Re: Langevin and Doppler effects...

Den 05.09.2024 23:45, skrev Richard Hachel:

 Paul B. Andersen :
 https://paulba.no/pdf/TwinsByDoppler.pdf
 Well.
There are often few numerical applications in the
developments of relativistic theorists, because this can pose some problems of contradiction, or even absurdity.
 We will return to the eternal problem, which has scared away all the contributors I have dealt with for 40 years, although the opposite is always said, and that I am the one who would be a wimp.
 So we return to the problem, which is nevertheless very simple.
 Stella goes into the stars for a journey of 24 light years.

 The speed is 0.8c on the way there (12 ly), and 0.8c on the way back (12 ly).
 We quickly obtain the following data by the Doppler effect.
 Stella and Terrence agree to beep every month.

You will find the answer here:
https://paulba.no/pdf/TwinsByDoppler.pdf
L = 12 ly
c = 1 ly/y
f = 12/y
v = 0.8c
γ = 5/3
But since you never read anything, I will repeat it here.

 It is clear that, for Stella, on the way there, as on the way back, she will beep
108 times (216 times in total). >
SO, Terrence will receive 216 beeps.

OK. Stella has aged Tₛ = 2L/vγ = 18 y
Number of received beeps Ns = Tₛ⋅f = 216

 For his part, Terrence will beep for 30 years. He will therefore send
360 beeps.

OK

 Except that if Terrence receives the same number of beeps on the way there as on the way back (108 each time), Stella receives many more beeps on the way back than on the way there.
 It is then interesting to know how many beeps Stella receives on the way there, and how many she receives on the way back.

Doppler shift when Stella is moving out  Do = √((c−v)/(c+v)) = 1/3
Doppler shift when Stella is moving back Db = √((c+v)/(c-v)) = 3
So number of received beeps Nt = Tₛ⋅f⋅Do/2 +  Tₛ⋅f⋅Db/2 = 36 + 324 = 360

 Once this is done, we can still progress in understanding things.
 We just have to not be afraid, and have a playful spirit.

Terrence will age C = 2L/v = 30y
Am I guessing wrong when I guess that you will present us with
the paradox that Terrence will receive
Tₜₛ⋅f⋅Do/2 + Tₜₛ⋅f⋅Db/2 = 60 + 540 = 600 beeps ? :-D
Hint:
https://paulba.no/pdf/TwinsByDoppler.pdf

 R.H.

--
Paul
https://paulba.no/