The case for True Length = Rest Length

[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! :tongue2:

 

[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! :tongue2:

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?

 

[rjbeery]

I've got to figure out some way of communicating to you the more fundamental aspect of the situation

Here's how I think of what you're trying to say: the longer the geometric path, the shorter the time. It's that simple, and I get it. The only way to make a longer path, though, is to accelerate. Saying that the Minkowski triangle inequality is more "fundamental" than the fact that what you're calling short-cuts cannot be exploited without acceleration is a bit of an arbitrary stance to take. I frankly don't even know what we're arguing about since we don't seem to be disagreeing on any objective measures.

 

[rjbeery]

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?

Actually I believe there is a mathematically complex method that can consider an accelerating observer to remain inertial, and I've also started to refer to non-inertial frames to include the possibility of of the introduction of large gravity sources rather than spatial travel, but rather than obfuscate the original point are you able to produce an explanation of the twin's paradox that does not involve acceleration and/or the presence of a gravity field for either twin?

 

[JesseM]

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

You said ... Einstein's convention of simultaneity was arbitrary.

That's a totally inaccurate paraphrase, please just quote my exact words rather than paraphrasing. What I actually said was:

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.

However, my understanding is that the 2nd postulate requires the 1-way speed of light to be c

The 2nd postulate says the 1-way speed of light must be c relative to inertial coordinate systems defined using Einstein's methods.

which arose from Maxwell's theory. Yes? If so, then the Einstein/Poincare convention would not have been arbitrarily selected.

As I stated above, the convention is decidedly non-arbitrary in the sense that the laws of physics (including Maxwell's law) take the same form in all inertial frames defined using this simultaneity convention, which makes this a particularly useful set of coordinate systems to use. But aside from this usefulness, there isn't any sense in which judgments made by inertial frames are more "true" than judgments made by non-inertial ones.

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?

The idea that the "local" speed of light is c is based on the idea of using a locally inertial coordinate system in your region, see the equivalence principle.

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.

What do you mean "integrated"? If you just want to map the coordinates of one frame to the coordinates of another frame there are no integrals involved, you just use the basic LT equations. Are you talking about calculating proper time along a worldline or something like that? If so that has nothing to do with mapping between frames, proper time is a frame-invariant quantity so it's just using the coordinates of one frame to determine the value of this quantity, which would be the same regardless of which frame you used to calculate it. And you aren't integrating the Lorentz transformations to find the proper time (I don't even know what it would mean to integrate a coordinate transformation), you're using the time dilation equation which tells you the rate that proper time $$\tau$$ is increasing relative to the coordinate time t in one frame, i.e. you're taking $$\frac{d\tau}{dt} = \sqrt{1 - v^2/c^2}$$ which can be rearranged as $$d\tau = \sqrt{1 - v^2/c^2} \, dt$$, and then you integrate that for an object with a varying velocity v(t), $$\int_{t_0}^{t_1} \sqrt{1 - v(t)^2/c^2} \, dt$$ to get the change in proper time between two coordinate times t0 and t1.

If you weren't talking about calculating proper time, can you be specific about what specifically is being integrated, and what you are trying to calculate with the integral? Exact equations would be helpful.

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?

Again I don't know what you're integrating, and the rest is confusingly worded, you say that the "departure point" and "turnabout point" "dilate more and more"--how can "points" dilate? Probably you mean that when B accelerates towards A, then "from B's perspective", A's clock is ticking very rapidly? If so, this would be a statement that's specifically about what happens during acceleration, in which case your subsequent comment that "This dilation ... is predicted by the LTs even in the all-inertial case" doesn't really make sense.

 

[ghwellsjr]

Actually I believe there is a mathematically complex method that can consider an accelerating observer to remain inertial, and I've also started to refer to non-inertial frames to include the possibility of of the introduction of large gravity sources rather than spatial travel, but rather than obfuscate the original point are you able to produce an explanation of the twin's paradox that does not involve acceleration and/or the presence of a gravity field for either twin?

Of course not, a traveling twin has to undergo acceleration to depart his twin and return. But that's not the issue. Let me rephrase the question. Do you equate "acceleration" with a "non-inertial frame"?

 

[ghwellsjr]

...I'm trying to illustrate something more fundamental than the blue guy watching the sequence of views of the red guy fly by...

...All the turnaround does is to give the round trip twin interesting variations in his view of the other twin's clock (as has already been pointed out in ealier posts). We can show the respective views each has of the other's clocks on the return trip if necessary (someone else could probably do that since I'm running out of steam).

Bob, have you got your steam back? I'd like to know exactly what you think each twin sees (with their own keen eyes) of the other twin and their clocks during the entire trip, please.

 

[Dale]

Acceleration does not cause time dilation. This has been tested to 10^18 g in muon storage rings. See the sticky on experimental basis of SR.

 

[rjbeery]

Do you equate "acceleration" with a "non-inertial frame"?

