological" issue about the "twin paradox" seems more interesting than the strict scientific - mathematical - new discussion of the question). That is to say, your are much likely right asserting that "in general" (or very often) people, physicists, even in published scientific papers, make confusion between these two very different concepts, BUT THERE IS NO CONFUSION AT ALL IN THE THEORY! As always, the only problem is to give precise DEFINITIONS, namely to do rigorous physics against poor, approximated physics...

* * * * *

SECTION 4 - ABOUT THE ONLY ONE POSSIBLE PRECISE MEANING OF "OBSERVED TIME" IN SR, AND AN (UNEXPECTED?!) CONSEQUENCE CONCERNING THE NON-EXISTENCE IN SOME CASE OF A TIME DILATION FOR THE "TRUE" OBSERVED TIME…

In the 2-dimensional Minkowski space-time M, we consider an inertial observer A, and we introduce a Lorentzian coordinate system R, with coordinates (x,t), for M such that A is represented by the equation x = 0. Then we consider another inertial observer B, not parallel with A, such that for instance A and B share the (unique) event (0,0). Let us suppose that x = vt is the equation representing B, and that v > 0 (this is not an additional true condition, since one could always change x in -x).

Well, given any event E in the life of A, this has coordinates (0,T), and T is indeed a proper time for A, the unique A's proper time T such that, at the event E0 = (0,0), T = 0 (one must not forget that proper time is defined just up to additive translations). Corresponding to this event E, there is a unique event H(E) in B's life, namely the only one event in B's life which is "judged" by A (by the coordinate system R associated to A) as simultaneous with E. Of course, H(E) has coordinates (vT,T), and we can ask what is the proper time of B at the event H(E). Of course, this proper time can be introduced only if we choose an "origin" for this B's proper time, but we ask the previous question intending that this proper time is defined as the only one which is equal to 0 in the event E0 = H(E0), exactly as in A's case. Well, it is well known that:

(1) proper time of B at the event H(E) = T*sqr(1-v^2) .

The previous identity (1) is the so-called relativistic time dilation, in the sense that B's "travel", from the event E0 to the event H(E) (of course in the case T > 0), lasts T from the point of view of the coordinate system R, or which is the same from the point of view of the COORDINATE TIME t, and T*sqr(1-v^2), from the point of view of B's proper clock.

Coming back to the main theme of our discussion, which does concern OBSERVED time, we can say that:

- (1) is the value which is OBSERVED as the time marked in B's clock by the "companion" A' of A, given by the equation x = vT, in the instant T of its clock, the clock of A' (we suppose of course that all observers "parallel" to A have made a synchronized choice of their proper times); A' is defined as the only one companion of A which sees B IN FRONT OF IT at the instant T.

So, in some sense, (1), which is a proper time for B, is even an observed time, BUT NOT FOR A, rather for this companion A' of A, or if we prefer for the coordinate time associated to the coordinate system R. But the point which we want to discuss now is: what is the indication of the clock of B, at the end of its travel, as TRULY OBSERVED by A?

Now we must be careful, because the answer to this question cannot be literally but the previous (1), imagining for instance the following situation. B is doing its (eternal, as far as the "past" is concerning) travel in space with respect to A; B has a clock; when B passes in front of A this travelling clock is marking 0 exactly as it is marking A's clock; then B goes away from A, and it crushes down against an obstacle at a distance L from A (the distance is obviously relative to A, a coordinate distance with respect to R). The crush happens of course at the instant L/v of the clock of A, which means that the aforesaid companion A' of A which is in front of the crush, at a distance L from A, sees that the crush happens at the time L/v of its own clock, the clock of A', while the clock of the poor B was marking (L/v)*sqr(1-v^2), the value given by (1). Of course, this value is even the value that will be observed by A as marked by B's clock in the very moment of the crush, but of course it will be seen by A not at the instant T of A's clock, but at the instant

L/v + L = T + vT = T*(1+v), this delay being due to the time L (= L/c, I always suppose that c is a pure number equal to 1), which a "backwards" photon (worldline: x-L = -t) spends in order to arrive from the event crush to the worldline of A. Of course, knowing that the "signal" of the event crush arrives with some delay, A could compute afterwards where (and when) the crush exactly happened.

