Twins Paradox: Why One Twin is Older When Reunited

[ghwellsjr]

I should have carried the explanation two steps further and applied the Principle of Relativity to point out that whatever Doppler ratio you see of the traveling observer's clock coming toward you compared to your clock, he sees the exact same Doppler ratio of your clock to his. And whatever Doppler ratio the traveling observer sees of the remote stationary clock to his, is the same that the remote observer sees of the traveling observer's clock to his own.

Yes, by invoking the principle of relativity along with the independence of light speed from the speed of the source, I agree that you can infer that the Doppler shift observed by the traveling twin on his outbound journey is the reciprocal of the shift on his return journey (with symmetrical speeds). And from this it follows that the total elapsed time for the traveling twin is less than for the stay-at-home twin. If I was to paraphrase your argument, I'd say you are considering three clocks a,b,c, with a and c mutually at rest, with clock b directly in between a and c, and moving away from a and toward c. Letting Da/Db denote the ratio of clock ticks from a reaching b to clock ticks of b, and so on, we have (Da/Db)(Db/Dc) = Da/Dc = 1, and by relativity we have Dc/Db = Db/Dc. Therefore, Da/Db is the reciprical of Dc/Db, and these represent the Doppler shift ratios for approaching and receding at the speed of b. Then, letting R denote the receding Doppler ratio, and 1/R the approaching Doppler ratio, you point out that for a symmetrical trip the average Doppler ratio for the whole trip is (R + 1/R)/2, which is strictly greater than 1 as R differs from 1. Hence the traveling twin ages less.

I'm glad you agree, thanks.

You must have understood my explanation enough to determine that there is no difference between Relativistic Doppler and Classical Doppler for what I described and I agree.

I don't follow that. Classical Doppler (for a theory in which the speed of the wave is independent of the speed of the source, like sound waves) would not lead to the same result, because classical Doppler gives different frequency ratio for the cases when the source or the receiver is moving.

I was referring just to the part in post #3, not #12. As long as the source, a, and the receiver, c, are moving identically so that they are in mutual rest, (Da/Db)(Db/Dc) = Da/Dc = 1. Do I need to draw a diagram?

Also I think this line of reasoning doesn't enable us to quantify the magnitude of the relativistic time dilation for any specific trip velocity, since it doesn't relate the speed to the Doppler ratios.

Correct, but it's irrelevant since everything applies at any velocity.

Keep in mind, I'm only trying to answer the OP's question: which twin ages more?

Yes, although I think it's debatable how much this explanation really clarifies why the twin who turns around ages less. It shows that if we impose the two principles, the turn-around twin must age less, but for a beginning student those two principles seem to be irreconcilable, and everyone knows we can derive infinitely many false conclusions from self-contradictory premises.

But these two principles are based on the history of experimental evidence and so they are not irreconcilable and so there is no danger of jumping to any false conclusions. I think it's important for a beginning student to grasp in a simple way that a traveling twin will return younger than the Earth twin and it's based on experimental evidence, facts, if you will and not merely theory. If we could easily do the experiment in a lab, they wouldn't find it so unusual or hard to grasp.

And especially when an OP asks the question the way this one did, we want to show him how the Principle of Relativity plus the second simple principle directly lead to the conclusion that the traveling twin ages less. He was thinking that the Principle of Relativity meant the twins would age the same.

So, for this reasoning to be persuasive, we need to explain why those two principles are not mutually contradictory (as they would be classically).

But the inverse nature of the two Dopplers is not classically contradictory as you have incorrectly concluded. Once you figure that out, maybe you'll have a different assessment.

But if that is not your concern, or even if it is, then this is one of those "why" questions that has no answer aside from "that's the way our world works".

And of course the light-speed principle applies only to inertial coordinates, so it comes back to the question of what makes one path inertial and the other not (if we imagine a purely relational world, for example, although the OP doesn't seem to be concerned about that - yet).

Yes, along with a slew of other assumptions that we take for granted but no one ever brings up when discussing other explanations of the Twin Paradox, so why bring them up for this one?

This, combined with the fact that this approach doesn't enable us to quantify the effect, may explain why this approach is not often followed.

Or, could it be that the approaches that are often followed are just repeated by people who don't really understand what they are talking about but this is one area of science where you're allowed to sound confusing and allowed to confuse your students and still be considered an expert?

This approach is not the end. It is just a beginning and needs to be followed up with more. I would follow it with a full explanation of Relativistic Doppler to state that what the twins observe is what a theory needs to explain and then show how Special Relativity fits perfectly (GR or gravity explanations are not needed) and use IRF's to show how they all explain all the observations and finally end with a non-inertial frame for the traveling twin based on radar measurements. All these frames identically support what each Twin sees, observes and measures and demonstrate the essential postulate of SR that light propagates at c. I would argue against the gravity, GR, multi-IRF or frame jumping explanations as promoting the Paradox instead of resolving it since they don't support what the twins see, observe or measure, nor do they demonstrate Einstein's second postulate of light propagating at c. At least that's my assessment and since I've not succeeded in getting anyone to prove me wrong, I'm sticking with it.