When I hear non-inertial frame I presume either the observer is under acceleration or he is in a gravity field but not freely falling. Does that answer your question? Also, why is that relevant?

Acceleration does not cause time dilation. This has been tested to 10^18 g in muon storage rings. See the sticky on experimental basis of SR.

Ahh that link looks very interesting, thanks. I don't have time to read it all right now but if your comment has anything to do with circular motion (which I presume based on the term "storage ring"), then there is undeniably http://en.wikipedia.org/wiki/Centripetal_force" involved.

 

[JesseM]

Ahh that link looks very interesting, thanks. I don't have time to read it all right now but if your comment has anything to do with circular motion (which I presume based on the term "storage ring"), then there is undeniably http://en.wikipedia.org/wiki/Centripetal_force" involved.

Yes, they were testing whether the acceleration due to circular motion had any effect on the time dilation (the experiment is described here). But note that what DaleSpam is saying is probably something you'd agree with, that the rate a clock ticks at any given moment relative to some inertial frame (i.e. $$d\tau/dt$$) is a function only of the clock's speed and not its acceleration, so a clock moving at speed v in a straight line will be slowed down (again relative to some inertial frame) by exactly the same amount as a clock moving at speed v in a circle. This need not conflict with the idea that acceleration is in some sense the "cause" of one twin aging less than the other in the twin paradox between meetings, though.

 

[ghwellsjr]

When I hear non-inertial frame I presume either the observer is under acceleration or he is in a gravity field but not freely falling. Does that answer your question? Also, why is that relevant?

I don't know if it answers my question. I'm trying to figure why you stated in post #178 (three lines up from the bottom):

I said because only by introducing non-inertial frames can one potentially take measurements which objectively prove that an age differential "actually" exists.​

when in fact, you said:

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.​

You just now stated:

When I hear non-inertial frame I presume either the observer is under acceleration or he is in a gravity field but not freely falling.​

But I didn't ask you about that. I'm asking about when you are thinking of an observer who is under acceleration, do you automatically equate that with a non-inertial frame, so strongly that you can use the terms interchangeably and expect other people to understand the same thing you are thinking?

If not, then why did you misquote yourself?

 

[bobc2]

Here's how I think of what you're trying to say: the longer the geometric path, the shorter the time. It's that simple, and I get it. The only way to make a longer path, though, is to accelerate. Saying that the Minkowski triangle inequality is more "fundamental" than the fact that what you're calling short-cuts cannot be exploited without acceleration is a bit of an arbitrary stance to take. I frankly don't even know what we're arguing about since we don't seem to be disagreeing on any objective measures.

So, I guess in that sense you could say that the blue car in the race against red below will win the race because he induced centripetal acceleration in turning onto the short cut to win the race.

"If the red and blue cars always travel the same speed along their direction of motion, I challenge you to find an example where any car could beat the red car to the finish line if he does not resort to acceleration, putting him in a position to take the short cut" (I'm just rephrasing your earlier comment about not being able to take the 4-D short cut without acceleration).

This is exactly analagous to your claim that acknowledges the shorter 4-D path for blue (Minkowski Inequality) but attributes the ability to take that path to blue's acceleration. Note red and blue both travel along their paths at the speed of light (for blue--even when he is going around the corner--and that is a very short world line path turning the corner).

My plots that tracked the proper time (proper distance) increments along each 4-D paths (blue and red) showed quite clearly there was no sudden increase in aging of the red guy (stay-at-home twin) during the time blue is accelerating. That's really the crucial point.

 

[ghwellsjr]

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 ...

Can you explain how you determined that "the rate of receipt of said pulses increase by a factor of gamma" or specifically what you mean by that, please?

 

[rjbeery]

But note that what DaleSpam is saying is probably something you'd agree with, that the rate a clock ticks at any given moment relative to some inertial frame (i.e. ) is a function only of the clock's speed and not its acceleration, so a clock moving at speed v in a straight line will be slowed down (again relative to some inertial frame) by exactly the same amount as a clock moving at speed v in a circle. This need not conflict with the idea that acceleration is in some sense the "cause" of one twin aging less than the other in the twin paradox between meetings, though.

You are right, I would not disagree that the "apparent" clock rate is purely a function of v. The objective clock rate, necessarily deduced after a reunion of parties involved, is determined by acceleration. This is true for the inertial muons as well. If the traveling twin "apparently" dies when his clock says noon, then he ACTUALLY died when HIS clock said noon; however the "rate of passage" of time between him and the distant observer has no objective meaning until they meet up again, and that depends on whether it is the distant observer or the dead body which is accelerated towards the other. Since this is difficult to do for a decayed muon, we're taking the information about "when" it decayed as having some sort of significance. If we were to speed off and catch up to the inertial muon prior to its decay I assure you we would not come to that same conclusion.

 

[rjbeery]

I'm asking about when you are thinking of an observer who is under acceleration, do you automatically equate that with a non-inertial frame, so strongly that you can use the terms interchangeably and expect other people to understand the same thing you are thinking?

