Re: Understanding the theory of special relativity

Den 18.01.2025 10:04, skrev Richard Hachel:

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

>
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.

It is the very same as this scenario:
https://paulba.no/pdf/TwinsByMetric.pdf
See: 2.2 B travels with constant speed and instant acceleration
And: 2.4 Concrete example
This is what SR predicts:
When Terrence (Twin A) is back, his watch show 23.664 y ≈ 23.7 y
When Stella is back (Twin B), her watch show 9.912y ≈ 9 y
(bad round off, but doesn't really matter)
The important point is:
When Stella is back, Terrence and Stella are co-located and
stationary to each other, and both can see both clocks which are
side by side.
Terrence can see that his watch shows tau_T
and Stella's watch shows tau_S.
Stella can see that her watch shows tau_S
and Terrence's clock shows tau_T.

 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.

You say tau_S = 9 year and tau_T = 13.5 year
I say   tau_S = 9 year and tau_T = 23.7 year

 Your mistake is to make a wrong integration thinking that it is right, and this distorts your To/Tr ratio

The actual numbers are not important in this case.
What is important is that when Stella is back, Terrence and Stella
are co-located and stationary to each other.
Terrence can see that his watch shows tau_T
and Stella's watch shows tau_S.
Stella can see that her watch shows tau_S
and Terrence's clock shows tau_T.
Do you agree? (yes or no, please. There is no third alternative)
--
Paul
https://paulba.no/