Again, thank you for your vote of confidence in this approach, even if it is only a beginning.

 

[Samshorn]

But the inverse nature of the two Dopplers is not classically contradictory as you have incorrectly concluded. Once you figure that out, maybe you'll have a different assessment.

This seems to be the central point. The disagreement isn't over the fact that (Da/Db)(Db/Dc) = Da/Dc = 1, but rather over the claimed equality (by relativity) Dc/Db = Db/Dc, which you use to claim that Da/Db and Dc/Db are recipricals. In other words, you rely on the relativistic fact that two inertial observers both see the same Doppler shift of signals from the other. Classically this equality doesn't unambiguously hold. Here's why:

We have two inertial clocks A and B with some mutual velocity v, and we assert the two principles of (1) relativity between inertial coordinate systems, and (2) independence of light speed from the speed of the source (again in terms of any inertial coordinate system). On this basis we can examine the relationship between the frequencies of light signals transmitted and received by the clocks in terms of (for example) the rest frame of A, in which the other clock B has velocity v.

Now, by the classical Doppler effect (meaning no relativistic time dilation of B relative to the time coordinates of A's rest frame), it's trivial to see that the light-speed principle implies that a signal sent from A with frequency fA will arrive at B with frequency fB where fB/fA = 1-v, whereas a signal sent from B with frequency FB will arrive at A with frequency FA where FA/FB = 1/(1+v). These two ratios are not equal, but the relativity principle requires them to be equal, so classically the two principles are irreconcilable.

The only way of reconciling those principles is that clock B must not advance equally with the coordinate time of A's rest frame. Specifically, clock B must advance more slowly by the time-dilation factor sqrt(1-v^2). Accounting for this difference in the rate of B, we find that the two ratios are

fB/fA = (1-v)/sqrt(1-v^2) and FA/fB = sqrt(1-v^2)/(1+v)

Both of these equal the relativistic Doppler factor sqrt[(1-v)/(1+v)], so we have satisfied the relativity requirement. Note that the time dilation factor applies regardless of whether A and B are approaching or receding from each other, and of course the Doppler effects in those cases are reciprocals.

The relativistic time dilation factor, which is necessary to logically reconcile the principles, automatically enables us to compute (quantitatively, not just qualitatively) the different ages of the twins, so nothing more is needed, and nothing less suffices. I think a student would accept your version of the Doppler explanation only if he failed to realize that the premises are mutually contradictory unless we posit relativistic time dilation, and once we do that, nothing else is needed. Also note that no third clock is needed. At best the Doppler considerations provide just one more derivation of the time dilation factor, but of limited applicability, and without clearly defining how that fits with the Lorentz transformation. As such, it will certainly seem unsatisfactory to the student when he realizes that not only must B run slow in terms of the standard time coordinates of A's rest frame, but A must run slow in terms of the standard time coordinates of B's rest frame. The only way to really understand this is to understand how inertial coordinate systems are related to each other, in full (i.e., the Lorentz transformation).

 

[ghwellsjr]

Was the math of post #2 within your grasp?

It's not just the math of post #2; the OP needs to understand why the math of post #2, as you presented it, only applies to the traveling twin, not the stay-at-home twin. There is also a corresponding equation that applies to the stay-at-home twin, with the same ratio R in it, but it differs from the one you gave in post #2 in a crucial respect. What that crucial respect is is not a question of math, it's a question of physics. Post #2 gives a hint at the answer to the question of physics, but you didn't give it explicitly.

I don't know if this is what you are referring to but if, or rather since, (R+1/R)/2 is the ratio of the stay-at-home twin's accumulated age compared to the traveling twin's accumulated age, then we can say its inverse, 2/(R+1/R), is the ratio of the traveling twin's accumulated age compared to the stay-at-home twin's age. In other words, its from the perspective of the stay-at-home twin looking at the traveling twin and we did this without knowing hardly anything at all.

But we do know something else which is that unlike the traveling twin who immediately sees a change from R to 1/R in the Doppler signal coming from the stay-at-home twin, the stay-at-home twin will not immediately see a change in the Doppler signal coming from the traveling twin but rather the Doppler will remain at R for a longer time than it remains at 1/R. And it's easy to calculate the exact fraction of time for R and 1/R.