If not, then why did you misquote yourself?

I already addressed this.

I've also started to refer to non-inertial frames to include the possibility of of the introduction of large gravity sources rather than spatial travel, but rather than obfuscate the original point are you able to produce an explanation of the twin's paradox that does not involve acceleration and/or the presence of a gravity field for either twin?

I apologize if my loose usage of terms has hindered your ability to comprehend my posts. Non-inertial frames, as a result of acceleration OR a gravity field, are necessary to break the reciprocal time dilation effects of SR. I changed my term from "acceleration" to the more encompassing "non-inertial frames" in anticipation of showing that the twin paradox can be demonstrated without high velocities whatsoever. I feel this would bolster my claim that it isn't the velocity that affects the age differential, don't you agree?

 

[bobc2]

The objective clock rate, necessarily deduced after a reunion of parties involved, is determined by acceleration.

rjbeery, If you still have your Penrose book (The Emperor's New Mind) look at his Figure 5.19 on page 256 (paper back version). Here is a quote of his caption for that figure:

"The so-called 'twin pradox' of special relativity is understood in terms of a Minkowski triangle inequality. (The Euclidean case is given for comparison.)"

His text reads:

"The world-line AC, represents the twin who stays at home while the traveller has a world-line composed of two segments AB and BC, these representing the outward and inward stages of the journey (see Fig. 5.19). The stay-at-home twin experiences a time measured by the Minkowski distance AC, while the traveller experiences a time given by the sum of the two Minkowski distances AB and BC. These times are not equal, but we find:

AC > AB + BC,

showing that indeed the time experienced by the stay-at-home is greater than that of the traveller."

 

[JesseM]

You are right, I would not disagree that the "apparent" clock rate is purely a function of v. The objective clock rate, necessarily deduced after a reunion of parties involved, is determined by acceleration.

Do you mean the objective elapsed time on the clock? "Rate" usually suggests a quantity that has an instantaneous value at each moment, like instantaneous velocity or the instantaneous value of $$\frac{d\tau}{dt}$$, whereas in the twin paradox the only objective comparison you can make is the total amount of time elapsed on each clock for the entire journey. And if you do mean elapsed time, although it's true that the one that accelerated will always have a smaller value, it's nevertheless also true that if you want to calculate the elapsed time using the coordinates of some inertial frame, the elapsed time is just a function of velocity. If the twins depart at t0 in some frame and reunite at t1, and a given twin has velocity as a function of time given by v(t) in that frame, then the elapsed time will be $$\int_{t_0}^{t_1} \sqrt{1 - v(t)^2/c^2} \, dt$$, an expression which doesn't involve acceleration. But if you evaluate this expression for both twins, you do find that the one with a v(t) whose value changed (the one that accelerated) will always have a smaller elapsed time than the one with a v(t) that was constant (the one that moved inertially).

Since this is difficult to do for a decayed muon, we're taking the information about "when" it decayed as having some sort of significance.

Well, frame-dependent results do have a sort of significance, just not the same as frame-independent ones. Certainly frame-dependent quantities can be very useful for making predictions about frame-independent ones, like using v(t) in some frame to calculate elapsed proper time above.

 

[JesseM]

I apologize if my loose usage of terms has hindered your ability to comprehend my posts. Non-inertial frames, as a result of acceleration OR a gravity field, are necessary to break the reciprocal time dilation effects of SR.

You don't need any non-inertial frames to analyze the twin paradox! You can calculate the proper time for both twins from the perspective of an inertial frame (which needn't be either twin's rest frame), you'll still reach the conclusion that the twin that accelerated will have aged less than the one that didn't. If you are under the impression that one cannot use inertial frames to analyze accelerated motion, that's incorrect, see here for example:

http://math.ucr.edu/home/baez/physics/Relativity/SR/acceleration.html

 

[rjbeery]

My plots that tracked the proper time (proper distance) increments along each 4-D paths (blue and red) showed quite clearly there was no sudden increase in aging of the red guy (stay-at-home twin) during the time blue is accelerating. That's really the crucial point.

"During"...that's a slippery word. According to my interpretation, red ages very quickly while blue ages very slowly (or not at all in your instant turn-around scenario) during blue's acceleration period.

In order to prove out any age differential that exists in any objective sense whatsoever, the fact remains that it takes a non-inertial frame to do so. Your "4D shortcut", as a matter of necessity based upon the math involved, DEMANDS a non-inertial frame. We are essentially arguing the same point and you are frustrated that I won't use your same terminology. Your interpretation is perfectly mathematically valid and I do not contend it, which is why we're going to have to agree to disagree on this.

 

[rjbeery]

You don't need any non-inertial frames to analyze the twin paradox! You can calculate the proper time for both twins from the perspective of an inertial frame (which needn't be either twin's rest frame), you'll still reach the conclusion that the twin that accelerated will have aged less than the one that didn't. If you are under the impression that one cannot use inertial frames to analyze accelerated motion, that's incorrect, see here for example:

http://math.ucr.edu/home/baez/physic...eleration.html

In my understanding, SR can handle acceleration of observed objects as long as the observer himself not being accelerated. I cannot read your link until tomorrow...keeping up with this many respondents is exhausting!

