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

Den 12.03.2024 20:42, skrev Richard Hachel:

Le 12/03/2024 à 20:19, "Paul B. Andersen" a écrit :

Den 12.03.2024 10:00, skrev Richard Hachel:

We know that in accelerated frames of reference the average speed is proportional to the instantaneous speed.
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Let Vrm=(1/2)Vri

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This is meaningless without definition of the entities.
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But I can guess:
You are not talking about "speed in an accelerated frame".
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You are talking about the speed of a stationary object
in an accelerated frame, measured in an inertial frame.
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If the object is instantly at rest at t = 0, and
the coordinate acceleration in the inertial frame is constant a,
Then the speed Vri(t) = at
and the average speed from t=0 to t=t is Vrm(t) = at/2.
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So  Vrm=(1/2)Vri

 That is what I am saying.

Yes. And I agree.
NOTE THIS:
  If the object is instantly at rest at t = 0, and
  the coordinate acceleration in the inertial frame is constant a,
  Then the speed Vri(t) = at

 Vr=a.Tr

Where Vr (Vri(t)) is the speed in the inertial frame,
and Tr (t) is the time coordinate in the inertial frame, and
----------------------------------------------------------------
a is the constant coordinate acceleration in the inertial frame!
================================================================
The consequence of this is that when t > c/a  v > c
which is impossible in SR.
That means that the coordinate acceleration a can't be constant!
==============================================================

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According to SR:
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For the above to be true, the coordinate acceleration must
be constant. This means that the proper acceleration must
increase with time.
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Let 1g = 1 c per year = 9.4998 m/s².
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when t ≥ 1.0 year the coordinate acceleration can't
be kept equal to 1 g.
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Generally:
The coordinate acceleration a can't be kept equal to n g
when t ≥ 1/n year.
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It is obviously normal to keep the proper acceleration constant,
and then Vrm  ≠ (1/2)Vri

As you said yourself:
"I beg you to understand something:
  in the rocket's frame of reference, the acceleration is constant."
That is the proper acceleration A is constant, and then
the coordinate acceleration a is NOT constant, it is
decreasing with time, Vri < at and Vrm > (1/2)Vri.
Se exact calculation below.

 I don't understand what you are saying.
 I think that between you and me, there is a different understanding of things.

Indeed!
Understand this:
What SR predicts is not a matter of opinion,
it is a matter of fact.
https://paulba.no/pdf/TwinsByMetric.pdf
see equation (38)
It is a FACT that according to SR, the speed of
an object with constant proper acceleration A is:
Vri(t) = A⋅t/√(1+(A⋅t/c)²)
Note that:
  Vri → A⋅t when t → 0
  Vri → c   when t → ∞
The average speed Vrm at the time t is:
Vrm = (integral from t=0 to t=t of Vri(t)dt)/t
Vrm = c²⋅(√(1+(A⋅t/c)²)-1)/A⋅t
Note that:
  Vrm → A⋅t/2 when t → 0
  Vrm → c     when t → ∞
So:
  Vrm/Vri  → 1/2  when t → 0
  Vrm/Vri  → 1    when t → ∞
So for any t > 0   Vrm > (1/2)Vri
It is not possible to make SR predict anything else!
====================================================
The coordinate acceleration a is:
  a = dVri/dt = A/(√(1+(A⋅t/c)²))³
where A is the proper acceleration
Note that:
  a → A when t → 0
  a → 0 when t → ∞

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But we know that relativistic physicists do not use this notion, and it is important to remind them of the relationship between real speed and observable speed.
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Vr=Vo/sqrt(1-Vo²/c²)

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This is nonsense.
According to "relativistic physicists" there is no difference
between the "real speed" and the "observable (measurable) speed"
in the inertial frame.

 That is what I am saying.
Physicists do not differentiate between Vo and Vr.

Because there is no such difference.
You are claiming that when you measure the speed of a passing object,
then you will always measure the real speed divided by γ.
Can you please explain how you arrived at this conclusion?
Show the maths, please.

It is really a shame that a large majority of physicists are not very intelligent, they understand nothing of what they are saying,
which is dramatic since their number serves as truth.

:-D
The "large majority of physicists" are obviously much more
intelligent than average. They have to be to pass the exams.
(I am not a physicist.)

A bit like in Nazi Germany everyone believed they were right because everyone held out their arm in front of Hitler.
Physicists are educated, but very few are intelligent.

A hysterical reaction!

 On usenet, there are perhaps five or six people with whom we can seriously discuss relativity, without immediately falling into hysterical reactions. (Julien Arlandis, Michel Talon, Paul B Anderson...). The others not only learned anything, but answer nonsense when asked a simple problem.
 

Above you have demonstrated that you belong to the group that
falls into hysterical reactions.
--
Paul
https://paulba.no/