# 四維矢量

（重定向自四維坐標

## 數學性質

${\displaystyle {x}^{\mu }\ {\stackrel {def}{=}}\ (ct,\,x,\,y,\,z)}$

「四維位移」定義為兩個事件之間的矢量差。在時空圖裏，四維位移可以用從第一個事件指到第二個事件的箭矢來表示。當矢量的尾部是坐標系的原點時，位移就是位置。四維位移 ${\displaystyle \Delta {x}^{\mu }}$  表示為

${\displaystyle \Delta {x}^{\mu }\ {\stackrel {def}{=}}\ (\Delta ct,\ \Delta x,\ \Delta y,\ \Delta z)}$

${\displaystyle {U}^{\mu }=\ ({U}^{0},\,{U}^{1},\,{U}^{2},\,{U}^{3})}$

${\displaystyle {U}_{\mu }=\ ({U}_{0},\,{U}_{1},\,{U}_{2},\,{U}_{3})=\ ({U}^{0},\,-{U}^{1},\,-{U}^{2},\,-{U}^{3})}$

${\displaystyle \eta _{\mu \nu }\ {\stackrel {def}{=}}\ \left({\begin{matrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{matrix}}\right)}$

${\displaystyle U_{\mu }=\eta _{\mu \nu }U^{\nu }}$

${\displaystyle \eta ^{\mu \nu }\ {\stackrel {def}{=}}\ \left({\begin{matrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{matrix}}\right)}$

### 勞侖茲變換

${\displaystyle \Lambda ^{\mu }{}_{\nu }=\ \left({\begin{matrix}\gamma &-\gamma \beta &0&0\\-\gamma \beta &\gamma &0&0\\0&0&1&0\\0&0&0&1\end{matrix}}\right)}$

${\displaystyle {\bar {x}}^{\mu }=\Lambda ^{\mu }{}_{\nu }\ x^{\nu }}$
${\displaystyle x^{\mu }={\bar {\Lambda }}^{\mu }{}_{\nu }\ {\bar {x}}^{\nu }}$

${\displaystyle {\bar {\Lambda }}^{\mu }{}_{\nu }=\ \left({\begin{matrix}\gamma &\gamma \beta &0&0\\\gamma \beta &\gamma &0&0\\0&0&1&0\\0&0&0&1\end{matrix}}\right)}$

${\displaystyle {\bar {x}}^{\mu }=\Lambda ^{\mu }{}_{\nu }\ x^{\nu }=\Lambda ^{\mu }{}_{\nu }\ {\bar {\Lambda }}^{\nu }{}_{\xi }\ {\bar {x}}^{\xi }}$

${\displaystyle \Lambda ^{\mu }{}_{\nu }\ {\bar {\Lambda }}^{\nu }{}_{\xi }=\delta ^{\mu }{}_{\xi }}$

${\displaystyle {\bar {\Lambda }}^{\mu }{}_{\nu }=\eta _{\alpha \nu }\ \eta ^{\beta \mu }\ \Lambda ^{\alpha }{}_{\beta }}$

${\displaystyle {\bar {U}}^{\mu }=\Lambda ^{\mu }{}_{\nu }\ U^{\nu }}$
${\displaystyle U^{\mu }={\bar {\Lambda }}^{\mu }{}_{\nu }\ {\bar {U}}^{\nu }}$

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

${\displaystyle {\frac {\mathrm {d} \tau }{\mathrm {d} t}}={\frac {1}{\gamma }}}$

### 閔考斯基內積

${\displaystyle U^{\mu }V_{\mu }\ {\stackrel {def}{=}}\ U^{0}V^{0}-U^{1}V^{1}-U^{2}V^{2}-U^{3}V^{3}}$

${\displaystyle U^{\mu }U_{\mu }=(U^{0})^{2}-(U^{1})^{2}-(U^{2})^{2}-(U^{3})^{2}}$

${\displaystyle \eta _{\mu \nu }\ {\stackrel {def}{=}}\ \left({\begin{matrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{matrix}}\right)}$

