- #1

Figaro

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- 7

## Homework Statement

I have encountered a problem in Sean Carroll's GR book, exercise 1.1

Consider an inertial frame S with coordinates ##~x^μ = (t, x, y ,z)~##, and a frame S' with coordinates ##x^{μ'}## related to S by a boost with velocity parameter ##v## along the y-axis. Imagine we have a wall at rest in S', lying along the line x' = -y'. From the point of view of S, what is the relationship between the incident angle of a ball hitting the mirror (traveling in the x-y plane) and the reflected angle? What about the velocity before and after?

## Homework Equations

Lorentz Transformation

Let w be the initial velocity and u be the final velocity ( v is the velocity parameter )

##β = \frac{v^2}{c^2}##

## The Attempt at a Solution

If I were to let the wall be parallel to the x-axis first and calculate the relationship between the incident angle and reflected angle as seen in the frame S, I would get,

Initial velocity can be broken down to:

##w_y = \frac{ w'_y + v}{ 1 + \frac{v w'_y}{c^2} }~~, ~~w_x = \frac{dx}{dt} = \frac{dx'}{ γ(dt' + \frac{v dy'}{c^2}) } = \frac{w'_x}{ γ(1 + \frac{v w'_y}{c^2}) }##

Final velocity can be broken down to:

##u_y = \frac{ -u'_y + v}{ 1 - \frac{v u'_y}{c^2} }~~, ~~u_x = \frac{dx}{dt} = \frac{dx'}{ γ(dt' - \frac{v dy'}{c^2}) } = \frac{u'_x}{ γ(1 - \frac{v u'_y}{c^2}) }##

Thus,

##tanθ_i = \frac{w'_x (1 - β^2)^½}{ w'_y + v } = \frac{ tanθ'_i (1 - β^2)^½}{ 1 + \frac{v}{w'_y} }##

##tanθ_f = \frac{u'_x (1 - β^2)^½}{ -u'_y + v } = \frac{ tanθ'_f (1 - β^2)^½}{ -1 + \frac{v}{u'_y} }##

Since ##~θ'_i = θ'_f~##, the relationship between ##~θ_i~## and ##~θ_f~## is,

##tanθ_i (1 + \frac{v}{w'_y} ) = tanθ_f (-1 + \frac{v}{u'_y} )##

Can somebody comment on my solution if it is correct?

Now what I'm not sure about is if the wall is inclined, I'm not sure on how should the ball approach the wall, because as I'm thinking, if it hits the wall at some incident angle, the y-component final velocity can be positive or negative depending on the size of the incident angle, if it is too big (with respect to the normal of course) then the ball will have a positive y-component final velocity, while if it is small then it can be negative. So how do I approach this problem?