# 参考系拖拽

## 参考系拖拽的数学推导

$c^{2} d\tau^{2} = \left( 1 - \frac{r_{s} r}{\rho^{2}} \right) c^{2} dt^{2} - \frac{\rho^{2}}{\Lambda^{2}} dr^{2} - \rho^{2} d\theta^{2} - \left( r^{2} + a^{2} + \frac{r_{s} r \alpha^{2}}{\rho^{2}} \sin^{2} \theta \right) \sin^{2} \theta \ d\phi^{2} + \frac{2r_{s} r\alpha}{\rho^{2}} d\phi dt$

$r_{s} = \frac{2GM}{c^{2}}$

$\alpha = \frac{J}{Mc}$
$\rho^{2} = r^{2} + \alpha^{2} \cos^{2} \theta\,\!$
$\Lambda^{2} = r^{2} - r_{s} r + \alpha^{2}\,\!$

$c^{2} d\tau^{2} = c^{2} dt^{2} - \frac{\rho^{2}}{r^{2} + \alpha^{2}} dr^{2} - \rho^{2} d\theta^{2} - \left( r^{2} + \alpha^{2} \right) \sin^{2}\theta d\phi^{2}$

$c^{2} d\tau^{2} = \left( g_{tt} - \frac{g_{t\phi}^{2}}{g_{\phi\phi}} \right) dt^{2} + g_{rr} dr^{2} + g_{\theta\theta} d\theta^{2} + g_{\phi\phi} \left( d\phi + \frac{g_{t\phi}}{g_{\phi\phi}} dt \right)^{2}$

$\Omega = -\frac{g_{t\phi}}{g_{\phi\phi}} = \frac{r_{s} \alpha r}{\rho^{2} \left( r^{2} + \alpha^{2} \right) + r_{s} \alpha^{2} r \sin^{2}\theta}$

$\Omega = \frac{r_{s} \alpha c}{r^{3} + \alpha^{2} r + r_{s} \alpha^{2}}$

$r_{inner} = \frac{r_{s} + \sqrt{r_{s}^{2} - 4\alpha^{2}}}{2}$

$r_{outer} = \frac{r_{s} + \sqrt{r_{s}^{2} - 4\alpha^{2} \cos^{2}\theta}}{2}$

## 參考系拖曳力的公式

$F = \frac{SJj}{r^{4}}$

J為中心質量自旋角動量 j為周圍質量公轉角動量 S為史氏常數

$S = \frac{2G}{c^{2}}$

$E = \frac{SJj}{r^{3}}$

## 参考文献

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