The case for True Length = Rest Length

[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.

 

[JesseM]

This presupposes that "having a velocity" has any absolute meaning.

I didn't mean to suggest that, obviously velocity is coordinate-dependent, but there is certainly a clear procedure for measuring velocity relative to any given inertial frame.

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.

Of course, you can evaluate the integral using v(t), t0 and t1 from different frames, and you'll always get the same answer for the elapsed proper time.

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.

Yes, I like that way of stating it.

 

[ghwellsjr]

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.

Yes, I like that way of stating it.

Not me. Relative velocities are caused by accelerations. The magnitude of the age difference is determined by their relative velocities and the time interval over which those velocities exist.

Acceleration is required to bring the two twins back to the same location so that when the clocks are compared in any frame of reference, they will have the same age difference as they would in any other frame of reference. But in and of itself, the acceleration has nothing to do with the age difference.

 

[rjbeery]

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.

Objectively, I'm taller than you AND you're taller than me. This is a nonsensical statement which contains a logical contradiction.

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?

Although I've said many times that a reunion is necessary to objectively determine anything, I'm now wondering if this is necessarily the case. If the accelerating twin ever visually sees the other's clock as being ahead of his own then he can conclude objectively that he is younger. I believe in all scenarios given this event occurs prior to their actual reunion. Whether or not this event occurs in your above situations depends on the particulars of your various scenarios.

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

That paper was referenced to explain my graph. I don't necessarily agree with the author's conclusions on everything. However, on the point you raised, he concludes with

Therefore, on outgoing and returning legs both traveling and stationary clocks seem to be going faster than each other, but the change of inertial frames at e constitutes a change of lines of simultaneity which results in a jump ahead between the times r and s as measured on the moving clocks with respect to the stationary clocks. The "missing time" between r and s becomes then the REASON for the differential aging.

Again, even with the 3 brothers version of the twin paradox, it isn't the relative velocity that causes the age differential; on both the outgoing and incoming legs the time dilation is reciprocal. It's in the 2 traveling brothers' change of inertial frames that "causes" what he calls the "missing time". Even though his conclusion is that it is STILL NOT VELOCITY (or speed) that caused the age differential, and therefore does not help your argument, this methodology of avoiding acceleration by passing information between moving frames makes me raise an eyebrow.

 

[rjbeery]

But in and of itself, the acceleration has nothing to do with the age difference.

Yes, except the existence of it.

 

[JesseM]

Not me. Relative velocities are caused by accelerations. The magnitude of the age difference is determined by their relative velocities and the time interval over which those velocities exist.

But presumably you'd agree that if I evaluate the integral \int_{t_0}^{t_1} \sqrt{1 - v(t)^2/c^2} \, dt for two paths between the same two points in spacetime, as a general rule if one path has a constant v(t) while the other has a varying v(t), that fact alone is enough to guarantee the one with the varying v(t) will have a smaller value when the integral is evaluated, without knowing any additional information about the specifics of the problem?

Acceleration is required to bring the two twins back to the same location so that when the clocks are compared in any frame of reference, they will have the same age difference as they would in any other frame of reference. But in and of itself, the acceleration has nothing to do with the age difference.

Suppose we draw two paths on a 2D plane between a pair of points A and B, one of them is a straight line while the other consists of two straight segments at different angles connected by a curved segment. Since we know a straight line is the shortest distance between points, we know the one with the curved segment will have a longer total length. Would you say that "in and of itself, the fact that one path has a curved segment has nothing to do with the difference in length"? If so I guess I don't know what you mean by "in and of itself" (and if you wouldn't say that, you'd be missing how closely statements about elapsed time in spacetime are analogous to statements about path length in space, see [post=2972720]this post of mine on the details of the geometric analogy[/post]).

 

[Dale]

Geometrically, attributing the difference in aging to the acceleration is the same as attributing the fact that the sum of two sides of a triangle are longer than the remaining side to the vertex between the two sides. While it is true that the vertex unambiguously occurs on the longer path the vertex itself has no length so it cannot be said to be the source of the extra length. The extra length is in the sides, not the vertex.