 

[JesseM]

In my understanding, SR can handle acceleration of observed objects as long as the observer himself not being accelerated. I cannot read your link until tomorrow...keeping up with this many respondents is exhausting!

In SR talking about what is measured by some "observer" is just a shorthand way of talking about what is measured in some frame, and you can use any frame you want to analyze any problem, as I said you can analyze the twin paradox from a frame where neither twin is at rest during any of the phases of the trip.

 

[GrayGhost]

JesseM,

Wrt integration, I mean only that each observer must sum his own proper time for each infitesimal over the interval, as he goes. Also, he must similarly sum the LT time solutions (of the other guy) for each infitesimal over the interval, as he goes.

So, Einstein's convention is the result of the 2nd postulate which stems from Maxwell's theory, and it's very useful because the 1st postulate is able to be upheld. You contend that his convention cannot be applied from the non-inertial POV. I just don't see why ...

Wrt twin B's use of the LTs, I just don't see what the problem is. Twin B measures light at c at his location. The LTs are kinematic. Now, I agree in that an inertial frame is required for referencing, and that twin B's POV (although inconvenient) is no less preferred. However, you've suggested that the LTs cannot be applied by twin B because he is non-inertial. In that I have to disagree. Twin B may apply the LTs, so long as he accounts for the configurational changes that arise in his surroundings due to changes in his own state of motion, which arise due to changes in his own orientation within the continuum as he undergoes proper acceleration. This particluar effect cannot be neglected by B, and only twin B has to deal with it (twin A does not). They are the very reason that the non-inertial POV is far less convenient, although no less preferred...

When you asked what I was talking about "wrt dilation between B's departure event and B's turnabout event", I was referring to these dynamic configurational changes. Remember, twin B travels across some proper length of the A-frame (an invariant), which per B must be contracted since he witnesses said length in motion. However, B's departure and turnabout events do not move per anyone, because events never move ... and their separation is dilated wrt the proper separation. So as twin B accelerates wildly, the separation between the 2 events changes wildly, and twin A advances or digresses wildly along his own worldline (per B, not per A or anyone else) because B's sense-of-simultaneity rotates rapidly. The more remotely located twin A is, the more dramatic the effect per B. When twin B runs the LTs for A, he cannot ignore this effect, one which exists in SR and is predicted by SR, but is not dynamic in SR.

GrayGhost

 

[JesseM]

JesseM,

Wrt integration, I mean only that each observer must sum his own proper time for each infitesimal over the interval, as he goes. Also, he must similarly sum the LT time solutions (of the other guy) for each infitesimal over the interval, as he goes.

I still don't get it, what does "sum the LT time solutions" mean? You can break down a worldline into a lot of short segments and calculate the proper time along each one, but again I don't see how this would involve the LT. If the endpoints of each segment had a spatial separation of dx and a time separation of dt and the velocity on that segment was v (all in whatever frame you were using), the proper time could either be calculated using the spacetime interval $$\sqrt{dt^2 - (1/c^2)dx^2}$$ or equivalently using the time dilation equation $$\sqrt{1 - v^2/c^2} dt$$. But in both cases we are only using the coordinates of a single frame, so we aren't using the LT which relates the coordinates of two different frames.

Do you have a clear mathematical procedure in mind or are you just basing this on a vague sense of how SR calculations work? If you have a clear idea, please spell out the equations; if not, consider the possibility that you may just be mistaken about how proper time along a worldline is calculated.

So, Einstein's convention is the result of the 2nd postulate which stems from Maxwell's theory, and it's very useful because the 1st postulate is able to be upheld. You contend that his convention cannot be applied from the non-inertial POV.

I never said you cannot use a simultaneity convention in a non-inertial frame that matches up at every moment with simultaneity in the instantaneous inertial rest frame of an accelerating object, in fact I definitely said you could do this. But the point is that in a non-inertial frame there is no longer anything particularly "useful" about this, since neither the 1st postulate nor the 2nd postulate can be expected to hold in a non-inertial frame with this sort of simultaneity convention. So why do you think a non-inertial frame with this sort of simultaneity convention is any better (or more consistent with SR) than a non-inertial frame with a different sort of simultaneity convention? Do you contend there is any concrete advantage or is it just that it has a greater aesthetic appeal to you?

Wrt twin B's use of the LTs, I just don't see what the problem is. Twin B measures light at c at his location. The LTs are kinematic.

I suspect you mean something different by "kinematic" then I would--can you define that word for me? I would say that the LT relate one purely inertial coordinate system covering all of spacetime to a different purely inertial coordinate system covering all of spacetime. The "v" that appears in the transformation equations must be a constant, not a variable which changes at different values of the time-coordinate, otherwise you are no longer dealing with the "Lorentz transformation" but rather some rather different coordinate transformation. Do you disagree?

