The case for True Length = Rest Length

[GrayGhost]

JesseM,

I'm trying to cut to the chase here ...

First, a ways back, a made a misleading statement that suggested that the visual experience of twin B would witness the "time jump" (no time is ever missing though) during B's rapid turnabout. In fact, B will only observer the rapid doppler shift, not the A-time-jump. The A-time-jump exists, but must be determined because it cannot be seen visually. I believe I corrected that mis-statement in subsequent posts. The LTs reveal to twin B that the A clock advanced wildly during B's own rapid turnabout, even though the light signals show only a doppler shift ... and the doppler shift requires that twin A jumped wildly across space per B. In addition to the rapid doppler shift, if twin A was emitting pulses at periodic intervals, B (upon completion of his rapid turnabout) would note that the rate of receipt of said pulses increase by a factor of gamma. OK, enough of that ...

I do realize that there exist various conventions of simultaneity for non-inertial POVs. Although they produce the correct net aging differential over the spacetime interval, they do not agree on how the A and B clocks related to each other during the interval before its end.

The inertial POV is convenient, and so twin A can build a database (with the use of the LTs) of collected variables for A & B including ... clock readout, separation, and momentary velocity ... for each infitesimal proper duration of A. Data over the entire roundtrip is collected. From this data, we can determine how B must have experienced it all. Not just "where A is per B at some B moment", but also what the visual effects would be after subsequent receipt of doppler shifted light signals. Bottom line, if we ran a completely controlled flight test, we should expect twin B to experience precisely what twin A predicts B should experience using the LTs. There's no mystery here IMO. In fact, I'd argue this ... if twin B experiences something different than that pedicted by twin A, then relativity theory is incorrect. Likewise, if you select some arbitrary convention of simultaneity for the twin B to use, one other than Einstein's, twin B will experience something different than what twin A predicts of B ... and so IMO the convention is not correct. It may well produce the final aging differential result, but it will fail to map "all points" in spacetime between the 2 systems (thus all points along the trek) in a way that both A & B agree.

From a practical POV, I do recognize why arbitrary conventions are desirable, because all the variables of the "other guy" may not be known. You referenced me to a convention that used "radar signals" for the twin B experience, whereby it is assumed that the EM's reflection event bisects the roundtrip duration ... however this is a bad assumption, as B is accelerating and the acceleration may not be steady at all. Again, the correct final aging differential may be obtained over the interval on the whole, however A & B would not agree on the mapping of spacetime between their systems for all the points during the trip after its start and before its end.

GrayGhost

 

[JesseM]

The LTs reveal to twin B that the A clock advanced wildly during B's own rapid turnabout

No they don't. The LTs deal with inertial frames only, there is no inertial frame where that is true. If you want to use a non-inertial coordinate system whose definition of simultaneity always matches that of B's instantaneous inertial rest frame, then the coordinate transformation used to map between B's non-inertial coordinate system and A's inertial one is obviously not the LT.

I do realize that there exist various conventions of simultaneity for non-inertial POVs. Although they produce the correct net aging differential over the spacetime interval, they do not agree on how the A and B clocks related to each other during the interval before its end.

Exactly, that's because simultaneity is relative to your choice of reference frame, and there is no objective physical reason to prefer one definition of simultaneity over another.

The inertial POV is convenient, and so twin A can build a database (with the use of the LTs) of collected variables for A & B including ... clock readout, separation, and momentary velocity ... for each infitesimal proper duration of A. Data over the entire roundtrip is collected. From this data, we can determine how B must have experienced it all.

What does "experienced" mean? Are you referring to the frame-invariant aspects of what happens to B, like what he sees visually (what his proper time is when the light from different events reaches him), or frame-dependent issues?

Not just "where A is per B at some B moment", but also what the visual effects would be after subsequent receipt of doppler shifted light signals.

All the frame-independent questions like what visual effects B sees can be calculated from the perspective of any non-inertial frame and they'll all agree.

In fact, I'd argue this ... if twin B experiences something different than that pedicted by twin A, then relativity theory is incorrect.

Only if you're talking about frame-independent facts like what B sees visually. If you're including things like simultaneity in what B "experiences" then you're just totally confused about what's a physical fact and what's merely a matter of human convention in relativity, I would have just as much right to define B's "experience" in terms of some non-inertial coordinate system with a different simultaneity convention.

Likewise, if you select some arbitrary convention of simultaneity for the twin B to use, one other than Einstein's

Einstein's convention was only part of the definition of inertial frames, he didn't claim it represented some sort of objective truth about what any given observer (even an inertial one) experiences, an idea so confused and ill-defined that I think it would be fair to call it not even wrong.

twin B will experience something different than what twin A predicts of B ... and so IMO the convention is not correct.

