On 5/18/21 10:38 AM, Tom Roberts wrote:
>
> Hmmm. There's still the ambiguity of which inertial frame to use: that
> of the observer or that of the distant person?
>
Suppose the "home twin" (she) and the "traveling twin" (he) are both
perpetually inertial, and have a relative speed of 0.866 ly/yr, so that
gamma = 2.0.
She can set up HER co-stationary grid of synchronized clocks, along with
a "helper observer" stationed at each of those clocks. He can also set
up HIS co-stationary grid of synchronized clocks, together with a
"helper observer" stationed at each clock. They each synchronize their
own grid of clocks by sending light signals between their clocks, and
make use of the fact that they know what the speed of light is, and that
it is constant.
He can eventually determine what her current age was when he was age t1,
by receiving a message from his helper observer who passed her when that
helper was aged t1. That message says that she was tau1 = t1 / 2 when
she and his helper were momentarily co-located.
When he receives that message, he completely believes it, because he
knows his grid of clocks are properly synchronized. And he also has
been able to confirm (by information from his helpers) that her grid of
clocks AREN'T synchronized.
Long ago, when he was age t1, he used the time dilation equation (TDE)
to tell him that she was ageing half as fast as he was (because their
relative speed v is 0.866 ly/yr, and so gamma = 2.0). So the TDE told
him THEN, when he was age t1, that she was age t1 / 2. And now, much
later, he has been able to confirm, by the message the receives from his
helper, that the answer he got from the TDE was indeed correct.
If instead (as you prefer), he uses the inertial reference frame in
which SHE is at rest to determine how old she is when he is t1 years
old, that result will disagree with what his own system of synchronized
clocks and helpers eventually tell him. If he is a physicist, he won't
be a very successful physicist if he disregards his own measurements.