haar-measure-lorentz | Parallel Transport

math/physics question (cc:@johncarlosbaez) Take the group SL(2,C), the double cover of the identity-connected component of the Lorentz group. As a manifold this group is homeomorphic to R3 × S3, where R3 are Lorentz boosts and S3 the group SU(2) (double cover of rotations). You can also view the R3 as a Riemannian hyperbolic space H3. Now H3 has a natural Lorentz invariant measure, and S3 has a natural rotation invariant measure. So the left-invariant Haar measure on SL(2,C) should be some positive function F times the measures on H3 and S3. Is there an explicit expression for this Haar measure in term of some/any coordinates on H3 and S3? (Asking here because I’m tired of the condescension on math/stack overflow towards explicit examples)

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Leo C. Stein 🦁

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Should this correspond to (possibly a multiple of) the Killing form, which in this case will have signature (—-+++)?

Oops sorry, you asked about a measure! So… maybe the measure associated to that Killing form?

Kartik Prabhu

It will. But that is also not explicitly given anywhere in terms of some nice coordinates.

William Donnelly

Choose your favourite coordinates on SL(2,C) and calculate the Killing form. You get a metric from which you can read off the measure. I have done this explicitly for SL(2,R) but I expect it would be similar for SL(2,C).

Yes, I know how to do that, which is why I asked a much more specific question.

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