Reflection of particle between Inertial frames in SR

[cpsinkule]

Homework Statement

Suppose a frame S' is related to S by a boost in the y direction by v. Imagine a wall is at rest in the S' frame along the line y'=-x'. Consider a particle moving in the x'y' plane that strikes the surface and is reflected by the usual law of reflection θ'_i=θ'_r. Find the relation between the incident and reflected angles in the S frame as well as the initial and final velocity of the particle in the S frame.

Homework Equations

Velocity addition, lorentz contraction

The Attempt at a Solution

I assumed for simplicity that, although the wall is in S' and moving, I would always be able to redifine the S origin so that the wall passes through the origin in both frames. Now the wall itself will be contracted in the y direction, so the equation of the wall in the S frame is y/β=-x (I'm using beta for the lorentz factor because gamma looks too similar to y). To find the angle of reflection and incidence in the S' frame I need a normal vector to the wall, which is easily obtained to be 1/√2*(i'+j'). Now the velocity of the particle in the S' frame is

u'=u_xi'+u_yj'
After striking the wall and being reflected, the particle's velocity will be reflected and reversed about the normal to the wall so that after contact the new velocity will be u=-u_yi'-u_xj'.
Now the incident and reflected angles can be calculated (and are equal in this frame) by cosθ=n*u/|u|
where n is the unit normal to the wall 1/√2*(i'+j')

To calculate the angles in the S frame we have to transform the initial and final velocities of the particles and then dot them with the NEW normal to the wall which can be obtained from y/β=-x, remembering that I'm using beta for the lorentz factor instead of gamma. The normal to the wall in this frame can be considered by a point on the perpendicular line y=x/β considering when x=1 y=1/β the normalized unit vector along this line is then

(i+1/βj)/((1/β)²+1)^1/2
Once I transform the above velocities via the addition formula and dot it with this normal vector, will I arrive at the correct answer? Is there an easier way to arrive at this result if it is correct? Forgive me for not typing the entire solution, but I feel like this is already in the realm of tldr.

 

[PeroK]

Deleted.

 

[Noctisdark]

By the principle of relativity θi = θr must be true, but θ may change, now the only difference between the two from is the tilt of the wall (let's forget about time for a while), I took at what you did and I totally agree with it, the great conclusion here is that angle is a frame dependent due to the contraction which itself is due to the relativity of sumiltaneity :)

 

[cpsinkule]

thank you :)