Conclusion about wave propagation in SR given L dot S = N = L' dot S'

In summary, the problem the student is having is trying to figure out what the results of wave prorogation in SR mean given that the number of waves crossing a section of spacetime in either frame is always the same.
  • #1
PhDeezNutz
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Homework Statement
I'm having trouble rendering latex on this website so I attached a typed up picture of my question and attempt at a solution (See below)
Relevant Equations
(See below)
Image 5-18-20 at 12.03 PM.jpg

Image 5-18-20 at 12.05 PM.jpg


The problem I am having is "What can you conclude about wave prorogation in SR given the results?". The best I can come up with is that the number of wave planes N crossing a section of spacetime in either frame is the same. The section may be bigger or smaller depending on which frame you're in but there's always N waves crossing regardless of which frame you're in.

I'm using intuitive 3-space notions of flux and applying it to minkowski spacetime and that may very well be a mistake.

Any hints on how to interpret L dot delta s = L' dot \delta s' = N?

Thank you in advanced for any help.
 
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  • #2
I think we can conclude (assuming S' is obtained from S through a Lorentz boost in the x-direction)

##L_0 c \Delta t = L'_0 \Delta t' , L_1 \Delta x = L'_1 \Delta x', L_2 \Delta y = L'_2 \Delta y', L_3 \Delta z = L'_3 \Delta z'##

Concentrating on the last 3 statements I think this result says that the component of the wave vector along the observation direction is the same in both frames. More details to follow.
 
  • #3
My previous post is definitely wrong. I think the answer is trivial...we were asked to show that L' can be obtained from L via a Lorentz transform and then what we could conclude from that. I think the answer is obvious (and kind of trivial)...only the component of the wave vector along the boost direction changes. That's the definition of a lorentz transform.

unless I was supposed to interpret L \dot S = L' \dot S' = N instead. I'm going to email my instructor and ask about this.
 
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  • #4
I asked my instructor said "the response is that the equation keeps the same form in any coordinate system. So it is written in covariant form including invariance to Lorentz transformations".

I'll have to think about all that.
 
  • #5
I guess the interpretation is that L dot \delta S is a true Lorentz scalar.
 

1. What is the significance of L dot S in wave propagation in SR?

L dot S represents the dot product of the angular momentum and spin of a particle. In wave propagation in special relativity, it is used to describe the direction of the wave vector and the polarization of the wave.

2. How does N play a role in wave propagation in SR?

N represents the total angular momentum of a system, which includes both the orbital and spin angular momentum. In wave propagation in special relativity, it is used to determine the energy and frequency of the wave.

3. What does L' dot S' indicate in wave propagation in SR?

L' dot S' represents the dot product of the primed angular momentum and spin of a particle. In wave propagation in special relativity, it is used to describe the direction and polarization of the wave in a different reference frame.

4. How does the equation L dot S = N = L' dot S' relate to wave propagation in SR?

This equation shows the conservation of angular momentum in wave propagation in special relativity. It states that the total angular momentum of the system is equal in all reference frames, regardless of the direction or polarization of the wave.

5. What conclusions can be drawn from the relationship between L dot S, N, and L' dot S' in wave propagation in SR?

The relationship between these quantities shows that angular momentum is conserved in wave propagation in special relativity. It also demonstrates the importance of considering both orbital and spin angular momentum in understanding the behavior of waves in different reference frames.

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