Proper orthochronous Lorentz group

Definition

For $\eta=\operatorname{diag}(-1,1,1,1)$ , the proper orthochronous Lorentz group is

SO^+(1,3) =\{\Lambda\in GL_4(\mathbb R): \Lambda^{\mathsf T}\eta\Lambda=\eta,\ \det\Lambda=1,\ \Lambda^0{}_0>0\}.

It is the identity component of the Lorentz group $O(1,3)$ . “Proper” means determinant $+1$ , and “orthochronous” means preserving the future cone.

Structure

The group is connected, noncompact, and has real dimension $6$ . Its Lie algebra is

\mathfrak{so}(1,3) =\{X\in M_4(\mathbb R):X^{\mathsf T}\eta+\eta X=0\}.

There is a double covering

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

with kernel $\{\pm I\}$ , and hence a real-Lie-group isomorphism

PSL(2,\mathbb C)_{\mathbb R}\cong SO^+(1,3).

The subscript $\mathbb R$ is essential: the target is a real Lie group, not a complex Lie group.

References

Gregory L. Naber, The Geometry of Minkowski Spacetime, 2nd ed., Springer, 2012, Chapters 1–2. Publisher record.
Roger Penrose and Wolfgang Rindler, Spinors and Space-Time, Vol. 1, Cambridge University Press, 1984, §1.2. Publisher record.