${\displaystyle U^{\mu }V_{\mu }=-U^{0}V^{0}+U^{1}V^{1}+U^{2}V^{2}+U^{3}V^{3}}$

${\displaystyle {U}^{\mu }{V}_{\mu }={\overline {\Lambda }}^{\mu }{}_{\alpha }\ {\overline {U}}^{\alpha }\ \eta _{\mu \beta }{V}^{\beta }={\overline {\Lambda }}^{\mu }{}_{\alpha }\ {\overline {U}}^{\alpha }\ \eta _{\mu \beta }\ {\overline {\Lambda }}^{\beta }{}_{\xi }\ {\overline {V}}^{\xi }={\overline {\Lambda }}^{\mu }{}_{\alpha }\ {\overline {U}}^{\alpha }\ \eta _{\mu \beta }\ {\overline {\Lambda }}^{\beta }{}_{\xi }\ \eta ^{\xi \zeta }\ {\overline {V}}_{\zeta }={\overline {\Lambda }}^{\mu }{}_{\alpha }\ {\overline {U}}^{\alpha }\ {\overline {\Lambda }}^{\zeta }{}_{\mu }\ {\overline {V}}_{\zeta }=\delta ^{\zeta }{}_{\alpha }\ {\overline {U}}^{\alpha }\ {\overline {V}}_{\zeta }={\overline {U}}^{\alpha }{\overline {V}}_{\alpha }}$

${\displaystyle U^{\mu }V_{\mu }={\overline {U}}^{\mu }{\overline {V}}_{\mu }}$

## 動力學實例

### 四維速度

${\displaystyle U^{\mu }\ {\stackrel {def}{=}}\ {\frac {\mathrm {d} x^{\mu }}{\mathrm {d} \tau }}={\frac {\mathrm {d} t}{\mathrm {d} \tau }}\ {\frac {\mathrm {d} x^{\mu }}{\mathrm {d} t}}=\left(\gamma c,\ \gamma \mathbf {u} \right)}$

${\displaystyle U^{\mu }}$  的空間部分與經典速度 ${\displaystyle \mathbf {u} }$  的關係為

${\displaystyle \left(U^{1},\,U^{2},\,U^{3}\right)=\gamma \mathbf {u} }$

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

${\displaystyle \left(c,0,0,0\right)_{MCRF}}$

### 四維加速度

${\displaystyle \alpha ^{\mu }\ {\stackrel {def}{=}}\ {\frac {\mathrm {d} U^{\mu }}{\mathrm {d} \tau }}=\left(\gamma {\dot {\gamma }}c,\,\gamma {\dot {\gamma }}\mathbf {u} +\gamma ^{2}{\dot {\mathbf {u} }}\right)}$

${\displaystyle {\dot {\gamma }}={\frac {\mathrm {d} \gamma }{\mathrm {d} t}}=\gamma ^{3}(\mathbf {u} \cdot \mathbf {a} )/c^{2}}$

${\displaystyle \alpha ^{\mu }=\left(\gamma ^{4}(\mathbf {u} \cdot \mathbf {a} )/c,\,\gamma ^{2}\mathbf {a} +\gamma ^{4}(\mathbf {u} \cdot \mathbf {a} )\mathbf {u} /c^{2}\right)}$

${\displaystyle \alpha _{\mu }U^{\mu }={\frac {1}{2}}{\frac {\mathrm {d} (U_{\mu }U^{\mu })}{\mathrm {d} \tau }}=0}$

${\displaystyle \alpha ^{\mu }=\left(0,\,\gamma ^{2}\mathbf {a} \right)_{MCRF}}$

### 四維動量

${\displaystyle P^{\mu }\ {\stackrel {def}{=}}\ mU^{\mu }=\left(\gamma mc,\,\gamma m\mathbf {u} \right)}$

${\displaystyle \mathbf {p} \ {\stackrel {def}{=}}\ m_{rel}\mathbf {u} =\gamma m\mathbf {u} }$

${\displaystyle \left(P^{1},\,P^{2},\,P^{3}\right)=\mathbf {p} }$

### 四維力

${\displaystyle F^{\mu }\ {\stackrel {def}{=}}\ {\frac {\mathrm {d} P^{\mu }}{\mathrm {d} \tau }}}$

