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

Den 14.03.2024 03:09, skrev Richard Hachel:

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!
====================================================