The math is leading to seemingly nonsensical results when plugging v=c into the above equations.
First Y can't equal infinity, it must always be cancelled out in any useful equation. I proved this was true previously in this thread when I did the math for the twin paradox with the impossible scenarios of the twin returning at -c or accelerating away at +c in post #551.
A brief summary of the result is that light travels at 2c with the velocity through space = c and the velocity through time also = c. Only for light is this true as a material object going near c would have its vt approach 0. Time does not stop for light. There is a discontinuity at v=c where the results for material objects are very different for light at v=c. The math further showed that as Alice approaches c away from Bob after travelling 3 ly, she will permanently age 2 yrs more than Bob. If she returned to Bob at -c, she will permanently age 2 yrs less than Bob. Relativity has no comment on these issues which it deems are out of bounds.
Normally the velocity through time is an observed illusion controlled by the equation
c2 = v2 +vt2
where vt = c/Y
So at v = .6c, vt = .8c.
But during the time of relative velocity imbalance, which lasts from the time a change in relative velocity is made to the time it is received, the velocity through time is no longer an illusion and is no longer controlled by vt=c/Y but by vht=cDSR (vht is the half speed of vt) . So what's happening is constant relative velocity has a constant Y but I don't know what form Y takes during an imbalanced relative velocity. Actually I do know what form it takes but I don't yet know the formula relating cDSR in terms of Y. It's possible the formula is
Yv(c-v) = cDSR or Yv(c+v) = c/DSR
Some other clues are that the lines of proper simultaneity have a rate of changing slope only during the velocity imbalance period. This rate of slope change may provide another significant formula. Another clue is the time lengths of the light signals multiplied by DSR yield Y.
In the train rear example, the pink light signal from the middle of the platform to the rear of the train takes .625 yrs from t=0 where DSR = 2 in that direction and 1/2 for the yellow light signal direction which has a length of 2.5 yrs. So .625 * 2 = Y = 2.5 * 1/2. This mathematical pattern also appears in the formula
Yww = (v +u)(YvYu) where Yw = YvYu(1 + vu/c2) and w =(v+ u)/(1 + vu/c2)
When you set u=c then w =c because that's the combo result of any v +c.
If w = c then Yw = Yu = Yc
So plugging in
Yv(c+v) = c/DSR
into the above equation and cancelling out Yw and Yu you get
c=c/DSR which is no longer an equation.
So something went wrong and it's probable that Yw, Yu and Yc are not equal when u=c. Just like in the example of Alice turning around at -c or changing velocity to +c away from Bob, Yc is not infinity and Yu has a finite value that is not equal to the finite value of Yw. I don't know yet but my math senses say this is where to look.
The length of the light signals changes depending on direction (and hence the sign of DSR) and the depiction of the relative velocity. In the Minkowski depiction, light takes longer to reach something receding from it than it does something going towards it. The very definition of relative velocity says these are two different combined relative velocities yet the MMX found no difference. Just as DSR gives Alice the power to go many times c through time during the velocity imbalance period, so too could DSR give light the same power to adjust its velocity through time mathematically to accommodate velocities relative to it to keep c constant.
The vt for c is manifest in its frequency which is affected by the DSR. It's possible that when light is blue shifted or red shifted, these are the manifestations of c having its vt adjusted by a velocity relative to it so that c remains constant.
Edited by ralfcis, 22 December 2019 - 02:54 PM.