"Not correct" by what standard? I'm sorry but no physicist would agree with you on this, the idea that there is a "true" definition of simultaneity for each observer is totally nonsensical and ill-defined as a scientific claim, since it doesn't yield a single testable prediction about the reading of any physical instrument which differs from the prediction that would be made using a non-inertial coordinate system with a different definition of simultaneity.

From a practical POV, I do recognize why arbitrary conventions are desirable

But you don't seem to understand that the simultaneity convention used in inertial frames is also just an "arbitrary convention", though obviously it is a very useful one since the laws of physics take the same form in all the inertial frames defined using this convention. Still it's not like there is any objective, non-conventional sense in which it is an "objective truth" that what is happening "now" for an inertial observer is the set of events that are simultaneous in his inertial rest frame. Again, that observer could choose a completely different convention and still get exactly the same predictions about all instrument-readings, what he will see visually, etc.

You referenced me to a convention that used "radar signals" for the twin B experience, whereby it is assumed that the EM's reflection event bisects the roundtrip duration ... however this is a bad assumption, as B is accelerating and the acceleration may not be steady at all.

So what? Why does this make it "bad"? The coordinate system is still completely well-defined and continuous for an observer with changing acceleration.

Again, the correct final aging differential may be obtained over the interval on the whole, however A & B would not agree on the mapping of spacetime between their systems for all the points during the trip after its start and before its end.

Sure they'd agree on the mapping, as long as they both knew how the other guy was defining his coordinate system. If B is using Marzke-Wheeler coordinates based on radar signals, do you think A can't find a coordinate transformation that maps between his inertial coordinates and the Marzke-Wheeler system used by B, provided he knows B's acceleration?

 

[rjbeery]

bobc2, there are many different ways to interpret the twin paradox. You seem to think that there is something I "don't get", but you are mistaken. You can choose to dismiss the acceleration involved but the fact remains that it is the acceleration which allows for what you're referring to as a shortcut through 4D space. I challenge you to find a twin paradox experiment wherein the younger twin, after they meet back together, has accelerated less than the older twin.

Also, you shouldn't reference "triangle inequalities" when you're discussing 4D time lines, as they are precisely the opposite of what you're pointing out. The LONGER the line in Minkowski light cone time line diagrams, the SHORTER the proper time.

 

[JesseM]

bobc2, there are many different ways to interpret the twin paradox. You seem to think that there is something I "don't get", but you are mistaken. You can choose to dismiss the acceleration involved but the fact remains that it is the acceleration which allows for what you're referring to as a shortcut through 4D space. I challenge you to find a twin paradox experiment wherein the younger twin, after they meet back together, has accelerated less than the older twin.

If both twins accelerate, it is quite possible that the one that accelerated "less" (less proper time spent acceleration, smaller value of G-force during acceleration) will have aged less, it just depends on the paths. For example, look at this spacetime diagram posted by DrGreg a while ago, in which twins A and B both spend exactly the same amount of time accelerating and with the same magnitude of acceleration, but because A spent more time between accelerations, B's path is closer to that of the inertial twin C and thus A will have aged significantly less:

It would be a simple matter to modify this diagram so that A's accelerations were slightly less than B's, and so A's velocities relative to C during the inertial phases of the trip were slightly smaller, but where A still aged less than either B or C.

Also, you shouldn't reference "triangle inequalities" when you're discussing 4D time lines, as they are precisely the opposite of what you're pointing out. The LONGER the line in Minkowski light cone time line diagrams, the SHORTER the proper time.

Yes, that's because the proper time involves a subtraction, i.e. \sqrt{dt^2 - dx^2}, whereas distance in space involved adding the two coordinates, i.e. \sqrt{dy^2 + dx^2}. Still there is a pretty perfect one-to-one mapping between statements about distances along paths through space and statements about proper times along paths through spacetime, so they can be considered very closely analogous even if a few signs are changed, I discussed the details of the analogy in [post=2972720]this post[/post].

 

[bobc2]

The LONGER the line in Minkowski light cone time line diagrams, the SHORTER the proper time.

I don't understand why you would say that. My diagram as well as the inequality statements for distance along world lines and proper times showed exactly what you are saying they did not show: The longer lines in the diagram (AB + BC) correspond to shorter Minkowski metric distances and also shorter proper times (AB/c + BC/c) than the shorter line on the diagram (AC).

Again, the triangular inequality is (AB + AC) is less than (AC)
or, (AB+AC) < AC

 

[bobc2]

If both twins accelerate, it is quite possible that the one that accelerated "less" (less proper time spent acceleration, smaller value of G-force during acceleration) will have aged less, it just depends on the paths. For example, look at this spacetime diagram posted by DrGreg a while ago, in which twins A and B both spend exactly the same amount of time accelerating and with the same magnitude of acceleration, but because A spent more time between accelerations, B's path is closer to that of the inertial twin C and thus A will have aged significantly less:

It would be a simple matter to modify this diagram so that A's accelerations were slightly less than B's, and so A's velocities relative to C during the inertial phases of the trip were slightly smaller, but where A still aged less than either B or C.

