Source: http://colinscosmos.com/wiki/boyer-lindquist-coordinates/
Timestamp: 2019-04-22 08:34:00+00:00

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Boyer-Lindquist coordinates are a description of a rotating (“Kerr”) black hole. They are a generalisation of Schwarzschild coordinates. They are the simplest coordinates for calculations, generally speaking, because the line element has just one cross-term in these coordinates. They were derived by Robert Boyer and Richard Lindquist, and published in a 1967 paper.
where (also written Σ by e.g. Frolov & Novikov), and (called a “discriminant” by Carter §4.3 in Wiltshire et al) are standard notation. Every source I have checked uses an equivalent expression to this one, apart from the original paper by Boyer & Lindquist (§2) which has a different sign for the coefficient of dt dφ, presumably due to Kerr’s original a, the sign of which was quickly changed by the community. The above metric is used by O’Neill §2.1, Visser §1.5, Teukolsky §2, and others listed shortly. With some algebra, the coefficient of may be rearranged to , so the version given by Frolov & Novikov §3.2.1 is equivalent.
which also turns out to be equivalent. He uses (§53) but a different definition of : (§54; also Frolov & Novikov label this A), for which we gave an identity above. [Check, earlier text reads: which we can show is also equal to ].
This is given by Frolov & Novikov §D.1.
Also given in Kerr \S2.6, but different.
These are especially useful for timelike geodesics.
When a→0, Schwarzschild coordinates result.
Canonical vector fields , . Then identities , , , , , , (O’Neill §2.1).
More properties of B-L coordinates: , (in Frolov for instance).
There is a “price to be paid for the algebraic simplicity that has made it the most widely known expression for the Kerr solution” — singular where Δ=0 (Carter §4.3 in Wiltshire et al).
Null tetrad (Frolov & Novikov §D.6).
These are given, for Boyer-Lindquist coordinates, in Frolov & Novikov §D.2, Mueller §2.14.1, and possibly O’Neill §2.
[This is formally identical to the Kerr coordinates case, apart from a couple of minus signs; also recall that φ‘s are defined differently].
Kerr published the discovery of the rotating black hole solution in 1963. Later, “In Papapetrou (1966) [http://adsabs.harvard.edu/abs/1966AnIHP…4…83P] there is a very elegant treatment of stationary axisymmetric Einstein spaces. He shows that if there is a real non-singular axis of rotation then the coordinates can be chosen so that there is only one off-diagonal component of the metric. We call such a metric quasi-diagonalizeable.” (Kerr §2.6 in Wiltshire et al 2009).
Boyer and Lindquist published their coordinates in 1967. Sadly, Boyer was killed 2 weeks after the editor received the submitted paper. Kerr claims that he and Ray Sachs also discovered this solution, but did not consider it: “Having derived this canonical form, we studied the metric for at least ten minutes and then decided that we had no idea how to introduce a reasonable source into a metric of this form, and probably would never have.” (ibid.) But they did not publish it, so Kerr appropriately credits Boyer & Lindquist.
The sign convention of Kerr’s original rotation parameter a was quickly changed by the community. Kerr credits this (in §2.5 of Wiltshire et al) to Boyer (see e.g. the comparison with Lense-Thirring precession in Boyer & Lindquist §2). Boyer & Lindquist termed their coordinates “S” coordinates, because they generalise Schwarzschild-Droste coordinates for a non-rotating black hole.

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