Let's call F the fraction of the total trip time as measured by the stay-at-home twin that he observes a Doppler of R coming from the traveling twin. This means that the fraction of the total time that he observes a Doppler of 1/R must be (1-F) because the total time must be the sum of the two fractional parts. And if we multiple each of these fractions by their respective Doppler factors and add them together, they must equal the total of 2/(R+1/R). Now we can write an equation and solve for F as a function of R:

FR + (1-F)/R = 2/(R+1/R)
(FR²+1-F)/R = 2R/(R²+1)
FR²+1-F = 2R²/(R²+1)
F(R²-1)+1 = 2R²/(R²+1)
F(R²-1) = 2R²/(R²+1)-1
F(R²-1) = (2R²-R²-1)/(R²+1)
F(R²-1) = (R²-1)/(R²+1)
F = 1/(R²+1)

The other fraction is merely (1-F):

1-F = 1-1/(R²+1) = (R²+1-1)/(R²+1) = R²/(R²+1)

And of course the sum of the two bold terms is 1.

This was pure math so I don't know if it is what you were driving at. Could you enlighten me?

 

[PeterDonis]

Let's call F the fraction of the total trip time as measured by the stay-at-home twin that he observes a Doppler of R coming from the traveling twin. This means that the fraction of the total time that he observes a Doppler of 1/R must be (1-F) because the total time must be the sum of the two fractional parts.

This is what I was referring to; the fraction F for the stay-at-home twin is different than it is for the traveling twin. For the latter it's just 1/2, since the Doppler changes when he turns around halfway through the trip. That's the key item of physics that I was talking about; the traveling twin observes the change in Doppler as soon as he turns around (because his turning around is what changes it), but the stay-at-home twin doesn't observe the change in Doppler until light from the turnaround point reaches him. This is a key aspect of the scenario that is not symmetric between the twins.

 

[Samshorn]

And it's easy to calculate the exact fraction of time for R and 1/R... And if we multiple each of these fractions by their respective Doppler factors and add them together, they must equal the total of 2/(R+1/R).

That isn't legitimate, because you're simply assuming your result, and then backing out whatever fraction is needed to give you that result. What you would really need to do is compute the result from the perspective of the stay-home twin from the same principles that are used to compute the result for the traveling twin, rather than simply assuming that the results are reciprocals of each other. This means you need to derive the fraction F from first principles, and then show this leads to the result 2/(R+1/R), rather than simply assuming the latter result and claiming that F just happens to have whatever value it needs to have in order to give that result. This will require you to actually quantify your reasoning about speeds, spatial distances, time intervals, and inertia, because ultimately that's the difference between the twins - one follows an inertial path and the other doesn't - so any valid explanation has to invoke the concept of inertia in some form.

 

[ghwellsjr]

But the inverse nature of the two Dopplers is not classically contradictory as you have incorrectly concluded. Once you figure that out, maybe you'll have a different assessment.

This seems to be the central point. The disagreement isn't over the fact that (Da/Db)(Db/Dc) = Da/Dc = 1, but rather over the claimed equality (by relativity) Dc/Db = Db/Dc, which you use to claim that Da/Db and Dc/Db are recipricals.

Good, I'm glad we got that detail ironed out.

In other words, you rely on the relativistic fact that two inertial observers both see the same Doppler shift of signals from the other.

And the relativistic fact that you are talking about here is the Principle of Relativity, not the Theory of Special Relativity, correct?

Classically this equality doesn't unambiguously hold.

If it doesn't, then classical concepts regarding the Principle of Relativity are inadequate, aren't they?

Here's why:

We have two inertial clocks A and B with some mutual velocity v, and we assert the two principles of (1) relativity between inertial coordinate systems, and (2) independence of light speed from the speed of the source (again in terms of any inertial coordinate system).

I make a distinction between inertial observers/clocks and inertial coordinate systems. If you're going to invoke inertial coordinate systems, then you need to assert another principle and I see at least three options:

1) Light propagates at c only in one detectable inertial frame.

2) Light propagates at c only in one nondetectable inertial frame.

3) Light propagates at c in any inertial frame.

If you pick option 1), then you have rejected the Principle of Relativity.

On this basis we can examine the relationship between the frequencies of light signals transmitted and received by the clocks in terms of (for example) the rest frame of A, in which the other clock B has velocity v.

Now, by the classical Doppler effect (meaning no relativistic time dilation of B relative to the time coordinates of A's rest frame), it's trivial to see that the light-speed principle implies that a signal sent from A with frequency fA will arrive at B with frequency fB where fB/fA = 1-v, whereas a signal sent from B with frequency FB will arrive at A with frequency FA where FA/FB = 1/(1+v). These two ratios are not equal, but the relativity principle requires them to be equal, so classically the two principles are irreconcilable.

No, the first principle is irreconcilable with one of the third optional principles, namely, 1), as I already pointed out. So don't pick a third principle and just leave the analysis as I have, concluding that the traveling twin will have accumulated less time than his brother, based only on the Principle of Relativity and the assertion that light propagates independently of the motion of its source (plus a slew of other assumptions that nobody ever bothers to bring up in other discussions about the Twin Paradox).