Remark - This set-up suggests quite a good Gedanken-Experiment which could (SHOULD) be done (no TWIN at all, which is much better!). Imagine that a clock is moving with some uniform speed v, with respect for instance to the Earth's surface, and then that it is made crushing against a wall. We can ask, which is the last value which is marked by this clock? (I mean that the clock stops working in the crush, and that everybody can see afterwards the last value that it marked). Knowing the (relative) speed v of the clock, and the (relative) length L of its travel, SR would foresee that this last value is exactly (1), namely (L/v)*sqr(1-v^2), and not the simple L/v, which any classical theory would have foreseen. Today I believe that SR is perhaps correct, in its prevision of the time dilation given by (1), when using atomic clocks on Earth's surface, but I would even bet on that:

- using a DIFFERENT clock (I mean a clock working with some other principle, for instance an old "mechanical" clock), the outcome of this same experiment should be very likely different (even if the clock is quite "good" as an atomic clock, when standing "still" on Earth's surface);

- even if the outcome of the experiment with atomic clocks would coincide with the relativistic prevision (1), then it would not be so when doing the same experiment on a train, or something else, effectively moving with respect to Earth's surface with some uniform speed w;

- the outcome of this experiment should depend on the height above the sea level, exactly as in the case of the repetitions of the Michelson-Morley experiment which has been performed by Miller.

Well, coming back to our argument, using ORDINARY LANGUAGE (which is always the main TRAP for physicists who believe that they can avoid the rigour of mathematics!) one could say that (1) is indeed an "observed time", in the sense that:

- it is of course an observed time for B, since it is a proper time for B;

- it is even an observed time for A' (at R-coordinate time T)

- it is an observed time for A (at A's proper time T(1+v); at the instant T of its proper time, A just sees the "past" of B, the crush has not happened yet in A's eyes),

and here it is the origin of all "confusions", between proper times, coordinate times, observed times...

As a matter of fact, we can also introduce a "TRUE" OBSERVED TIME for A with respect to B's clock, meaning by that the value that A sees marked in B's clock at any instant T of its own proper time, that is to say A's proper time (we do not need now to introduce any companion A' of A, any coordinate system, and so we could use this expression without ambiguity even in the case of accelerated observers, either A or B or both - in the general case one should only be careful about possible events in the life of A in which B is not seen at all!).

This true observed time (I repeat it, the time of B as observed by A in some instant T of the proper time of A - I now make strict use of the scenario specified at the beginning, in which an origin for all proper times was defined, see the Figure II in the attachment) is not equal to (1), but to:

(2) [T*sqr(1-v^2)]/(1+v) .

As a consequence of this discussion, we can remark now how apparently the "same" question could have three different answers in SR. If we take two events in the life of B, marked by two R-coordinate times t0 and t1 (that is to say, the two events are given, in the coordinates (x,t), by (vt0,t0) and (vt1,t1) respectively), how much it lasts the "travel" of B (of course the travel is only relative to A, B is quite standing still from its point of view), from the first event to the second?

One could say that this duration is t1-t0, from a coordinate time point of view, and that it is (t1-t0)*sqr(1-v^2), from the point of view of B's clock (B-proper time's point of view). Or even (t1-t0)*sqr(1-v^2)/(1+v) from the point of view of the "observed time" of A, in the sense that this is the difference between the values that A sees marked by B's clock at the instant t0 (A's proper time!) and at the instant t1.

Remark - Of course, what we have just said does not mean that, if the beginning and the end of the travel are marked by some events like flashing, or crushing, then A sees the flashing or the crushing in the instants t0 and t1! If the beginning of (the part of) the travel of B is given by a flashing, then it arrives at A at the instant t0(1+v), in which A sees the proper time of B t0(1+v)*sqr(1-v^2)/(1+v) = t0*sqr(1-v^2), and the same holds for the end of the travel, the crush, which means that in this sense the length of the travel, as seen by A, would coincide exactly with (t1-t0)*sqr(1-v^2), which is in fact the "proper length" of the travel (as for that, this is a common value for all observers parallel to A, or even for all inertial observers).

