洛伦兹变换

（重定向自勞侖茲變換

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

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

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

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

洛伦兹变换的四维形式

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

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

${\displaystyle \beta ={\frac {v}{c}},\quad \gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}$，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${\displaystyle [K\to K^{\prime }]}$${\displaystyle K}$${\displaystyle K^{\prime }}$。那對任意三個参照系${\displaystyle [K\to K^{\prime \prime }]=[K\to K^{\prime }][K^{\prime }\to K^{\prime \prime }]}$
2. 組合律${\displaystyle [K\to K^{\prime }]\left([K^{\prime }\to K^{\prime \prime }][K^{\prime \prime }\to K^{\prime \prime \prime }]\right)=\left([K\to K^{\prime }][K^{\prime }\to K^{\prime \prime }]\right)[K^{\prime \prime }\to K^{\prime \prime \prime }]}$
3. 單位元：存在保留參照系的單位轉换${\displaystyle [K\to K]}$
4. 逆元：對任何參照系轉换${\displaystyle [K\to K^{\prime }]}$都有返回原本參照系的轉换${\displaystyle [K^{\prime }\to K]}$

符合群公理的轉换矩陣

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

${\displaystyle {\begin{pmatrix}t^{\prime }\\0\end{pmatrix}}={\begin{pmatrix}\Lambda _{11}&\Lambda _{12}\\\Lambda _{21}&\Lambda _{22}\end{pmatrix}}{\begin{pmatrix}t\\vt\end{pmatrix}}}$

${\displaystyle \Lambda _{21}+v\,\Lambda _{22}=0}$

${\displaystyle {\begin{pmatrix}t^{\prime }\\-vt^{\prime }\end{pmatrix}}={\begin{pmatrix}\Lambda _{11}&\Lambda _{12}\\\Lambda _{21}&\Lambda _{22}\end{pmatrix}}{\begin{pmatrix}t\\0\end{pmatrix}}}$

${\displaystyle \Lambda _{21}+v\,\Lambda _{11}=0}$

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

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

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

${\displaystyle \gamma ^{2}+\Lambda _{12}v\gamma =1}$

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

${\displaystyle \kappa \equiv {\frac {\Lambda _{12}}{v\gamma }}={\frac {\Lambda _{12}^{\prime }}{v^{\prime }\gamma ^{\prime }}}}$

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

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

伽利略轉换

${\displaystyle \kappa =0}$得伽利略轉換矩陣：

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

勞侖茲變換

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

${\displaystyle c}$是在所有參照系內不變的速度上限。

洛伦兹变换的推论

${\displaystyle u'_{x}={\frac {u_{x}-v}{1-{\frac {vu_{x}}{c^{2}}}}}}$
${\displaystyle u'_{y}={\frac {u_{y}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}{1-{\frac {vu_{x}}{c^{2}}}}}}$
${\displaystyle u'_{z}={\frac {u_{z}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}{1-{\frac {vu_{x}}{c^{2}}}}}}$

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

洛伦兹变换的几何理解

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

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

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

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

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

{\displaystyle {\begin{aligned}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{aligned}}}

參考資料

1. ^ 朗道, 列夫; 栗弗席兹. 經典場論. 理論物理教程 第二卷.