圖科斯基方程

圖科斯基方程（英文：Teukolsky equation）是康奈爾大學的索爾·圖科斯基（Saul Teukolsky）於二十世紀七十年代創立的克爾度規下的廣義相對論引力場方程^[1]。方程的基本思想是在克爾幾何的框架下應用微擾數值求解愛因斯坦場方程，其適用範圍包括各種微擾場：

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

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

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

其中s叫做自旋權重（spin weight），是一個與微擾場的自旋有關的量，在引力場的微擾下${\displaystyle s=\pm 2\,}$；方程中其他物理量的含義請參考克爾度規。

參考資料

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]. （原始內容存檔於2017-11-14）.