 

[rjbeery]

In other words...

geometrically, attributing the difference in aging to the acceleration is the same as attributing the fact that the sum of two sides of a triangle are longer than the remaining side to the vertex between the two sides. While it is true that the vertex unambiguously occurs on the longer path the vertex itself has no length so it cannot be said to be the source of the extra length. The extra length is in the sides, not the vertex.

=

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.

 

[JesseM]

Perhaps it's the ambiguous meaning of the phrase "caused by" that's the source of disagreement, you could instead say something like "the fact that twin B accelerated is a necessary and sufficient condition for him to have aged less than the inertial twin A when they reunite".

 

[rjbeery]

I would be amenable to that, reiterating that we're ignoring gravity.

 

[ghwellsjr]

OK rjbeery, let's see if we can find some common ground.

Do you agree with this statement:

When a clock accelerates, its tick rate changes.​

 

[JesseM]

OK rjbeery, let's see if we can find some common ground.

Do you agree with this statement:

When a clock accelerates, its tick rate changes.​

Again coordinate-dependent, in a non-inertial coordinate system a clock can be experiencing proper acceleration (or coordinate acceleration) but having an unchanging tick rate. Would be better to say "when a clock accelerates, its tick rate changes relative to all inertial frames."

 

[rjbeery]

If we're trying to speak strictly in objective terms then I am unable to agree with that statement. The holder of the clock would say no while all others would say yes.

 

[DrGreg]

Would be better to say "when a clock accelerates, its tick rate changes relative to all inertial frames."

Actually, even that statement need not be true. A clock moving at constant speed round a circular path is accelerating, but relative to someone permanently at the centre of the circle, the clock rate is unchanging!

How about "when a clock properly accelerates in a straight line, its tick rate changes relative to all inertial frames."?

 

[bobc2]

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.

 

[rjbeery]

How about "when a clock properly accelerates in a straight line, its tick rate changes relative to all inertial frames."?

Agreed.

A clock moving at constant speed round a circular path is accelerating, but relative to someone permanently at the centre of the circle, the clock rate is unchanging!

I like this! Of course, it also occurs to me that if Twin B circles Twin A he will clearly age less but, because their relative speed is zero, it's another counter-example to ghwellsjr's statement below.

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.

 

[rjbeery]

bobc2, in the Minkowski diagrams how does an object do such things as create a wavy timeline or rotate its timeline by 90 degrees so that it may "take a shortcut"?

 

[DrGreg]

Of course, it also occurs to me that if Twin B circles Twin A he will clearly age less but, because their relative speed is zero, it's another counter-example to ghwellsjr's statement below.

Er, no, because "speed" means scalar magnitude of vector velocity, not rate of change of scalar separation. The relative speed is not zero in this case.

 

[rjbeery]

Er, no, because "speed" means scalar magnitude of vector velocity, not rate of change of scalar separation. The relative speed is not zero in this case.

Maybe I was hasty; I was considering relative speed to be change in relative distance over time...let's try another tack: Twin A is the one with relative speed from the orbiting Twin B's perspective, yet they both agree that B is aging more slowly...does this suggest that it isn't speed but acceleration that is the cause?

 

[bobc2]

bobc2, in the Minkowski diagrams how does an object do such things as create a wavy timeline or rotate its timeline by 90 degrees so that it may "take a shortcut"?

Answer: The same way it does in the 3-D world race between blue and red cars (my post #224). The blue car accelerated to turn the corner for the short cut. The blue guy traveling twin in the 4-D twin paradox example did the same thing.

That's the whole point. Sure, you need the acceleration to rotate in 4-D space, but that doesn't mean the acceleration caused the path to be shorter, any more than the acceleration for the blue 3-D race car "caused" the short cut (even though you needed to accelerate in order to take the shorter path).

