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

Den 14.03.2024 17:13, skrev Richard Hachel:

Le 14/03/2024 à 15:02, "Paul B. Andersen" a écrit :

Your opinion of SR is irrelevant.
>
SR is a consistent theory, and the issue is:
"Does SR predict that accelerated objects will behave as
  you claim they do?"

 No, YOU are saying that my concepts are irrelevant.

I am saying that SR does not predict that accelerated objects
will behave as you claim they do.

As for accelerated frames of reference, I said that things are poorly explained by physicists and that certain equations are false.

I suppose you are referring to my "false" equations:
Speed:            Vr(t) = a⋅t/√(1+(a⋅t/c)²)
Average speed     Vm(t) = c²⋅(√(1+(a⋅t/c)²)-1)/a⋅t
where your "correct" equations are:
Speed:  Vr(t) = a⋅t
Average speed  Vm(t) = a⋅t/2
:-D
----------------------------------------------
It seems that you still haven't the contradicting facts.
So let us yet again review your claims and the contradicting facts.
PLEASE READ THEM FOR THE FIRST TIME IN 40 YEARS!
A rocket is accelerating at the constant proper acceleration a.
An inertial frame of reference K(x,t) is at the time t = 0
instantly co-moving with the rocket.
You claim:
According to SR the speed of the rocket in K is Vr(t) = a⋅t
===========================================================
Note that this means that Vr > c when t > c/a
which according to SR is impossible.
Contradicting fact:
-------------------
So this is wrong.
You can see the correct derivation here:
https://paulba.no/pdf/TwinsByMetric.pdf
See chapter 2.3, equation (15)
Vr(t) = a⋅t/√(1+(a⋅t/c)²)
Note that:
  Vr → a⋅t when t → 0
  Vr → c   when t → ∞
Your problem is that you do not understand the difference
between proper acceleration of the rocket, and the rocket's
coordinate acceleration in the inertial frame.
If A is the coordinate acceleration in K, we have:
A = dVr/dt = a/(√(1+(a⋅t/c)²))³
Note that:
  A → a when t → 0
  A → 0 when t → ∞
So  Vr(t) = ∫(from 0 to t)A⋅dt = a⋅t/√(1+(a⋅t/c)²)
You claim:
According to SR is the average speed of the rocket Vm(t) = Vr(t)/2
=====================================================================
Contradicting fact:
-------------------
This is wrong.
Vr(t) = a⋅t/√(1+(a⋅t/c)²)
The average speed Vm at the time t is:
Vm = (integral from t=0 to t=t of Vr(t)dt)/t
Vm = c²⋅(√(1+(a⋅t/c)²)-1)/a⋅t
Note that:
  Vm → a⋅t/2 when t → 0
  Vm → c     when t → ∞
So:
  Vm/Vr  → 1/2  when t → 0
  rm/Vr  → 1    when t → ∞
So for any t > 0   Vm > Vr/2
It is not possible to make SR predict anything else!
====================================================
--
Paul
https://paulba.no/