Twins Paradox: Why One Twin is Older When Reunited

[georgir]

The two of you are just using the words "see" in different ways. You appear to be talking about the coordinate assignments made by the comoving inertial coordinate systems. He appears to be talking about the light that reaches the twins.

In fact his formulation is more suitable for "see"...
Still, the Lorentz transformations that give rise to the whole paradox, and hence the original post's question, apply to the actual coordinate system distances and rates, not to radar measurements.

 

[ghwellsjr]

ghwellsjr's first post is total bull, on the reverse trip each brother still sees the other's clock as slower, by exactly the same ratio if the speed difference is the same...

the key to this paradox is understanding the different definition of "now", or lines/planes/spaces of simultaneoty on a spacetime diagram, for different frames of reference. more specifically, the 'now' of the twin right before he reverses is much different than his 'now' after he reverses, from which comes the greater age of his distant twin despite he always seeing the rate of his clock slower.

draw a spacetime diagram and it is very obvious.

The two of you are just using the words "see" in different ways. You appear to be talking about the coordinate assignments made by the comoving inertial coordinate systems. He appears to be talking about the light that reaches the twins.

In fact his formulation is more suitable for "see"...
Still, the Lorentz transformations that give rise to the whole paradox, and hence the original post's question, apply to the actual coordinate system distances and rates, not to radar measurements.

Does this mean you have down graded your assessment of my first post from "total bull" to just "bull"?

Keep in mind, the OP never asked about Time Dilation or simultaneity issues. He just wanted to know which of the two brothers would be older since motion is relative. And I gave him an answer to that question.

But since you suggest that spacetime diagrams make it very obvious, I will provide some. First, for the Inertial Reference Frame (IRF) in which one brother (depicted in blue) remains at rest and the other brother (depicted in black) travels away at 0.6c for 12 months and then returns at the same speed for the same length of time for a total time of 24 months and finds his other brother has aged by 30 months. The dots on each brother's worldline mark off 1-month intervals and signals traveling at the speed of light are sent by each brother towards the other brother:

In this diagram, it is very obvious that the blue brother's time is not dilated whereas the traveling black brother's time is dilated. His dots marking off 1-month intervals of time are stretched out compared to the Coordinate Time of the IRF by a factor of 1.25 which is the value of gamma at a speed of 0.6c.

The diagram also clearly shows the symmetric Relativistic Doppler that both brothers observe of the other brother's time--it takes 2 months before they each see the other ones clock turn over 1 month.

At the moment of turn-around the Coordinate Time is at 15 months and the traveling black brother immediately sees the Doppler factor change from 1/2 to its inverse, 2. Up to this point, he has seen his blue brother age by 6 months but after this point, he sees his brother age by 24 months for a total of 30 months. However, the blue brother at rest does not see anything differently but continues to see the Doppler factor at 1/2 until month 24 and at this point he sees his brother turn around and start heading for home. Now he sees the Doppler factor change from 1/2 to 2 for the remaining 6 months. So the blue brother sees his black brother age by one year during two years of his time and then he sees his brother age by another year during just one-half year of his own time for a total of two years or 24 months.

Next is the IRF for the rest state of the traveling brother during the first half of his trip which we obtain by using the Lorentz Transformation on the coordinates of events in the original IRF to get a new set of coordinates:

Now we can see that the blue brother, who is at rest in the first IRF is traveling at 0.6c to the left in this IRF and so his clock is Time Dilated by the 1.25 factor during the entire trip. However, neither brother has any awareness of this, they continue to see everything the same as it was depicted in the first IRF. And what do they see? The black brother sees his blue brother age by six months during the first half of his journey and by 24 months during the second half of his journey. And the blue brother sees his black brother age by 12 months during 24 months of his own progress and then by another 12 months during the last 6 months of his own Proper Time. Note that there is no new information depicted in this IRF.

For the last IRF, we have the traveling black brother at rest during the last half of his trip and this time the blue brother is traveling to the right at 0.6c and his clock is Time Dilated during the entire trip by the gamma factor of 1.25:

And once again, although the Coordinate Time is 25.5 months at the turn-around event, this is not observable by either brother but all the other observations mentioned for the first two IRF's apply exactly the same in this IRG. Just read the previous descriptions while looking at this IRF and you will see that this is the case.

At the end of your last post, you mentioned that radar measurements did not apply to the OP's question. I hope you realize that I made no mention of radar during the course of answering the OP's question and it's strange that you would speak disparagingly of it because that is exactly Einstein's convention for synchronizing remote clocks in an IRF. However, we can carry the process one step further and use radar to build a non-inertial rest frame for the traveling brother and it will look like this:

Notice how the Coordinate Time for the turn-around event is 12 months, although it is not apparent in this drawing because we depict the black brother as stationary. Also note that the explanations given for the previous IRF diagrams apply exactly the same in this non-IRF and we could use this (or any of the IRF diagrams) to show how the blue twin could use radar to construct the first IRF in which he is at rest.

