Boyer–Lindquist coordinates

In the mathematical description of general relativity, the Boyer–Lindquist coordinates^[1] 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.

The Hamiltonian for particle motion in Kerr spacetime is separable in Boyer–Lindquist coordinates. Using Hamilton–Jacobi theory one can derive a fourth constant of the motion known as Carter's constant.^[2]

The 1967 paper introducing Boyer–Lindquist coordinates by Richard W. Lindquist^[1] was also a posthumous publication for Dr. Robert Hamilton Boyer, who was killed in the 1966 University of Texas tower shooting at age 33.^[3][4]

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

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

or equivalently

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

where

${\displaystyle \Delta =r^{2}-2Mr+a^{2}+Q^{2},}$ called the discriminant,

${\displaystyle \rho ^{2}=r^{2}+a^{2}\cos ^{2}\theta ,}$

and

${\displaystyle a={\frac {J}{M}},}$ called the Kerr parameter.

Note that in geometrized units ${\displaystyle M}$, ${\displaystyle a}$, and ${\displaystyle Q}$ all have units of length. This line element describes the Kerr–Newman metric. Here, ${\displaystyle M}$ is to be interpreted as the mass of the black hole, as seen by an observer at infinity, ${\displaystyle J}$ is interpreted as the angular momentum, and ${\displaystyle Q}$ 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 ${\displaystyle r}$, ${\displaystyle \theta }$, ${\displaystyle \phi }$ to Cartesian coordinates ${\displaystyle x}$, ${\displaystyle y}$, ${\displaystyle z}$ is given (for ${\displaystyle M\to 0}$) by:^[5]${\displaystyle {\begin{aligned}x&={\sqrt {r^{2}+a^{2}}}\sin \theta \cos \phi \\y&={\sqrt {r^{2}+a^{2}}}\sin \theta \sin \phi \\z&=r\cos \theta \end{aligned}}}$

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

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

${\displaystyle \sigma ^{1}={\frac {\rho }{\sqrt {\varepsilon \Delta }}}dr}$

${\displaystyle \sigma ^{2}=\rho \,d\theta }$

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

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

${\displaystyle ds^{2}=-\varepsilon (\sigma ^{0})^{2}+\varepsilon (\sigma ^{1})^{2}+(\sigma ^{2})^{2}+(\sigma ^{3})^{2}。}$

In the region outside the outer horizon,

$\displaystyle ds^{2}=\sigma ^{a}\otimes \sigma ^{b}\eta _{ab}}$

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

ねじれのない回転接続$\displaystyle \omega ^{ab}}$定義される

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

ねじれテンソルは、ねじれのある接続と、ねじれのない対応する接続​​との差を表します。慣例として、リーマン多様体は常にねじれのない幾何学で指定されます。ねじれは、同等の平坦な幾何学を指定するためによく用いられます。

スピン接続は、曲率2形式を計算するための中間的な中継点を提供するので有用である。

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

これはスピノル場との結合を記述するのに最も適した形式であり、ツイスター形式への道を開くものである。

スピン接続の 6 つのコンポーネントはすべてゼロではありません。これらは次のとおりです。^{[ 6 ]}

${\displaystyle \omega ^{01}={\frac {1}{\rho ^{3}}}\left[{\frac {-2Mr^{2}+2rQ^{2}+a^{2}[M+r+(M-r)\cos 2\theta ]}{2{\sqrt {\Delta }}}}\,\sigma ^{0}+ra\sin \theta \,\sigma ^{3}\right]}$

${\displaystyle \omega ^{02}={\frac {a\cos \theta }{\rho ^{3}}}\left[a\sin \theta \,\sigma ^{0}+{\sqrt {\Delta }}\,\sigma ^{3}\right]}$

${\displaystyle \omega ^{03}={\frac {a}{\rho ^{3}}}\left[r\sin \theta \,\sigma ^{1}-{\sqrt {\Delta }}\cos \theta \,\sigma ^{2}\right]}$

${\displaystyle \omega ^{12}={\frac {1}{\rho ^{3}}}\left[a^{2}\sin \theta \cos \theta \,\sigma ^{1}+r{\sqrt {\Delta }}\,\sigma ^{2}\right]}$

${\displaystyle \omega ^{13}={\frac {r}{\rho ^{3}}}\left[a\sin \theta \,\sigma ^{0}+{\sqrt {\Delta }}\,\sigma ^{3}\right]}$

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

リーマンテンソルを完全に書き出すと非常に冗長になります。それはFrè ^{[ 6 ]}に記載されています。リッチテンソルは対角形式をとります。

${\displaystyle {\mbox{Ric}}={\frac {Q^{2}}{\rho ^{4}}}{\begin{bmatrix}1&0&0&0\\0&-1&0&0\\0&0&1&0\\0&0&0&1\\\end{bmatrix}}}$

マイナス1のエントリの位置に注目してください。これは完全に電磁気的寄与から来ています。つまり、電磁応力テンソルが${\displaystyle F_{ab}}$非消滅成分は2つしかない。${\displaystyle F_{01}}$そして${\displaystyle F_{23}}$すると、対応するエネルギー・運動量テンソルは次の形になる。

${\displaystyle T^{\mbox{Maxwell}}={\frac {F_{01}^{2}+F_{23}^{2}}{4}}{\begin{bmatrix}1&0&0&0\\0&-1&0&0\\0&0&1&0\\0&0&0&1\\\end{bmatrix}}}$

これを重力場のエネルギー・運動量テンソルと等しいとすると、カー・ニューマンの電気真空解が得られる。

• シャピロ、SL; テウコルスキー、SA (1983).ブラックホール、白色矮星、中性子星：コンパクト天体の物理学. ニューヨーク：ワイリー. p.  357. ISBN 9780471873167。