Re: [SR] Usefulness of real velocities in accelerated relativistic frames of reference.

Le 18/03/2024 à 22:12, "Paul B. Andersen" a écrit :

Quite.
 We can now review the journey to Tau Ceti.
 Both Earth and Tau Ceti are considered to be inertial.
 A rocket is stationary on  Earth, When its clock show  τ = 0 and
the Earth clock show t = 0 the rocket engine starts an give
the rocket a constant proper acceleration a = 10 m/s².
 a = 10 m/s² = 1.05265 ly/y/y   c = 1 ly/y  d = 12 ly

  Tout cela est entièrement exact.

 According to your equations v = a⋅t and vₘ = a⋅t/2:

 Please note : Vr=a.Tr
               Vrm=a.Tr/2

===================================================
 d =  ∫a⋅t⋅dt + 0 ly = a⋅t²/2  => t = √(2⋅d/a)
 The rocket will pass Tau Ceti at the terrestrial time:
  t = √(2⋅d/a) = 4.7764 y

 No, no, no, no !
 In terrestrial time, To=(x/c)/sqrt(1+2c²/ax)  To=12.92 years.

The proper time of the rocket when it passes Tau Ceti is:
  τ = √(2⋅d/a) = 4.7764 y

 Absolutely.

The speed of the rocket in the terrestrial frame
when it passes Tau Ceti is:  v = a⋅t = 5.2860 ly/y

 Yes, Vr=5.0245c
 BUT: Vo=0.980c
 

 The average speed is: vₘ = a⋅t/2 = 2.6430 ly/y

 Vrm=2.51c if Vr=5.0245c
 Observable speeds Vo is not real speeds Vr, apparent speeds Vapp is not observable speeds Vo.  R.H.