The rest of your post shows what you can do by adopting either option 2) or option 3) but is not required to answer the OP's question.

The only way of reconciling those principles is that clock B must not advance equally with the coordinate time of A's rest frame. Specifically, clock B must advance more slowly by the time-dilation factor sqrt(1-v^2). Accounting for this difference in the rate of B, we find that the two ratios are

fB/fA = (1-v)/sqrt(1-v^2) and FA/fB = sqrt(1-v^2)/(1+v)

Both of these equal the relativistic Doppler factor sqrt[(1-v)/(1+v)], so we have satisfied the relativity requirement. Note that the time dilation factor applies regardless of whether A and B are approaching or receding from each other, and of course the Doppler effects in those cases are reciprocals.

The relativistic time dilation factor, which is necessary to logically reconcile the principles, automatically enables us to compute (quantitatively, not just qualitatively) the different ages of the twins, so nothing more is needed, and nothing less suffices. I think a student would accept your version of the Doppler explanation only if he failed to realize that the premises are mutually contradictory unless we posit relativistic time dilation, and once we do that, nothing else is needed. Also note that no third clock is needed. At best the Doppler considerations provide just one more derivation of the time dilation factor, but of limited applicability, and without clearly defining how that fits with the Lorentz transformation. As such, it will certainly seem unsatisfactory to the student when he realizes that not only must B run slow in terms of the standard time coordinates of A's rest frame, but A must run slow in terms of the standard time coordinates of B's rest frame. The only way to really understand this is to understand how inertial coordinate systems are related to each other, in full (i.e., the Lorentz transformation).

Hopefully, a student would learn and understand that there is a difference between the observable differential aging between two observers/clocks and the unobservable time dilation explanation offered by a theory such as LET or SR. Maybe this difference might motivate a student to study the subject matter in a coherent way.

 

[ghwellsjr]

And it's easy to calculate the exact fraction of time for R and 1/R... And if we multiple each of these fractions by their respective Doppler factors and add them together, they must equal the total of 2/(R+1/R).

That isn't legitimate, because you're simply assuming your result, and then backing out whatever fraction is needed to give you that result.

I didn't assume my result--I had no idea what it was going to be--and I was surprised at what it turned out to be.

What you would really need to do is compute the result from the perspective of the stay-home twin from the same principles that are used to compute the result for the traveling twin, rather than simply assuming that the results are reciprocals of each other. This means you need to derive the fraction F from first principles, and then show this leads to the result 2/(R+1/R), rather than simply assuming the latter result and claiming that F just happens to have whatever value it needs to have in order to give that result. This will require you to actually quantify your reasoning about speeds, spatial distances, time intervals, and inertia, because ultimately that's the difference between the twins - one follows an inertial path and the other doesn't - so any valid explanation has to invoke the concept of inertia in some form.

I don't understand your objection at all. If I can correctly determine that A sees B's clock accumulate Q times the amount of time on his own clock, then there can be no other answer than B sees A's clock accumulate 1/Q times the time on his own clock. Why is this controversial?

 

[Samshorn]

Quote by ghwellsjr: "But the inverse nature of the two Dopplers is not classically contradictory as you have incorrectly concluded."
Quote by Samshorn: "By the classical Doppler effect (meaning no relativistic time dilation of B relative to the time coordinates of A's rest frame)... a signal sent from A with frequency fA will arrive at B with frequency fB where fB/fA = 1-v, whereas a signal sent from B with frequency FB will arrive at A with frequency FA where FA/FB = 1/(1+v). These two ratios are not equal, but the relativity principle requires them to be equal, so classically the two principles are irreconcilable."
Quote by ghwellsjr: "...then classical concepts regarding the Principle of Relativity are inadequate, aren't they?"

Indeed, which is why you were mistaken when you claimed above that the reciprocalness of the two Doppler effects is not classically contradictory. Hopefully that's clear now.

I make a distinction between inertial observers/clocks and inertial coordinate systems. If you're going to invoke inertial coordinate systems, then you need to assert another principle...

It isn't just me who is invoking inertial coordinate systems, you are invoking them too (albeit tacitly and with denials). The two principles that you explicitly cited as your foundation are both expressed in terms of inertial coordinate systems. For example, Einstein expresses the relativity principle by saying the laws of physics are not affected "whether they be referred to the one or the other of two systems of coordinates in uniform translatory motion". No other principle is needed for this, provided of course that you've given some operationally meaningful definition of "inertial coordinate system" (which Einstein did in the first sentence of the first paragraph of Part I of his EMB paper). Likewise the lightspeed principle (or any proposition about 'speeds') has meaning only in the context of operationally meaningful measures of space and time.