Excellent point, JesseM.

 

[rjbeery]

I don't understand why you would say that. My diagram as well as the inequality statements for distance along world lines and proper times showed exactly what you are saying they did not show: The longer lines in the diagram (AB + BC) correspond to shorter Minkowski metric distances and also shorter proper times (AB/c + BC/c) than the shorter line on the diagram (AC).

My point was that the phrase "http://en.wikipedia.org/wiki/Triangle_inequality" " has a distinct meaning in geometry which directly contradicts what you're saying. I've never heard of that term used in the context of proper time line charting, but it's not really pertinent one way or the other as I was not disputing your math.

JesseM, that graph clearly shows I misspoke and I thank you for it! If I may use it to further my original point (ignoring the initial and final accelerations for simplicity), charting the lines of simultaneity from the travelers' perspectives on their journey will reflect legs of reciprocity before and after their periods of acceleration.

It is only DURING their periods of acceleration that this reciprocity is broken. The sections I've labeled d1 and d2 show the sections of the graphs that encompass the differential. If we were to remove these sections the world lines of A and C, or B and C, respectively, would be of equal length. My point is that it is the acceleration which causes the aging differential. Your very valid supplemental point was that the timing of that acceleration determines the extent of the effect, and viewing the effect on this graph makes it very obvious why -- the further out on the graph the acceleration occurs, the longer the "d" segment will be on C's world line.

 

[Mentz114]

In my earlier analysis of the barn and pole scenario one of my conclusions was incorrect, so just for the record I'm posting a correction.

It seems there is an observer, in the middle of the barn who will see the pole completely contained in the barn, contrary to my assertion otherwise.

In the space-time diagram attached, the observer in the middle of the barn sees 1) the back of the pole enter the barn, 2) the front of the pole going past the back of the barn, 3) the door closing.

So, at any time between events A and B, this observer will see the pole inside the barn. We can conclude that whether the pole is seen to be in the barn is frame dependent, as JesseM pointed out some time ago.

Well, I have to work these things out for myself ...

Thanks to DaleSpam for pointing this out, and to PAllen for a PM response.

 

[JesseM]

JesseM, that graph clearly shows I misspoke and I thank you for it! If I may use it to further my original point (ignoring the initial and final accelerations for simplicity), charting the lines of simultaneity from the travelers' perspectives on their journey will reflect legs of reciprocity before and after their periods of acceleration.

But in what sense are those "lines of simultaneity from the traveler's perspective"? The traveler is a non-inertial observer, so if you want to talk about their "perspective" you have to construct a non-inertial frame for them. And as I said to GrayGhost, you can construct a non-inertial frame which has the property that its definition of simultaneity at any point on the traveler's worldline matches that of the traveler's instantaneous inertial rest frame at that point, that is not the only type of non-inertial frame you could construct for the traveler, and there are no "preferred" non-inertial frames in relativity.

I've labeled d1 and d2 show the sections of the graphs that encompass the differential. If we were to remove these sections the world lines of A and C, or B and C, respectively, would be of equal length.

What do you mean by "remove those sections"? If you calculate the proper time only along the blue inertial sections of A's worldline, leaving out the proper time along the red accelerating sections, the sum is still going to be considerably less than the proper time along C's worldline.

My point is that it is the acceleration which causes the aging differential.

"Causes" is a little ambiguous. Again I would make an analogy with geometry--if we have two paths in space between points P1 and P2, and one path is a straight line with constant slope (relative to some cartesian coordinate system) while the other has a bend in it (changing slope), then since a straight line is the shortest distance between two points it is guaranteed that the path which had the segment with changing slope will have a greater overall length. Would you say in this case that the segment with changing slope "caused" the distance differential, even if the bent path consisted of two long straight segments with a very short bent segment joining them, like a "V" with a slightly rounded bottom? Obviously in this case most of the extra length of the bent path is on the straight segments which each have constant slope, not the curved segment with a changing slope, but it is true that the bent segment is what allows the straight segments to go in different directions and thus have a greater length than the other path which goes straight from P1 to P2. So if you would describe the bent segment as having "caused" the distance differential here, I have no problem with your using the same language in the twin paradox; but if you wouldn't, then I don't think you should in the twin paradox either, since as I explain at length in [post=2972720]this post[/post] I think the two situations are perfectly analogous.

 

[GrayGhost]

JesseM,

In all-inertial scenarios, A & B use the LTs to predict the other's experience. They each use the very same convention of simultaneity, ie Einstein's. That convention assumes the 1-way speed of light equals the 2-way speed of light equals invariant c.

