# Reflection of particle between Inertial frames in SR

• cpsinkule
In summary, according to the principle of relativity, the angle of incidence and reflection is the same in both frames, however, the angle may change due to the contraction of the wall.
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.

## 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'
=uxi'+uyj'
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=-uyi'-uxj'.
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.

Deleted.

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 dependant due to the contraction which itself is due to the relativity of sumiltaneity :)

thank you :)

## 1. How does the reflection of particles between inertial frames in Special Relativity (SR) work?

In SR, the laws of physics are the same in all inertial frames, meaning that the relative motion between two frames does not affect the outcome of experiments. This includes the reflection of particles, which follows the same rules regardless of the frame of reference.

## 2. What is the significance of the speed of light in the reflection of particles between inertial frames in SR?

The speed of light is the same in all inertial frames, according to the principles of SR. This means that the velocity of a reflected particle will be the same in all frames, regardless of the relative motion between them.

## 3. Can particles be reflected between non-inertial frames in SR?

No, the laws of physics in SR only apply to inertial frames, which are frames of reference that are moving at a constant velocity. Non-inertial frames, which are accelerating or rotating, have different laws of physics and do not follow the same rules for the reflection of particles.

## 4. How does the reflection of particles between inertial frames in SR affect the concept of time dilation?

Time dilation, which is the slowing of time for a moving object, is also a principle of SR. In the context of reflection of particles, this means that the time it takes for a particle to be reflected between two inertial frames will be the same, regardless of the relative motion between the frames.

## 5. Can the reflection of particles between inertial frames in SR violate the principle of causality?

No, the principle of causality, which states that an effect cannot occur before its cause, is still valid in SR. The reflection of particles between inertial frames does not violate this principle, as the cause (the particle's initial motion) always precedes the effect (its reflection). This is true in all frames of reference, regardless of their relative motion.

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