In that I have to disagree. Twin B may apply the LTs, so long as he accounts for the configurational changes that arise in his surroundings due to changes in his own state of motion, which arise due to changes in his own orientation within the continuum as he undergoes proper acceleration.

I have no idea what it would mean to "account for configurational changes" when you "apply the LTs". Here are the Lorentz transformation equations:

t' = gamma*(t - vx/c^2)
x' = gamma*(x - vt)
y'=y
z'=z

with gamma = 1/sqrt(1 - v^2/c^2)

They're pretty straightforward, if you know the coordinates t,x,y,z of some event in the unprimed frame, you plug those coordinates into these equations to get the coordinates t',x',y',z' in the primed frame. And again, v is a constant in these equations. So can you explain mathematically, in terms of these equations, what it means to "account for configurational changes" and how that could alter the value of t',x',y',z' for an event with a known t,x,y,z? Do you just mean that at different times B would have a different rest frame so at one time he might be interested in the coordinates t',x',y',z' of frame #1 but at a different time he might be interested in the coordinates t'',x'',y'',z'' of a different frame #2?

This particluar effect cannot be neglected by B

Why can't it be? Any observer is free to use any frame they want to, their own state of motion does not obligate them to use a particular frame, it's simply a matter of convention that for inertial observers we typically define what each one "observes" in terms of their rest frame. But even if I am an inertial observer, nothing would stop me from ignoring this convention and making all my measurements and calculations from the perspective of an inertial frame which is moving relative to me at 0.6c, for example. Do you disagree?

They are the very reason that the non-inertial POV is far less convenient, although no less preferred...

I don't understand what you mean by "no less preferred". Usually a "preferred" frame or set of frames is one where the equations of the laws of physics take some "special" form that they don't in other frames, and in this sense all inertial frames are "preferred" when compared to non-inertial ones in SR, since commonly-used useful equations such as Maxwell's laws or the time dilation equation only work in inertial frames.

When you asked what I was talking about "wrt dilation between B's departure event and B's turnabout event", I was referring to these dynamic configurational changes.

Again, don't know what "dynamic configurational changes" means. It's starting to seem like a lot of your argument is based on technobabble, vaguely technical-sounding phrases which in fact have no well-defined meaning. Please either use standard terms in the standard way, or if you're going to make up your own non-standard terminology, please define it in precise mathematical terms.

Remember, twin B travels across some proper length of the A-frame (an invariant)

Don't know what you mean by "proper length" here, usually proper length/proper distance refers to the distance along some particular spacelike worldline, although sometimes proper length also refers to the rest length of some rigid object moving inertially. I don't see how either meaning would make sense here.

which per B must be contracted since he witnesses said length in motion.

How can a "length" be contracted? I can understand what it means for the length of a rigid object to be contracted, but not a free-floating "length" which doesn't seem to be the length of any particular object (or the distance between two objects). And didn't you just say this "length" was an "invariant", meaning it should be the same in every frame? Again, please try not to speak in vague technobabble, give me something like a specific numerical example where we can actually calculate a value for whatever "length" you're talking about.

However, B's departure and turnabout events do not move per anyone, because events never move

Don't know what you mean by "move". Certainly there is no coordinate system where a given event has shifting positions at different times, since each event is instantaneously brief and only happens at a particular instant in time. But the position coordinates of the events may of course be different in different frames.

... and their separation is dilated wrt the proper separation.

When you say "their separation" you talking about their time separation (difference in time coordinates $$\Delta t$$ between the two), their distance separation (difference in position coordinates $$\Delta x$$ between the two), or the invariant spacetime interval ($$\sqrt{\Delta t - (1/c^2)\Delta x}$$? And likewise what does "proper separation" mean? Again I would request that you give some simple numerical example where you give specific values for the terms you use.

So as twin B accelerates wildly, the separation between the 2 events changes wildly, and twin A advances or digresses wildly along his own worldline (per B, not per A or anyone else) because B's sense-of-simultaneity rotates rapidly.

Objects don't have a "sense-of-simultaneity", again it is simply a matter of convention what coordinate system we associate with what object. As I said, even if I am an inertial observer I am perfectly free to use an inertial coordinate system moving at 0.6c relative to me, this goes against the most common convention for what is meant by the words "my perspective" but as long as I explain what I'm doing there is no physical reason why I am "wrong" to use a frame other than my rest frame. Do you disagree?

 

[ghwellsjr]

...Non-inertial frames, as a result of acceleration OR a gravity field, are necessary to break the reciprocal time dilation effects of SR. I changed my term from "acceleration" to the more encompassing "non-inertial frames" in anticipation of showing that the twin paradox can be demonstrated without high velocities whatsoever. I feel this would bolster my claim that it isn't the velocity that affects the age differential, don't you agree?