Just leave the analysis as I have, concluding that the traveling twin will have accumulated less time than his brother, based only on the Principle of Relativity and the assertion that light propagates independently of the motion of its source...

Again, those two principles are expressed in terms of inertial coordinate systems. So you can't claim to be invoking these principles while denying that you are treating with inertial coordinate systems. Also, even if we overlook the logical inconsistency of your analysis, it doesn't allow you to conclude the traveling twin ages less, because it fails to quantity Q, which could be 1.

Hopefully, a student would learn and understand that there is a difference between the observable differential aging between two observers/clocks and the unobservable time dilation explanation offered by a theory such as LET or SR. Maybe this difference might motivate a student to study the subject matter in a coherent way.

Special relativity is not incoherent, whereas the "explanation" that you propose as an alternative to special relativity is incoherent, for the reasons I've explained. Again, you invoke principles whose meanings are explicitly given in terms of inertial coordinates, and yet you claim to do without any reference to inertial coordinates. That isn't logically consistent. Also, you yourself admit that your approach can never be quantitative, and for the same reason, i.e., inability to connect with any operationally defined measures of space and time.

I didn't assume my result--I had no idea what it was going to be--and I was surprised at what it turned out to be.

No, you simply assumed your result, i.e., you assumed that the age ratios of the twins are reciprocal, and from this assumption you backed out what value F would need to have in order for your assumption to be true. That is backwards from what's needed to demonstrate why the twins get reciprocal results.

I don't understand your objection at all. If I can correctly determine...

Well, it's certainly possible to correctly determine the result according to special relativity, but one of my points is that you haven't correctly determined anything. Again, without providing any operational meaning to the concepts of speed (as in the 'speed' of light and the motions of the twins) you have no warrant to even expect any Doppler shift at all (e.g., your unquantified Q could always be 1), not to mention that you can't apply either of the principles you claim to be applying, which are based on the same operational meanings. All you've done is appropriated a result from special relativity and forfeited its quantitative content and claimed to be able to explain it qualitatively by some fuzzy hand-waving (with the mistaken notion that classical Doppler is all that's required) with no logical coherence. For example, you claim time dilation is not observable, which of course is false, and in fact time dilation is a necessary consequence of your premises, as explained previously.

...that A sees B's clock accumulate Q times the amount of time on his own clock, then there can be no other answer than B sees A's clock accumulate 1/Q times the time on his own clock. Why is this controversial?

It's called the twins "paradox" precisely because beginning students suspect that if we worked out the result from the other twin's perspective, applying the same principles, we would get the symmetrical (not reciprocal) result, typically because they think of relativity in purely kinematical terms, without understanding the dynamical foundations in the principle of inertia. Hence they suspect that special relativity is logically inconsistent. To explain why special relativity is logically consistent, and why the other twin sees 1/Q, it makes no sense to simply assume that special relativity is logically consistent and that therefore the twin must see 1/Q. This is the very thing that we need to show, by carrying out the analysis from the other twin's point of view. This requires us to explain why the two points of view are dynamically asymmetrical, even though they are kinematically symmetrical, and this requires us to define operationally meaningful measures of space and time - the very thing you fail to do.

 

[ghwellsjr]

Indeed, which is why you were mistaken when you claimed above that the reciprocalness of the two Doppler effects is not classically contradictory. Hopefully that's clear now.

I hope you are not purposely misrepresenting what I said. I will assume that you never understood what I said. So I will draw a diagram to show how even under classical Doppler the explanation that I offered in the second paragraph of post #3 works:

The horizontal axis is distance and the vertical axis is time.

Here is that explanation:

Let's say that you are looking at two observers with clocks. One of them is a fixed distance away from you so you will see his clock ticking at the same rate as your own. They may have different times on them but we don't care about that, we only care about tick rates. The second observer is traveling at a constant speed from the first one directly toward you. Therefore, you will see his clock ticking faster than yours. You also know that you are seeing time progress on the traveler's clock compared to the first observer's clock just as the traveler is seeing them but delayed in time, provided that the light from both clocks travels towards you at the same speed. Therefore the ratio of the tick rates that you see of the distant fixed clock to the traveler's clock is some ratio that we called R. But the ratio of the tick rates that you see of the traveler's clock compared to your own (which is the same as the distant fixed clock) is the inverse of R.

You are the green observer C. You are looking at the clocks of the red observer A and the blue observer B. The red observer A is stationary with respect to you but both of you are traveling in the medium at the same speed. The blue observer, B, is traveling at some unknown, arbitrary speed from observer A towards you, the green observer C. The tick of each clock is represented by a colored dot.

In this example, you see blue's clock ticking three times faster than your own as depicted by the pale blue signal lines coming from his tick dots toward you. But you receive three of his for every one of your tick dots.

