# 四維頻率

${\displaystyle f^{\mu }\ {\stackrel {def}{=}}\ \left(f,\,f\mathbf {n} \right)}$

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

${\displaystyle \omega ^{\mu }\ {\stackrel {def}{=}}\ \left(\omega ,\,\omega \mathbf {n} \right)}$

${\displaystyle \omega ^{\mu }=2\pi f^{\mu }}$

${\displaystyle {k}^{\mu }=\left(k,\,\mathbf {k} \right)}$

## 勞侖茲變換

${\displaystyle \Lambda _{\nu }^{\mu }={\begin{bmatrix}\gamma &-\beta _{x}\,\gamma &-\beta _{y}\,\gamma &-\beta _{z}\,\gamma \\-\beta _{x}\,\gamma &1+(\gamma -1){\frac {\beta _{x}^{2}}{\beta ^{2}}}&(\gamma -1){\frac {\beta _{x}\beta _{y}}{\beta ^{2}}}&(\gamma -1){\frac {\beta _{x}\beta _{z}}{\beta ^{2}}}\\-\beta _{y}\,\gamma &(\gamma -1){\frac {\beta _{y}\beta _{x}}{\beta ^{2}}}&1+(\gamma -1){\frac {\beta _{y}^{2}}{\beta ^{2}}}&(\gamma -1){\frac {\beta _{y}\beta _{z}}{\beta ^{2}}}\\-\beta _{z}\,\gamma &(\gamma -1){\frac {\beta _{z}\beta _{x}}{\beta ^{2}}}&(\gamma -1){\frac {\beta _{z}\beta _{y}}{\beta ^{2}}}&1+(\gamma -1){\frac {\beta _{z}^{2}}{\beta ^{2}}}\\\end{bmatrix}}}$

${\displaystyle \mathbf {E} =E_{0}e^{-i(k^{\mu }x_{\mu })}{\hat {\boldsymbol {\eta }}}_{1}}$
${\displaystyle \mathbf {B} =B_{0}e^{-i(k^{\mu }x_{\mu })}{\hat {\boldsymbol {\eta }}}_{2}}$

${\displaystyle {\overline {\mathbf {E} }}={\overline {E}}_{0}e^{-i({\overline {k}}^{\mu }{\overline {x}}_{\mu })}{\hat {\boldsymbol {\eta }}}_{1}}$
${\displaystyle {\overline {\mathbf {B} }}={\overline {B}}_{0}e^{-i({\overline {k}}^{\mu }{\overline {x}}_{\mu })}{\hat {\boldsymbol {\eta }}}_{2}}$

${\displaystyle {\overline {k}}^{\mu }=\Lambda _{\nu }^{\mu }{k}^{\nu }}$

${\displaystyle {\overline {k}}={\overline {k}}^{0}=\gamma (k-\beta _{x}k_{x}-\beta _{y}k_{y}-\beta _{z}k_{z})=k^{\mu }v_{\mu }/c}$

${\displaystyle f^{\mu }=ck^{\mu }/2\pi }$

${\displaystyle {\overline {f}}={\overline {f}}^{0}=f^{\mu }v_{\mu }/c}$

## 參考文獻

1. ^ Jackson, John David, Classical Electrodynamic 3rd., USA: John Wiley & Sons, Inc.: pp. 543–548, 1999, ISBN 978-0-471-30932-1
• Woodhouse, N.M.J. Special Relativity. London: Springer-Verlag. 2003: 84–90. ISBN 1852334266.
• Griffiths, David J. Introduction to Electrodynamics (3rd ed.). Prentice Hall. 1998: pp. 477–543. ISBN 0-13-805326-X.