# Derivation of Kerr metric, is there any reference?

In studying general relativity, many text deals with the derivation of Schwarzschild metric starting from generic metric form. After that impose static, spherical symmetry and obtain the desired Schwarzschild metric.

But I haven't find any reference for above process in Kerr metric. (Add a condition of rotation.) In many textbooks on General relativity and black hole textbook they just state the form of Kerr metric, and do some calculation.

Is there any reference (textbook or paper) contain explicit derivation of Kerr metric?

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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. – Robin Ekman Dec 4 '14 at 15:39
• @Jim and other editors: Before using the spec. ref. tag please check its tag wiki. – Qmechanic Dec 4 '14 at 16:01

## 2 Answers

The best reference I know of is in the book The Kerr Spacetime, edited by Matt Visser. David Wiltshire and Susan Scott.

The introduction by Matt Visser contains a lot of additional info on the original paper, and the subsequent chapter by Kerr contains a detailed account of everything that motivated him to look for the metric and the steps that don't appear in the original paper.

A summary of the idea is this: in the null tetrad approach start with an algebraically special metric (see Petrov Classification) and use the Goldberg-Sachs theorem to simplify the tetrad. Then assume the existence of two groups of isometry, namely that the spacetime is stationary and axisymmetric. Then impose asymptotic flatness. Write the Cartan structure equations, that with these assumptions should reduce to ODEs, the solution of which gives a two-parameter family of metrics, the Kerr family. The book spells it out in considerable detail.

You should know there is another way of getting the Kerr metric though, with the use of the Newman-Janis algorithm. The paper by Drake and Szekeres "An explanation of the Newman-Janis Algorithm" is a good starting point as any. The idea of the NJ algorithm is that by starting with a non-rotating solution of the Einstein Equation (Schwarzschild in this case) you can obtain the rotating generalization by means of a complex substitution. It seems almost magical because if gives the Kerr metric with little effort (at least comparing with Kerr's original derivation), and it works for tons of other cases too. The trade-off is that it seems obscure at first why the algorithm should work at all, but the aforementioned paper does a good job at explaining it. This method is the one Newman used to get the Kerr-Newman metric (the charged rotating case) from Reissner-Nordstrom.

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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).

• 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.

• I'm surprised no one mentioned Chapter 6 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 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!