It is perhaps instructive to remark another interesting feature of formula (2): it holds only for T > 0, in which case one has to introduce indeed BACKWARDS photons. Things change completely if T < 0 (see again the Figure II in attachment), and formula (2) becomes:

(3) [T*sqr(1-v^2)]/(1-v) ,

which means, I shall repeat it, that at the instant T (T < 0) of A's proper time, A sees the clock of B marking the value indicated by (3).

In some sense, the difference between (2) and (3) is exactly the same phenomenon which holds for Doppler effect, one has a different outcome in the case the observer B is approaching A, or B is at the contrary driving away from A. As a matter of fact, when A sees B approaching it, then A sees B farther as it is "really"; when B is driving away, A sees b (B's clock) CLOSER than it is really.

All this has a curious consequence. Imagine for instance the following set-up: A observes the travel of B in two different instants t0 and t1 of its own proper time (A's clock), and wants to establish the connection between the difference (t1-t0), which it could say is the travel's lenght from its point of view, and the length which he SEES as the real travel's length for B, having defined this concept as the difference between the values marked by B's clock as A sees it directly at the given instant t0 and t1. Well, if t0 and t1 are both positive, that is to say, if B is always driving away from A, then we must use formula (2), and get for this length:

(4) [(t1-t0)*sqr(1-v^2)]/(1+v) = (t1-t0)*sqr[(1-v)/(1+v)],

which gives still a "time dilation". That is to say, B SEEMS to have spent less time for its travel than it should have to.

But, if t0 and t1 are both negative, we must instead use formula (3), and we get:

(5) [(t1-t0)*sqr(1-v^2)]/(1-v) = (t1-t0)*sqr[(1+v)/(1-v)],

which is NOT anymore a time dilation, but a "time contraction".

It seems interesting even to examine the "mixed case", in which t0 < 0 and t1 > 0. Suppose for instance that A looks at B's clock in some instant -T when B is approaching, and that A looks again at B's clock in the instant T when A is now driving away. Then, in front of a time

T-(-T)=2T of its own clock, A observes an elapsed time in the clock of B equal to:

(6) [T*sqr(1-v^2)]/(1+v) - [(-T)*sqr(1-v^2)]/(1-v) = 2T/sqr(1-v^2)

which is exactly the INVERSE of the usual relativistic time dilation!

All this implies, in my opinion, that the usual treatment of the twin argument has nothing to do, in principle, with the "observed time", as it was on the contrary said in your point 2. These observed times could be computed indeed, but they give a result which is completely different, even from a qualitative point of view, from the one which we have found in the previous Section 2.

I shall do it immediately, but let me finish this Section saying that I hope that all this would be able to show how one must be very careful when using ordinary language terms which would seem to have an univocal meaning in SR. In we give precise definitions, if we exactly state what we are looking for, then everything meaningful could be asked, and the theory will always give a unique correct answer. Troubles and paradoxes appear only when mixing different concepts which SEEM the same thing.

* * * * *

SECTION 5 - THE SECOND RECONCILIATION USING OBSERVED TIME IS A SIMPLE APPLICATION OF THE FORMULAE GIVEN IN THE PREVIOUS SECTION

We intend to give now the Tables corresponding to the previous Tables 2, 3 using the observed time instead than the proper time.

TABLE 2bis - Observed time of B from the point of view of A

We must always start from the decomposition of the travel of B defined by the events E0, E1, E3, E4. E0, and E4 are "common" events for both "twins". When B is in E1, a photon which starts from B towards A, will be seen by A in the instant L/v - e + L - ev (this value defines an event H1 in A's life). When B is in E3, a photon starting again from B towards A, will be seen by A in the instant L/v + e + L - ev , and this value defines another event in A's life, call it H3.

As we know, A sees the clock of B marking (L/v - e)*sqr(1-v^2) when A is in the event H1, and A sees the clock of B marking

(L/v - e)*sqr(1-v^2) + T(e) when A is in the event H3.