Now, I'm beginning to see the source of our difference in preferred way of interpreting the twin paradox. You like to stick close to the phenomenological approach--just concentrate on the observations and don't make too much of ideas that are not solidly supported by well identified postulates and theorems (although I'm not convinced that even from that point of view you are on solid ground with your acceleration causality idea--but I'll let the others sort that out). My problem is I tend to come down on the side of the realists and would like to have a "real" (whatever that means) objective 4-dimensional space with 4-D objects (it has serious implications about the interpretation of causality--and I certainly would not accept the idea that acceleration "causes" the affine space connection to the laws of physics. I tend to want physics to be expressed geometrically. However I am ambivalent about that as well, because I know quite well the implications that follow--and at a subjective level I don't like those implications at all. (I'm sure Ben Crowell and JesseM would have some pretty tough questions for those green men from hyperspace in the sketch below.)

Wow! I've got to hand it to you, RJBeery. You sure have your hands full and are doing a gallant job of fending off everyone. Stick to your guns.

 

[JesseM]

Answer: The same way it does in the 3-D world race between blue and red cars (my post #224). The blue car accelerated to turn the corner for the short cut. The blue guy traveling twin in the 4-D twin paradox example did the same thing.

That's the whole point. Sure, you need the acceleration to rotate in 4-D space, but that doesn't mean the acceleration caused the path to be shorter, any more than the acceleration for the blue 3-D race car "caused" the short cut (even though you needed to accelerate in order to take the shorter path).

Wow! I've got to hand it to you, RJBeery. You sure have your hands full and are going a gallant job of fending off everyone.

This whole argument seems to turn on semantics--how are you defining "caused"? I don't think it's a word that has any formal technical definition in physics. Presumably there are situations where X and Y are two physical facts, and X is a necessary and sufficient condition for Y, but you would not agree that "X caused Y"?

 

[Dale]

In other words...

=

No. I think it is weird to talk of length being "caused" by a bend. Even with the caveat about the magnitude.

 

[harrylin]

Maybe I was hasty; I was considering relative speed to be change in relative distance over time...let's try another tack: Twin A is the one with relative speed from the orbiting Twin B's perspective, yet they both agree that B is aging more slowly...does this suggest that it isn't speed but acceleration that is the cause?

The cause of what? If you check out the equations, time dilation is only a function of the velocity - more precisely, of the velocity wrt to any inertial reference system. For the Twin case, we are dealing with two velocities and acceleration is the cause of the asymmetry, by changing one of the velocities. This was understood from the very start when the example was given.

The time dilation equation implies that clock rate is not affected by acceleration, and this is correct for many clocks. Tests with accelerator rings have confirmed this for elementary particles. See "The Clock Hypothesis" in http://www.phys.ncku.edu.tw/mirrors/physicsfaq/Relativity/SR/experiments.html

Harald

 

[bobc2]

This whole argument seems to turn on semantics--how are you defining "caused"? I don't think it's a word that has any formal technical definition in physics. Presumably there are situations where X and Y are two physical facts, and X is a necessary and sufficient condition for Y, but you would not agree that "X caused Y"?

You always seem to come up with the relevant observations, JesseM. I don't think I could disagree with you. Also, I originally thought RJBerry's posts were a little off the wall, but I've gained much more respect for his ideas (not that what I think about his ideas is relevant).

 

[rjbeery]

This whole argument seems to turn on semantics--how are you defining "caused"?

I've been thinking about the semantics issue and how it may be resolved...if we look at the logical definition of causality, there are 3 categories of causes:

Necessary causes:
If x is a necessary cause of y, then the presence of y necessarily implies the presence of x. The presence of x, however, does not imply that y will occur.
Sufficient causes:
If x is a sufficient cause of y, then the presence of x necessarily implies the presence of y. However, another cause z may alternatively cause y. Thus the presence of y does not imply the presence of x.
Contributory causes:
A cause may be classified as a "contributory cause," if the presumed cause precedes the effect, and altering the cause alters the effect. It does not require that all those subjects which possesses the contributory cause experience the effect. It does not require that all those subjects which are free of the contributory cause be free of the effect. In other words, a contributory cause may be neither necessary nor sufficient but it must be contributory.

