Lorentz group

The Lorentz group is the group of all linear transformations on Minkowski spacetime that preserve the Minkowski metric, #m/def/group/lorentz i.e.

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

Proof of group

That is a group is obvious, since it consists of invertible matrices which preserve a certain property Let . Then clearly

for all . Hence by One step subgroup test 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 with , 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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