imported>JohnJeremyGoodman: The coefficient of d\phi dt in the 2nd form of the Boyer-Lindquist line element needed to be doubled. See eq. (57) of Visser (arXiv:0706.0622v3)

2025-04-09T06:11:20Z

The coefficient of d\phi dt in the 2nd form of the Boyer-Lindquist line element needed to be doubled. See eq. (57) of Visser (arXiv:0706.0622v3)

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{{short description|Coordinate system used in general relativity for the Kerr metric}}

In the mathematical description of [[general relativity]], the '''Boyer–Lindquist coordinates'''<ref name="boyer_lindquist_1967">{{cite journal | last1 = Boyer | first1 = Robert H. | last2=Lindquist | first2 = Richard W. | year = 1967 | title = Maximal Analytic Extension of the Kerr Metric | journal = Journal of Mathematical Physics | volume = 8 | issue = 2 | pages = 265–281 | doi=10.1063/1.1705193 |bibcode = 1967JMP.....8..265B }}</ref> are a generalization of the [[Coordinate system|coordinates]] used for the [[metric (general relativity)|metric]] of a [[Schwarzschild black hole]] that can be used to express the metric of a [[Kerr black hole]].

The [[Hamiltonian mechanics|Hamiltonian]] for particle motion in Kerr spacetime is separable in Boyer–Lindquist coordinates. Using [[Hamilton–Jacobi equation|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>

The 1967 paper introducing Boyer–Lindquist coordinates<ref name="boyer_lindquist_1967"/> was a posthumous publication for Robert H. Boyer, who was killed in the 1966 [[University of Texas tower shooting#Victims|University of Texas tower shooting]].<ref>{{cite web |url=http://behindthetower.org/the-victims/ |title=The Victims|website=Behind the Tower |date=15 July 2016 |access-date=2 November 2022}}</ref><ref>{{cite journal |title=Robert Hamilton Boyer |journal=Physics Today |date=September 1966 |volume=19 |issue=9 |page=121 |doi=10.1063/1.3048457 |doi-access=free }}</ref>

==Line element==
The [[line element]] for a black hole with a total [[Mass–energy equivalence|mass equivalent]] <math>M</math>, angular momentum <math>J</math>, and charge <math>Q</math> in Boyer–Lindquist coordinates and [[geometrized units]] (<math>G=c=1</math>) is
:<math> ds^2 = -\frac{\Delta}{\rho^2}\left(dt - a \sin^2\theta \,d\phi \right)^2 +\frac{\sin^2\theta}{\rho^2}\Big(\left(r^2+a^2\right)\,d\phi - a \,dt\Big)^2 + \frac{\rho^2}{\Delta}dr^2 + \rho^2 \,d\theta^2 </math>
or equivalently
:<math> ds^2 = -\left(1-\frac{2Mr}{\rho^2}\right)dt^2 + \frac{\rho^2}{\Delta}dr^2 + \rho^2 d\theta^2 + \left[r^2 + a^2 + \frac{2Mr a^2 \sin^2\theta}{\rho^2}\right]\sin^2\theta \,d\phi^2 - \frac{4Mr a \sin^2\theta}{\rho^2} \,d\phi dt</math>
where
:<math>\Delta = r^2 - 2Mr + a^2 + Q^2,</math> called the ''discriminant'',
:<math>\rho^2 = r^2 + a^2 \cos^2\theta,</math>
and
:<math>a = \frac{J}{M},</math> called the ''Kerr parameter''.
Note that in geometrized units <math>M</math>, <math>a</math>, and <math>Q</math> all have units of length. This line element describes the [[Kerr–Newman metric]]. Here, <math>M</math> is to be interpreted as the [[mass]] of the black hole, as seen by an observer at infinity, <math>J</math> is interpreted as the [[angular momentum]], and <math>Q</math> the [[electric charge]]. These are all meant to be constant parameters, held fixed. The name of the discriminant arises because it appears as the discriminant of the quadratic equation bounding the time-like motion of particles orbiting the black hole, ''i.e.'' defining the ergosphere.

The coordinate transformation from Boyer–Lindquist coordinates <math>r</math>, <math>\theta</math>, <math>\phi</math> to Cartesian coordinates <math>x</math>, <math>y</math>, <math>z</math> is given (for <math>m\to 0</math>) by:<ref name="Visser">Matt Visser, arXiv:0706.0622v3, eqs. 60-62</ref>
<math>\begin{align}
x &= \sqrt {r^2 + a^2} \sin\theta\cos\phi \\
y &= \sqrt {r^2 + a^2} \sin\theta\sin\phi \\
z &= r \cos\theta
\end{align}</math>