In the classic twins scenario, only twin B goes non-inertial, and so he ages less over the roundtrip interval. Twin A applies his convention of simultaneity in the same way done in SR, as defined by Einstein. He can do this whether the remote twin B is undergoing proper acceleration or not. Although Twin B is accelerating, we may consider each individual point of his trek as a momentary frame of reference. At any point, B may apply the Einstein convention of simultaneity as done in the all-inertial case. That's what Rindler diagrams do. Then, the resultant mapping of spacetime between the 2 systems (over the interval) remains consistent with that defined by SR, as opposed to using some other differing convention of simultaneity for twin B which diverges from SR.

JesseM, let me ask you ... WHY should we use an altogether different convention of simultaneity for twin B and diverge from SR, when we in fact do not have to? Why not remain consistent with that as defined by the special theory, in scenarios of acceleration?

I fully realize that the convention of simultaneity is arbitrary. I don't see that it was arbitrary in OEMB though. The Einstein convention was required, because light speed was invariant in any and all frames. So why arbitrarily choose one for twin B that differs from Einstein's convention, when it's not necessary?

GrayGhost

 

[JesseM]

JesseM,

In all-inertial scenarios, A & B use the LTs to predict the other's experience.

You still have not defined "experience". There is no sense in which the inertial definition of simultaneity is an intrinsic part of what A & B "experience" as I would understand that term (i.e. what they will see or what will be registered on any physical instruments they carry with them), it is just a common convention for defining what coordinate system A and B should use.

Although Twin B is accelerating, we may consider each individual point of his trek as a momentary frame of reference.

Sure, at any "individual point" he has a "momentary (inertial) frame of reference", but as soon as you start talking about his opinion of the rate A's clock is ticking according to B's changing definition of simultaneity during the acceleration, you are talking about a non-inertial frame.

Then, the resultant mapping of spacetime between the 2 systems (over the interval) remains consistent with that defined by SR

If by "defined by SR" you mean "using the Lorentz transformation", then no, there is no mapping between an inertial frame and a non-inertial frame according to the Lorentz transformation. The LTs only map between all-inertial frames, not some hodgepodge where you use the definition of simultaneity from different inertial frames at different points on B's worldline.

as opposed to using some other differing convention of simultaneity for twin B which diverges from SR.

Your use of phrases like "diverges from SR" just doesn't agree with the standard way that physicists talk. "SR" is understood to be a theory of physics, not a convention about coordinate systems. If you use a non-inertial coordinate system in flat spacetime, and all your predictions about frame-independent facts are identical to the predictions of an inertial coordinate system, no physicist would say you have "diverged from SR" here, you are just using the theory of SR in a non-inertial coordinate system.

I fully realize that the convention of simultaneity is arbitrary. I don't see that it was arbitrary in OEMB though. The Einstein convention was required, because light speed was invariant in any and all frames.

It was only supposed to be invariant in all inertial frames of the type defined by Einstein. I didn't mean "arbitrary" to suggest there was no advantage to using this definition of inertial frames, in fact I explicitly said otherwise in my comment 'you don't seem to understand that the simultaneity convention used in inertial frames is also just an "arbitrary convention", though obviously it is a very useful one since the laws of physics take the same form in all the inertial frames defined using this convention'. But when dealing with non-inertial frames there is no such benefit to defining your simultaneity convention to always match that of the instantaneous inertial rest frame, since the speed of light will not be constant in this type of non-inertial frame nor will the laws of physics take the same form in non-inertial frames defined this way for different observers.

So why arbitrarily choose one for twin B that differs from Einstein's convention, when it's not necessary?

Einstein's "convention" doesn't say anything about non-inertial frames of accelerating observers, it certainly doesn't say they must use a simultaneity convention that matches that of their instantaneous inertial frame at every point. The advantages to Einstein's convention for inertial observers were the ones mentioned above (constant speed of light in all inertial frames, same laws of physics in all inertial frames), but these specific advantages disappear when you move to non-inertial frames, regardless of whether or not you choose to define their simultaneity to match that of the instantaneous inertial rest frame at each point.

 

[rjbeery]

JesseM, looking back I mangled my last post pretty well. Let me do a follow-up with some math that will show what I'm trying to get across.

 

[Mike_Fontenot]

JesseM, let me ask you ... WHY should we use an altogether different convention of simultaneity for twin B and diverge from SR, when we in fact do not have to? Why not remain consistent with that as defined by the special theory, in scenarios of acceleration?

