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Le 09/04/2024 à 21:14, "Paul B. Andersen" a écrit :If the velocity v is a long the line of sight, then the angular
diameter of the Earth will increase proportionally with time,
but the Earth will appear to be stationary, so its apparent
speed is zero.
But if the Earth is approaching Stella at an angle ϕ from
the line of sight, she can measure the angular velocity,
and when the distance is known, she can calculate
the apparent transverse velocity, which indeed may be
higher than c.
Vapp = v⋅sin(ϕ)/(1 - (v/c)*cos(ϕ))
For example, if ϕ = π/4 and v = 0.8c, Vapp = 1.3c
If Vapp ≠ 0, Stella will never reach the Earth, obviously.
So this apparent speed is of no interest for the twin 'paradox',
even if it is interesting for 'superluminal jets'.
No, you're the one in the wrong here.No apparent phenomenon can have physical consequences.
Apparent speeds are, precisely, of capital interest for the Langevin traveler.
Moreover, if we properly understand the stroke of genius of live-live vision for any direct observer, as is the case with protaginists who experience EVERYTHING live-live from start to finish, and the tremendous upheaval conceptually, we see where Langevin's paradox was nestled, that is to say in the spatial zoom effect that we accept for Terrence but not for Stella. However, we must accept both.THE END.
"The effects of physics are reciprocal by permutation of frame of reference, including apparent effects!".
It is this formidable equation that you come up against:
D'=D.sqrt(1-Vo²/c²)/(1+cosµ.Vo/c)
You accept it for Terrence looking at Stella and her rocket.
But the terrible counter-intuitiveness of the reciprocal effect has disconcerted physicists for 120 years. They do not understand that the same, strictly the same effect occurs for the opposite speaker, and that Stella does not cross 7.2al twice, but twice 36al forwards, and twice 4 al backwards.
In total, she will have crossed 72al/18years=4c forward,
and 8al/18ans=0.4444c towards the rear of its rocket.
This is very important to understand, if we want to understand the paradox. And this is the fundamental key that no one had the trigger to understand.
R.H.
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