Ok, so the math seems to be too much for anyone to decipher so let's go off road a bit. If a 4yr old was looking at my STD, she'd ask why is the yellow line longer than the pink line. The people in the choo-choo see the light travel from the center at 1ly per yr in both directions so how can the people on the platform see the light travel 2.5ly per 2.5 yrs from the center to the front of the train and only .625 ly per .625 yr to the back of the train. (Please notice c is constant from all perspectives without any of Einstein's wizardry.) The answer, little girl, is the time on the train is dilated from the platform's perspective. Just like the train's velocity can be expressed in v=x/t or v'=Yv=x/t' if t' is the train's dilated time, so can the light's velocity be expressed within the train's frame from the platform's perspective as c'=Yc where Y=1.25 at .6c. Within the train's frame from the train's perspective Y=1 so c'=c and v'=v=0 because the train is stationary from its own perspective.
But the youngster would respond, "but time dilation is the same for both perspectives so there's still an inequality in light travel time and distance travelled between the two halves from the platform's perspective that the people in the train don't experience." Wow, I'd say, you're much smarter than anyone I've met on a physics forum. The answer is incomprehensible to anyone else listening in. Relative velocity has 3 problems: 1. it can't be depicted on an STD, 2. the doppler shift ratio, not time dilation, is directly related to relative velocity and 3. relative velocity depends on direction and a change in sign results in an inverse of DSR. So the pink line going backward would have a DSR =2 while the one going forward would have a DSR= .5. The depicted length of the yellow line =2.5 but it's relative velocity length, if that could be drawn, is multiplied by the DSR to equal 1.25 and dividing by Y would equal 1 from the platform's perspective. The depicted length of the pink line is .625 but multiplied by its DSR=2 is also 1.25 or 1 when divided by Y. So you see, with a little math and STD understanding, both lengths are the same from the platform's perspective and the train's. Of course I've spent a lot of time previously proving why this is so and since no one understood that, they won't understand this.
PS. If you're having trouble with the concept of Yv and Yc, it is the distance travelled (invariant from any perspective) in perspective time. So if Alice takes off from Earth at .8c and travels 1 ly, using earth's perspective of her dilated time where Y = 5/3 at .8c, her Yv = 4/3 c. But she is not covering that distance faster than light would because light's Yc using earth's perspective of that frame's dilated time is 5/3c. Yv or Yc can't be seen from any perspective but can be calculated when Alice sees she has travelled 1 ly in only 3/4 yr of her time from earth's perspective. Light has travelled that same distance in 3/5 yr of her time from earth's perspective. Length contraction was brought in to try to explain why Alice isn't really travelling faster than c but there's no need for that as Yv is not the same thing as v. Only v is required to remain below c.
Edited by ralfcis, 09 May 2019 - 03:30 PM.