# How to obtain Kerr Metric via Spinors (N-P Formalism)

How to obtain Kerr Metric via Spinors (Newman-Penrose Formalism)?

I am a bit confused with Ray d'Inverno's Book.

Why perform the coordinates transformation:

2r-1 -> r-1 + r*-1

I am bit confused of it.

And I am a bit confused, too, of how to write out null tetrad...

Is there any resource rendering in-detail discussion of it?

WannabeNewton
Does D'Inverno talk about spinors? I would be thankful if you could point me to where in the text itself if he does.

Does D'Inverno talk about spinors? I would be thankful if you could point me to where in the text itself if he does.

Page 250...

A null tetrad is something of a ruse. Two of its legs are real null vectors whose inner product is 1, while the other two are formed by the introduction of a spurious "i" and have components that are real, orthogonal spacelike vectors:

(s1 + i s2)^2 = (s1^2 - s2^2) + 2i (s1.s2) = 0

..so you could take s1 = (1,0,0,0) and s2=(0,1,0,0) in the rest frame and boost them up to non-zero momentum. So a typical null tetrad is up to normalization,

[1,i,0,0], [1,-i,0,0], [0,0,1,1], [0,0,-1,1]

If one now wants the Pauli algebra (2nd-rank mixed spinorial) form of these basis vectors, you get up to normalization,

[0,1;0,0] [0,0;1,0] [1,0;0,0] [0,0;0,1]

This is what one is really after, and the tetrad itself is a side show.

Ad-hoc introduction of "i" in this way always leads to confusion of spacetime covariants. It is to be avoided.

-drl

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Bill_K
How to obtain Kerr Metric via Spinors
Why perform the coordinates transformation:
2r-1 -> r-1 + r*-1

Going to complex coordinates seems like a strange idea at first, but for the Kerr metric it turns out to be useful, because there's a natural interpretation in which the mass m is located not at the origin but at a distance 'a' along the imaginary axis, where 'a' is the Kerr parameter.

the mass m is located not at the origin but at a distance 'a' along the imaginary axis, where 'a' is the Kerr parameter.

Why is that?

How can it affect the extension of r

Can someone explain it?

Thanks.