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

On 03/13/2024 02:00 PM, Paul B. Andersen wrote:

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

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That is what I am saying.

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Yes. And I agree.
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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
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Vr=a.Tr

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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!
================================================================
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The consequence of this is that when t > c/a  v > c
which is impossible in SR.
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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

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As you said yourself:
"I beg you to understand something:
  in the rocket's frame of reference, the acceleration is constant."
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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.
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Se exact calculation below.
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I don't understand what you are saying.
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I think that between you and me, there is a different understanding of
things.

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

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

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

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:-D
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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.)
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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.

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A hysterical reaction!
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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.
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Above you have demonstrated that you belong to the group that
falls into hysterical reactions.
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It's sort of so that accelerating has a "first start"
and that decelerating has a "last stop", yet
it is "non-standard", because our model is "real-valued",
and the real numbers have a standard definition,
and in it, there is no smallest positive real number (iota).
It's much after atomism, whether there "are",
"fundamental particles", in fact it's very much
the same sort of conceit, the concession the idea,
the "infinitely-divisible" or "non-atomic", divisible, and
the "infinitely-divided" or "atomic", non-divisible.
So, the idea of "smooth acceleration" vis-a-vis
"constant acceleration", really sort of requires
a combined "standard", and "non-standard",
together "super-standard", view, or model,
in the mathematical model, for the physical model,
"real numbers or the real-valued or the mechanics
of continuous domains: continuum mechanics,
dynamical modeling", "physics:  parameterized by t".
Newton kind of keeps things simple,
"it's all encapsulated to an instant with
an instantaneous exchange of kinetic
energy as the juxtaposition of two
non-deformable bodies geometrically
in perfect in-elastic collision", ..., "their
equal and opposite reactions".
So is the theory, ....
This is after Galileo keeps things simple,
"rest at rest motion at motion".
Then Einstein keeps things simple,
"the above is the classical limit, though
motion involves contraction and relaxation
of the space of the bodies that they are in,
and the space of the bodies that they are".
So, mathematics _owes_ physics, a proper
model of "super-standard change".
Dynamics, ....
It's a continuum mechanics, ....