Метрика Керра
Метрика и некоторые св-ва
В координатах Бойера-Линдквиста \[\dd{s}^2=-\left(r^2+\alpha^2\sin^2\theta\right)\left(\frac{\dd{r}^2}{r^2+\alpha^2-(2Mr-q^2)}+\dd{\theta}^2\right)+\frac{r^2+\alpha^2-(2Mr-q^2)}{r^2+\alpha^2\sin^2\theta}\left(\dd{t}-\alpha\cos^2\theta\dd{\varphi}\right)^2-\frac{\cos^2\theta}{r^2+\alpha^2\sin^2\theta}\left((r^2+\alpha^2)\dd{\varphi}-\alpha\dd{t}\right)^2\] или \[\dd{s}^2=\left(1-\frac{2Mr-q^2}{r^2+\alpha^2\sin^2\theta}\right)\dd{t}^2-\frac{r^2+\alpha^2\sin^2\theta}{r^2+\alpha^2-(2Mr-q^2)}\dd{r}^2-\left(r^2+\alpha^2\sin^2\theta\right)\dd{\theta}^2-\left(r^2+\alpha^2+\frac{(2Mr-q^2)\alpha^2\cos^2\theta}{r^2+\alpha^2\sin^2\theta}\right)\cos^2\theta\dd{\varphi}^2+2\frac{2Mr-q^2}{r^2+\alpha^2\sin^2\theta}\alpha\cos^2\theta\dd{t}\dd{\varphi}\]
Случай \(M=0,\;\alpha=0,\;q=0\)
\[\dd{s}^2=\dd{t}^2-\dd{r}^2-r^2\dd{\theta}^2-r^2\cos^2\theta\dd{\varphi}^2\]
Случай \(\alpha=0,\;q=0\)
\[\dd{s}^2=\left(1-\frac{2M}{r}\right)\dd{t}^2-\frac{r}{r-2M}\dd{r}^2-r^2\dd{\theta}^2-r^2\cos^2\theta\dd{\varphi}^2\]
Случай \(M=0,\;q=0\)
\[\dd{s}^2=\dd{t}^2-\frac{r^2+\alpha^2\sin^2\theta}{r^2+\alpha^2}\dd{r}^2-\left(r^2+\alpha^2\sin^2\theta\right)\dd{\theta}^2-\left(r^2+\alpha^2\right)\cos^2\theta\dd{\varphi}^2\]
Случай \(M=0,\;\alpha=0\)
\[\dd{s}^2=\left(1+\frac{q^2}{r^2}\right)\dd{t}^2-\frac{r^2}{r^2+q^2}\dd{r}^2-r^2\dd{\theta}^2-r^2\cos^2\theta\dd{\varphi}^2\]
Случай \(q=0\)
\[\dd{s}^2=\left(1-\frac{2Mr}{r^2+\alpha^2\sin^2\theta}\right)\dd{t}^2-\frac{r^2+\alpha^2\sin^2\theta}{r^2+\alpha^2-2Mr}\dd{r}^2-\left(r^2+\alpha^2\sin^2\theta\right)\dd{\theta}^2-\left(r^2+\alpha^2+\frac{2Mr\alpha^2\cos^2\theta}{r^2+\alpha^2\sin^2\theta}\right)\cos^2\theta\dd{\varphi}^2+2\frac{2Mr}{r^2+\alpha^2\sin^2\theta}\alpha\cos^2\theta\dd{t}\dd{\varphi}\]
Случай \(\alpha=0\)
\[\dd{s}^2=\left(1-\frac{2Mr-q^2}{r^2}\right)\dd{t}^2-\frac{r^2}{r^2-(2Mr-q^2)}\dd{r}^2-r^2\dd{\theta}^2-r^2\cos^2\theta\dd{\varphi}^2\]
Случай \(M=0\)
\[\dd{s}^2=\left(1+\frac{q^2}{r^2+\alpha^2\sin^2\theta}\right)\dd{t}^2-\frac{r^2+\alpha^2\sin^2\theta}{r^2+\alpha^2+q^2}\dd{r}^2-\left(r^2+\alpha^2\sin^2\theta\right)\dd{\theta}^2-\left(r^2+\alpha^2-\frac{q^2\alpha^2\cos^2\theta}{r^2+\alpha^2\sin^2\theta}\right)\cos^2\theta\dd{\varphi}^2-2\frac{q^2}{r^2+\alpha^2\sin^2\theta}\alpha\cos^2\theta\dd{t}\dd{\varphi}\]
Случай \(q=0,\;\theta=0,\;\alpha=M\)
\begin{array} \dd{s}^2 & = \left(1-\frac{2M}{r}\right)\dd{t}^2-\frac{r^2}{(r-M)^2}\dd{r}^2-\left(r^2+M^2+\frac{2M^3}{r}\right)\dd{\varphi}^2+2\frac{2M^2}{r}\dd{t}\dd{\varphi}\\ & = -\frac{r^2}{(r-M)^2}\dd{r}^2+\frac{(r-M)^2}{r^2}\left(\dd{t}-M\dd{\varphi}\right)^2-\frac{1}{r^2}\left((r^2+M^2)\dd{\varphi}-M\dd{t}\right)^2 \end{array}
Случай \(q=0,\;\theta=0,\;r=R,\;\varphi=\frac{V}{R} t\)
\[\dd{s}^2=\left(1-\frac{2M}{R}+\frac{4M\alpha}{R^2}V-\left(1+\frac{\alpha^2}{R^2}+\frac{2M\alpha^2}{R^3}\right)V^2\right)\dd{t}^2\dot=Fdt^2\]
\[A=\frac{\sqrt{\left(1+\frac{\alpha^2}{R^2}\right)\left(1+\frac{\alpha^2}{R^2}-\frac{2M}{R}\right)}}{F}\left(-\frac{M}{R^2}+\frac{V^2}{R}+\frac{2M\alpha V}{R^3}-\frac{M\alpha^2V^2}{R^4}\right)\]
Для Земли \[\dd{s}^2\approx\left(1-1.39\cdot10^{-9}+2.68\cdot10^{-21}-2.41\cdot10^{-12}-9.25\cdot10^{-25}-1.29\cdot10^{-33}\right)\dd{t}^2\] \[A\approx\]
Случай \(q=0,\;\theta=\frac{\pi}{2},\;r=R,\;\varphi=0\)
\[\dd{s}^2=\left(1-\frac{2MR}{R^2+\alpha^2}\right)\dd{t}^2\]
\[A=\frac{M}{R^2+\alpha^2-2MR}\frac{R^2-\alpha^2}{R^2+\alpha^2}\sqrt{\frac{R^2+\alpha^2-2MR}{R^2+\alpha^2}}\]
Для Земли \[\dd{s}^2\approx\left(1-\frac{2M}{R}+\frac{2M\alpha^2}{R^3}\right)\dd{t}^2\approx\left(1-1.39\cdot10^{-9}+5.34\cdot10^{-22}\right)\dd{t}^2\] \[A\approx\]