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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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    $\begingroup$ 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. $\endgroup$
    – Robin Ekman
    Commented Dec 4, 2014 at 15:39
  • $\begingroup$ @Jim and other editors: Before using the spec. ref. tag please check its tag wiki. $\endgroup$
    – Qmechanic ♦
    Commented Dec 4, 2014 at 16:01
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    $\begingroup$ Section 8.3 of Norbert Straumann's General Relativity (Second Edition) $\endgroup$
    – Sean
    Commented Oct 5, 2022 at 11:22

2 Answers 2

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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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  • $\begingroup$ Do the references you mention include a derivation of the "equatorial symmetry" mention in O'Neill's store.doverpublications.com/0486493423.html section 2.3? I have a model in mind that would violate that. In particular, I would like to prove/disprove that such an aberration is consistent with the overall theory. i.e. would such an unbalance get radiated away; destroy the t Killing vector. $\endgroup$
    – rrogers
    Commented Jul 30, 2020 at 14:40
  • $\begingroup$ @rrogers, I don't have O'Neill's book with me, so I can't be sure exactly what this equatorial symmetry is, but skimming through the book I mentioned, I see no derivation of symmetries, rather symmetries are assumed and used to simplify Cartan equations. But those symmetries are only stationary and axisymmetric killing vectors $\endgroup$
    – cesaruliana
    Commented Jul 30, 2020 at 16:56
  • $\begingroup$ To paraphrase: The top and bottom halves are reflections. To be exact: "There is a (unique) isometry $\epsilon:K\rightarrow K$ called the equatorial isometry whose restriction to each Boyer-Lindquist block sends $\vartheta$ to $\pi-\vartheta$ , leaving the others coordinates unchanged." Where $\vartheta$ is the latitude coordinate; as judged from far off; $\vartheta = 0$ being the equator. $\endgroup$
    – rrogers
    Commented Jul 30, 2020 at 18:20
  • $\begingroup$ sigh: $\vartheta =\pi / 2 $ being the equator. $\endgroup$
    – rrogers
    Commented Jul 30, 2020 at 19:25
  • $\begingroup$ @rrogers, reading chapter 2, mentioned in the answer, Kerr does not explicitly talks about equatorial symmetry, neither proving or assuming it. What I do think is equivalent is that he says that the metric being of Petrov type D implies the existence of a finite symmetry that reverses time or the axis of rotation. Reversing the axis of rotation seems to me to be equivalent with reversing points above and below the equator. If that is the case, Kerr claims that this follows from algebraic type $\endgroup$
    – cesaruliana
    Commented Jul 31, 2020 at 4:52
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