Segment E0-H1

elapsed time for A: L/v - e + L - ev

elapsed time for B as observed by A: (L/v - e)*sqr(1-v^2)

Segment H1-H3

elapsed time for A: 2e

elapsed time for B as observed by A: T(e)

Segment H3-E4

elapsed time for A: L/v - e - L + ev

elapsed time for B as observed by A: (L/v - e)*sqr(1-v^2)

Total time elapsed for A:

L/v - e + L - ev + 2e + L/v - e - L + ev = 2L/v, OK.

Total time elapsed for B:

(L/v - e)*sqr(1-v^2) + T(e) + (L/v - e)*sqr(1-v^2) = DT(B), OK.

TABLE 3bis - Observed time of A from the point of view of B

When B is in the events E0, E1, E3, E4, he receives a unique photon from A, which started from events K0 = E0, K1, K3, K4 = E4 in the life of A. An easy computation shows that the event K1 is then defined by the time L/v - e - L + ev (proper time of A, the value which B sees marked by A's clock when B is in E1, when B is starting its deceleration), while the event K3 is defined by the time L/v + e - L + ev, which is once again the proper time of A which B sees marked in A's clock when B is in E3, the end of its acceleration towards A.

Segment E0-K1

elapsed time for A as observed by B: L/v - e - L + ev

corresponding time elapsed for B (E0-E1): (L/v - e)*sqr(1-v^2)

Segment K1-K3

elapsed time for A as observed by B: 2e

corresponding time elapsed for B (E1-E3): T(e)

Segment K3-E4

elapsed time for A as observed by B: L/v - e + L - ev

corresponding time elapsed for B (E3-E4): (L/v - e)*sqr(1-v^2)

Total time elapsed for A:

L/v - e - L + ev + 2e + L/v - e + L - ev = 2L/v, OK.

Total time elapsed for B:

(L/v - e)*sqr(1-v^2) + T(e) + (L/v - e)*sqr(1-v^2) = DT(B), OK.

ANALYSIS OF TABLE 2BIS - In the segment E0-H1, A has spent a time equal to: L/v - e + L - ev, while A observes that the clock of B has spent a time equal to: (L/v - e)*sqr(1-v^2). This is perfectly coherent with the previous formula (4) (Section 4), since B is now driving away from A, and we must get a time dilation expressed by:

[(L/v - e + L - ev)*sqr(1-v^2)]/(1+v) = (L/v - e)*sqr(1-v^2) !

In the segment H1-H3, A has spent a time 2e, while A observes that the clock of B has spent a time equal to T(e), in this case both times are indeed infinitesimal! Moreover, since T(e) < 2e*sqr(1-v^2) < 2e, we still have a "time dilation" (the time spent by the travelling twin with respect to the "stay-at-home" twin is LESS).

At last, in the segment H3-E4, A has spent a time equal to:

L/v - e - L + ev, while A observes that the clock of B has spent a time equal to: (L/v - e)*sqr(1-v^2). This is now perfectly coherent with the previous formula (5) (Section 4), since B is now approaching A, and we get a TIME CONTRACTION expressed by:

[(L/v - e - L + ev)*sqr(1-v^2)]/(1-v) = (L/v - e)*sqr(1-v^2) !

In other words, in the first segment now A sees indeed B's clock going slower, in the second segment (the "acceleration period", which is now infinitesimal for both A and B proper times) A sees it again going slower, but in the third and final segment, when B is approaching, A sees B's clock going FASTER. Summing up, the conclusion is always the one which is well known, the outcome is ASYMMETRIC, and from a quantitative point of view A ascertains that B's clock has GLOBALLY gone SLOWER.

ANALYSIS OF TABLE 3BIS - In the segment E0-E1, the clock of B has spent a time equal to: (L/v - e)*sqr(1-v^2), while B observes A going from E0 to K1, spending a time equal to: L/v - e - L + ev. This is once again perfectly coherent with the previous formula (4) (Section 4), since A is now driving away from B (we are now discussing B's observations), and we must get an observed time dilation (for A's clock with respect to B's clock) expressed by:

[[(L/v - e)*sqr(1-v^2)]*sqr(1-v^2)]/(1+v) = (L/v - e)(1-v) =

= L/v - e - L + ev !!

