# 洛伦兹变换

## 洛伦兹变换的提出

19世纪后期建立了麦克斯韦方程组，标志着经典电动力学取得了巨大成功。然而麦克斯韦方程组在经典力学伽利略变换下并不是协变的。

## 洛伦兹变换的数学形式

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

$\begin{cases} x = \frac{x'+vt'}{\sqrt{1-\frac{v^2}{c^2}}} \\ y = y' \\ z = z' \\ t = \frac{t'+\frac{v}{c^2}x'}{\sqrt{1-\frac{v^2}{c^2}}} \end{cases}$

$\begin{cases} x' = x-vt \\ y' = y \\ z' = z \\ t' = t \end{cases}$

## 洛伦兹变换的四维形式

$\begin{cases} x^{0} = ct \\ x^{\prime}{}^{0} = ct^{\prime} \end{cases}$

$\begin{bmatrix}x^{\prime}{}^{0}\\x^{\prime}{}^{1}\\x^{\prime}{}^{2}\\x^{\prime}{}^{3}\end{bmatrix} = \begin{bmatrix} \gamma&-\beta\gamma&0&0\\ -\beta\gamma&\gamma&0&0 \\ 0&0&1&0\\ 0&0&0&1\\ \end{bmatrix}\begin{bmatrix}x^{0}\\x^{1}\\x^{2}\\x^{3}\end{bmatrix}$

$\beta = \frac{v}{c}, \quad \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$，称为洛伦兹因子

## 勞侖茲變換的推導

### 从群論出發的推導

1. 閉合：两個參照系轉换叠加得另外一轉换。以$[K \to K^\prime]$$K$$K^{\prime}$。那對任意三個参照系$[K \to K^{\prime\prime}] = [K \to K^\prime] [K^{\prime} \to K^{\prime\prime}]$
2. 組合律$[K \to K^{\prime}] \left([K^{\prime} \to K^{\prime\prime}] [K^{\prime\prime} \to K^{\prime\prime\prime}]\right) = \left([K \to K^{\prime}] [K^{\prime} \to K^{\prime\prime}]\right) [K^{\prime\prime} \to K^{\prime\prime\prime}]$
3. 單位元：存在保留參照系的單位轉换$[K \to K]$
4. 逆元：對任何參照系轉换$[K \to K^\prime]$都有返回原本參照系的轉换$[K^\prime \to K]$

### 符合群公理的轉换矩陣

$\begin{pmatrix}t^\prime \\ z^\prime\end{pmatrix} = \begin{pmatrix} \Lambda_{11} & \Lambda_{12} \\ \Lambda_{21} & \Lambda_{22} \end{pmatrix} \begin{pmatrix} t \\ z \end{pmatrix}$

$\begin{pmatrix}t^\prime \\ 0\end{pmatrix} = \begin{pmatrix} \Lambda_{11} & \Lambda_{12} \\ \Lambda_{21} & \Lambda_{22} \end{pmatrix} \begin{pmatrix} t \\ v t \end{pmatrix}$

$\Lambda_{21}+v \, \Lambda_{22}=0$

$\begin{pmatrix}t^\prime \\ -v t^\prime \end{pmatrix} = \begin{pmatrix} \Lambda_{11} & \Lambda_{12} \\ \Lambda_{21} & \Lambda_{22} \end{pmatrix} \begin{pmatrix} t \\ 0 \end{pmatrix}$

$\Lambda_{21} + v \, \Lambda_{11}=0$

$\begin{pmatrix}t^\prime \\ z^\prime\end{pmatrix} = \begin{pmatrix} \gamma & \Lambda_{12} \\ -v \gamma & \gamma \end{pmatrix} \begin{pmatrix} t \\ z \end{pmatrix}$

$\begin{pmatrix}t \\ z\end{pmatrix} = \frac{1}{\gamma^2 + \Lambda_{12} v \gamma} \begin{pmatrix} \gamma & -\Lambda_{12} \\ v \gamma & \gamma \end{pmatrix} \begin{pmatrix} t^\prime \\ z^\prime \end{pmatrix}$

$\frac{1}{\gamma^2 + \Lambda_{12} v \gamma} \begin{pmatrix} \gamma & -\Lambda_{12} \\ v \gamma & \gamma \end{pmatrix} =\begin{pmatrix} \gamma & -\Lambda_{12} \\ v \gamma & \gamma \end{pmatrix}$

