# 四维速度

## 经典力学的情形

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

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

## 相对论理论

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

### 时间膨胀

$t = \gamma \tau \,$

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

$u\,$是经典速度矢量的欧几里德模

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

### 四维速度的定义

$\mathbf{U} = \frac{\mathrm{d}\mathbf{x}}{\mathrm{d} \tau}$

### 四维速度的分量

$x^0 = ct = c \gamma \tau \,$

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

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

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

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

### 四维速度和加速度

$a^{\mu} = \frac{d U^{\mu}}{d\tau}$

$0=\frac{d}{d\tau} U^{\mu} U_{\mu} = 2 U_{\mu} \frac{d U^{\mu}}{d\tau}$

$U_{\mu} a^{\mu} = 0\,$

## 注释

$U_\mu U^\mu = -c^2 \,$

## 参考文献

• Einstein, Albert; translated by Robert W. Lawson. Relativity: The Special and General Theory. New York: Original: Henry Holt, 1920; Reprinted: Prometheus Books, 1995. 1920.
• Rindler, Wolfgang. Introduction to Special Relativity (2nd). Oxford: Oxford University Press. 1991. ISBN 0-19-853952-5.