Using these definitions we can begin labeling things.

IGNORING GRAVITY: Relative velocity, on its own, is necessary but not sufficient for absolute time dilation. Acceleration, on its own, is also necessary but not sufficient. Only taken together, and only under certain circumstances, are velocity AND acceleration necessary and sufficient for absolute time dilation. (How do we account for the "only under certain circumstances"? I'm thinking of the times in which both twins experience a combination of velocity and acceleration such that the time dilation effects are nullified.)

WITH GRAVITY: Relative velocity is now neither necessary nor sufficient for absolute time dilation, but now falls under the contributory cause definition. Acceleration WITH Relative velocity, as above, continues to be sufficient. Gravity, on its own, is also sufficient (again, "only under certain circumstances"). There is also what I consider to be a dubious example of the twin paradox involving 3 brothers that is devoid of either acceleration or gravity which can be found http://chaos.swarthmore.edu/courses/PDG/AJP000384.pdf". The author concludes that the "cause" of time dilation here is the change of inertial frames, which I should note, also occur under the other two mentioned sufficient causes. Presuming this encompasses ALL possible sufficient causes for absolute time dilation, we have now identified a single globally valid necessary cause: a change in inertial frames for at least one of the twins.

 

[GrayGhost]

JesseM,

Sorry, my PC has been choking a bit as late. I'll review your prior recent posts again before responding. My prior illustration was initially drafted for the all-inertial case, and I tweeked it (quickly) for the twin B POV only for sake of point. Attached is an enhancement of the prior illustration in an attempt to resolve your complaints about it.

I designate numbers (0 thru 8) on the illustration, which should not be taken as "an order of events". They are merely to have references in any discussion of the illustration.

When twin B executes a virtually-instant-acceleration, he must hold (per the LTs) that twin A moves rapidly from point 1 to point 3. The twin A clock at point 1 is (virtually) the same time readout as at point 2. Therefore, during B's rapid acceleration, the A clock must wildly advance 75% of the A-time from point 2 to point 4 (given v = 0.866c). Doppler effects do not directly show a "time jump", however indirectly they do after the doppler effects are accounted for and light transit time (from twin A) is negated.

GrayGhost

 

[JesseM]

Sorry, my PC has been choking a bit as late. I'll review your prior recent posts again before responding.

OK, I hope you will pay particular attention to the end of my previous post to you, and answer the questions I asked you there.

When twin B executes a virtually-instant-acceleration, he must hold (per the LTs) that twin A moves rapidly from point 1 to point 3.

No, because there is no single inertial frame where such a rapid jump occurs. Of course it's true that if you use the LT to find the distance from A to B immediately before acceleration in the inertial frame where B was at rest before acceleration, and then use the LT to find the distance from A to B immediately after the acceleration in the frame where B was at rest after the acceleration, the two distances are different. But each of these two distinct frames cover all of spacetime, both the times before acceleration and the times after, and in neither frame does A's position make any sudden jump. It's only if you make a Frankenstein's monster frame by stitching together the first frame's coordinates for events before the acceleration and the second frame's coordinates for events after (i.e., make a non-inertial frame whose judgments about simultaneity and distance at each point on B's worldline matches with B's instantaneous inertial rest frame at that point) that you can say that A's position "jumped" as in your diagram. But the Lorentz transformation does not deal with Frankenstein's monster frames created by stitching together regions of different inertial frames, it only deals with mappings between one inertial frame and another inertial frame. If you use the LT to map from the x,t coordinates in A's frame to some set of x',t' coordinates in any other frame, you'll never come up with a single coordinate system where the x'-coordinate of A jumped by a large amount over an arbitrarily brief interval of coordinate time dt'.

 

[GrayGhost]

No, because there is no single inertial frame where such a rapid jump occurs.

