Twins Paradox: Why One Twin is Older When Reunited

[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.