GrayGhost, you are on the right track.

https://www.physicsforums.com/showpost.php?p=2845669&postcount=47

https://www.physicsforums.com/showpost.php?p=3106767&postcount=38

Mike Fontenot

 

[bobc2]

GrayGhost, you are on the right track.

https://www.physicsforums.com/showpost.php?p=2845669&postcount=47

https://www.physicsforums.com/showpost.php?p=3106767&postcount=38

Mike Fontenot

Mike, have you been looking at some of the other approaches to the acceleration problem (I had not, until noticing some of JesseM's posts). For example:



From “M¨ARZKE-WHEELER COORDINATES FOR ACCELERATED
OBSERVERS IN SPECIAL RELATIVITY”

By M. PAURI1 AND M. VALLISNERI

“Finally, we have discussed how to use the notion of M¨arzke-Wheeler simultaneity
to elucidate the relativistic paradox of the twins, by establishing
a continuous correspondence between the lapses of proper time experienced
by the twins. It is possible to attribute the differential aging of the twins
to distinct segments of their world-lines, where we can conclude that one
twin is aging faster. Although this attribution is not unique, it is justified
physically by recourse to generalized Einstein synchronization, and it
is not possible with other definitions of simultaneity (such as a na¨ıve use of
instantaneous Lorentz frames).”

I think they are referring to your instantaneous Lorentz frames with their comment about the naive use of those.

 

[Mike_Fontenot]

Mike, have you been looking at some of the other approaches to the acceleration problem (I had not, until noticing some of JesseM's posts). For example:

From “M¨ARZKE-WHEELER COORDINATES FOR ACCELERATED
OBSERVERS IN SPECIAL RELATIVITY”

By M. PAURI1 AND M. VALLISNERI

I just took a very brief look at the above paper, and it appears to be the same as the Dolby&Gull simultaneity that has been discussed previously on this forum.

Dolby&Gull, like ALL alternatives to my "CADO" simultaneity, suffers from the fatal flaw that it contradicts the accelerating observer's own elementary calculations using his own elementary measurements.

In addition, Dolby&Gull suffers from an additional fatal flaw: it is non-causal, in that it requires that the accelerating observer's CURRENT conclusions, about the current age of a distant person, depend upon whether or not that observer will CHOOSE to accelerate IN HIS DISTANT FUTURE.

Here are some additional previous posts of mine that are pertinent:

https://www.physicsforums.com/showpost.php?p=3114946&postcount=16

https://www.physicsforums.com/showpost.php?p=2812867&postcount=50

https://www.physicsforums.com/showpost.php?p=2978931&postcount=75

Mike Fontenot

 

[Dale]

Dolby&Gull, like ALL alternatives to my "CADO" simultaneity, suffers from the fatal flaw that it contradicts the accelerating observer's own elementary calculations using his own elementary measurements.

No they don't. Repeating a lie doesn't make it true. All reference frames will agree on the predicted results from any experimental measurement.

 

[JesseM]

Dolby&Gull, like ALL alternatives to my "CADO" simultaneity, suffers from the fatal flaw that it contradicts the accelerating observer's own elementary calculations using his own elementary measurements.

As DaleSpam says, all coordinate systems predict the same thing about "measurements" in the sense of readings on any physical instrument, and you have never defined what you mean by "elementary calculations" despite being repeatedly asked about this.

In addition, Dolby&Gull suffers from an additional fatal flaw: it is non-causal, in that it requires that the accelerating observer's CURRENT conclusions, about the current age of a distant person, depend upon whether or not that observer will CHOOSE to accelerate IN HIS DISTANT FUTURE.

This is not a standard definition of "non-causal", causality in physics is normally a physical idea, it doesn't refer to the properties of coordinate systems. Anyway, why is this "fatal"? As long as the surfaces of simultaneity in a coordinate are spacelike, it's always the case that when an observer assigns a coordinate to an event, he only does so after the event has occurred according to his own definition of simultaneity, due to the fact that light will take some time to get from the event to him and he won't know about the event until he receives a signal from it. Since Dolby-Gull/Marzke-Wheeler coordinates are based on radar signals, in these coordinates one can likewise assign a coordinate to an event at the moment one first receives a signal from that event.

Anyway, if you don't like non-inertial coordinate systems where the definition of simultaneity at each point on your worldline depends on your future motion, that still doesn't mean the only option left is to have simultaneity match the observer's instantaneous inertial rest frame. For example, we could define a non-inertial frame where the definition of simultaneity always matches that of a single inertial frame, such as the frame of the "stay-at-home twin" in the twin paradox. It could still be the case in this non-inertial frame that the accelerating twin has a constant position coordinate, and that the time-coordinate of events on his worldline matches up with his own proper time.

 

[rjbeery]

JesseM, I've changed the graphic a bit to hopefully help make my point. Below are the world lines of twins in the "twins paradox" (it's basically just the graphic you provided with the B participant removed). I've added "clock ticks" on each time line, corresponding to approximately 1 year each. Twin C ages 21 years in this scenario and Twin A ages 13.

Now when I ineloquently said

If we were to remove these sections the world lines of A and C, or B and C, respectively, would be of equal length.

I was referring to the length of each observer's proper time line, NOT the geometrical length of their time line on the graph. Viewing the twin's paradox in this manner, and removing the section associated with A's acceleration, both twins would have aged 11 years. My point is that this, to me, implies that the age differential is due to the acceleration of twin A. Does that help?

