# 四維矢量

## 數學性質

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

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

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

${U}_{\mu}=\ ({U}_0,\, {U}_1,\, {U}_2,\, {U}_3)=\ ({U}^0,\, - {U}^1,\, - {U}^2,\, - {U}^3)$

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

$U_{\mu} =\eta_{\mu \nu} U^{\nu}$

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

### 勞侖茲變換

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

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

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

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

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

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

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

$\Delta t=\gamma\Delta \tau$

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

### 閔考斯基內積

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

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

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

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

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

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

## 動力學實例

### 四維速度

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

$U^{\mu}$的空間部分與經典速度矢量$\mathbf{u}$的關係為

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

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

### 四維加速度

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

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

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

$\alpha_\mu U^\mu = \frac{1}{2} \frac{\partial (U_\mu U^\mu)}{\partial \tau}=0$

### 四維動量

$P^\mu\ \stackrel{def}{=}\ m U^\mu=\left(\gamma m c,\, \gamma m\mathbf{u} \right)$

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

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

### 四維力

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

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

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

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

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

## 物理內涵

### 質能方程式

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

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

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

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

$K=\gamma m c^2+K_0$

$K=\gamma m c^2 - m c^2$

$E = \gamma m c^2$

$E=m_{rel} c^2$

### 能量-動量關係式

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

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

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

$E^2 = (pc)^2 + m^2 c^4$

## 電磁學實例

### 四維電流密度

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

$J^{\mu}=\rho_0 U^{\mu}$

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

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

### 電磁四維勢

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

$\Box A^\mu = \mu_0 J^\mu$ ;

### 四維頻率和四維波矢量

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

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

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

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

## 參考文獻

• 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.