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

Den 18.03.2024 10:38, skrev Richard Hachel:

Le 17/03/2024 à 14:43, "Paul B. Andersen" a écrit :

Den 16.03.2024 15:29, skrev Richard Hachel:

 

So the coordinate transformation equations are:
  x' = x - v⋅t
  y' = y
  z' = z
  t' = t
>
Which IS Galilean relativity.

 Absolutly, this is galilean relativity.
But that's not what I'm talking about.

Yes, that's what you are talking about.
Since your equations:
   Speed of rocket in inertial frame:             Vr=a.Tr
   Average speed of rocket in the inertial frame: Vrm=(1/2)Vr
   are valid _only_  in Newtonian Mechanics with Galilean relativity,
   then the theory which is consistent with said equations
   is Newtonian Mechanics.
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
According to your equations v = a⋅t and vₘ = a⋅t/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
The proper time of the rocket when it passes Tau Ceti is:
  τ = √(2⋅d/a) = 4.7764 y
The speed of the rocket in the terrestrial frame
when it passes Tau Ceti is:  v = a⋅t = 5.2860 ly/y
The average speed is: vₘ = a⋅t/2 = 2.6430 ly/y
Note that d/vₘ = 4.5403 y < t    Why :-D
According to SR:
================
The rocket will pass Tau Ceti at the terrestrial time:
  t = √((d/c)²+2⋅d/a) = 12.9156 y
The proper time of the rocket when it passes Tau Ceti is:
  τ = (c/a)⋅arsinh(a⋅t) = 3.13894 y
The speed of the rocket in the terrestrial frame
when it passes Tau Ceti is:
   v = a⋅t/√(1 + (a⋅t/c)²) = 0.9973 ly/y
The average speed is:
   vₘ = c²⋅(√(1+(a⋅t/c)²)-1)/a⋅t = 0.9291 ly/y
Note that  vₘ/v > 1/2
===================================================================
The equations below are not consistent with your your equations
v = a⋅t and vₘ = a⋅t/2.

I'm talking about special relativity (Hachel sauce).
The transformations of Hachel (that's me) concerning the Galilean environments are as follows:
x'=(x-Vo.To)/sqrt(1-Vo²/c²)
y'=y
z'=z
To'=[To-(x.Vo/c²)]/sqrt(1-Vo²/c²)]
 The opposite becoming:
  x=(x'+Vo.To')/sqrt(1-Vo²/c²)
y=y'
z=z'
To=[To'+(x'.Vo/c²)]/sqrt(1-Vo²/c²)]
 I never said anything else.
 R.H.

--
Paul
https://paulba.no/