Re: [SR] Their proper times will necessarily be equal

Le 20/04/2024 à 19:07, Python a écrit :

On 4/13/24 1:36 AM, Richard Hachel wrote:

"If two different observers travel an identical path in equal observable times, then their proper times will necessarily be equal.

 Yes, of course.
 My response differs from others because I interpret your context
differently.

 You interpret path as a path in space-time while Hachel is considering
path in space only (so it is true in only ONE reference frame)
 For instance, he claims that if two travelers would go from Earth
to Tau Ceti, one at constant velocity w.r.t Earth the other one
with a constant acceleration both will measure the same (proper)
time for their trips.

Yes, this is entirely correct.
If two different protagonists use different frames of reference to join Tau Ceti (12al), and if we notice that in the terrestrial frame of reference the travel times are similar and the path is similar (direct path), then the two proper times will be similar.
This is the case of a trip by Stella in Galilean mode at constant speed Vo=0.929c, and of a trip in accelerated mode by Bella (10m/s²=1.052ly/y²).
The duration of the trip, which is also very well calculated by my relativistic theoretician friends around the world, will be To=x/Vo in the first case, and To=(x/c).sqrt(1+2c²/ax) in the second case, i.e. 12,915 years.
But this duration will also be the same for the two rockets with regard to their own time. Tr=4.776 years.
Physicists tell me that while this is true for Stella, it is not true for Bella.
However, here, it is they who are mistaken and misuse space-time concepts in an accelerated environment.
On the other hand, be careful, for this to be true, and for it to remain true, two clarifications:
1. You must only use one reference frame during the entire trip.
2. the accelerated twin must go into “native” state, i.e. at rest.
R.H.