Re: Well...

We set To²=Tr²+Et²
 It’s great, life is above all very easy.
 We take Stella, we calculate her proper time (Tr=4.776 years), her improper time (validated by Terrence, To=12.915 years), her constant speed Vo=0.929c, her distance traveled to Tau Ceti (12al).
 We take Bella, in an accelerated reference frame at 1.052 ly/year², we calculate its proper time, its improper time, and the distance traveled.
 In RRH (Hachel's special relativity), if we place the proper time in y, the distance traveled in x,
and the improper tense on the LENGTH of the hypotenuse, all this still works very well.
   Even in accelerated frames of reference as long as you pay close attention to what you do.
   Furthermore, what is the slope of this hypotenuse for Stella? Unlike the speed Vr.
   But what can we say for Bella, who will trace a curve since her movement is accelerated?
   What becomes of To for her?
 It is the length of the line which links, at each moment, the origin O
  and the point we find ourselves on the curve. That's it, To. Bad weather.
   And what happens to Vr at each moment? It is the inverse of the slope of the tangent at the given point.
   This may seem very strange.
   But it is nevertheless mathematical.
   The most complex thing to understand (it took me decades to find the tilt) is that in the case
of the accelerated twin, the line which links the origin to the chosen point of the curve IS To, and that this line is obviously NOT the curve itself.
   And that on the other hand the real instantaneous speed (Vr) is given by the inverse of the slope of the tangent to the curve, speed which must then be transcribed into Vo. Vo=Vr/sqrt(1+Vr²/c²)
   Everything then enters into a sumptuous logic and a great conceptual beauty.
   We then obtain the following two equations, which physicists nevertheless deny, because they do not understand the beauty of reasoning, and persist in posing totally false equations.
   For the instantaneous speed of the accelerated mobile as a function of x and a:
  Voi/c=[1+c²/2ax]^-(1/2)
   For the improper time of the particle or the mobile:
  To=(x/c).sqrt(1+2c²/ax) Validated by physicists that.
   For the proper time Tr=sqrt(2x/a) quite simply.
   There, a terrible incomprehension can shake the biggest minds in physics, because they will ask:
Voi=Δx/ΔTo
   However, as obvious as it may seem, it is completely false.
   Something that would drive the greatest theoreticians crazy if they're not careful.
   Because Voi, it is given by the inverse of the SLOPE of the tangent to the curve at the given point,
  (and Voi transcribed from Vri), and not by the slope of the line which joins the origin to this point, the only use of which is the length which corresponds for each point to To.
   Breathe, breathe...
   Grab three cups of coffee, edit this post, and think for an hour or two about what I'm trying to explain to you in a few words.
   R.H.