Just remember, there is no preferred IRF (or non-inertial frame) not even one in which an observer is at rest and any frame will depict exactly the same information that any other frame will (if drawn correctly without eliminating any significant details).

 

[Samshorn]

Keep in mind, the OP never asked about Time Dilation or simultaneity issues. He just wanted to know which of the two brothers would be older since motion is relative. And I gave him an answer to that question.

I think the reason for the disagreement is that your "answer" was expressed in purely kinematic terms (which are obviously not adequate to distinguish between the twins), and on the claim that "there is no difference between Relativistic Doppler and Classical Doppler for what I described". In previous posts, including post #47 (to which you never responded) I tried to explain why that is not true.

In general, I think you're confusing two very different things: (1) Showing that the Doppler effects implied by special relativity are self-consistent (something which no one disputes), and (2) Claiming that a naive kinematic view of the Doppler effect, without even distinguishing between classical and relativistic Doppler, and without invoking time dilation, the principle of inertia, and some operationally meaningful definition of motion, somehow "explains" which brother would be older. If all you saying is (1), then I don't think anyone disagrees, although it doesn't really answer the OP's question. But you seem to be saying (2), which is flat out wrong. You keep drawing pictures, but you seem determined to never acknowledge that those pictures have meaning only if we grant the very conceptual premises that you claim to be dispensing with. And if you ever accept those premises, the pictures become superfluous - except as redundant demonstrations of (1), which no one disputes anyway.

 

[georgir]

Radar measurements just don't give a sufficient picture when the radar itself is moved... Your last graphic is particularly indicative about that with how nonsensical it is. The vertical section of the blue line is completely random, indicating that the blue twin traveled in parallel to the black twin for no particular reason. You could just as well make it curved, zig-zag or make it spell your name in that segment.

 

[ghwellsjr]

Radar measurements just don't give a sufficient picture when the radar itself is moved... Your last graphic is particularly indicative about that with how nonsensical it is. The vertical section of the blue line is completely random, indicating that the blue twin traveled in parallel to the black twin for no particular reason. You could just as well make it curved, zig-zag or make it spell your name in that segment.

There is only one way for the non-inertial twin to use the radar measurements to generate the diagram. It's not random. It's not nonsensical. And there is no way he could have made that portion curved, zig-zag or spell any name. If you don't know what the process is, you can see how it is done on my posts on this page:

https://www.physicsforums.com/showthread.php?t=644948&highlight=triplets&page=6

But you're welcome to show me, if you are so sure about your position, how you would follow the radar method to spell out "geo". Show me, don't just make an idle claim.

 

[Dale]

Radar measurements just don't give a sufficient picture when the radar itself is moved.

The radar itself is always moving in some frame.

Your last graphic is particularly indicative about that with how nonsensical it is. The vertical section of the blue line is completely random, indicating that the blue twin traveled in parallel to the black twin for no particular reason. You could just as well make it curved, zig-zag or make it spell your name in that segment.

None of that is true. Radar coordinates are a perfectly well defined coordinate system. Furthermore, they are a natural generalization of Einsteins synchronization convention. Radar coordinates reduce to standard inertial coordinates for inertial radars and to Rindler coordinates for uniformly accelerating radars.

 

[georgir]

But you do not have a uniformly accelerating radar nor an inertial radar. You have instant acceleration - i.e. infinitely strong acceleration for zero duration here. You can approximate it with a uniform acceleration for a short time, but then the blue line should not have any angles in it, it will be a gradual curve. It will be a perfect hyperbola if you assume constant acceleration. It will be any shape you want if you assume some other type of acceleration profile.

EDIT: ok, not "any shape you want". maybe it will not be possible to spell your name with hyperbola fragments. but still can vary quite a bit and showing it as a straight line is quite random.

EDIT2: actually, please do explain to me, how do you 'radar' something beyond the apparent event horizons that appear when you are accelerating? as even if you approximate the instant acceleration with a gradual acceleration, it is still a very large acceleration and the event horizons will be terribly close...

 

[Dale]

But you do not have a uniformly accelerating radar nor an inertial radar. You have instant acceleration - i.e. infinitely strong acceleration for zero duration here.

Which is fine. The radar coordinate convention is adapted to arbitrarily accelerating observers. It is merely a nice feature that it reduces to the Rindler and Minkowski coordinates for the appropriate observers, not a restriction.

