Lorentz group

The group of linear transformations preserving the Minkowski bilinear form.

Lorentz group

Fix the Minkowski bilinear form on $\Bbb R^n$ of signature $(1,n-1)$ , represented (in a standard basis) by the matrix

\eta=\mathrm{diag}(-1,1,\dots,1).

Definition

The Lorentz group in dimension $n$ is the subgroup

O(1,n-1)=\{A\in \mathrm{GL}(n,\Bbb R)\mid A^{T}\eta A=\eta\}.

It is an instance of the orthogonal group in an indefinite signature . The case $n=4$ is the classical Lorentz group of special relativity.

Two commonly used subgroups are:

$SO(1,n-1)=\{A\in O(1,n-1)\mid \det A=1\}$ (the “special” Lorentz group),
the identity component $SO^{+}(1,n-1)$ (often called “proper orthochronous”), consisting of matrices preserving both orientation and time orientation.

Lie algebra

Its Lie algebra is the indefinite orthogonal Lie algebra

\mathfrak{so}(1,n-1)=\{X\in \mathfrak{gl}(n,\Bbb R)\mid X^{T}\eta+\eta X=0\},

an instance of orthogonal Lie algebras .

Context

The Lorentz group acts linearly on Minkowski space, and adjoining translations yields the Poincaré group , the full isometry group of Minkowski spacetime.