« on: 28/11/2019 22:04:14 »
The formula you give is for Doppler shift do to ordinary motion ( rather than Cosmological Redshift) In this case, then v/c has to be determined using the Relativistic addition of velocities as noted by Halc. This gives a value of 0.882... for v/c and ~0.7288 for the Doppler shift. B would see a frequency of 0.7288 hz. If you try to use 1.2 for v/c, you get (1-1.2)/(1+1.2) = -0.909 under the radical, which leaves you with trying to get the square root of a negative number.
Think of it this way, If You were at rest with respect to the start point, after 1 sec, A and B would be 1.2 light sec apart. ( assume A move to the left and B to the right)
A sends a signal. That signal travels at c relative to you. After other second the front of that signal is 1 light sec from where it
was emitted. A has moved 0.6 ls to the left and B another 0.6 ls to the right. Distance between signal and B is 1.8-1 = .8 light sec. In one more sec the signal has moved another 1 ls to to right from and B 0.6 ls to the right. New distance = 1.4-1 = 0.4 ls In one more sec, the light travels 1 ls, B .06 ls, and the distance is 1-1 =0. the light has caught up with B. 3 sec after emission, the light reached B.
Perfectly true, but irrelevant. The light reaching B will be Doppler shifted so that
fB = fA√(1-β)/(1+β) where β = v/c = 1.2
So if A transmits, say, a 1 GHz pulse of radio waves, B will be able to calculate their relative speed as the received pulse will be at 0.4 GHz.
And indeed we have observed astronomical objects with extreme redshifts.
The conundrum is resolved by remembering that you only require infinite energy to accelerate a massive object from rest to c with respect to its starting point. There is no reason why two objects shouldn't travel in opposite directions at > 0.5c relative to their origin.