You can also see the same thing that the blue observer sees when he looks at the red observer's clock compared to his own, just delayed in time, namely that blue sees red's clock ticking at 1/3 the rate of his own clock. This is depicted by the black lines coming from each of red's tick dots and impinging on blue's tick dots but only every third one.

The two Doppler effects that I was talking about are your observations of B's clock compared to your own (3, in this example) and A's clock compared to B's clock (1/3, in this example). You can pick any other example and you will get the same result that the two Doppler factors are inverses of each other. And it doesn't matter if you consider classical Doppler or Relativistic Doppler or if you consider that the observers are moving with respect to a medium or stationary in it.

OK, got that? Please do not misrepresent what I said again. Of course if you think I am mistaken, then you're going to have to offer some proof--a diagram would be good.

It isn't just me who is invoking inertial coordinate systems, you are invoking them too (albeit tacitly and with denials). The two principles that you explicitly cited as your foundation are both expressed in terms of inertial coordinate systems. For example, Einstein expresses the relativity principle by saying the laws of physics are not affected "whether they be referred to the one or the other of two systems of coordinates in uniform translatory motion". No other principle is needed for this, provided of course that you've given some operationally meaningful definition of "inertial coordinate system" (which Einstein did in the first sentence of the first paragraph of Part I of his EMB paper). Likewise the lightspeed principle (or any proposition about 'speeds') has meaning only in the context of operationally meaningful measures of space and time.

The Principle of Relativity has experimental evidence which does not require the formulation of a specific set of laws and a transformation that embodies that principle. This is what Einstein was talking about in his 1920 book on relativity. For example, in at the end of chapter 5 which is about the Principle of Relativity, he says:

However, the most careful observations have never revealed such anisotropic properties in terrestrial physical space, i.e. a physical non-equivalence of different directions. This is a very powerful argument in favour of the principle of relativity.

Or in chapter 7 where he says:

Prominent theoretical physicists were therefore more inclined to reject the principle of relativity, in spite of the fact that no empirical data had been found which were contradictory to this principle.

So I'm talking about the "careful observations" and "empirical data" that support the Principle of Relativity without considering what specific laws or transformation is involved.

When MMX was performed, did they have to establish a reference frame or a coordinate system involving spatial coordinates and an establishment of coordinate time throughout that spatial coordinate system? In the same way, my thought experiment involving three observers and their clocks, which is separate from the twin scenario, does not establish any coordinate system. It does make a lot of assumptions which are always made in any discussion of this type but are rarely enunciated, so I don't know why we have to get any more specific than just stating the normal and usual way of discussing a thought experiment.

So in my three-observer/clock thought experiment, I stated that if the Principle of Relativity is assumed to be true, then whatever Doppler effect each observer determines for any other observer has a symmetrical relationship. This is the part that I said does not work under classical Doppler but that because classical Doppler does not conform to the Principle of Relativity in the way that we are concerned about here.

Again, those two principles are expressed in terms of inertial coordinate systems. So you can't claim to be invoking these principles while denying that you are treating with inertial coordinate systems. Also, even if we overlook the logical inconsistency of your analysis, it doesn't allow you to conclude the traveling twin ages less, because it fails to quantity Q, which could be 1.

Once again, you are misrepresenting what I said. What I said was:

If I can correctly determine that A sees B's clock accumulate Q times the amount of time on his own clock, then there can be no other answer than B sees A's clock accumulate 1/Q times the time on his own clock. Why is this controversial?

This is true even if Q = 1, isn't it? And it's true when Q ≠ 1, isn't it. It's not controversial, is it?

Special relativity is not incoherent, whereas the "explanation" that you propose as an alternative to special relativity is incoherent, for the reasons I've explained. Again, you invoke principles whose meanings are explicitly given in terms of inertial coordinates, and yet you claim to do without any reference to inertial coordinates. That isn't logically consistent. Also, you yourself admit that your approach can never be quantitative, and for the same reason, i.e., inability to connect with any operationally defined measures of space and time.

Again, a misrepresentation of what I said. I didn't say or imply that SR was incoherent. Why do you imply things like this? And if you properly understood my approach, you would see that it is perfectly coherent and logically consistent and has nothing to do with the fact that it has limitations.

No, you simply assumed your result, i.e., you assumed that the age ratios of the twins are reciprocal, and from this assumption you backed out what value F would need to have in order for your assumption to be true. That is backwards from what's needed to demonstrate why the twins get reciprocal results.

You make it sound like my assumption is invalid. How can it not be valid? You objection doesn't make any sense at all.

