The case for True Length = Rest Length

[GrayGhost]

JesseM,

I never said there is an inertial frame where A's clock suddenly starts moving forward very rapidly when B accelerates. I simply said "from the B POV" this is what must happen. I know that there is no gravitational dilation in the twins scenario, but since "that's what they labeled it" on their own graph, I merely referred to it since it supports what I've been saying here.

As you know, SR can be used to analyse the classic twins scenario. Also, Rindler diagrams do not alter the LTs in any way. They simply present the lines of simultaneity for accelerating worldlines.

The twins scenario may well be a pre-planned controlled roundtrip flight test if desired (not that it matters). Via the LTs, and for any moment, twin A can determine (via integration) not only what time the moving B clock must then read, but also the separation between A and B, per B. From this data, we can easily determine what B determines of A's clock wrt time and range, for any instant. The twin B experience must agree with twin A's LT predictions of B. Again, I'm not talking doppler effects here. I'm talking about the mapping each point in spacetime between the 2 systems.

If you happen to choose a different sense-of-simultaneity that differs from what relativity requires, then although you may obtain the same end result (final relative age differential), I do not see that you can obtain correct results for any arbitrary point along twin B's trek. Again, what B holds of the A clock at any instant (and its relative range), must precisely match "what twin A predicts the B experience to be" using the LTs.

You disagree?

GrayGhost

 

[PAllen]

JesseM,

I never said there is an inertial frame where A's clock suddenly starts moving forward very rapidly when B accelerates. I simply said "from the B POV" this is what must happen. I know that there is no gravitational dilation in the twins scenario, but since "that's what they labeled it" on their own graph, I merely referred to it since it supports what I've been saying here.

As you know, SR can be used to analyse the classic twins scenario. Also, Rindler diagrams do not alter the LTs in any way. They simply present the lines of simultaneity for accelerating worldlines.

The twins scenario may well be a pre-planned controlled roundtrip flight test if desired (not that it matters). Via the LTs, and for any moment, twin A can determine (via integration) not only what time the moving B clock must then read, but also the separation between A and B, per B. From this data, we can easily determine what B determines of A's clock wrt time and range, for any instant. The twin B experience must agree with twin A's LT predictions of B. Again, I'm not talking doppler effects here. I'm talking about the mapping each point in spacetime between the 2 systems.

If you happen to choose a different sense-of-simultaneity that differs from what relativity requires, then although you may obtain the same end result (final relative age differential), I do not see that you can obtain correct results for any arbitrary point along twin B's trek. Again, what B holds of the A clock at any instant (and its relative range), must precisely match "what twin A predicts the B experience to be" using the LTs.

You disagree?

GrayGhost

I disagree with a lot of the above, and know that JesseM is making important points. I would like to ask a few other questions for you to think about. Note that ghwellsjr has pointed out that B never see (through a telescope, for example) A's clock rapidly advance. You have argued that this is a visual effect and that B 'must' interpret A's clock to rapidly advance (instantaneously in the instant turnaround scenario). I would like you to outline exactly how B should interpret some measurements they make to justify this without resorting to 'must interpret reality according to LT of A's perspective via B's instantaneous velocity'.

Consider the following:

1) At the moment of turnaround, parallax and subtended angle suggest A has instantly moved further away. Interpreting this via lightspeed delay, 'must' B conclude they are suddenly re-seeing a blue version of events they've already seen? This would follow from a constant lightspeed interpretation A being suddenly much farther away. Except that images contradict this - they are not re-seeing older events. Further, instant travel is precluded.

2) On the other hand, if B factors out blueshift, they still see an increase in luminosity of A (see relativistic beaming). So they should instead conclude that A has jumped closer, rather than farther? This leads to inverse paradoxes from (1).

3) If they in fact use the assumption of two way lightspeed and one way lightspeed being constant, and continuously bounced signals off of A, they would see interpret something completely different (see Dolby and Gull re radar coordinates). Here, they would interpret a very smooth simultaneity history, factoring in that B knows he hasn't been 'eternally' moving at his current velocity relative to A (and LTs are specifically derived for eternally inertial observers).

There is, in fact, no reasonable interpretation of direct measurements made by B that leads to the Fermi-Walker coordinates that you are proposing as the only valid choice. Note that there is no wrong physical prediction resulting from using Fermi-Walker coordinates correctly, but there is also no simple physical interpretation of measurements that supports them for this scenario. (Actually, there are imaginary measurements that support them: maintaining set of clocks with Born rigid relationships between them; however, this has little relation to achievable measurements).

 

[rjbeery]

The participation on this forum is fantastic, and the feedback is of a high caliber, but it's also exhausting.

DaleSpam: There is no point beyond me giving you something to ponder. Speaking of arbitrary, I find it arbitrary to NOT correct for SR-related length contraction when we DO correct for size at a distance, width at an angle, sound pitch of an engine speeding by, color for binary rotating stars, and Doppler effects of Gendanken space-walkers with clocks. I'm just seeking some logical consistency really, but the convention itself bears no consequences on Science one way or the other.

ghwellsjr and bobc2: Holy crap, you guys win by attrition I guess. I don't know where you find the time to write all that...

