Lorentz group

Orthochronous Lorentz group

The orthochronous Lorentz group is the group of all Lorentz transformations preserving the direction of time, lorentz i.e.

since is forbidden (see discussion below), this consists of all Lorentz transformations with positive .

Discussion

First note that by definition for all ,¹

and taking adjoints

the component of which yield

implying the properties

the former yields equality iff for the latter yields equality iff for each . Thus in particular iff for each , in which case . Now since , which splits into disconnected components based on the sign of .

Proof of defining property

Let . Consider a timelike vector , i.e. implying . Now if , then

Now let , and be timelike with . It follows is timelike with and , thus

Hence an arbitrary preserves the direction of time iff and reverses the direction of time iff .

From this it immediately follows that is a group, since if and preserve the direction of time so too must and . Furthermore it follows that is a Normal subgroup since for any and we the result of will preserve the direction of time.

Alternate proof of normal subgroup

Define . Let . If then applying the inequalities in ^eq1 and the Cauchy-Schwarz inequality

and indeed since otherwise it would have null determinant. Similarly if

and by the same since otherwise it would have null determinant. Hence is a group homomorphism and its kernel is a normal subgroup.

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Footnotes

1. 2018. From the Lorentz Group to the Celestial Sphere, §1.3, p. 9 ↩