In this thread, I originally gave a resolution of the twin paradox that was purely qualitative, not quantitative. That was intentional, because some people's eyes glaze over when they see any equation. I gave an argument that allowed the traveling twin to INFER that (according to him) she must instantaneously age by a large amount when he instantaneously reverses his velocity at the his turnaround.
In what follows, for continuity I'm going to first repeat what I said in that first post. But then, I'll follow that with a quantitative description, and finally end with a specific numerical example.
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Qualitative Description:
The important thing in understanding the twin paradox is to start with the most trusted result of special relativity: the time dilation equation (TDE) for two inertial people moving at a relative speed "v". From that alone, we can easily determine how old she and he are at their reunion (by looking at it from HER perspective, since she is always inertial). He will be younger than she is, by a definite and calculable fraction.
THEN, we try to get the same answer, but this time using HIS perspective instead of hers. He can also use the TDE for EACH of the inertial legs of his trip for his analysis, and so he will say that she will be younger than he is at their reunion (by the same calculable fraction that she used). Something is clearly wrong: they MUST agree about their respective ages at their reunion, because they are standing right next to each other then, and looking at each other.
The only thing he could be leaving out in his analysis is that he has (perhaps unconsciously) been assuming that nothing happens to her age during his instantaneous velocity change. That's the only remaining time in his life where additional ageing by her could have occurred. So he then KNOWS that during his instantaneous velocity change, she MUST have instantaneously gotten older by exactly the amount that is required to make them agree about their respective ages when they are reunited.
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Quantitative Description:
What is the exact cause of the fact that the home twin is older than the traveling twin at their reunion?
It is caused by the change in velocity of the traveling twin (he), when he is separated from the home twin (she).
The simplest case is when his change of velocity is instantaneous (but the outcome is similar when his acceleration is finite for a finite duration). According to him, when he instantaneously changes his velocity with respect to her by delta_v, he says her age instantaneously changes by the quantity
- L * delta_v,
where "L" is their (positive) separation then, according to HER, and
delta_v = v(inbound) - v(outbound).
Velocities are taken as positive when the twins are moving apart, so delta_v is negative in the standard twin paradox scenario.
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Numerical Example:
I'll give a specific example of how the above equation is used. Let the relative speed of the twins be 0.866, using units of lightyears and years. That gives a gamma factor of exactly 2. So we immediately know that, when he is not changing his speed (and she never does), they each conclude that the other is ageing half as fast as they themselves are. Suppose she says that he goes outbound for 40 years (of her time). So she says that he is 20 years old at his turnaround. His turnaround is an EVENT, so everyone (including him) must agree that he is 20 years old then.
Similarly, according to her, she ages 40 more years while he is returning home. He ages by 20 years during his return (and both she and he agree about that, because the turnaround and the reunion are both EVENTS, so everyone agrees about his age then). So at the reunion, she is 80 years old, and he is 40 years old. They HAVE to agree about that, because they are standing side-by-side and looking that each other at the reunion.
But while he was going at a constant speed on his outbound leg, HE says (using the TDE) that she was aging at half his rate, so HE says she only got 10 years older on his outbound leg, and likewise for his inbound leg. So he concludes that she should only be 20 years old when he gets home. But she's not ... she's 80 ... he can see that. So where did she age the additional 60 years, according to him? There's only one place that could have happened: she HAD to have aged (according to him) by 60 years during his instantaneous turnaround.
And that's exactly what the simple equation in the above quantitative description section gives. Their distance apart at the turnaround (according to her) is
L = 40 * v = 40 * 0.866 = 34.64 ly.
In the qualitative section, I said
"According to him, when he instantaneously changes is velocity with respect to her by delta_v, he says her age instantaneously changes by the quantity
- L * delta_v,
where delta_v is his velocity change at the turnaround (and it's negative, so her age change is positive). We know from the known results at the reunion that he must conclude that she instantaneously gets OLDER when he accelerates TOWARD her.
Specifically,
delta_v = -0.866 - (+0.866) = -1.732.
So he says she instantaneously gets OLDER during his instantaneous turnaround by
- L * delta_v = (-34.64) * (-1.732) = 60 years.
Therefore she is 80 years old at their reunion, according to his calculations (and according to what he sees with his eyes then).
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