# 四維頻率

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

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

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

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

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

## 勞侖茲變換

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

$\mathbf{E}=E_0 e^{ - i(k^{\mu}x_{\mu})}\hat{\boldsymbol{\eta}}_1$
$\mathbf{B}=B_0 e^{ - i(k^{\mu}x_{\mu})}\hat{\boldsymbol{\eta}}_2$

$\overline{\mathbf{E}}=\overline{E}_0 e^{ - i(\overline{k}^{\mu}\overline{x}_{\mu})} \hat{\boldsymbol{\eta}}_1$
$\overline{\mathbf{B}}=\overline{B}_0 e^{ - i(\overline{k}^{\mu}\overline{x}_{\mu})} \hat{\boldsymbol{\eta}}_2$

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

$\overline{k}=\overline{k}^0=\gamma(k - \beta_x k_x - \beta_y k_y - \beta_z k_z)=k^{\mu}v_{\mu}/c$

$f^{\mu}=c k^{\mu}/2\pi$

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

## 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.