# 廣義相對論中的數學

## 狹義相對論

${\displaystyle x'={\frac {x-vt}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}$
${\displaystyle y'=y}$
${\displaystyle z'=z}$
${\displaystyle t'={\frac {t-{\frac {v}{c^{2}}}x}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}$

${\displaystyle {c^{2}}{t^{2}}-x^{2}-y^{2}-z^{2}={c^{2}}{t'^{2}}-x'^{2}-y'^{2}-z'^{2}}$

${\displaystyle {c^{2}}{dt^{2}}-dx^{2}-dy^{2}-dz^{2}={c^{2}}{dt'^{2}}-dx'^{2}-dy'^{2}-dz'^{2}}$

${\displaystyle \mathbf {p} =m\mathbf {v} }$
${\displaystyle E=mc^{2}}$

${\displaystyle m={\frac {m_{0}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}$ ，:${\displaystyle m_{0}}$ 為物質在靜止下的質量

${\displaystyle E^{2}={\left(pc\right)}^{2}+{\left(m_{0}c^{2}\right)}^{2}}$

## 廣義相對論

### 數學形式

${\displaystyle ds^{2}=g_{\mu \nu }{dx^{\mu }}{dx^{\nu }}}$

${\displaystyle g_{\mu \nu }}$ 度規張量

${\displaystyle {dx^{\mu }}}$ ${\displaystyle {dx^{\nu }}}$ 逆變張量

${\displaystyle {\frac {d^{2}x^{\mu }}{ds^{2}}}+\Gamma _{\nu \sigma }^{\mu }{\frac {dx^{\nu }}{ds}}{\frac {dx^{\sigma }}{ds}}=0}$

${\displaystyle \Gamma _{\nu \sigma }^{\mu }}$ 克里斯多福符號

${\displaystyle R_{\nu \rho \sigma }^{\beta }={\frac {\partial \Gamma _{\nu \sigma }^{\beta }}{\partial x^{\rho }}}-{\frac {\partial \Gamma _{\nu \rho }^{\beta }}{\partial x^{\sigma }}}+\Gamma _{\nu \sigma }^{\alpha }\Gamma _{\alpha \rho }^{\beta }-\Gamma _{\nu \rho }^{\alpha }\Gamma _{\alpha \sigma }^{\beta }}$