Of course, but it must occur per he who transitions said inertial frames. He is B. Twin B is actively transitioning inertial frames of reference. It seems to me that the LTs still tell us what should happen, by extrapolation ...

We may consider a number of inertial observers all momentarily colocated at twin B's departure event from twin A. One at 0.1c, another at 0.2c, 0.3c, etc thru 0.866c. If B's acceleration is virtually instant, virutally no time passes for he or anyone else momentarily colocated at the departure event. Twin B transitions the frames in almost no time at all, and his final frame is 0.866c. In the limit as acceleration approaches instant, at completeion of B's acceleration, twin B and the always inertial 0.866c observer are essensitally one in the same far as their POV goes. When he completes the rapid acceleration, he should record (just about) the same of the planet X clock that the always inertial 0.866c observer does.

The Fermi Walker cooridnates appear to do the very same thing. Look at the animation of the right side of this link ... and note the wild jumps during proper acceleration ...

http://en.wikipedia.org/wiki/Fermi%E2%80%93Walker_transport"

GrayGhost

 

[JesseM]

Of course, but it must occur per he who transitions said inertial frames. He is B. Twin B is actively transitioning inertial frames of reference.

What does "transitions inertial frames of reference" mean? You seem to have some totally over-concrete notion that objects are just naturally "in" some frame or another, as opposed to it being a mere human convention to link particular objects to particular frames. Nothing is stopping B from continuing to use his old inertial rest frame (or any other frame) even when he is no longer at rest in it, there is no reason for him to "transition" unless he chooses to do so!

Twin B transitions the frames in almost no time at all, and his final frame is 0.866c. In the limit as acceleration approaches instant, at completeion of B's acceleration, twin B and the always inertial 0.866c observer are essensitally one in the same far as their POV goes. When he completes the rapid acceleration, he should record (just about) the same of the planet X clock that the always inertial 0.866c observer does.

Again, the coordinates you "record" for distant objects and events (including the coordinate distance to a distant planet) is a matter of what frame you choose to use, it's not like you are "naturally" forced to record things in the inertial reference frame where you are currently at rest. You don't seem to understand this point about the frame associated with a given observer being purely a matter of choice/convention as opposed to something natural, which is exactly why I asked you the following question in post #201, then repeated it in #204, then again asked you to address it in post #236:

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?

If you aren't willing to address this question I don't see much point in continuing this conversation.

Also, consider the following quote from p. 43 of https://www.amazon.com/dp/0716723271/?tag=pfamazon01-20 by John Wheeler and Edwin Taylor, about how events (and by extension, worldlines composed of a series of events) are not naturally "in" any particular frame:

'A rocket carries a firecracker. The firecracker explodes. Does this event--the explosion--take place in the rocket frame or in the laboratory frame? Which is the "home" frame for the event? A second firecracker, originally at rest in the laboratory frame, explodes. Does this second event occur in the laboratory frame or in the rocket frame?

'Events are primary, the essential stuff of Nature. Reference frames are secondary, devised by humans for locating and comparing events. A given event occurs in both frames--and in all possible frames moving in all possible directions and with all possible constant relative speeds though the region of spacetime in which the event occurs. The apparatus that "causes" the event may be at rest in one free-float frame; Another apparatus that "causes" a second event may be at rest in a second free-float frame in motion relative to the first. No matter. Each event has its own unique existence. Neither is "owned" by any frame at all.

'A spark jumps 1 millimeter from the antenna of Mary's passing spaceship to a pen in the pocket of John who lounges in the laboratory doorway (Section 1.2). The "apparatus" that makes the spark has parts riding in different reference frames--pen in laboratory frame, antenna in rocket frame. The spark jump--in which frame does this event occur? It is not the property of Mary, not the property of John--not the property of any other observer in the vicinity, no matter what his or her state of motion. The spark-jump event provides data for every observer.