 

[JesseM]

JesseM, I've changed the graphic a bit to hopefully help make my point. Below are the world lines of twins in the "twins paradox" (it's basically just the graphic you provided with the B participant removed). I've added "clock ticks" on each time line, corresponding to approximately 1 year each. Twin C ages 21 years in this scenario and Twin A ages 13.

But if the yellow dots are supposed to be clock ticks which are an equal amount of proper time apart for both twins, then this diagram is just wrong, successive clock ticks of each twin would not be simultaneous in the traveling twin's frame when he is moving inertially. In the traveling twin's frame while moving inertially, the Earth twin is moving at relativistic speed, so the Earth twin's clock is running slow due to time dilation--for example if their relative speed is 0.6c, then if you pick two events which occur one year apart for the traveling twin and draw lines of simultaneity in the traveling twin's frame, the points where these lines of simultaneity intersect the Earth twin's worldline will only be 0.8 years apart for the Earth twin.

I was referring to the length of each observer's proper time line, NOT the geometrical length of their time line on the graph. Viewing the twin's paradox in this manner, and removing the section associated with A's acceleration, both twins would have aged 11 years. My point is that this, to me, implies that the age differential is due to the acceleration of twin A. Does that help?

That doesn't work though, even if you remove the section of C's worldline that is "associated with A's acceleration" (i.e. remove all points on C's worldline between where the two thick yellow lines intersect it), the proper time on the remainder of C's worldline would not add up to the proper time along the blue segments of A's worldline, in fact it would be significantly less than the proper time along the blue segments of A's worldline.

 

[bobc2]

JesseM, I've changed the graphic a bit to hopefully help make my point. Below are the world lines of twins in the "twins paradox" (it's basically just the graphic you provided with the B participant removed). I've added "clock ticks" on each time line, corresponding to approximately 1 year each. Twin C ages 21 years in this scenario and Twin A ages 13.

Now when I ineloquently said

I was referring to the length of each observer's proper time line, NOT the geometrical length of their time line on the graph. Viewing the twin's paradox in this manner, and removing the section associated with A's acceleration, both twins would have aged 11 years. My point is that this, to me, implies that the age differential is due to the acceleration of twin A. Does that help?

rjbeery, JesseM is certainly correct. You're sketch really doesn't demonstrate the situation correctly. You have not shown the proper time markers on each observer's world line. Consider using the hyperbolic calibration curves for the traveling twin; only then will you see the actual comparison of metric distance traveled (and proper times lapsed). I don't have time right at the moment to work up the sketches but will get back to it if you haven't had a chance to work them up.

 

[rjbeery]

You have not shown the proper time markers on each observer's world line. Consider using the hyperbolic calibration curves for the traveling twin; only then will you see the actual comparison of metric distance traveled (and proper times lapsed). I don't have time right at the moment to work up the sketches but will get back to it if you haven't had a chance to work them up.

I'm not sure how you can make that statement if the velocities and acceleration rates involved have not even been given! Are you both suggesting that the "clock ticks" would not look something like I have illustrated? If you do your sketches I believe you will find that what I've done is accurate.

successive clock ticks of each twin would not be simultaneous in the traveling twin's frame when he is moving inertially.

As far as the "lines of simultaneity" go, there are a many ways to calculate them (e.g. from C's perspective, from A's perspective, using only what is observed, using Einstein's method, correcting for Doppler, correcting for Doppler and SR...). I've chosen to draw them from a hyper-privileged perspective such that a tick on each twin's clock has a one-to-one correspondence with the other. In other words, they have been corrected for ALL effects including time of light travel. My contention is that these are areas of reciprocal time passage, and that the area between the thick yellow lines contains a break in symmetry due to the acceleration of Twin A.

 

[JesseM]

As far as the "lines of simultaneity" go, there are a many ways to calculate them (e.g. from C's perspective, from A's perspective, using only what is observed, using Einstein's method, correcting for Doppler, correcting for Doppler and SR...). I've chosen to draw them from a hyper-privileged perspective such that a tick on each twin's clock has a one-to-one correspondence with the other. In other words, they have been corrected for ALL effects including time of light travel. My contention is that these are areas of reciprocal time passage, and that the area between the thick yellow lines contains a break in symmetry due to the acceleration of Twin A.

OK, I assumed they were supposed to be from A's instantaneous inertial frame, but it sounds like you were actually drawing them from two frames (different on the outbound voyage and the inbound voyage) where A and C were moving with equal speeds in opposite directions (away from each other on the outbound voyage, towards each other on the inbound one), in which case both clocks would tick at equal rates in this frame. But hopefully you'd agree that these aren't areas of "reciprocal time passage" in any objective sense, that the rates are only equal in this one frame?

 

[rjbeery]

But hopefully you'd agree that these aren't areas of "reciprocal time passage" in any objective sense, that the rates are only equal in this one frame?

