Lorentz transformation for velocity?

Question

Hi,

A distant quasar is found to be moving away from the earth at 0.5c. A galaxy closer to the earth and along the same line of sight is moving away from us at 0.4c. What is the recessional speed of the quasar as measured by astronomers in the other galaxy?

It should be less than 0.5c right? I am thinking about using Lorentz transformation for velocity:

u' = (u - v) / (1 - uv/c^2). My question with that is why are there 3 velocites: u', u, and v? Those are all velocities, right?


Thanks!

Answer 1

You got it. u and v are the respective velocities of the two objects as viewed from an outside frame. u' is the measured relative velocity of one as seen by the other. In your case, u = 0.5 and v = 0.4.

Answer 2

For your assigned problem, if you accept the underlying principle of special relativity that the laws of physics appear to be the same for all observers moving at uniform velocity, it must be true for any CORRECT version of the transformation remains the same same with the primed and unprimed reference frames switched. Imagine two identical spaceships A and B approaching each other travelling at constant velocities on a direct collision course. An observer of A sees that B is aproaching at speed vB. An observer on B sees A is approaching at speed vA. If vA and vB are different, the two observers would predict that different things would happen during and after the collision, which is impossible. Therefore the obvserved values of vA and vB must be the same. The question is asking you to verify that the Lorentz transformation is "correct" (or to be more pedantic, "not obviously wrong") because it DOES make the two velocities the same - just as the Newtonian definition of relative velocities would also make them the same.

Answer 3

u is the velocity of the quasar as seen from earth (0.5c)
v is the velocity of the galaxy as seen from earth (0.4c)
u' is the velocity of the quasar as seen from the galaxy.

Answer 4

You are correct.

The answer is less than 0.5c.

Your formula is correct.

u' is what you want (the relative velocity) in terms of u and v are the given velocities. Do the math.