Well, it's certainly possible to correctly determine the result according to special relativity, but one of my points is that you haven't correctly determined anything. Again, without providing any operational meaning to the concepts of speed (as in the 'speed' of light and the motions of the twins) you have no warrant to even expect any Doppler shift at all (e.g., your unquantified Q could always be 1), not to mention that you can't apply either of the principles you claim to be applying, which are based on the same operational meanings. All you've done is appropriated a result from special relativity and forfeited its quantitative content and claimed to be able to explain it qualitatively by some fuzzy hand-waving (with the mistaken notion that classical Doppler is all that's required) with no logical coherence. For example, you claim time dilation is not observable, which of course is false, and in fact time dilation is a necessary consequence of your premises, as explained previously.

Another misrepresentation of what I said. I never said "that classical Doppler is all that's required". You need to read carefully all that I have written and understand it instead of misrepresenting it and accusing me of having mistake notions and characterizing my posts as "fuzzy hand-waving" and having "no logical coherence".

It's called the twins "paradox" precisely because beginning students suspect that if we worked out the result from the other twin's perspective, applying the same principles, we would get the symmetrical (not reciprocal) result, typically because they think of relativity in purely kinematical terms, without understanding the dynamical foundations in the principle of inertia. Hence they suspect that special relativity is logically inconsistent. To explain why special relativity is logically consistent, and why the other twin sees 1/Q, it makes no sense to simply assume that special relativity is logically consistent and that therefore the twin must see 1/Q. This is the very thing that we need to show, by carrying out the analysis from the other twin's point of view. This requires us to explain why the two points of view are dynamically asymmetrical, even though they are kinematically symmetrical, and this requires us to define operationally meaningful measures of space and time - the very thing you fail to do.

I made very clear that I was not talking about Special Relativity. I was talking about the Principle of Relativity. The OP did not ask about Special Relativity, he asked:

In the twin's paradox, why one of the brothers is older when they meet again?. if movement is relative. What determines which of them ends up older?

Movement being relative is an issue of the Principle of Relativity and I wanted to show that it's not because of any particular theory about relativity, but just because of that Principle and that light propagates independently of it the motion of its source, that we can determine which twin ends up older, just like he asked.

By the way, none of the ideas that I presented in this thread are unique to me. They have come up on this forum and in other references by other people in the past.

 

[phyti]

If this example is correct, it shows B seeing a slower clock rate for A than A sees for B. That would mean the classical doppler is not symmetrical.

https://www.physicsforums.com/attachments/58741

 

[ghwellsjr]

If this example is correct, it shows B seeing a slower clock rate for A than A sees for B. That would mean the classical doppler is not symmetrical.

https://www.physicsforums.com/attachments/58741

Actually, I didn't show A watching B but I have added it into this drawing:

It turns out that A sees B's clock ticking at 1/3 his own just like B sees A's clock so in this case, it is symmetrical but that's because A and B are both traveling at the same speed in opposite directions with respect to the classical medium and so the ratio is 1 as your document states.

Later on, I'll upload some more examples where A and B are traveling at different speeds so the ratio of their Dopplers will not be 1 and yet B's Doppler observation of A is the inverse of C's Doppler observation of B. That, after all, was the point of the drawing, not any symmetry or non-symmetry anywhere else.

 

[Samshorn]

Please do not misrepresent what I said again. Of course if you think I am mistaken, then you're going to have to offer some proof--a diagram would be good.

I think your mistake has already been explained, and the exact equations were presented to prove it (i.e., the classical Doppler equations). Actually there are two classical Doppler models: The classical emission theory satisfies the relativity principle but not the independence of light-speed from the source, whereas the classical wave model satisfies the independence principle but not the relativity principle. Neither of these would imply any asymmetric ages for the re-united twins.

The Principle of Relativity has experimental evidence which does not require the formulation of a specific set of laws and a transformation that embodies that principle.

The point is that I quoted the statement of the relativity principle from Einstein's EMB paper, and it is explicitly expressed in terms of inertial coordinate systems. If that's the principle you are invoking, then you can't claim to not be referring to inertial coordinate systems. Conversely, if you claim your approach does not rely in any way on the concept of inertial coordinate systems, then you can't claim to be invoking the principle of relativity. If you want to propose some other principle, you're free to do so, but you shouldn't call it the principle of relativity, because that name is already taken.

I'm talking about the "careful observations" and "empirical data" that support the Principle of Relativity without considering what specific laws or transformation is involved.

Again, you need to clearly state your principles. My contention is that you can't - at least not without abandoning your approach and adopting special relativity.

When MMX was performed, did they have to establish a reference frame or a coordinate system involving spatial coordinates and an establishment of coordinate time throughout that spatial coordinate system?

If the experimenters had tried to do that themselves, they would have gotten it wrong, but nature did it for them. Physical entities in any uniform state of motion configure themselves in equilibrium in such a way that their descriptions are the same in terms of co-moving inertial coordinates. This is why the MMX returns null.