EDIT: ok, not "any shape you want". maybe it will not be possible to spell your name with hyperbola fragments. but still can vary quite a bit and showing it as a straight line is quite random.

Not random at all. Random means that for the same input the output is unpredictable. The radar coordinates are not random at all, they always produce the same output for a given input.

They are well defined and ghwellsjr has correctly drawn them. Each successive radar pulse that is sent and received prior to the acceleration is longer than the previous, so the radar distance is increasing. Each successive radar pulse that is sent and received after the acceleration is shorter than the previous, so the radar distance is decreasing. Each successive radar pulse that is sent before and received after the acceleration is the same duration as the previous, so the radar distance is constant. As shown by ghwellsjr.

See figure 9 here:
http://arxiv.org/abs/gr-qc/0104077

You might want to re-think your objection. What really bothers you about it? You probably assume that the inertial twin's worldline must have a single bend in it, but why did you assume that? Perhaps that assumption is not as solid as you might think.

EDIT2: actually, please do explain to me, how do you 'radar' something beyond the apparent event horizons that appear when you are accelerating?

You stop accelerating. The horizons only exist if you undergo constant acceleration forever.

 

[ghwellsjr]

But you do not have a uniformly accelerating radar nor an inertial radar. You have instant acceleration - i.e. infinitely strong acceleration for zero duration here. You can approximate it with a uniform acceleration for a short time, but then the blue line should not have any angles in it, it will be a gradual curve.

This is a little misleading. What you should say is the blue line will still be made up of three straight line segments, as I have drawn it, but with two gradual curves interconnecting them (at about 7.5 and 16.5 months). But why offer this legitimate complaint on just this one drawing when it applies equally to all Twin Paradox explanations that assume instant acceleration?

It will be a perfect hyperbola if you assume constant acceleration. It will be any shape you want if you assume some other type of acceleration profile.

When you say "It" you imply that the entire shape of the line between 7.5 and 16.5 months can be any shape you want which is not true. You should have said "They" because there are two very small interconnecting curves located separately at 7.5 and 16.5 months and they are not independent, if you make one be "any shape you want" then the other one cannot be any other shape you want. And unless you make the acceleration take a very long time, like many days, this shape that you want will be entirely covered up by the dots at 7.5 and 16.5 months and the diagram will end up looking exactly like I drew it. You would have to draw a separate "zoomed in" diagram to show this detail that you want.

EDIT: ok, not "any shape you want". maybe it will not be possible to spell your name with hyperbola fragments. but still can vary quite a bit and showing it as a straight line is quite random.

Now you are changing your terminology. First you said, "the blue line should not have any angles in it, it will be a gradual curve." That could be interpreted as a true statement if you meant, as I explained earlier that there are three straight line segments interconnected with gradual curves rather that sharp angles. But now you are objecting to my straight line which I presume can only be interpreted as the straight line going from 7.5 to 16.5 months. This segment can be nothing but a straight line unless you want to completely change the problem and eliminate all inertial motion from the traveling twin. But that's not what you said you wanted to do. You simply wanted to change the "instant acceleration" between the two inertial segments of the traveling twin to be "a uniform acceleration for a short time".

EDIT2: actually, please do explain to me, how do you 'radar' something beyond the apparent event horizons that appear when you are accelerating? as even if you approximate the instant acceleration with a gradual acceleration, it is still a very large acceleration and the event horizons will be terribly close...

In order to see the effect of an event horizon, the acceleration must last for as long as from the time the radar signal was sent until the time it was received. This is probably the issue that is confusing you. The reason why there is a straight line segment between 7.5 months and 16.5 months is that during that interval the radar was sent at different times during the inertial outgoing portion of the traveler's trip and received at different times during the inertial returning portion of the traveler's trip.

Another way of looking at this is that the signal that the traveler receives before, during or after his acceleration was sent at his 3-month mark and received at his 12-month mark. His own acceleration, whether it be instant, uniform or "gradual" will not significantly change his 12-month mark. If the acceleration lasts less than one day, it can only change the marks on the diagram by less than one day or light-day, not enough to notice on any diagram that would fit on one page.

 

[georgir]

Dalespam, thank you for that arxiv article. I finally understand what 'radar coordinates' mean and why my last two posts were wrong.

That said, I still don't view "radar coordinates" as a sensible alternative to standard coordinate systems. Coordinates of distant objects depend on the observer's past and future movement, and that makes them rather useless. Surely, when someone says "now", he means the simultanety space orthogonal to his worldline at the given point, regardless of his past or future acceleration.

But that is indeed just a matter of semantics. If your radar coordinates somehow help the OP figure things out easier, that is good. I can attest that for a long time they only served to confuse me specifically, but hey, maybe I'm just slow. Or maybe you should have posted that link sooner...