'Drive a steel stake into the ground to mark the corner of a plot of land. Is this a "Daytime stake" or a "Nighttime stake"? Neither! It's just a stake, marking a location in space, the arena of surveying. Similarly an event is neither a "laboratory event" nor a "rocket event." It is just an event, marking a location in spacetime, the arena of science.'

 

[Mike_Fontenot]

[...]
Nothing is stopping B from continuing to use his old inertial rest frame (or any other frame) even when he is no longer at rest in it, there is no reason for him to "transition" unless he chooses to do so!
[...]

The reason for him to "transition", is that if he bases his conclusions about simultaneity on the conclusions of an inertial frame in which he is not at rest, he will then be forced to ignore his own elementary calculations involving his own elementary measurements.

Here is a pertinent post that addresses that issue:

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

The issue that you (JesseM and GrayGhost) are both "dancing around" (the elephant in the room, really), is this: Whenever a person is NOT accelerating for some segment of his life, WHEN in that segment can he legitimately be considered to be an inertial observer? I.e., when is he allowed to use the Lorentz Equations to determine simultaneity at a distance, just as a perpetually-inertial observer is allowed to do?

The correct answer to that question is that he can legitimately be considered to be inertial during the ENTIRE segment. In fact, he MUST be considered to be inertial during the entire segment, in order to maintain consistency with his own elementary calculations.

Other alternatives for the (sometimes accelerating) traveler's simultaneity have been previously endorsed by others on this forum. All of those alternatives answer the above question differently from the answer I gave above ... i.e., they all maintain that a traveler who is not PERPETUALLY inertial CANNOT always use Lorentz simultaneity during segments of their lives in which they are unaccelerated.

For example, Fredrik is a proponent of Dolby&Gull simultaneity:

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

Another alternative, endorsed by Passionflower and Dalespam, is Minguizzi simultaneity:

https://www.physicsforums.com/showpost.php?p=2965909&postcount=72 .

For proponents of these (or any other alternative) simultaneities, EVERY discussion of the standard time-dilation result SHOULD be begun by declaring that neither observer has EVER accelerated in the past. And for some of the alternatives (like Dolby&Gull), it is also necessary to begin by declaring that neither observer will EVER accelerate in the future. Otherwise, those alternatives maintain that the time-dilation result cannot be used at arbitrarily large distances.

Requiring PERPETUALLY-INERTIAL observers, in order to use the Lorentz equations, opens a BIG can-of-worms: Can it really be said (even in principle) that ANY person has NEVER accelerated? We are all composed of elements (and/or their constituents) which have existed since the big bang ... have none of those elements EVER accelerated?

Mike Fontenot

 

[ghwellsjr]

Mike, you don't seem to understand the difference between observers and frames, even after Jesse's very clear description of it.

 

[Dale]

The reason for him to "transition", is that if he bases his conclusions about simultaneity on the conclusions of an inertial frame in which he is not at rest, he will then be forced to ignore his own elementary calculations involving his own elementary measurements.

No he won't. He can use any frame he likes whether or not he is at rest and whether or not the frame is inertial. All frames will predict the same results for all measurements. That is required by the first postulate. Your deliberate and persistent ignorance on this topic is simply astounding.

 

[GrayGhost]

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?

Well, so long as the eventual receipt of light signals validates the convention, then I don't see that it should matter, also including the prediction and validation of doppler effects. Consider the twins ... (during a controlled flight test) if EM transmitted from twin A contains his own time readout and the relative range of B from A at that time (per A), then the spacetime predictions made by twin B (of A) must comply, after the light transit time is negated and doppler effects accounted for. This must be true no matter what convention B uses, assuming the convention matches reality.

If you use 0.6c as your frame of reference, this requires that you must have the added burden of transforming via the composition of velocities formula. So, it's less convenient. Either that, or the relativistic effects predicted for 0.6c must be defined as your standard (non-proper) POV for all relative motion, which again is inconvenient. Similarly, and worse yet, would be the use of a non-inertial frame for your reference for all motion, given you are always-inertial. Can any of these be done? yes. Would anybody want to? Probably not.

