Special relativity MOC

Lorentz group

The Lorentz group 𝐿 =O(3,1) is the group of all linear transformations on Minkowski spacetime that preserve the Minkowski metric, lorentz i.e.

O(3,1)={Λ∈GL4(ℝ):𝜂(Λ𝑥,Λ𝑦)=𝜂(𝑥,𝑦)∀𝑥,𝑦∈ℝ4}={Λ∈GL𝟜(ℝ):Λ𝖳𝜂Λ=𝜂}={(Λ𝜇𝜈)∈GL4(ℝ):Λ𝜇𝜅𝜂𝜇𝜈Λ𝜈𝜆=𝜂𝜅𝜆}

where 𝜂 is the Minkowski metric tensor.¹ The Semidirect product acting on the translation group ℝ4 forms the Poincaré group.

Proof of group

That O(3,1) is a group is obvious, since it consists of invertible matrices which preserve a certain property Let Λ,Λ′ ∈O(3,1). Then clearly

𝜂(Λ′Λ−1𝑥,Λ′Λ−1)=𝜂(Λ−1𝑥,Λ−1𝑦)=𝜂(ΛΛ−1𝑥,ΛΛ−1𝑦)=𝜂(𝑥,𝑦)

for all 𝑥,𝑦 ∈ℝ4. Hence by One step subgroup test O(3,1) is a group.

This forms a 6-dimensional Lie group.

Subgroups

The most important subgroups are

• The Proper Lorentz group is the group of Lorentz transformations of determinant 1, i.e. preserving the orientation of space.
• The Orthochronous Lorentz group is the group of Lorentz transformations (Λ𝜇𝜈) with Λ00 >0, i.e. preserving the direction of time.
• The Proper orthochrounous Lorentz group is proper and orthochronous, and is the path connected subgroup

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Footnotes

1. 2018. From the Lorentz Group to the Celestial Sphere, §1.2.3, p. 8 ↩