# TIL: 关于《星际穿越》里面的一些物理相关的计算：1小时等于7年的时间差可能吗？etc

Caveat lector：本文有影响阅读的数学术语，和《星际穿越》的情节。

## 引力不同的地方时间的流逝的快慢

${\displaystyle \gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}={\frac {1}{\sqrt {1-\beta ^{2}}}}={\frac {dt}{d\tau }}}$

$v \approx 0.9999999998672 c$

$ds^{2}_{\mathrm{KN}}=\left(1-\frac{2Mr-Q^2}{R^2}\right)du^{2}+2du\,dr+2\frac{a\sin^{2}\theta}{R^2}\left(2Mr-Q^2\right) du\,d\phi \\ -2a\sin^{2}\theta \,dr\,d\phi-R^{2}d\theta^{2}+\frac{\sin^{2}\theta}{R^2}\left(\Delta a^{2}\sin^{2}\theta-(a^2+r^2)^2\right)\,d\phi^{2}.$

$\rho^{2}\equiv r^{2}+a^2\cos^{2}\theta, \qquad \Delta\equiv r^{2}+a^2-2Mr+Q^2.$

$d s^{2}=\left(1-\frac{r_{s} r}{\rho^{2}}\right) c^{2} d t^{2}-\frac{\rho^{2}}{\Delta} d \tau^{2}-\rho^{2} d \theta^{2}-\left(r^{2}+\alpha^{2}+\frac{r_{s} r \alpha^{2}}{\rho^{2}} \sin ^{2} \theta\right) \sin ^{2} \theta d \phi^{2}+\frac{2 r_{s} r \alpha \sin ^{2} \theta}{\rho^{2}} c d t d \phi \tag{1}$

$v_{\mathrm{escape}}=c=\sqrt{\frac{2 G M}{r_{s}}} \tag{2}.$

$F=\frac{G M m}{\left(r_{s}+x\right)^{2}}=m a=m \frac{d v}{d t}=m \frac{d x}{d t} \frac{d v}{d x}=m v \frac{d v}{d x}$

$\left(\frac{d \tau}{d t}\right)^{2}=1-\frac{r_{s} r}{\rho^{2}}$

$\frac{2 G M r}{r^{2}+J^{2} \cos ^{2} \theta / M^{2}}=1-\left(\frac{d \tau}{d t}\right)^{2} \tag{3}$

$r = 3.5\times 10^{11} \mathrm{m}.$

$r_{+}=\frac{G M}{c^{2}}+\frac{1}{c} \sqrt{\left(\frac{G M}{c}\right)^{2}-a^{2}}$

$r_{+} \approx \frac{G M}{c^{2}}=\frac{1}{2} r_{s}$

## 落入黑洞之后怎么办？

$d s^{2}=-\mathrm{d} t^{2}+\mathrm{d} \tau^{2}+\mathrm{d} z^{2}-\sinh ^{2} r\left(\sinh ^{2} r-1\right) \mathrm{d} \phi^{2}+\sqrt{2} \sinh ^{2} r(\mathrm{d} \phi \mathrm{d} t+\mathrm{d} t \mathrm{d} \phi)$

#### 参考资料：

• The Kerr spacetime: A brief introduction, M. Visser, [url]http://arxiv.org/pdf/0706.0622.pdf.[/url]
• Wikipedia关于Kerr Metric和Kerr-Newman metric的条目.
• Ikjyot Singh Kohli的博客：ikjyotsinghkohli24.wordpress.com .
• The Science of Interstellar, K. Thorne.
• Exploring Black Holes: Introduction to General Relativity, E. Taylor, J. Wheeler.

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