==Vierbein==
The [[Tetrad (general relativity)|vierbein]] [[one-form]]s can be read off directly from the line element:
:<math>\sigma^0 = \frac{\sqrt{\varepsilon\Delta}}{\rho}\left(dt - a \sin^2\theta \,d\phi \right)</math>
:<math>\sigma^1 = \frac{\rho}{\sqrt{\varepsilon\Delta}}dr</math>
:<math>\sigma^2 = \rho \,d\theta</math>
:<math>\sigma^3 = \frac{\sin\theta}{\rho}\Big(\left(r^2+a^2\right)\,d\phi - a \,dt\Big)</math>
where <math>\varepsilon = \mathrm{sgn}(\Delta) </math> which is <math>1</math> outside the outer horizon, <math>-1</math> between the inner and outer horizons, and <math>1</math> inside the inner horizon. The line element is given by
:<math>ds^2=-\varepsilon (\sigma^0)^2 + \varepsilon (\sigma^1)^2 + (\sigma^2)^2 + (\sigma^3)^2.</math>
In the region outside the outer horizon,
:<math>ds^2=\sigma^a\otimes\sigma^b \eta_{ab}</math>
where <math>\eta_{ab}</math> is the flat-space [[Minkowski metric]].

==Spin connection==
The [[torsion tensor|torsion-free]] [[spin connection]] <math>\omega^{ab}</math> is defined by
:<math>d\sigma^a + \omega^{ab} \wedge \sigma^c \eta_{bc}=0</math>

The [[contorsion tensor]] gives the difference between a connection with torsion, and a corresponding connection without torsion. By convention, Riemann manifolds are always specified with torsion-free geometries; torsion is often used to specify equivalent, flat geometries.

The spin connection is useful because it provides an intermediate way-point for computing the [[Riemann curvature tensor|curvature two-form]]:
:<math>R^{ab}=d\omega^{ab}+\omega^{ac}\wedge\omega^{db}\eta_{cd}</math>
It is also the most suitable form for describing the coupling to [[spinor]] fields, and opens the door to the [[twistor theory|twistor formalism]].

All six components of the spin connection are non-vanishing. These are:<ref name="Fre">Pietro Giuseppe Frè, "Gravity, a Geometrical Course, Volume 2: Black Holes, Cosmology and Introduction to Supergravity", (2013) Springer-Verlag</ref>
:<math>\omega^{01}=\frac{1}{\rho^3}
\left[\frac{-2Mr^2+2rQ^2+a^2[M+r+(M-r)\cos 2\theta]}{2\sqrt{\Delta}}\,\sigma^0
+ra\sin\theta\,\sigma^3\right]</math>

:<math>\omega^{02}=\frac{a\cos\theta}{\rho^3}
\left[a\sin\theta\,\sigma^0+\sqrt{\Delta}\,\sigma^3\right]</math>

:<math>\omega^{03}=\frac{a}{\rho^3}
\left[r\sin\theta\,\sigma^1-\sqrt{\Delta}\cos\theta\,\sigma^2\right]</math>

:<math>\omega^{12}=\frac{1}{\rho^3}
\left[a^2\sin\theta\cos\theta\,\sigma^1+r\sqrt{\Delta}\,\sigma^2\right]</math>

:<math>\omega^{13}=\frac{r}{\rho^3}
\left[a\sin\theta\,\sigma^0+\sqrt{\Delta}\,\sigma^3\right]</math>

:<math>\omega^{23}=\frac{\cot\theta}{\rho^3}
\left[a\sqrt{\Delta}\sin\theta\,\sigma^0+(r^2+a^2)\,\sigma^3\right]</math>

==Riemann and Ricci tensors==
The Riemann tensor written out in full is quite verbose; it can be found in Frè.<ref name="Fre"/> The [[Ricci tensor]] takes the diagonal form:
:<math>\mbox{Ric}=\frac{Q^2}{\rho^4}
\begin{bmatrix}
1 & 0 & 0 & 0\\
0 & -1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{bmatrix}</math>
Notice the location of the minus-one entry: this comes entirely from the electromagnetic contribution. Namely, when the [[electromagnetic stress tensor]] <math>F_{ab}</math> has only two non-vanishing components: <math>F_{01}</math> and <math>F_{23}</math>, then the corresponding [[energy–momentum tensor]] takes the form

:<math>T^\mbox{Maxwell}=\frac{F_{01}^2+F_{23}^2}{4}
\begin{bmatrix}
1 & 0 & 0 & 0\\
0 & -1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{bmatrix}</math>
Equating this with the energy–momentum tensor for the gravitational field leads to the [[Kerr–Newman metric|Kerr–Newman electrovacuum solution]].

==References ==
{{reflist|1}}
* {{cite book|last1=Shapiro |first1=S. L. |last2=Teukolsky |first2=S. A. |title=Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects |location=New York |publisher=Wiley |page=357 |date=1983 |isbn=9780471873167}}

{{DEFAULTSORT:Boyer-Lindquist coordinates}}
[[Category:Black holes]]
[[Category:Coordinate charts in general relativity]]

Boyer–Lindquist coordinates - Revision history

imported>JohnJeremyGoodman: The coefficient of d\phi dt in the 2nd form of the Boyer-Lindquist line element needed to be doubled. See eq. (57) of Visser (arXiv:0706.0622v3)