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From: "Paul B. Andersen" <relativity@paulba.no>
Newsgroups: sci.physics.relativity
Subject: Re: [SR] Usefulness of real velocities in accelerated relativistic
 frames of reference.
Date: Fri, 15 Mar 2024 15:13:54 +0100
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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
>> ===========================================================
>> 
>> Note that this means that Vr > c when t > c/a
>> which according to SR is impossible.
>> 
>> 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!
>> ====================================================
> 
> 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?

-- 
Paul

https://paulba.no/