Lorentz Factor for relative velocities

In summary, the homework statement states that the Lorentz factor for relative velocity w is given by \gamma(w)=\gamma(u) \gamma(v) (1-\textbf{u.v})
  • #1
Hirdboy
2
0

Homework Statement


Two particles have velocities u, v in some reference frame. The Lorentz factor for their relative velocity w is given by
[itex]\gamma(w)=\gamma(u) \gamma(v) (1-\textbf{u.v})[/itex]
Prove this by using the following method:
In the given frame, the worldline of the first particle is [itex] X =(ct,\textbf{u}t)[/itex] Transform
to the rest frame of the other particle to obtain
[itex] t' = \gamma_v t (1-\textbf{u.v}/c^2) [/itex]
Obtain [itex] dt'/dt [/itex] and use the result that [itex] dt/d\tau = \gamma [/itex]

Homework Equations


[itex] ct' = \gamma (ct-v/c) [/itex]
[itex] x' = \gamma (x-vt) [/itex]
-Define Lorentz Transform as L
[itex] dt/d\tau = \gamma [/itex]


The Attempt at a Solution


Firstly we are in the frame where the two particles velocities are u and v.
The first step comes from applying LX to give: [itex] t' = \gamma_v t (1-\textbf{u.v}/c^2) [/itex]

Differentiating the result gives [itex] dt'/dt = \gamma_v (1-\textbf{u.v}/c^2) [/itex]
I think that then may be equal to [itex] \gamma_u [/itex] but cannot see how that will help me solve it. Very grateful to all suggestions thank you.
 
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  • #2
Welcome to PF!

Hirdboy said:
[itex] dt'/dt = \gamma_v (1-\textbf{u.v}/c^2) [/itex]

This looks good. You'll now need to "use the result that [itex] dt/d\tau = \gamma [/itex]".

Can you see a way to conjure the proper time ##d\tau## into ##dt'/dt##? Hint: chain rule.
 
Last edited:
  • #3
Thank you,
This means (I think):

That I'd be right in saying

[itex] \frac{dt'}{d\tau} = \gamma_w [/itex]
and [itex] \frac{dt}{d\tau} = \gamma_u [/itex]

We know [itex] \frac{dt'}{dt} [/itex] = [itex] \gamma_v (1-\textbf{u.v}/c^2) [/itex]
and [itex] dt'/d\tau = \frac{dt'}{dt} \frac{dt}{d\tau} [/itex]

Subbing in gives the desired result [itex]\gamma_w=\gamma_u \gamma_v (1-\textbf{u.v}/c^2)[/itex]

Finding it quite confusing working out what [itex] \gamma [/itex] relates to which velocity, so thank you so much for all your help!
 

What is the Lorentz Factor for relative velocities?

The Lorentz Factor, denoted by the symbol γ, is a term used in special relativity to describe the relationship between an object's velocity and its observed mass, length, and time. It is defined as γ = 1/√(1-v²/c²), where v is the relative velocity between two objects and c is the speed of light.

Why is the Lorentz Factor important?

The Lorentz Factor is important because it helps us understand the effects of relative velocities on physical quantities. It allows us to calculate the differences in mass, length, and time between objects moving at different speeds, and explains phenomena such as time dilation and length contraction.

How does the Lorentz Factor affect time?

The Lorentz Factor is directly related to time dilation, which is the phenomenon where time appears to pass slower for objects moving at high speeds. As the relative velocity between two objects increases, the Lorentz Factor also increases, resulting in a slower passage of time for the moving object.

Can the Lorentz Factor be greater than 1?

Yes, the Lorentz Factor can be greater than 1. In fact, it approaches infinity as the relative velocity approaches the speed of light. This means that as an object's speed increases, its observed mass, length, and time become infinitely large, and the laws of classical physics no longer hold.

How is the Lorentz Factor used in practical applications?

The Lorentz Factor is used in many practical applications, such as in particle accelerators, GPS systems, and satellite communications. It is also used in the design of high-speed vehicles and in the correction of time differences in satellite navigation systems due to time dilation effects.

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