# 四维速度

## 经典力学的情形

${\displaystyle \mathbf {x} =x^{i}(t)={\begin{bmatrix}x^{1}(t)\\x^{2}(t)\\x^{3}(t)\\\end{bmatrix}}}$

${\displaystyle {\mathbf {u} }\equiv {\mathrm {d} \mathbf {x} \over \mathrm {d} t}}$

${\displaystyle {\mathbf {u} }=(u^{1},u^{2},u^{3})={\mathrm {d} x^{i} \over \mathrm {d} t}=\left({\frac {\mathrm {d} x^{1}}{\mathrm {d} t}}\;,{\frac {\mathrm {d} x^{2}}{\mathrm {d} t}}\;,{\frac {\mathrm {d} x^{3}}{\mathrm {d} t}}\right)}$

## 相对论的情形

${\displaystyle \mathbf {X} =x^{\mu }(\tau )={\begin{bmatrix}x^{0}(\tau )\\x^{1}(\tau )\\x^{2}(\tau )\\x^{3}(\tau )\\\end{bmatrix}}={\begin{bmatrix}ct\\x^{1}(t)\\x^{2}(t)\\x^{3}(t)\\\end{bmatrix}}}$

### 时间膨胀

${\displaystyle t=\gamma \tau \,}$

${\displaystyle \gamma ={\frac {1}{\sqrt {1-{\frac {u^{2}}{c^{2}}}}}}}$

${\displaystyle u\,}$是经典速度矢量的欧几里德模

${\displaystyle u=||\mathbf {u} ||={\sqrt {(u^{1})^{2}+(u^{2})^{2}+(u^{3})^{2}}}}$.

### 四维速度的定义

${\displaystyle \mathbf {U} ={\frac {\mathrm {d} \mathbf {X} }{\mathrm {d} \tau }}}$

### 四维速度的分量

${\displaystyle x^{0}=ct=c\gamma \tau \,}$

${\displaystyle x^{0}}$对固有时${\displaystyle \tau \,}$求导数，可得四维速度${\displaystyle U^{\mu }\,}$${\displaystyle \mu =0\,}$的分量：

${\displaystyle U^{0}={\frac {\mathrm {d} x^{0}}{\mathrm {d} \tau \;}}=c\gamma }$

${\displaystyle U^{i}={\frac {\mathrm {d} x^{i}}{\mathrm {d} \tau }}={\frac {\mathrm {d} x^{i}}{\mathrm {d} x^{0}}}{\frac {\mathrm {d} x^{0}}{\mathrm {d} \tau }}={\frac {\mathrm {d} x^{i}}{\mathrm {d} x^{0}}}c\gamma ={\frac {\mathrm {d} x^{i}}{\mathrm {d} (ct)}}c\gamma ={1 \over c}{\frac {\mathrm {d} x^{i}}{\mathrm {d} t}}c\gamma =\gamma {\frac {\mathrm {d} x^{i}}{\mathrm {d} t}}=\gamma u^{i}}$

${\displaystyle \mathbf {u} =u^{i}={dx^{i} \over dt}}$

${\displaystyle U=\gamma \left(c,\mathbf {u} \right)}$

### 四维速度和加速度

${\displaystyle A^{\mu }={\frac {dU^{\mu }}{d\tau }}}$

${\displaystyle 0={\frac {d}{d\tau }}(U^{\mu }U_{\mu })=2U_{\mu }{\frac {dU^{\mu }}{d\tau }}}$

${\displaystyle U_{\mu }A^{\mu }=0\,}$

## 注释

${\displaystyle U_{\mu }U^{\mu }=-c^{2}\,}$

## 参考文献

1. ^ Bernard Schutz. A First Course in General Relativity. Cambridge University Press. 14 May 2009. ISBN 978-0-521-88705-2.