Re: Understanding the theory of special relativity

Le 17/01/2025 à 13:30, "Paul.B.Andersen" a écrit :

Den 17.01.2025 02:37, skrev Richard Hachel:
Stella is in a rocket and is co-located with Terrence when
she starts her rocket engine and accelerates at 1 c/year
away from Terrence for 2.25 year on her clock.
Then she turns her rocket around and accelerates at 1 c/year
towards Terrence for 4.5 years.
Then she turns her rocket around and accelerates (brakes) at 1 c/year
away from Terrence for 2.25 years.
 When Stella is back at Terrence both stop their watches.
They are now co-located and stationary to each other.
Their clocks are side by side and can easily be compared.
Terrence clock shows 23.7 years.
Stella's watch shows 9 years.

In the problem you pose:
A rocket leaves the earth and accelerates (a=1ly/year).
This during a proper time Tr (or tau) = 2.25 years.
Which gives a total of Tr=9 years.
But there is an error in the way you transpose time into observable time in the terrestrial frame of reference. How old will Terrence be in this case?
The first thing is to cut the journey into four, since the four segments will give Tr=2.25 years.
For the first segment we will have To=Tr/sqrt(1+(1/4)Vr²/c²) if you have followed what Dr. Hachel says.
Here we can remain Newtonian and set Vr=a.Tr without any problem.
Let To=Tr/sqrt(1+(1/4)a².Tr²/c²)
And, To=3.3867 years
The phenomenon is reproduced four times:
To(final)=13.5468 years
Terrence will be a little over 13 and a half years old and Stella 9 years old.
Your mistake is to make a wrong integration thinking that it is right, and this distorts your To/Tr ratio
R.H.