The coherent account provided by special relativity of all the available experimental results is based solidly on the empirical agreement between inertial isotropy (represented roughly by what Einstein called "coordinate systems in which Newton's laws of mechanics hold good") and light-speed isotropy. It is an empirical fact that the one-way speed of light (in vacuum) is c in terms of any such system of coordinates. Without understanding this, you can't really give a coherent account of special relativity.

This is the part that I said does not work under classical Doppler but that's because classical Doppler does not conform to the Principle of Relativity in the way that we are concerned about here.

No, you began by insisting that classical Doppler was adequate for your explanation. I'm glad to see you're not claiming that any more. (As a reminder, you said "there is no difference between Relativistic Doppler and Classical Doppler for what I described".)

I didn't say or imply that SR was incoherent. Why do you imply things like this?

Fortunately we have a verbatim record of what you said: "Hopefully, a student would learn and understand that there is a difference between the observable differential aging between two observers/clocks and the unobservable time dilation explanation offered by a theory such as LET or SR. Maybe this difference might motivate a student to study the subject matter in a coherent way."

Thus you claim your explanation differs from that offered by LET or SR because yours involves only observables, whereas LET or SR involve "unobservable (sic) time dilation", and you hope that when the student sees this he may be motivated to study the subject in a coherent way, by which you mean the way you have proposed based on "observables" rather than the way SR explains things in terms of "unobservable (sic) time dilation". You happen to be wrong, because time dilation IS observable, and the explanations given by special relativity ARE coherent, whereas your explanation is incoherent, for the reasons I've explained several times already. (You base yourself on a principle whose very terms you reject.)

This is true even if Q = 1, isn't it? And it's true when Q ≠ 1, isn't it. It's not controversial, is it?

Special relativity is valid and not controversial. As mentioned previously, what you've done is to appropriate one particular result from special relativity, denude it of its quantitative content (by disconnecting it from any physically meaningful conception of speed), and then proposed an inadequate and incoherent "explanation" of that result - and for good measure you've insinuated that special relativity is incoherent. And of course since your reasoning allows Q=1, it doesn't even unambiguously predict any asymmetric ages for the re-united twins. Yes, your claims are controversial.

The horizontal axis is distance and the vertical axis is time.

If so, then you are making use of a space-time coordinate system, and presumably you accept that your "distance" and "time" have some operational significance, since otherwise your drawings are meaningless. It's good that you now (at least tacitly) acknowledge that you are reasoning with space-time coordinates (contrary to your claim). If pressed, you would probably even admit (eventually) that you are dealing with a special class of space-time coordinates, specifically, a system in terms of which (as Einstein put it, somewhat imprecisely) "the equations of Newtonian mechanics hold good". But this contradicts your claim to be doing without inertial space-time coordinate systems, as well as your claim that time dilation is unobservable. Notice that you say the vertical axis is "time", and yet it cannot be "time" for the moving twin. (If it was, your principles are violated, as explained by the Doppler equations in a previous post.) Hopefully you can see that your diagram illustrates why "time" for the moving twin cannot be the "time" of your vertical axis.

You make it sound like my assumption is invalid. How can it not be valid?

Already explained.

Movement being relative is an issue of the Principle of Relativity and I wanted to show that it's not because of any particular theory about relativity, but just because of that Principle and that light propagates independently of it the motion of its source, that we can determine which twin ends up older, just like he asked.

Again, I quoted the statement of the principle of relativity from Einstein's EMB paper, which is explicitly expressed in terms of inertial coordinate systems, and yet you claim that you can apply this principle without any consideration of or reference to inertial coordinate systems. That is plainly illogical. It also explains why you think time dilation is unobservable, and why you don't understand the empirical content of the light-speed principle.

By the way, none of the ideas that I presented in this thread are unique to me. They have come up on this forum and in other references by other people in the past.

That's unfortunately true.

 

[ghwellsjr]

If this example is correct, it shows B seeing a slower clock rate for A than A sees for B. That would mean the classical doppler is not symmetrical.

https://www.physicsforums.com/attachments/58741

Actually, I didn't show A watching B but I have added it into this drawing:

It turns out that A sees B's clock ticking at 1/3 his own just like B sees A's clock so in this case, it is symmetrical but that's because A and B are both traveling at the same speed in opposite directions with respect to the classical medium and so the ratio is 1 as your document states.

Later on, I'll upload some more examples where A and B are traveling at different speeds so the ratio of their Dopplers will not be 1 and yet B's Doppler observation of A is the inverse of C's Doppler observation of B. That, after all, was the point of the drawing, not any symmetry or non-symmetry anywhere else.

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

https://www.physicsforums.com/attachment.php?attachmentid=58761&stc=1&d=1368690498​

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:

https://www.physicsforums.com/attachment.php?attachmentid=58762&stc=1&d=1368690498​

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.

 

[georgir]

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.

 

[Fredrik]

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 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.

 

[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.