It is the relative speed over a period of time that affects the age differential. The greater the relative speed AND the greater the period of time, the greater the age differential. Acceleration only creates a change in the relative speed and unless the acceleration continues over a long time, it will have an insignificant effect on the age differential. Remember, there's three accelerations involved, one for taking off, one for turning around and one for landing.

PLEASE NOTE: I had prepared a MUCH longer response because there are so many things in your statement that I disagree with but it got to be quite lengthy so I'm just dealing here with the anwer to your question.

 

[GrayGhost]

JesseM,

I understand the LTs very well. I must admit, discussing the details from an accelerating POV is not the easiest thing to do. It's all rather clear in my mind, and I've discussed it with others who didn't have a problem. Let me draw you a picture ...

see attached thumbnails.

The figure shows the twin B POV,and the twin A POV. It shows only the outbound leg. The blue dots show twin B's clock as the A-frame clock passes by. There are 5 clocks strung between twin A and B's turnabout point, of the A-frame, all synchronised with each other. I intentionally omitted the clock readings as not to clutter the diagram. The point is to show the wild shifts in A clocks as B does a virtual immediate acceleration at departure, and a virtual immediate decelration (back into the A frame) at the turnabout point. These wild swings (per B) cannot be neglected by B when making predictions of A. These configurational changes are required per extrapolation of SR to the accelrational case.

The wild swings are denoted by the green arrows, which denote the shift in A clock readout and A locations at commencement and end of the virtually immediate twin B proper accelration.

Does that help any?

GrayGhost

 

[JesseM]

JesseM,

I understand the LTs very well. I must admit, discussing the details from an accelerating POV is not the easiest thing to do.

If you use an accelerating coordinate system, you are not using the Lorentz transformation. If you map from A's frame to a series of different inertial frames where B is at rest at different moments, then you are using the Lorentz transformation for each mapping from A's frame to any given frame in the series, but a series of different inertial frames is not the same as a single non-inertial coordinate system where the definition of simultaneity is different at different time-coordinates.

The figure shows the twin B POV,and the twin A POV. It shows only the outbound leg. The blue dots show twin B's clock as the A-frame clock passes by. There are 5 clocks strung between twin A and B's turnabout point, of the A-frame, all synchronised with each other. I intentionally omitted the clock readings as not to clutter the diagram. The point is to show the wild shifts in A clocks as B does a virtual immediate acceleration at departure, and a virtual immediate decelration (back into the A frame) at the turnabout point. These wild swings (per B) cannot be neglected by B when making predictions of A. These configurational changes are required per extrapolation of SR to the accelrational case.

The wild swings are denoted by the green arrows, which denote the shift in A clock readout and A locations at commencement and end of the virtually immediate twin B proper accelration.

I don't see any wild swings, in fact your diagram appears to show only two inertial frames, one where A is moving at a constant velocity of 0.866c and another where A is at rest. The frame where A is moving at 0.866c has lines of simultaneity shown in gray, while A's rest frame has lines of simultaneity shown as dotted lines. The angle between the gray lines of simultaneity and the dotted lines of simultaneity never changes.

You could draw a diagram showing the lines of simultaneity in the traveling twin's instantaneous inertial rest frame at each point on his worldline, like this diagram from the last section of the twin paradox FAQ:

But obviously the Lorentz transformation cannot be used to obtain a single coordinate system where each of the blue lines represents a surface of constant t'-coordinate.

Anyway, could you please answer at least a few of the questions I asked in my previous post? If you just keep restating your claims without ever answering questions I don't see how this discussion can go anywhere. For example, I would very much like to know if you agree or disagree that objects do not have any intrinsic "sense-of-simultaneity" as I said here:

Objects don't have a "sense-of-simultaneity", again it is simply a matter of convention what coordinate system we associate with what object. As I said, even if I am an inertial observer I am perfectly free to use an inertial coordinate system moving at 0.6c relative to me, this goes against the most common convention for what is meant by the words "my perspective" but as long as I explain what I'm doing there is no physical reason why I am "wrong" to use a frame other than my rest frame. Do you disagree?

Likewise I would also like to see some kind of mathematical definition of what you mean when you talk about "integrating the LT" to obtain elapsed proper time, as I asked at the start of the previous post.

 

[rjbeery]

It is the relative speed over a period of time that affects the age differential.

OK, now consider a twin's paradox where twin A remains on Earth and twin B moves locally to a greater gravity well (say, the surface of the sun). Twin B shall age less over time and there is no relative velocity to speak of that can account for it. There is, however, an analogy to acceleration caused by the stronger gravity field.

Also, the problem with your statement is that speed is relative, as you said, which would also imply that any age differential is relative, which it is (and therefore does not exist in any objective sense until acceleration becomes involved).

 

[ghwellsjr]

OK, now consider a twin's paradox where twin A remains on Earth and twin B moves locally to a greater gravity well (say, the surface of the sun). Twin B shall age less over time and there is no relative velocity to speak of that can account for it. There is, however, an analogy to acceleration caused by the stronger gravity field.

