Lorentz Group

The Lorentz group is the Group $L$ of time-preserving linear Isometries of Minkowski Space $\Bbb{R}^4$ with the pseudo-Riemannian metric

$$d\tau^2 = -dt^2 + dx^2 + dy^2 + dz^2.$$

It is also the Group of Isometries of 3-D Hyperbolic Space. It is time-preserving in the sense that the unit time Vector $(1,0,0,0)$ is sent to another Vector $(t,x,y,z)$ such that $t>0$.

A consequence of the definition of the Lorentz group is that the full Group of time-preserving isometries of Minkowski $\Bbb{R}^4$ is the Direct Product of the group of translations of $\Bbb{R}^4$ (i.e., $\Bbb{R}^4$ itself, with addition as the group operation), with the Lorentz group, and that the full isometry group of the Minkowski $\Bbb{R}^4$ is a group extension of $\Bbb{Z}_2$ by the product $L\otimes\Bbb{R}^4$.

The Lorentz group is invariant under space rotations and Lorentz Transformations.

See also Lorentz Tensor, Lorentz Transformation

References

Arfken, G. ``Homogeneous Lorentz Group.'' §4.13 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 271-275, 1985.