Boyer–Lindquist coordinates

The line element for a black hole with a total mass equivalent $M$, angular momentum $J$, and charge $Q$ in Boyer–Lindquist coordinates and geometrized units ($G=c=1$) is

:$ds^2\; =\; -\backslash frac\{\backslash Delta\}\{\backslash rho^2\}\backslash left(dt\; -\; a\; \backslash sin^2\backslash theta\; \backslash ,d\backslash phi\; \backslash right)^2\; +\backslash frac\{\backslash sin^2\backslash theta\}\{\backslash rho^2\}\backslash Big(\backslash left(r^2+a^2\backslash right)\backslash ,d\backslash phi\; -\; a\; \backslash ,dt\backslash Big)^2\; +\; \backslash frac\{\backslash rho^2\}\{\backslash Delta\}dr^2\; +\; \backslash rho^2\; \backslash ,d\backslash theta^2$

or equivalently

:$ds^2\; =\; -\backslash left(1-\backslash frac\{2Mr\}\{\backslash rho^2\}\backslash right)dt^2\; +\; \backslash frac\{\backslash rho^2\}\{\backslash Delta\}dr^2\; +\; \backslash rho^2\; d\backslash theta^2\; +\; \backslash left[r^2\; +\; a^2\; +\; \backslash frac\{2Mr\; a^2\; \backslash sin^2\backslash theta\}\{\backslash rho^2\}\backslash right]\backslash sin^2\backslash theta\; \backslash ,d\backslash phi^2\; -\; \backslash frac\{4Mr\; a\; \backslash sin^2\backslash theta\}\{\backslash rho^2\}\; \backslash ,d\backslash phi\; dt$

where

:$\backslash Delta\; =\; r^2\; -\; 2Mr\; +\; a^2\; +\; Q^2,$ called the discriminant,

:$\backslash rho^2\; =\; r^2\; +\; a^2\; \backslash cos^2\backslash theta,$

and

:$a\; =\; \backslash frac\{J\}\{M\},$ called the Kerr parameter.

Note that in geometrized units $M$, $a$, and $Q$ all have units of length. This line element describes the Kerr–Newman metric. Here, $M$ is to be interpreted as the mass of the black hole, as seen by an observer at infinity, $J$ is interpreted as the angular momentum, and $Q$ 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 $r$, $\backslash theta$, $\backslash phi$ to Cartesian coordinates $x$, $y$, $z$ is given (for $m\backslash to\; 0$) by:Matt Visser, arXiv:0706.0622v3, eqs. 60-62

$\backslash begin\{align\}$

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}

The vierbein one-forms can be read off directly from the line element:

:$\backslash sigma^0\; =\; \backslash frac\{\backslash sqrt\{\backslash varepsilon\backslash Delta\}\}\{\backslash rho\}\backslash left(dt\; -\; a\; \backslash sin^2\backslash theta\; \backslash ,d\backslash phi\; \backslash right)$

:$\backslash sigma^1\; =\; \backslash frac\{\backslash rho\}\{\backslash sqrt\{\backslash varepsilon\backslash Delta\}\}dr$

:$\backslash sigma^2\; =\; \backslash rho\; \backslash ,d\backslash theta$

:$\backslash sigma^3\; =\; \backslash frac\{\backslash sin\backslash theta\}\{\backslash rho\}\backslash Big(\backslash left(r^2+a^2\backslash right)\backslash ,d\backslash phi\; -\; a\; \backslash ,dt\backslash Big)$

where $\backslash varepsilon\; =\; \backslash mathrm\{sgn\}(\backslash Delta)$ which is $1$ outside the outer horizon, $-1$ between the inner and outer horizons, and $1$ inside the inner horizon. The line element is given by

:$ds^2=-\backslash varepsilon\; (\backslash sigma^0)^2\; +\; \backslash varepsilon\; (\backslash sigma^1)^2\; +\; (\backslash sigma^2)^2\; +\; (\backslash sigma^3)^2.$

In the region outside the outer horizon,

:$ds^2=\backslash sigma^a\backslash otimes\backslash sigma^b\; \backslash eta\_\{ab\}$

where $\backslash eta\_\{ab\}$ is the flat-space Minkowski metric.

The torsion-free spin connection $\backslash omega^\{ab\}$ is defined by

:$d\backslash sigma^a\; +\; \backslash omega^\{ab\}\; \backslash wedge\; \backslash sigma^c\; \backslash eta\_\{bc\}=0$

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 curvature two-form:

:$R^\{ab\}=d\backslash omega^\{ab\}+\backslash omega^\{ac\}\backslash wedge\backslash omega^\{db\}\backslash eta\_\{cd\}$

It is also the most suitable form for describing the coupling to spinor fields, and opens the door to the twistor formalism.

All six components of the spin connection are non-vanishing. These are:Pietro Giuseppe Frè, "Gravity, a Geometrical Course, Volume 2: Black Holes, Cosmology and Introduction to Supergravity", (2013) Springer-Verlag

:$\backslash omega^\{01\}=\backslash frac\{1\}\{\backslash rho^3\}$

\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]

:$\backslash omega^\{02\}=\backslash frac\{a\backslash cos\backslash theta\}\{\backslash rho^3\}$

\left[a\sin\theta\,\sigma^0+\sqrt{\Delta}\,\sigma^3\right]

:$\backslash omega^\{03\}=\backslash frac\{a\}\{\backslash rho^3\}$

\left[r\sin\theta\,\sigma^1-\sqrt{\Delta}\cos\theta\,\sigma^2\right]

:$\backslash omega^\{12\}=\backslash frac\{1\}\{\backslash rho^3\}$

\left[a^2\sin\theta\cos\theta\,\sigma^1+r\sqrt{\Delta}\,\sigma^2\right]

:$\backslash omega^\{13\}=\backslash frac\{r\}\{\backslash rho^3\}$

\left[a\sin\theta\,\sigma^0+\sqrt{\Delta}\,\sigma^3\right]

:$\backslash omega^\{23\}=\backslash frac\{\backslash cot\backslash theta\}\{\backslash rho^3\}$

\left[a\sqrt{\Delta}\sin\theta\,\sigma^0+(r^2+a^2)\,\sigma^3\right]

The Riemann tensor written out in full is quite verbose; it can be found in Frè. The Ricci tensor takes the diagonal form:

:$\backslash mbox\{Ric\}=\backslash frac\{Q^2\}\{\backslash rho^4\}$

\begin{bmatrix}

1 & 0 & 0 & 0\\

0 & -1 & 0 & 0 \\

0 & 0 & 1 & 0 \\

0 & 0 & 0 & 1 \\

\end{bmatrix}

Notice the location of the minus-one entry: this comes entirely from the electromagnetic contribution. Namely, when the electromagnetic stress tensor $F\_\{ab\}$ has only two non-vanishing components: $F\_\{01\}$ and $F\_\{23\}$, then the corresponding energy–momentum tensor takes the form

:$T^\backslash mbox\{Maxwell\}=\backslash frac\{F\_\{01\}^2+F\_\{23\}^2\}\{4\}$

\begin{bmatrix}

1 & 0 & 0 & 0\\

0 & -1 & 0 & 0 \\

0 & 0 & 1 & 0 \\

0 & 0 & 0 & 1 \\

\end{bmatrix}

Equating this with the energy–momentum tensor for the gravitational field leads to the Kerr–Newman electrovacuum solution.

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• {{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}}

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Category:Black holes

Category:Coordinate charts in general relativity