# Understanding Lorentz Factor: Proving and Explaining Its Invariance

• CassiopeiaA
In summary, the relative Lorentz factor is given by:γ(r)=γ(1)γ(2)(1−v1.v2)where it is Lorentz invariant.
CassiopeiaA
If two particles have velocities v1 and v2 in a rest frame, how do we prove that the relative lorentz factor is given by :
γ(r)=γ(1)γ(2)(1−v1.v2)

and why is this quantity lorentz invariant

Did you try simply checking that it is Lorentz invariant? It obviously reduces to the relative Lorentz factor for frames where one of the particles is at rest, so if it is Lorentz invariant it is the relative Lorentz factor.

Do you know how to use 4-vectors? (In particular 4-velocities.)

CassiopeiaA said:
If two particles have velocities v1 and v2 in a rest frame, how do we prove that the relative lorentz factor is given by :
γ(r)=γ(1)γ(2)(1−v1.v2)

and why is this quantity lorentz invariant
The original derivation is in §5 of https://www.fourmilab.ch/etexts/einstein/specrel/www/

Here's an analogous identity.
##\cos(B-A)=\cos B\cos A(1+\tan B\tan A)##,
which doesn't depend on the axis used to measure angles.

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Ok, let's have particles with three-velocities ##\vec{v}_1## and ##\vec{v}_2## wrt. to an inertial reference frame. The first step is to translate this to the appropriate covariant objects. In this case that's the four-velocities of the (on-shell) particles which are the four-velocities
$$u_j^{\mu}=\gamma_j (1,\vec{v}_j), \quad \gamma_j=\frac{1}{\sqrt{1-\vec{v}_j^2}}, \quad c=1, \quad j \in \{1,2 \}$$
They are related to the four-momenta of the particles by
$$p_j^{\mu}=m_j u_j^{\mu},$$
where ##m_j## are the invariant (or rest) masses of the particles, but this we don't need.

The relative velocity is now defined as the velocity of particle 2 in the rest frame of particle 1. Thanks to the covariant tensor formalism you don't need to perform the Lorentz transform here (although that's a nice exercise, you should do for yourself, but it's also a bit tedious to type here in the forum). The trick is to express everything covariantly. In the rest frame of particle 1 (in the following written with a tilde over the components of four-vectors) you have
##\tilde{u}_1^{\mu}=(1,0,0,0), \quad \tilde{u}_2^{\mu} = \tilde{\gamma}_2 (1,\vec{\tilde{v}_2}).##
From this it's clear that
##\tilde{\gamma}_2=u_2^0=\tilde{u}_{1,\mu} \tilde{u}_2^{\mu}.##
But now, this is a scalar expression (the Minkowski product of two four-vectors) and thus you have
##\tilde{\gamma}_2=u_1 \cdot u_2=\gamma_1 \gamma_2 (1-\vec{v}_1 \cdot \vec{v}_2),##
which is the formula you wanted to derive in posting #1.

One should keep in mind that the Lorentz transform is at the heart of deriving the Minkowski space as the space-time model, establishing the four-tensor formalism. For practical calculations you usually don't need Lorentz transforms, if everything is formulated in covariant form, and you can express the quantities you want to know in terms of covariant equations, which is the case for all physically interesting quantities. Sometimes one has to define appropriate quantities in a covariant way (e.g., temperature and other thermodynamic quantities and material constant, where usually you define them in the (local) rest frame of the matter and then express this definition in a manifestly covariant way or the invariant cross sections for processes in high-energy physics).

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## 1. What is the Lorentz factor and why is it important?

The Lorentz factor is a mathematical term used to describe the relationship between an object's velocity and its mass. It is an important concept in special relativity, as it helps to explain phenomena such as time dilation and length contraction.

## 2. How is the Lorentz factor calculated?

The Lorentz factor is calculated using the equation γ = 1/√(1 - v^2/c^2), where v is the velocity of the object and c is the speed of light. This equation shows that as an object's velocity approaches the speed of light, its Lorentz factor approaches infinity.

## 3. What does it mean for the Lorentz factor to be invariant?

The invariance of the Lorentz factor means that it remains the same regardless of the observer's frame of reference. In other words, the Lorentz factor is a universal constant that is not affected by the relative motion of different observers.

## 4. How is the invariance of the Lorentz factor proven?

The invariance of the Lorentz factor can be proven using mathematical equations and experimental evidence. One of the most famous experiments that supports the invariance of the Lorentz factor is the Michelson-Morley experiment, which demonstrated that the speed of light remains constant regardless of the observer's frame of reference.

## 5. Can you give an example of the invariance of the Lorentz factor in action?

One example of the invariance of the Lorentz factor can be seen in the concept of time dilation. When an object is moving at a high velocity, time passes more slowly for that object compared to a stationary observer. This is due to the Lorentz factor remaining constant, regardless of the observer's frame of reference.

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