Complex tetrad vs Real metric

Question

I have a question on the relationship between the complex tetrad in general relativity and the metric. All the papers I've sen so far just usually state the metric and the (null) tetrad without discussing the relation between the two.

My question is: clearly, null tetrad can be complex. Some of these complex tetrads do give a real metric, but not all. For example, the Kinnersley tetrad

$l=(0,1,0,0)$

$n=\rho\overline{\rho}(r^{2}+a^{2},-\frac{1}{2}(s^{2}-2ms+a^{2}),0,a)$

$m=-\frac{1}{\sqrt{2}}\overline{\rho}(ia\sin x,0,1,i\csc x)$

where $\rho=-\frac{1}{s-ia\cos x}$, is supposed to give the Kerr metric. The Kerr metric is of course real. But when I substitute this null tetrad directly, I get a metric with complex entries. Just simply taking the real part of it doesn't do the trick, i.e. I don't get Kerr.

So how do I get a real metric from a complex tetrad? Thank you for any help.

This post imported from StackExchange MathOverflow at 2016-01-19 22:02 (UTC), posted by SE-user GregVoit

Comments

Can you please give a reference where a complex tetrad is actually used, so that one can see the context?