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

Den 14.03.2024 17:18, skrev Richard Hachel:

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

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
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Note that this means that Vr > c when t > c/a
which according to SR is impossible.
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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:
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So this is wrong.
You can see the correct derivation here:
https://paulba.no/pdf/TwinsByMetric.pdf
See chapter 2.3, equation (15)
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Vr(t) = a⋅t/√(1+(a⋅t/c)²)
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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.
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If A is the coordinate acceleration in K, we have:
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A = dVr/dt = a/(√(1+(a⋅t/c)²))³
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Note that:
  A → a when t → 0
  A → 0 when t → ∞
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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
=====================================================================
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Contradicting fact:
-------------------
This is wrong.
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Vr(t) = a⋅t/√(1+(a⋅t/c)²)
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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
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Note that:
  Vm → a⋅t/2 when t → 0
  Vm → c     when t → ∞
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So:
  Vm/Vr  → 1/2  when t → 0
  rm/Vr  → 1    when t → ∞
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So for any t > 0   Vm > Vr/2
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It is not possible to make SR predict anything else!
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>
You don't understand anything I'm telling you...

 I do indeed understand that you telling me:
  Vr(t) = a⋅t
and:
  Vm(t) = Vr(t)/2
 And I do indeed understand that what you are telling me is wrong.
And it is a very naive and elementary blunder!
 

>
In these conditions, it is very difficult to discuss.

 I do understand that you find it difficult to defend your own words.
So that's why you don't even try, right?