图科斯基方程

图科斯基方程（英文：Teukolsky equation）是康奈尔大学的索尔·图科斯基（Saul Teukolsky）于二十世纪七十年代创立的克尔度规下的广义相对论引力场方程^[1]。方程的基本思想是在克尔几何的框架下应用微扰数值求解爱因斯坦场方程，其适用范围包括各种微扰场：

${\displaystyle \left[{\frac {r^{2}+a^{2}}{\Delta }}-a^{2}\sin ^{2}\theta \right]{\frac {\partial ^{2}\psi }{\partial t^{2}}}+{\frac {4Mar}{\Delta }}{\frac {\partial ^{2}\psi }{\partial t\partial \phi }}+\left[{\frac {a^{2}}{\Delta }}-{\frac {1}{\sin ^{2}\theta }}\right]{\frac {\partial ^{2}\psi }{\partial \phi ^{2}}}-\Delta ^{-s}{\frac {\partial }{\partial r}}\left(\Delta ^{s+1}{\frac {\partial \psi }{\partial r}}\right)}$

${\displaystyle -{\frac {1}{\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial \psi }{\partial \theta }}\right)-2s\left[{\frac {a(r-M)}{\Delta }}+{\frac {i\cos \theta }{\sin ^{2}\theta }}\right]{\frac {\partial \psi }{\partial \phi }}}$

${\displaystyle -2s\left[{\frac {M\left(r^{2}-a^{2}\right)}{\Delta }}-r-ia\cos \theta \right]{\frac {\partial \psi }{\partial t}}+s\left[s\cot ^{2}\theta -1\right]\psi =4\pi \Sigma {\mathcal {T}}}$

其中s叫做自旋权重（spin weight），是一个与微扰场的自旋有关的量，在引力场的微扰下${\displaystyle s=\pm 2\,}$；方程中其他物理量的含义请参考克尔度规。

参考资料

1. Teukolsky S A. Perturbations Of A Rotating Black Hole. 1. Fundamental Equations For Gravitational Electromagnetic, And Neutrino Field Perturbations. Astrophys. J. 1973, 185: 635 [2008-02-12]. （原始内容于2017-11-14）.