Yes, it's a arbitrary analysis but aren't they all?. I could just as easily continue to draw the one-to-one correspondence through the area of acceleration and be left with an age differential at the end of Twin A's trip but it wouldn't be illustrative of much; I'm just explaining how I think about the problem. This method visually accounts for age differentials of the Twins involved in any amount of acceleration by either one of them.

In the end can we "really" say that acceleration causes the age differential? Probably not. It's a bit subjective, really. But the fact that the age differential cannot exist without a frame change due to acceleration is sound enough logic for me to make that claim.

 

[bobc2]

I'm not sure how you can make that statement if the velocities and acceleration rates involved have not even been given! Are you both suggesting that the "clock ticks" would not look something like I have illustrated? If you do your sketches I believe you will find that what I've done is accurate.

Our sketches don't seem to look the same, rjbeery. My hyperbolic calibration curve is not precise, but is still good enough to communicate the concept we are talking about. I've marked off proper time intervals for each of the twins. Each observer moves along his world line at the speed of light.

Don't worry about the turnaround. I can have the twin travel as far out into the galaxies as needed to make the point. Of course that's why his view of the stay-at-home twin changes so rapidly during the turnaround. But my focus is on the proper times and distances. The traveling twin indeed takes a shortcut through spacetime. And be sure to start a new origin with a new hyperbolic calibration curve from which to measure proper distances when the twin changes directions.

By the way, you can find a comparison of the Euclidean triangle inequality to the Minkowski triangle inequality on page 256 of Penrose's "The Emperor's New Mind" (paper back).

 

[rjbeery]

Our sketches don't seem to look the same, rjbeery.

They are the same, or rather they show the same thing. Draw a line from each twin's proper time intervals (1 to 1, 2 to 2, 3 to 3, 4 to 4, and 5 to 5), up to the point of turn-around. Now do the same moving backwards from their reunion (12 to 9, 11 to 8, 10 to 7, 9 to 6, and 8 to 5). What you have outlined is a triangled area that represents the break in symmetry. The static twin's triangle segment is 3 time interval units long, representing exactly the age differential between the two twins upon their reunion. The same conclusion can be drawn from both of our sketches.

By the way, you can find a comparison of the Euclidean triangle inequality to the Minkowski triangle inequality on page 256 of Penrose's "The Emperor's New Mind" (paper back).

That's actually one of my favorite physic's books! Anyway I didn't mean to nitpick but you said "remember the triangle inequality" and I was just pointing out that the unqualified phrase is associated with "Euclidean triangle inequality", or the precise opposite of what you intended.

 

[bobc2]

They are the same, or rather they show the same thing. Draw a line from each twin's proper time intervals (1 to 1, 2 to 2, 3 to 3, 4 to 4, and 5 to 5), up to the point of turn-around. Now do the same moving backwards from their reunion (12 to 9, 11 to 8, 10 to 7, 9 to 6, and 8 to 5). What you have outlined is a triangled area that represents the break in symmetry. The static twin's triangle segment is 3 time interval units long, representing exactly the age differential between the two twins upon their reunion. The same conclusion can be drawn from both of our sketches.

O.K. Below I followed your instructions for connecting the proper times for the outgoing trip. However, your instructions for the second half of the trip did not make sense (why would you arbitrarily put a proper time gap in the middle?), so I have continued with the proper time sequence of connecting the corresponding times, which leaves the proper time gap at the end instead of the middle. By the way, the lines connecting proper times are not the same as lines corresponding to the sequence of blue X1 coordinates (which is what you have probably been trying to use).

But why the worry about gaps? Let's just show the mapping of the proper times onto the spacetime manifold and quit trying to read some kind of causal effect into the gaps? I could have a lot more to say about the proper time "gap" in my plot below, but we better not go there at this point.

That's actually one of my favorite physic's books! Anyway I didn't mean to nitpick but you said "remember the triangle inequality" and I was just pointing out that the unqualified phrase is associated with "Euclidean triangle inequality", or the precise opposite of what you intended.

Good. So, I guess we agree on the Minkowski inequality. Thus, we have the traveling twin in the example above taking the 10 unit shortcut through 4-dimensional space as compared to the stay-at-home twin traveling the 13 units (I don't care whether you regard them as proper time differentials or proper distance differentials).

 

[GrayGhost]

JesseM,

... the related post link is above.

If I may clarify a few statements you made (in paraphrase) ...

You said ... Einstein's convention of simultaneity was arbitrary. However, my understanding is that the 2nd postulate requires the 1-way speed of light to be c, which arose from Maxwell's theory. Yes? If so, then the Einstein/Poincare convention would not have been arbitrarily selected.

You said that ... the speed of light is not constant in a non-inertial frame. However, is it not true that twin B would measure light at c when measured at his own location, just as light is measured at c locally in a gravity well?

