Mirror 4-Momentum: Find Frequency Change on Reflection

[SHawking01]

Homework Statement

A mirror moves perpendicular to its plane with speed βc. A light ray is incident on the mirror from the “forward” direction (i.e., vm · vl < 0, where vm is the mirror’s 3-velocity and vl is the light ray’s 3-velocity) with incident angle θ (measured with respect to the mirror’s normal vector). (a) Find cos φ, where φ is the angle of reflection. (b) By what factor does the frequency of the light change upon reflection?

Homework Equations

None that I know of.

The Attempt at a Solution

P~ in = E c (sin θin , − cos θin , 0 , 1)
P~ ′ in = E c [γ(sin θin − β) , − cos θin , 0 , γ(1 − β sin θin)]
I don't know what else.

 

[Orodruin]

What happens in the mirror rest frame when the light is reflected?

 

[SHawking01]

In the rest frame of the mirror the angle of reflection equals the angle of incidence and there is no frequency shift.

 

[Orodruin]

So what will be the 4-momentum of the light after reflection?

 

[SHawking01]

Well, I went to my teacher and found the answer. But thanks for your help anyways!