Lorentz group - Knowlpedia

Fix the Minkowski bilinear form on $\mathbb R^n$ with one negative and $n-1$ positive directions, represented by

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

Thus the signature notation $(1,n-1)$ here lists negative directions first. This is the convention used in Minkowski vector space and the Hermitian determinant model.

The Lorentz group in dimension $n$ is the subgroup

O(1,n-1)=\{A\in \mathrm{GL}(n,\mathbb R)\mid A^{\mathsf 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 $SO^{+}(1,n-1)$ , consisting of matrices preserving both orientation and a chosen time orientation.

The identity component is also called the proper orthochronous Lorentz group (with the linked page treating the four-dimensional case).

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.

Remarks

The Lorentz group acts linearly on Minkowski space, and adjoining translations yields the Poincaré group, the full isometry group of Minkowski spacetime. In four dimensions, the identity component has the spin covering

SL(2,\mathbb C)_{\mathbb R}\longrightarrow SO^+(1,3)

and the induced real-Lie-group isomorphism $PSL(2,\mathbb C)_{\mathbb R}\cong SO^+(1,3)$ . The full group $O(1,3)$ has four connected components; the spin cover above does not include parity or time reversal.

References

Barrett O'Neill, Semi-Riemannian Geometry With Applications to Relativity, Academic Press, 1983, Chapter 5. Publisher record.
Gregory L. Naber, The Geometry of Minkowski Spacetime, 2nd ed., Springer, 2012, Chapters 1–2. Publisher record.