GrayGhost

 

[JesseM]

Well, so long as the eventual receipt of light signals validates the convention, then I don't see that it should matter, also including the prediction and validation of doppler effects. Consider the twins ... (during a controlled flight test) if EM transmitted from twin A contains his own time readout and the relative range of B from A at that time (per A), then the spacetime predictions made by twin B (of A) must comply, after the light transit time is negated and doppler effects accounted for. This must be true no matter what convention B uses, assuming the convention matches reality.

If you use 0.6c as your frame of reference, this requires that you must have the added burden of transforming via the composition of velocities formula. So, it's less convenient. Similarly, and worse yet, would be the use of a non-inertial frame for your reference for all motion, given you are always-inertial. Can it be done? yes. Would anybody want to? Probably not.

OK, so you agree there is no natural sense in which changing velocities causes you to "transition" from one inertial frame to another, that it is just a matter of it being more convenient? Well, that's why I've been objecting to your repeated claims that "according to the LTs" the surface of simultaneity swings around during acceleration, the LTs don't dictate what coordinate system an accelerating observer should use, that's a matter of human choice based on considerations like convenience. I also think that in the case of an accelerating observer, it'd be nonsense to say it's most "convenient" to use a coordinate system where the definition of simultaneity at any given instant always matches the definition in the instantaneous inertial rest frame...calculating this from observations would actually be fairly tricky (because B has to figure out at what point in his past his velocity was such that the surface of simultaneity in his inertial rest frame at that moment would intersect with the event of A sending the signal that B is receiving at this moment), whereas something like the Wheeler-Marzke system would be fairly simple, you just constantly send out radar signals and assign the event of the signal bouncing off A a time-coordinate halfway between the time on your clock when that signal was sent and the time on your clock when it returned to you.

 

[ghwellsjr]

Well, so long as the eventual receipt of light signals validates the convention, then I don't see that it should matter, also including the prediction and validation of doppler effects. Consider the twins ... (during a controlled flight test) if EM transmitted from twin A contains his own time readout and the relative range of B from A at that time (per A), then the spacetime predictions made by twin B (of A) must comply, after the light transit time is negated and doppler effects accounted for. This must be true no matter what convention B uses, assuming the convention matches reality.

If you use 0.6c as your frame of reference, this requires that you must have the added burden of transforming via the composition of velocities formula. So, it's less convenient. Either that, or the relativistic effects predicted for 0.6c must be defined as your standard (non-proper) POV for all relative motion, which again is inconvenient. Similarly, and worse yet, would be the use of a non-inertial frame for your reference for all motion, given you are always-inertial. Can any of these be done? yes. Would anybody want to? Probably not.

GrayGhost

All that you say in terms of convention being validated and the convention matching reality for twin A's rest frame is also true for any other inertial frame. That's the problem. What would you think of Michelson and Morley conducting their experiment one time and announcing to the world that they had discovered the absolute ether rest frame because they detected no ether wind? But we know better, don't we? Every inertial frame will behave exactly like an absolute ether rest frame. You cannot use that "evidence" to claim that a particular inertial rest frame is preferred, just because the convention is validated by reality or because the math is more convenient.

You can use any frame to determine things that are frame invariant, such as the measurements that each twin will make or the things that they see with their own keen eyes because these things leave in the light transit time. But you are presumming to "know" the light transit time in an absolute sense so that you can back it out of the measurement and determine the actual time that the traveling twin can deduce of the stationary twin's clock. This determination is not frame invariant and so no argument based on reality or convenience is valid.

What you can do legitimately is build a scenario where the two twins agree to determine everything from a frame in which they are both at rest to start out and when the trip is finished (is that what you meant by "a controlled flight test"?) but that will completely undermine they "paradox" in the Twin Paradox.

 

[ghwellsjr]

How about "when a clock properly accelerates in a straight line, its tick rate changes relative to all inertial frames."?

Agreed.

OK, good, rjbeery, now do you agree with this statement?