 

[GrayGhost]

PAllen,

I'll think carefully about your post before responding. Thanx.

GrayGhost

 

[JesseM]

I never said there is an inertial frame where A's clock suddenly starts moving forward very rapidly when B accelerates. I simply said "from the B POV" this is what must happen.

Yes, and I explained that the question of what coordinate system we choose to represent as the "POV" of a non-inertial observer is purely a matter of convention.

As you know, SR can be used to analyse the classic twins scenario. Also, Rindler diagrams do not alter the LTs in any way. They simply present the lines of simultaneity for accelerating worldlines.

There are other commonly-used non-inertial coordinate systems which give a different view of simultaneity with distant clocks than the one in the instantaneous inertial frame, like Marzke-Wheeler coordinates where simultaneity is based on the use of radar signals. (If the non-inertial observer sends out a signal at time T1 according to their clock, and the event of it bouncing off some distant object is labeled E, and the reflected signal returns to the observer at time T2 on his clock, then he will say that E occurred at a time halfway between T1 and T2. The linked paper shows on p. 10 what this imples for simultaneity in a twin paradox scenario.) Would you say that these coordinate systems "alter the LTs"? I don't even understand the meaning of that phrase, since the idea that light has a coordinate speed of c is built into the LT but apparently it doesn't bother you that light can have a variable coordinate speed in Rindler coordinates.

If you happen to choose a different sense-of-simultaneity that differs from what relativity requires

Relativity doesn't "require" anything about what definition of simultaneity is used in a non-inertial frame, this is a totally imaginary requirement you have invented.

then although you may obtain the same end result (final relative age differential), I do not see that you can obtain correct results for any arbitrary point along twin B's trek.

This sounds like a circular argument, are you just defining "correct results" to mean agreement with simultaneity in the instantaneous inertial rest frame at each point? Again this is a requirement that you have just invented, all that physicists care about when using non-inertial coordinate systems is that they agree with inertial ones about coordinate-invariant facts, not coordinate-dependent ones like simultaneity or the coordinate speed of light.

Again, what B holds of the A clock at any instant (and its relative range), must precisely match "what twin A predicts the B experience to be" using the LTs.

Nope, there is no recognized requirement that this "must" match, this is just a requirement you have invented for no good reason that I can see.

 

[Dale]

There is no point beyond me giving you something to ponder. Speaking of arbitrary, I find it arbitrary to NOT correct for SR-related length contraction when we DO correct for size at a distance, width at an angle, sound pitch of an engine speeding by, color for binary rotating stars, and Doppler effects of Gendanken space-walkers with clocks. I'm just seeking some logical consistency really, but the convention itself bears no consequences on Science one way or the other.

We do that correction, and call the corrected value the "rest length". And you didn't answer the question:

Given that the concept you label "true length" is exactly the SAME as the concept everyone else labels "rest length", why do you feel the need to CHANGE the term from "rest length" to "true length"?

 

[bobc2]

ghwellsjr and bobc2: Holy crap, you guys win by attrition I guess. I don't know where you find the time to write all that...

I don't know that there are any winners here. But, here's just one more offering on behalf of shortcuts. The lower right depicts red and blue moving together at relativistic speed (relative to the black rest system). They start together at point A, then at point B they are still together, but the red guy heads for point C.

Now, tell me, when they are side by side at point B, which guy is closer to point C? Does changing direction make the red guy somehow closer than the blue guy?

 

[rjbeery]

Now, tell me, when they are side by side at point B, which guy is closer to point C? Does changing direction make the red guy somehow closer than the blue guy?

I had considered myself done with this thread but...I don't get your point here. What are you saying?

 

[bobc2]

I had considered myself done with this thread but...I don't get your point here. What are you saying?

I'm saying that it is a 4-dimensional universe. The 4-D distance from point B to point C in this 4-D space is the same for the blue guy and the red guy. Then, the blue guy keeps going in the same direction and the red guy changes direction in 4-D space. Just because the red guy changes direction does not change the 4-D distance from point B to point A.

It's the path that red takes in 4-D space that determines the total 4-D distance traveled from point A to point B to point C. And he moves along the 4-D path at the speed of light.

Thus, it is the path through 4-D space that determines the age of the red guy when he arrives at point C. Sure, he had to accelerate to change direction at point B, but that had nothing to do with the 4-D distance from B to C--the chosen path determined his age.

 

[bobc2]

rjbeery, you are probably bothered by the seemingly instantaneous acceleration at the turnaround. It is not instantaneous at all (see sketch below). You just can't see the small path length of curved world line on the astronomic scale of the sketch. If you have questions about proper distance, time, world lines and hyperbolic calibration curves, etc., I'm sure someone on the forum could probably explain it better than I have. Nevertheless, the aging inequality as shown below clearly explains the twin paradox as a result of two different world line paths taken through spacetime. The acceleration is not relevant to the aging, other than providing the means for the traveler to change direction of motion in 4-D space.

 

[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.