 

[Dale]

I still don't view "radar coordinates" as a sensible alternative to standard coordinate systems.

There are no standard coordinate systems for general non inertial observers.

 

[PAllen]

I think the reason for the disagreement is that your "answer" was expressed in purely kinematic terms (which are obviously not adequate to distinguish between the twins), and on the claim that "there is no difference between Relativistic Doppler and Classical Doppler for what I described". In previous posts, including post #47 (to which you never responded) I tried to explain why that is not true.

In general, I think you're confusing two very different things: (1) Showing that the Doppler effects implied by special relativity are self-consistent (something which no one disputes), and (2) Claiming that a naive kinematic view of the Doppler effect, without even distinguishing between classical and relativistic Doppler, and without invoking time dilation, the principle of inertia, and some operationally meaningful definition of motion, somehow "explains" which brother would be older. If all you saying is (1), then I don't think anyone disagrees, although it doesn't really answer the OP's question. But you seem to be saying (2), which is flat out wrong. You keep drawing pictures, but you seem determined to never acknowledge that those pictures have meaning only if we grant the very conceptual premises that you claim to be dispensing with. And if you ever accept those premises, the pictures become superfluous - except as redundant demonstrations of (1), which no one disputes anyway.

There are very simple sets of assumptions from which Doppler implies differential aging. For example, it is sufficient to assume:

- Doppler directly obeys the principle of relativity
- The speed of light in vacuo is not affected by motion of it source

The first of these is consistent with a Galilean corpuscular light model, but not a naive aether theory. The second is consistent with an aether theory, but not a Galilean corpuscular theory. Adopt both, and differential aging follows.

This gets at the ambiguity of what is meant by pre-relativistic Doppler. Bradley implicitly used a corpuscular model of aberration, which explains many later scientists dissatisfaction with it (since so much evidence established a wave model), despite its empirical success. Einstein (so far as I know) provided the first fully satisfactory derivation of aberration in 1905.

[Edit: I see you agree with much of the above in your #14, but don't find it satisfactory. I find it interesting to get at the ambiguity of what it is meant by pre-relativistic Doppler, and that there was already a contradiction observed - Bradley derivation of aberration versus waves in aether. One could imagine an alternate reality in which experiments forced adoption of these axioms, leading to SR.]

[Edit 2: Note that source motion independence of light speed correlates with Doppler related observation: if this were not true, you would expect a sudden turnaround distant object to have both red and blue shifted images for a period of time. Stating that distant object motion change never produces double images is sufficient along with relativity of Doppler to derive differential aging]

 

[ghwellsjr]

I am redoing post #48 because I recently noticed that the two diagrams fell through the cracks and did not appear originally:

I have two more examples. Here's the first one:

In this example, A and C are stationary with respect to the medium and B is traveling at one-half the speed of a signal in the medium. You can see that B observes A's clock ticking at one-half the rate of his own and C can also observe this. You can see that C observes that B's clock is ticking at double the rate of his own. These two ratios are inverses of each other which is the point that I have been making, for all Doppler ratios, whether classical or relativistic, as long as the speed of the signal is independent of the source.

Now to the issue that you brought up, B observes A's clock ticking at one-half the rate of his own clock but A observes B's clock ticking at two-thirds the rate of his own clock.

A second example:

In this example, A is traveling to the left at 40% of the speed of signals in the medium and B is traveling to the right at 12.5%. As a result, B sees A's clock ticking at 5/8 of the rate of his own clock. And C sees B's clock ticking at 8/5 the rate of his own.

And A observes B's clock ticking at 8/15 of his own clock.

Does this all make sense? Can you see that the only time A and B have a symmetrical measurement of the other ones clock with respect to their own is when they are both traveling in the medium at the same speed.

 

[Nugatory]

That said, I still don't view "radar coordinates" as a sensible alternative to standard coordinate systems. Coordinates of distant objects depend on the observer's past and future movement, and that makes them rather useless. Surely, when someone says "now", he means the simultaneity space orthogonal to his worldline at the given point, regardless of his past or future acceleration.

Or at least that's what he means until he encounters the Andromeda paradox, which shows that there are conditions under which interpreting "now" as "the simultaneity space orthogonal to his world line at the given point" doesn't yield a particularly sensible picture of what's going on.

And that's what happens in the easy case of an inertial observer. For a non-inertial observer, the same event may lie in the simultaneity space (defined by using momentarily co-moving inertial frames) of multiple points on the observer's world line, so the "orthogonal at this point" definition is no definition at all.

Really, there's only one criterion for whether a coordinate system is "sensible" or not: does it make a particular situation easier to analyze? If it does, then it is "sensible" to use it in the analysis of that particular problem.