When an object properly accelerates in a straight line, its length along that line changes relative to all inertial frames.​

 

[JesseM]

OK, good, rjbeery, now do you agree with this statement?

When an object properly accelerates in a straight line, its length along that line changes relative to all inertial frames.​

By "properly accelerates" are you specifically talking about Born rigid acceleration? Because it is quite possible for an object to accelerate in such a way that its length in a given inertial frame doesn't change, it all depends on the timing of when different points on the object start accelerating and what proper acceleration they experience (if they all start accelerating simultaneously in a given inertial frame, and they all experience identical proper acceleration at each moment of time in that frame, then the object's length won't change in that frame--see the http://en.wikipedia.org/wiki/Bell's_spaceship_paradox]Bell[/PLAIN] spaceship paradox).

 

[ghwellsjr]

Actually, I only added the word "properly" because DrGreg added it to his modification of your modification of my original statement about an accelerated clock to rjbeery and it's the version that rjbeery agreed to. I really don't care but if it matters concerning the length of an object, why didn't you complain about DrGreg's addition of the word with regard to an accelerated clock?

 

[JesseM]

Actually, I only added the word "properly" because DrGreg added it to his modification of your modification of my original statement about an accelerated clock to rjbeery and it's the version that rjbeery agreed to. I really don't care but if it matters concerning the length of an object, why didn't you complain about DrGreg's addition of the word with regard to an accelerated clock?

Ideal clocks are usually imagined to be point particles, since proper time is well-defined along the worldline of a point particle, so for an ideal clock there is no issue with different parts of the clock having different acceleration profiles. And even if you're dealing with an extended object, as long as the distance between ends (in any inertial frame) measured in light-seconds is very small compared to the time in seconds between the beginning and end of its motion (or the beginning and end of the time window you wish to consider), I think it should be the case that proper time experienced by different points on the object will differ by a very small amount compared to the total proper time experienced by anyone point. In contrast the change in length from the beginning to the end could be quite large compared to the total length at the beginning, if the object is not accelerating in a Born-rigid way.

 

[ghwellsjr]

Would be better to say "when a clock accelerates, its tick rate changes relative to all inertial frames."

Actually, even that statement need not be true. A clock moving at constant speed round a circular path is accelerating, but relative to someone permanently at the centre of the circle, the clock rate is unchanging!

How about "when a clock properly accelerates in a straight line, its tick rate changes relative to all inertial frames."?

DrGreg--why did you insert the word "properly" into JesseM's suggested improvement to my original statement?

 

[DrGreg]

DrGreg--why did you insert the word "properly" into JesseM's suggested improvement to my original statement?

Because what everyone has been calling "acceleration" in this thread should, strictly speaking, be called "proper acceleration" i.e. acceleration measured by a comoving inertial observer, or equivalently as measured by an accelerometer. More generally "acceleration" can be measured relative to any frame, including a non-inertial frame, so without the word "proper", different observers could disagree whether something is accelerating or not. But everyone agrees what "proper acceleration" is, it's frame-invariant.

Actually this has nothing whatsoever to do with the point JesseM raised in post #246. The point here is when you are talking about an extended object rather than a point-like particle, it is possible to (properly) accelerate different parts of the object by different amounts. It would seem that when you put forward the proposition

When an object properly accelerates in a straight line, its length along that line changes relative to all inertial frames​

I would assume you meant that the object's length as measured by itself (or to be more precise as measured in the comoving inertial frame) remains constant over time. This is described as "Born rigid acceleration". And then your proposition is correct and is the space-equivalent of the statement about time previously made. But in general objects might not accelerate in a Born rigid way, and then the proposition might fail.

(For example, if you start to accelerate an object by starting to push from the back, it takes time for the force to travel through the object and the front won't start to accelerate until some time later and the object will necessarily have compressed in its own rest frame, never mind any other frame. This compression has nothing to do with relativity. If instead of pushing the back, you pulled the front, you'd have stretched the object.)