# Null tetrad for a metric with most of the metric components non zero

I am working on a metric which is basically $$g = g_0 + h_{\mu \nu}$$ where $$g_0$$ is the Kerr metric up to order a (taking $$a^2 = 0$$) and $$h_{\mu \nu}$$ denotes perturbed metric. Hence the complete metric g is a complicated metric having 14 nonzero components and only two metric components are zero. The perturbed metric is also in terms of spherical harmonics $$Y(l,m)$$. I am trying to construct a null tetrad for this complete metric which is a complicated thing to do. Can anyone suggest to me any method for this or give any reference?

I tried to solve it by the method of aligning the tetrad vector $$l$$ along the null radial geodesic. For this, I choose $$l_r$$ to be 1 and from that calculated other components of vector $$l$$. But I am not getting the correct answer with this as norm $$l$$ is not coming out to be zero with this.

I also searched on how to find null tetrad vectors for the Kerr metric, but couldn't find any material on this. It would be helpful if someone suggest any reference for this also.

• Is this useful? physics.stackexchange.com/a/305574/226902 ..of course, the actual calculations in your case will be nasty. Possibly useful reference (discussing null tetrad for Kerr-Newman): arxiv.org/abs/1410.6626 Nov 16, 2021 at 14:34
• @Quillo In both these examples, the form of the metric is similar to Schwarzschild metric in Eddington Finkelstein coordinates, where most of the metric components are zero. I tried this but it didn't help. I am looking for constructing the null tetrad for a metric with with most of the elements nonzero.
– apk
Nov 17, 2021 at 3:56