Also, the problem with your statement is that speed is relative, as you said, which would also imply that any age differential is relative, which it is (and therefore does not exist in any objective sense until acceleration becomes involved).

I guess I should have posted the longer response that I had compiled before I posted the shorter one. Here it is:

...Non-inertial frames, as a result of acceleration OR a gravity field, are necessary to break the reciprocal time dilation effects of SR. I changed my term from "acceleration" to the more encompassing "non-inertial frames" in anticipation of showing that the twin paradox can be demonstrated without high velocities whatsoever. I feel this would bolster my claim that it isn't the velocity that affects the age differential, don't you agree?

I do not agree with hardly anything in this statement. I cannot just give a simple answer. Let me take each sentence one at a time:

Non-inertial frames, as a result of acceleration OR a gravity field, are necessary to break the reciprocal time dilation effects of SR.​

First off, when using SR, we ignore gravity, so you shouldn't be including it.

So now your sentence should be:

Non-inertial frames, as a result of acceleration, are necessary to break the reciprocal time dilation effects of SR.​

1) Acceleration of an object or observer does not cause or create a non-inertial frame. You get to pick any arbitrary frame to define/specify/illustrate/demonstrate/analyze your entire scenario including all objects and observers. You can choose an inertial frame or a non-inertial frame but if you pick a non-inertial frame, you're on your own, I don't like torture, especially optional self-inflicted torture. I like to pick the easiest frame to analyze a problem in.

2) The reciprocal time dilation effects are not broken by any frame that you choose. Any two clocks/observers in relative motion will observe time dilation in the other one. When you select an inertial frame, you also specify motions in absolute terms, which assigns absolute time dilations to each clock/observer but this does not show you what those clocks/observer can see and observe.

Remember, the whole point of the Twin Paradox is that each twin sees the other one as experiencing time dilation, throughout the entire trip, except for the brief inconsequential moments of acceleration.

I changed my term from "acceleration" to the more encompassing "non-inertial frames" in anticipation of showing that the twin paradox can be demonstrated without high velocities whatsoever.​

1) You are conflating "acceleration" and "non-inertial frames" when they are not in any sense equivalent. "Acceleration" might be related to "non-inertial" but when you tack on the word "frame" you are changing the issue.

2) We never needed high velocities to demonstrate the twin paradox and you should be using the word "speed" rather than "velocity". You do know, I hope, that velocity means some speed in some direction and it doesn't matter what the direction is when we are considering time dilation. If two twins are sitting in the living room and one of them gets up and goes to another room and comes back, he will have aged a smaller amount than his stationary twin and we can calculate exactly the amount of age difference if we know exactly how he moved.

I feel this would bolster my claim that it isn't the speed that affects the age differential, don't you agree?​

1) It's a relative speed between two observers/clocks that creates a relative time dilation.

2) The longer the relative speed is in effect, the greater the age difference so it's not just the speed that matters, it's the time the twin travels at a given speed that matters.

 

[rjbeery]

First off, when using SR, we ignore gravity, so you shouldn't be including it.

Are we seeking the cause of time dilation or not? My original point was that inertial frames of the twins restricted to SR were incapable of explaining the twin paradox. If you want to make the claim that it is relative velocity, and my counter-example invoking GR contradicts your assertion, it seems a bit dubious of you to restrict the area of conversation.

Also, the problem with your statement is that speed is relative, as you said, which would also imply that any age differential is relative, which it is (and therefore does not exist in any objective sense until acceleration becomes involved).

This statement is supported by the the following excerpt from the link provided by JesseM

The difference between general and special relativity is that in the general theory all frames of reference including spinning and accelerating frames are treated on an equal footing. In special relativity accelerating frames are different from inertial frames. Velocities are relative but acceleration is treated as absolute.

Call it semantics, ghwellsjr, but you simply cannot prove that the age differential exists because of velocity (or speed) alone. If we restrict the scope to SR, then ONLY acceleration is absolute, and relative velocities are incapable of announcing any objective age differential whatsoever. If we INCLUDE GR, then we are capable of producing objective age differentials without relative velocities whatsoever. It is neither necessary, nor sufficient to have velocity as a criterion for the twins' time dilation. What IS necessary is that one of the twins experiences relative velocity with acceleration, OR they reside in a gravity field. To me, acceleration of an observer and a local gravity field to the observer have one thing in common, which is that they alter his non-inertial state. Therefore, I conclude that altering the non-inertial state of one of the twins is necessary to prove an age differential.

 

[JesseM]

Call it semantics, ghwellsjr, but you simply cannot prove that the age differential exists because of velocity (or speed) alone.

It's ambiguous what you mean by "because of velocity (or speed) alone", it can certainly be calculated as a function of speed alone...you never responded to my post #195, do you disagree with any of the following?

