There are two stars A and B relatively at rest in the universe with the proper length of 12 ly.
A person is traveling at the speed 0.8c relative to the stars along the extended line of stars A & B (towards A).
When the person gets to the midpoint between A & B, he sees a light pulse from stars A and B respectively.
How many years ago does he claim the Star A emits the light pulse? (18 years)
Using Lorenz's transformation equation, t = G [ t' + vx'/c2 ]
What was the time difference for A when the person's time has elapsed these 18 years?
The Attempt at a Solution
After length contraction, 12 ly => 7.2 ly for the person.
3.6 ly / (c-0.8c) = 18 years.
By using the Lorenz's transformation equation from the person's R.F. to A's R.F., the result is not 6 ly / c = 6 years as I expected.
Primed variables are with respect to the person, while the unprimed ones are to the Star A.
t = G [ t' - vx/c2 ]
tf = 0.6 [ t'f - 0.8c*3.6ly/c2 ] = 0.6t'f - 1.728y
ti = 0.6 [ t'i - 0.8c*( 3.6ly + 0.8c*18y)/c2 ] = 0.6t'i - 8.64y
tf - ti = 0.6( t'f - t'i ) - 1.728 + 8.64
= 0.6*18y + 6.912y = 17.712y (Relative to Star A.)
Star A would claim that the light pulse only travels 6 years from himself to the person who's in the midpoint.
Why is it not 6ly/c= 6y?? (Relative to Star A.)
This is the discrepancy I'm asking about.