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 - The original reference is Kerr, R.P. *Gravitational field of a spinning mass as an example of algebraically special metrics*. [Phys. Rev. Lett. **11**, 237 (1963)](https://doi.org/10.1103/PhysRevLett.11.237).

 - you can also read this paper , it is very useful: [A Possible Intuitive Derivation of the Kerr Metric in Orthogonal Form Based On Ellipsoidal Metric Ansatz](http://arxiv.org/abs/1210.5922).

 - I'm surprised no one mentioned [Chapter 6](https://books.google.com/books?id=LBOVcrzFfhsC&pg=PA273) of Chandrasekhar's *Mathematical Theory of Black Holes*. Then again, I personally found his derivation so opaque that I wrote my own de novo [derivation](http://analyticphysics.com/General%20Relativity/A%20(Relatively)%20Simple%20Derivation%20of%20the%20Kerr%20Metric.htm) in Boyer-Lindquist coordinates using the Cartan formalism.

 - I think if we follow the Ray D' Inverno book intitled Introducing Einstein Relativity then we get easily this metric. on chapter 19 it is explanied in a very simple method

 - Here is a review of the process of discovery by the man himself: icra.it/icranet_report/08Kerr.pdf The OP might like to read this and see if he still has the fortitude for the task!