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| In the mathematical description of [[general relativity]], the '''Boyer–Lindquist coordinates''' are a generalization of the coordinates used for the metric of a [[Schwarzschild black hole]] that can be used to express the metric of a [[Kerr black hole]].
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| The coordinate transformation from Boyer–Lindquist coordinates <math>r, \theta</math>, <math>\phi</math> to cartesian coordinates x, y, z is given by
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| ::<math>{x} = \sqrt {r^2 + a^2} \sin\theta\cos\phi</math>
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| ::<math>{y} = \sqrt {r^2 + a^2} \sin\theta\sin\phi</math>
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| ::<math>{z} = r \cos\theta \quad </math>
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| The line element for a black hole with mass <math>M</math>, angular momentum <math>J</math>, and charge <math>Q</math> in Boyer–Lindquist coordinates and [[natural units]] (<math>G=c=1</math>) is
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| ::<math> ds^2 = -\frac{\Delta}{\Sigma}\left(dt - a \sin^2\theta d\phi \right)^2 +\frac{\sin^2\theta}{\Sigma}\Big((r^2+a^2)d\phi - a dt\Big)^2 + \frac{\Sigma}{\Delta}dr^2 + \Sigma d\theta^2 </math> | |
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| where
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| ::<math> \Delta = r^2 - 2Mr + a^2 + Q^2 </math>
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| ::<math> \Sigma = r^2 + a^2 \cos^2\theta </math>
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| ::<math> a = J/M </math>, the angular momentum per unit mass of the black hole
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| Note that in natural units <math>M</math>, <math>a</math>, and <math>Q</math> all have units of length. This line element describes the [[Kerr-Newman metric]].
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| The Hamiltonian for test particle motion in Kerr spacetime was separable in Boyer–Lindquist coordinates. Using Hamilton-Jacobi theory one can derive a fourth constant of the motion known as [[Carter constant|Carter's constant]].<ref name="carter_1968">{{cite journal | last = Carter | first = Brandon | authorlink = Brandon Carter | year = 1968 | title = Global structure of the Kerr family of gravitational fields | journal = Physical Review | volume = 174 | issue = 5 | pages = 1559–1571 | bibcode=1968PhRv..174.1559C | doi=10.1103/PhysRev.174.1559}}</ref>
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| ==References ==
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| {{reflist|1}}
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| *Boyer, R. H. and Lindquist, R. W. ''Maximal Analytic Extension of the Kerr Metric''. J. Math. Phys. 8, 265-281, 1967.
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| *Shapiro, S. L. and Teukolsky, S. A. ''Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects''. New York: Wiley, p. 357, 1983.
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| {{DEFAULTSORT:Boyer-Lindquist coordinates}}
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| [[Category:Black holes]]
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| [[Category:Coordinate charts in general relativity]]
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| {{relativity-stub}}
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