$\gamma^2 + \Lambda_{12} v \gamma = 1$

$\begin{pmatrix} \gamma^\prime & \Lambda_{12}^\prime \\ -v^\prime \gamma^\prime & \gamma^\prime \end{pmatrix} \begin{pmatrix} \gamma & \Lambda_{12} \\ -v \gamma & \gamma \end{pmatrix} = \begin{pmatrix} \gamma^\prime\gamma - \Lambda_{12}^\prime v\gamma & \gamma^\prime\Lambda_{12} + \gamma\Lambda_{12}^\prime \\ -\gamma^\prime\gamma(v+v^\prime) & \gamma^\prime \gamma-v^\prime \gamma^\prime \Lambda_{12}\end{pmatrix}$

$\kappa \equiv \frac{\Lambda_{12}}{v \gamma} = \frac{\Lambda_{12}^\prime}{v^\prime \gamma^\prime}$

$\gamma = \frac{1}{\sqrt{1 + \kappa v^2}}$

$\frac{1}{\sqrt{1+\kappa v^2}} \begin{pmatrix} 1 & \kappa v \\ -v & 1 \end{pmatrix}$

### 伽利略轉换

$\kappa=0$得伽利略轉換矩陣：

$\begin{pmatrix}t^\prime \\ z^\prime \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ -v & 1\end{pmatrix} \begin{pmatrix} t \\ z \end{pmatrix}$

### 勞侖茲變換

$\begin{pmatrix}t^\prime \\ z^\prime \end{pmatrix} = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \begin{pmatrix} 1 & -\frac{v}{c^2} \\ -v & 1\end{pmatrix} \begin{pmatrix} t \\ z \end{pmatrix}$

$c$是在所有參照系內不變的速度上限。

## 洛伦兹变换的推论

$u'_x = \frac{u_x-v}{1 - \frac{v u_x}{c^2}}$
$u'_y = \frac{u_y\sqrt{1-\frac{v^2}{c^2}}}{1-\frac{vu_{x}}{c^2}}$
$u'_z = \frac{u_z\sqrt{1-\frac{v^2}{c^2}}}{1-\frac{vu_{x}}{c^2}}$

$u'_{x} = u_{x} - v$
$u'_{y} = u_{y}$
$u'_{z} = u_{z}$

## 洛伦兹变换的几何理解

$\begin{bmatrix}x^{\prime}\\y^{\prime}\end{bmatrix} =\begin{bmatrix}\cos\theta & \sin\theta \\ -\sin\theta & \cos\theta\end{bmatrix} \begin{bmatrix}x\\y\end{bmatrix}$

$\begin{bmatrix}x^{\prime\prime}\\y^{\prime\prime}\end{bmatrix} =\begin{bmatrix}\cos(\theta+\phi) & \sin(\theta+\phi) \\ -\sin(\theta+\phi) & \cos(\theta+\phi)\end{bmatrix} \begin{bmatrix}x\\y\end{bmatrix}$

$\begin{bmatrix}x^\prime{}^0\\ x^\prime{}^1\end{bmatrix} =\begin{bmatrix}\cosh w & -\sinh w\\ -\sinh w & \cosh w\end{bmatrix} \begin{bmatrix}x^0\\ x^1\end{bmatrix}$

$(x^\prime{}^0)^2-(x^\prime{}^1)^2=(x^0)^2-(x^1)^2$

$\begin{bmatrix}x^{\prime\prime}{}^0\\ x^{\prime\prime}{}^1\end{bmatrix} =\begin{bmatrix}\cosh (w_{21} + w_{32}) & -\sinh (w_{21} + w_{32})\\ -\sinh (w_{21} + w_{32}) & \cosh (w_{21} + w_{32})\end{bmatrix} \begin{bmatrix}x^0\\ x^1\end{bmatrix}$

\begin{align} w_{31} &= w_{21} + w_{32} \\ \tanh w_{31} &= \tanh (w_{21} + w_{32}) = \frac{\tanh w_{21} + \tanh w_{32}}{1+\tanh w_{21} \tanh w_{32}} \\ \beta_{31} &= \frac{\beta_{21} + \beta_{32}}{1+\beta_{21} \beta_{32}} \end{align}