You said that ... there is no mapping between an inertial frame and a non-inertial frame according to the Lorentz transformation. However, the LTs are kinematic. The LT solns are integrated in the all-inertial case too, although the integration is way easier given the all linear motion. I don't see why they cannot be integrated in the case of acceleration, so long as an inertial frame is referencable. IMO, the LTs apply to the twins scenario. The extra caveate is that from the B POV, the twin B departure point (from A frame) and the turnabout point (of A frame) dilate more and more with increased B proper acceleration. This dilation cannot be ignored, is predicted by the LTs even in the all-inertial case, and causes the extra aging of twin A relative to B. You disagree?

GrayGhost

 

[rjbeery]

your instructions for the second half of the trip did not make sense (why would you arbitrarily put a proper time gap in the middle?)

From a few posts ago...

Yes, it's a arbitrary analysis but aren't they all?. I could just as easily continue to draw the one-to-one correspondence through the area of acceleration and be left with an age differential at the end of Twin A's trip

Actually, if you insist of drawing it in this manner then I just make the claim that the slope of the simultaneity line is constant until one of the twins accelerates, at which point reciprocity is broken. Either way, the reciprocity is broken due to the acceleration.

But why the worry about gaps? Let's just show the mapping of the proper times onto the spacetime manifold and quit trying to read some kind of causal effect into the gaps?

Because without acceleration there are no gaps, and that's my entire point (or...to use your diagram, without acceleration the slope of the simultaneous line remains constant for eternity). GHWellsJr asked me the following

How about we talk about time dilation now since you said you wanted to. Do you have the same attitude about the rate at which clocks at rest tick versus moving clocks? Do you make the claim that the tick rate of a moving clock is an illusion and that the true tick rate is that of the rest tick rate?

And my response was, restricted to the context of SR, YES. He then asked "why restrict it to SR?" and I said because only by introducing non-inertial frames can one potentially take measurements which objectively prove that an age differential "actually" exists. His question has led this thread so far out into left-field that I actually had to go back to the OP to remember what the hell started all of this!

 

[bobc2]

From a few posts ago...

Actually, if you insist of drawing it in this manner then I just make the claim that the slope of the simultaneity line is constant until one of the twins accelerates, at which point reciprocity is broken. Either way, the reciprocity is broken due to the acceleration.

I fully understand the sequence of cross-section views of the 4-D universe for the blue twin (in my diagram). I full understand that his continuous sequence of cross-sections of the universe change rapidly as he follows the curved path along his world line (corresponding to deceleration and then acceleration) in turning to take the short cut back to the red twin. I'm trying to illustrate something more fundamental than the blue guy watching the sequence of views of the red guy fly by as he (blue guy) rotates his view (faster than the speed of light if we set up a short proper path length going around the 4-D curve segment). It is no more significant than you observing a laser beam sweep across the face of the moon faster than the speed of light just because you can rotate your laser device sitting here on the Earth pointing the beam at the moon on a dark night).

I've got to figure out some way of communicating to you the more fundamental aspect of the situation--that we have two different world line paths followed by the twins--one is a shorter path than the other. If you simply think of these twins as 4-dimensional objects, their long length strung out for millions of miles along their respective 4th dimensions, then the question is manifestly, which object is longer (being careful to remember the Minkowski Triangle Inequality). Surely you would see this as objectively more fundamental than talking about how fast the light beam (blue's view) sweeps across the middle gap (sure, I've never argued there was not what you call a "gap").

After your last post I thought you had grasped the significance of the Minkowski Triangle Inequality that I referenced in Penrose's book. But, that doesn't seem to have any affect on your view of the twin paradox problem. How do you account for the Minkowski Triangle inequality in this example?

 

[ghwellsjr]

Because without acceleration there are no gaps, and that's my entire point (or...to use your diagram, without acceleration the slope of the simultaneous line remains constant for eternity). GHWellsJr asked me the following

How about we talk about time dilation now since you said you wanted to. Do you have the same attitude about the rate at which clocks at rest tick versus moving clocks? Do you make the claim that the tick rate of a moving clock is an illusion and that the true tick rate is that of the rest tick rate?

And my response was, restricted to the context of SR, YES. He then asked "why restrict it to SR?" and I said because only by introducing non-inertial frames can one potentially take measurements which objectively prove that an age differential "actually" exists. His question has led this thread so far out into left-field that I actually had to go back to the OP to remember what the hell started all of this!

Here is the actual exchange that you are referring to:

Why do you say "restricted to SR"? Are you leaving open a loop-hole through which you can explain the Twin Paradox?

It's because SR effects produce measurements that are apparently contradictory and reciprocal (i.e. each party concludes the other's watch is slower), similar to mutual foreshortening. When you involve acceleration you break that reciprocity.

Not only did you not mention "non-inertial frames" in your immediate response, you never used that expression until this very quote. It must be that you think when an observer is accelerating, SR requires a non-inertial frame, correct?