(be careful about the sign of the speed: the speed of A with respect to B, better to B', is -v, but only if we leave the orientation of the x-axis unchanged; in the formulae of Section 4, I always supposed instead the speed to be positive, and the only one thing which really matters is the difference between the approaching case and the driving away case).

In the segment E1-E3, B has spent a time T(e), while B observes that the clock of A has spent a time equal to 2e, once again in this case (the observed time case) both times are indeed infinitesimal. But, since T(e) < 2e, we do not have anymore a "time dilation" (the time spent by the travelling twin, which is now A, with respect to the stay-at-home twin, which is now B, is MORE, and not LESS).

At last, in the segment E3-E4, B has spent a time equal to:

(L/v - e)*sqr(1-v^2), while B observes that the clock of A has spent a time equal to: L/v - e + L - ev. This is once again perfectly coherent with the previous formula (5) (Section 4), since A is now approaching B, and we must get a TIME CONTRACTION expressed by:

[[(L/v - e)*sqr(1-v^2)]*sqr(1-v^2)]/(1-v) = (L/v - e)(1+v) =

= L/v - e + L - ev !!

As before, in the first segment now B sees A's clock going slower, in the second segment (the "acceleration period", which is now infinitesimal for both B and A proper times) B sees it going faster, and in the third and final segment, when A is approaching, B sees A's clock continuing to go FASTER. Summing up, the conclusion is always the same, the outcome is ASYMMETRIC, but now from a quantitative point of view B ascertains that A's clock has GLOBALLY gone FASTER.

* * * * *

SECTION 6 - CONCLUSION

I hope that the previous Tables 1, 2, 3, 2bis, 3bis will be able to make you completely persuaded that there are no problems at all for SR arising from the discussion of the "twin paradox". The "reconciliation" you are looking for in your point 3, is quite easy to be achieved in all cases, and Dingle unfortunately (I say that thinking of all anti-relativistic party, to which I completely belong, which cannot be but damaged by Dingle's and similar others misinterpretations of relativity) was quite wrong. As most people believe, and in this case I belong to the MAJORITY side, the twin paradox is really a DEAD ISSUE (even you are much likely right pointing out that this fact does not necessarily imply that the majority of people has a full consciousness of the correct treatment of the question too!).

At last, I would frankly say that there is really no great advantage in going on with this discussion, but if you wish to do that please, instead of forcing me to answer to YOUR questions, state clearly where I did possibly make mistakes in my previous considerations. If this is the case, I shall of course acknowledge my mistakes, and I shall correct them, but I am sure A PRIORI that in any case the situation with respect to SR will not change! My mistakes would not be SR mistakes!! This is not a challenge between ME and YOU, but between a precise mathematical theory and its common misinterpretations, which all arise from the fact that the mathematical theory is "absurd" from the ordinary rationality point of view (a statement about which EVERYBODY agree, in this case we have no question of majority and minority), exactly absurd from the point of view of that "common sense" which some physicists seem unable to let apart, even when discussing phenomena in the relativistic space-time (they are continuing to use even in the new set-up the "ordinary Physics" language).

I repeat once more that the only point which would be PHYSICALLY interesting to discuss is whether SR is correct or wrong from the experimental point of view. As far as this question is concerning, ALMOST NOBODY today would bet on an experimental failure of SR (the so-called breakdown of Lorentz-invariance) in its field of competence. Now I belong to the MINORITY side, since I believe that SR could be shown indeed experimentally wrong (it is not an entirely conventional theory, as somebody, like Max Jammer for instance, is asserting), and much more that it is very likely wrong, all its experimental "confirmations" until now notwithstanding.

Once again best regards from yours most sincerely

Umberto Bartocci

P.S. Many thanks for the interesting paper which you have sent to me, which just arrived by ordinary mail (as I already told you, I do not know too much literature)…

P.P.S. Of course you were quite right in your "quick post script" of 16th July. I was just joking, exaggerating, provoking, but the SUBSTANCE of matters does not change: putting forward any kind of "theoretical objections" against SR is the same thing as putting forward objections against a circle, or a sphere! (these are concepts A PRIORI, as the whole Mathematics is). One can discuss, challenge SR only from the experimental point of view (and this could be done only A POSTERIORI - as you have surely well realized during our discussions, I like very much Kant)...