And if you do mean elapsed time, although it's true that the one that accelerated will always have a smaller value, it's nevertheless also true that if you want to calculate the elapsed time using the coordinates of some inertial frame, the elapsed time is just a function of velocity. If the twins depart at t0 in some frame and reunite at t1, and a given twin has velocity as a function of time given by v(t) in that frame, then the elapsed time will be $$\int_{t_0}^{t_1} \sqrt{1 - v(t)^2/c^2} \, dt$$, an expression which doesn't involve acceleration. But if you evaluate this expression for both twins, you do find that the one with a v(t) whose value changed (the one that accelerated) will always have a smaller elapsed time than the one with a v(t) that was constant (the one that moved inertially).

 

[rjbeery]

and a given twin has velocity as a function of time

This presupposes that "having a velocity" has any absolute meaning. If twin B is the traveler that later returns, we could equally analyze the situation by saying that A was the initial traveler, and B greatly accelerated to catch up with him at a later point. Mathematically, you will arrive at the same answer. Also, when you say

if you want to calculate the elapsed time using the coordinates of some inertial frame, the elapsed time is just a function of velocity

We're requiring an objective answer which demands a reunion. A reunion necessitates acceleration of at least one of the twins. Let me make a statement that maybe we can both agree on:

Ignoring gravity, the existence of an objective age differential in the Twins Paradox is caused by acceleration while its magnitude is determined by their relative velocities.

 

[ghwellsjr]

Are we seeking the cause of time dilation or not?

No, we are trying to disabuse you of the notion that length contraction and time dilation are illusions.

My original point was that inertial frames of the twins restricted to SR were incapable of explaining the twin paradox.

Well, at least I now understand what you meant in post #101 when you said:

Restricted to SR, which is the scope of what we're discussing, the appearance of moving clocks ticking slowly is an illusion. Proof of this is that the effect is reciprocal, in the same way that if you and I are not facing squarely to each other we could both make the claim that the other guy is narrower. It's a bit nonsensical to assign any true or intrinsic value to a measured property if it leads to a logical contradiction.​

I would like you to come to an understanding that your proof is defective, reciprocal time dilation (and length contraction) are not nonsensical, and they don't lead to any logical contradiction. Do you believe me when I say that I understand all this? Wouldn't you like to understand this, too, instead of believing it is nonsensical and leads to a logical contradiction?

If you want to make the claim that it is relative velocity, and my counter-example invoking GR contradicts your assertion, it seems a bit dubious of you to restrict the area of conversation.

You (and I) agree that we should not be including GR in this discussion. And I didn't claim that it is relative velocity that leads to an age difference. I said it is a relative speed over a period of time that leads to an age difference.

This statement is supported by the the following excerpt from the link provided by JesseM

The difference between general and special relativity is that in the general theory all frames of reference including spinning and accelerating frames are treated on an equal footing. In special relativity accelerating frames are different from inertial frames. Velocities are relative but acceleration is treated as absolute.​

Acceleration is treated as an absolute because it can always be measured, independently of any reference frame or any reference to another object. In other words, it is not relative. If one object is accelerating away from another object, that second object is not accelerating away from the first object. We always know which object is undergoing the acceleration. But with speed, there is no point in merely saying that an object is traveling at any particular speed. We always have to say it is traveling at some speed with reference to a defined frame or another object or to itself if we are talking about before and after it accelerated.

Call it semantics, ghwellsjr, but you simply cannot prove that the age differential exists because of velocity (or speed) alone. If we restrict the scope to SR, then ONLY acceleration is absolute, and relative velocities are incapable of announcing any objective age differential whatsoever. If we INCLUDE GR, then we are capable of producing objective age differentials without relative velocities whatsoever. It is neither necessary, nor sufficient to have velocity as a criterion for the twins' time dilation. What IS necessary is that one of the twins experiences relative velocity with acceleration, OR they reside in a gravity field. To me, acceleration of an observer and a local gravity field to the observer have one thing in common, which is that they alter his non-inertial state. Therefore, I conclude that altering the non-inertial state of one of the twins is necessary to prove an age differential.

Can you prove an age differential exists if only one twin accelerates but never comes back to the first twin? For example, the traveling twin accelerates away from the first twin. Does that mean he's younger? After awhile the traveling twin decelerates so that he is at rest with respect to his brother but far away. Does that mean he's younger? The traveling twin now accelerates back toward his brother. Does that mean he's younger? The traveling twin now decelerates when he's only half way back so that he is once more at rest with respect to his brother but at half the distance than the first time he stopped. Does that mean he's younger?

You should read and study the paper you suggested I read in your post #123:

http://chaos.swarthmore.edu/courses/PDG/AJP000384.pdf

You will find in there an explanation of the Twin Paradox that does not involve acceleration at all. It's on the second page, second column. It references a similar graph you put in your post and advised me to read for more analysis. And I suggest you read section VI THE ROLE OF ACCELERATION CRITICIZED. Remember, this is the paper you suggested that I read for more analysis. Maybe you should follow your own advice.