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

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.
 Let Vrm=(1/2)Vri

This is meaningless without definition of the entities.
But I can guess:
You are not talking about "speed in an accelerated frame".
You are talking about the speed of a stationary object
in an accelerated frame, measured in an inertial frame.
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.
So  Vrm=(1/2)Vri

 We are talking about real relativistic speeds (Richard Verret's hobby).

According to SR:
For the above to be true, the coordinate acceleration must
be constant. This means that the proper acceleration must
increase with time.
Let 1g = 1 c per year = 9.4998 m/s².
To keep the coordinate acceleration a equal to  1 g
the proper acceleration A must be:
at t = 0   year   A =   1.00 g
at t = 0.8 year   A =   1.65 g
at t = 0.9 year   A =   3.33 g
at t = 0.99 year  A =  33.4  g
at t = 0.999 year A = 333.5  g
at t = 1.000 year A = infinite
when t ≥ 1.0 year the coordinate acceleration can't
be kept equal to 1 g.
Generally:
The coordinate acceleration a can't be kept equal to n g
when t ≥ 1/n year.
It is obviously normal to keep the proper acceleration constant,
and then Vrm  ≠ (1/2)Vri

 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.
 Vr=Vo/sqrt(1-Vo²/c²)

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

 This immediately leads us to Vom/sqrt(1-Vom²/c²)=(1/2)Voi/sqrt(1-Voi²/c²)
 and therefore to:
 Vom²(1-Voi²/c²)=(1/4)Voi²(1-Vom²/c²)
 Let Vom²=(1/4)Voi²+(3/4)Voi².Vom²/c²
 Hence Vom=(1/2)Voi/sqrt(1-(3/4)Vom²/c²) and, conversely, Voi=2.Vom/sqrt(1+3Vom²/c²)
  Thank you for your kind attention.
 R.H.

--
Paul
https://paulba.no/