${\displaystyle F^{\mu }=m{\frac {\mathrm {d} U^{\mu }}{\mathrm {d} \tau }}=m\alpha ^{\mu }}$

${\displaystyle F^{\mu }=m\left(\gamma ^{4}(\mathbf {u} \cdot \mathbf {a} )/c,\,\gamma ^{2}\mathbf {a} +\gamma ^{4}(\mathbf {u} \cdot \mathbf {a} )\mathbf {u} /c^{2}\right)}$

${\displaystyle \mathbf {f} \ {\stackrel {def}{=}}\ {\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}}}$

${\displaystyle \left(F^{1},\,F^{2},\,F^{3}\right)=\gamma \mathbf {f} }$

## 物理內涵

### 質能方程式

${\displaystyle \mathrm {d} W=\mathbf {f} \cdot \mathrm {d} \mathbf {x} }$

${\displaystyle \mathrm {d} K=\mathrm {d} W=\mathbf {f} \cdot \mathrm {d} \mathbf {x} }$

${\displaystyle {\frac {\mathrm {d} K}{\mathrm {d} t}}=\mathbf {f} \cdot {\frac {\mathrm {d} \mathbf {x} }{\mathrm {d} t}}=\mathbf {f} \cdot \mathbf {u} }$

${\displaystyle {\frac {\mathrm {d} K}{\mathrm {d} t}}=m\gamma ^{3}(\mathbf {u} \cdot \mathbf {a} )=mc^{2}{\frac {\mathrm {d} \gamma }{\mathrm {d} t}}}$

${\displaystyle K=\gamma mc^{2}+K_{0}}$

${\displaystyle K=\gamma mc^{2}-mc^{2}}$

${\displaystyle E=\gamma mc^{2}}$

${\displaystyle E=m_{rel}c^{2}}$

### 能量-動量關係式

${\displaystyle P^{\mu }=\left({\frac {E}{c}},\,\mathbf {p} \right)}$

${\displaystyle P^{\mu }P_{\mu }={\frac {E^{2}}{c^{2}}}-(p)^{2}}$

${\displaystyle P^{\mu }P_{\mu }=m^{2}U^{\mu }U_{\mu }=m^{2}c^{2}}$

${\displaystyle E^{2}=(pc)^{2}+m^{2}c^{4}}$

## 電磁學實例

### 四維電流密度

${\displaystyle J^{\mu }\ {\stackrel {def}{=}}\ (\rho c,\,\mathbf {j} )}$

${\displaystyle J^{\mu }=\rho _{0}U^{\mu }}$

${\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot \mathbf {j} =0}$

${\displaystyle {\frac {\partial J^{\mu }}{\partial x^{\mu }}}=0}$

### 電磁四維勢

${\displaystyle A^{\mu }\ {\stackrel {def}{=}}\ (\phi /c,\,\mathbf {A} )}$

${\displaystyle \Box A^{\mu }=\mu _{0}J^{\mu }}$  ;

### 四維頻率和四維波矢量

${\displaystyle {\nu }^{\alpha }\ {\stackrel {def}{=}}\ (f,\,f\mathbf {n} )}$

${\displaystyle {\nu }^{\alpha }{\nu }_{\alpha }=(f)^{2}(1-n^{2})=0}$

${\displaystyle {K}^{\alpha }\ {\stackrel {def}{=}}\ \left({\frac {2\pi f}{c}},\,\mathbf {k} \right)}$

${\displaystyle {K}^{\alpha }={\frac {2\pi {\nu }^{\alpha }}{c}}}$

## 參考文獻

1. Bernard Schutz. A First Course in General Relativity. Cambridge University Press. 14 May 2009. ISBN 978-0-521-88705-2.
2. ^ Carver A. Mead. Collective Electrodynamics: Quantum Foundations of Electromagnetism. MIT Press. 2002: 37–38. ISBN 9780262632607.
• Griffiths, David J. Introduction to Electrodynamics (3rd ed.). Prentice Hall. 1998: pp. 477–543. ISBN 0-13-805326-X.
• Rindler, W. Introduction to Special Relativity (2nd edition). Clarendon Press Oxford. 1991. ISBN 0-19-853952-5.