Derive Particle Speed in Terms of Invariant U.V | Relative 4-velocities Homework

In summary: Some people like to denote the same vector in different coordinate systems using primes for some reason ... I guess mostly for when writing in components instead of putting the prime on the indices.In summary, the conversation discusses the derivation of an expression for the speed of a particle relative to an observer in terms of the invariant U · V. The approach involves using the rest frame of the observer to obtain an expression for the relative velocity, and then evaluating the U · V product in any frame. Some people use primes to denote the same vector in different coordinate systems.
  • #1
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Homework Statement


In a particular inertial frame of reference, a particle with 4-velocity V is observed by an
observer moving with 4-velocity U. Derive an expression for the speed of the particle
relative to the observer in terms of the invariant U · V

Homework Equations



##U.V=U'.V'##[/B]

The Attempt at a Solution


[/B]
I used the relation the ##U.V=U'.V'## because U.V is invariant.

Using the rest frame of the observer I obtained

##U'.V' = -\gamma(u_R)c^2##

Then rearranging to find ##u_R## I obtained

##u_R = c*sqrt(1-(c^4/(U.V)^2))##

I'm just wondering if I used the correct method and got the correct result for ##u_R##

Any insight is appreciated
 
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  • #2
scottJH said:
I'm just wondering if I used the correct method and got the correct result for ##u_R##

Any insight is appreciated

Yes, your approach and result seem reasonable.
 
  • #3
As a non-expert in relativity (far from it), I'm confused as to why a primed frame even needs to be introduced in this problem, since U and V are individually frame invariant. So the dot product of U and V can be calculated using the components of these vectors as reckoned with respect to any convenient reference frame. This is what scottH actually did. So why the need for the primes?

Chet
 
  • #4
The point is that the rest frame of the observer is this convenient frame where you get an expression for the relative velocity in terms of the inner product. Naturally, once the derivation is done and the expression for the relative velocity in terms of the product is known, you can choose to evaluate ##U\cdot V## in any frame.
 
  • #5
Orodruin said:
The point is that the rest frame of the observer is this convenient frame where you get an expression for the relative velocity in terms of the inner product. Naturally, once the derivation is done and the expression for the relative velocity in terms of the product is known, you can choose to evaluate ##U\cdot V## in any frame.
That's what I thought. So why the need for the primes?

Chet
 
  • #6
Some people like to denote the same vector in different coordinate systems using primes for some reason ... I guess mostly for when writing in components instead of putting the prime on the indices.
 
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Likes Chestermiller

1. How do you derive the particle speed in terms of invariant U.V?

To derive the particle speed in terms of invariant U.V, we use the Lorentz transformation equations and the formula for relative velocity. This allows us to calculate the particle's speed in terms of the relative velocities of the observer and the particle.

2. What is the significance of using invariant U.V in this derivation?

Invariant U.V, also known as the invariant four-velocity, is a crucial concept in special relativity. It represents the four-dimensional momentum of a particle, which remains constant regardless of the observer's frame of reference. By using invariant U.V in our derivation, we can determine the particle's speed in a way that is consistent across all frames of reference.

3. Can you explain the concept of relative 4-velocities?

Relative 4-velocities refer to the four-dimensional velocities of two objects as measured by an observer in a particular frame of reference. It takes into account both the spatial and temporal components of velocity and allows us to compare the velocities of two objects in different frames of reference.

4. Are there any assumptions made in this derivation?

Yes, there are several assumptions made in the derivation of the particle speed in terms of invariant U.V. These include assuming that the particles are moving in a straight line, that they are not accelerating, and that we are working in a vacuum with no external forces acting on the particles.

5. How is this derivation relevant in the field of physics?

This derivation is relevant in the field of physics as it helps us understand the behavior of particles at high speeds, which is crucial in areas such as particle physics and astrophysics. It also demonstrates the principles of special relativity and how they can be applied to real-world scenarios.

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