Special theory of Relativity

Michaelson-Morley Experiment: 방향에 따른 빛의 속력의 차이가 관찰되지 않음 -> 매질 에테르가 존재한다는 가정에 모순

 

Galilean Transform (classical dynamics)

$$\begin{align}x&=x^\prime+ut \newline y&=y^\prime \newline z&=z^\prime \newline t&=t^\prime \newline v&=v^\prime+u\end{align}$$

 

Lorenz Transform (modern dynamics)

$$\begin{align}\gamma &= \frac{1}{\sqrt{1-u^2/c^2}}\newline x^\prime&=\gamma(x-ut)\newline t^\prime&=\lambda(t-\frac{ux}{c^2})\newline v_x^\prime&=\frac{dx-udt}{dt-udx/c^2}=\frac{v_x-u}{1-uv_x/c^2}\newline \vec{p}&=\frac{m\vec{u}}{\sqrt{1-u^2/c^2}}=\gamma m\vec{u}\newline \vec{F}&=\frac{d\vec{p}}{dt}=\frac{m(du/dt)}{(1-u^2/c^2)^{3/2}}\newline K&=\int\frac{dp}{dt}udt=\gamma mc^2-mc^2\newline E&=K+mc^2\newline \left(\frac{E}{mc^2}\right)^2&=\frac{1}{1-v^2/c^2}\left(\frac{P}{mc}\right)^2\newline E^2&=(mc^2)^2+(pc)^2\end{align}$$

 

Space-Time Coordinate : $(ict,x,y,z)$

 

Momentum

$$\begin{align}P_\mu &= (E/c,P_x,P_y,P_z)\newline\begin{bmatrix}P_0^\prime\newline P_1^\prime\newline P_2^\prime\newline P_3^\prime \end{bmatrix} &= \begin{bmatrix} \gamma & -i\gamma\beta & 0 & 0 \newline i\gamma\beta & \gamma 0 & 0 \newline 0 & 0 & 1 & 0 \newline 0 & 0 & 0 & 1\end{bmatrix}\begin{bmatrix}P_0\newline P_1\newline P_2\newline P_3 \end{bmatrix}\end{align}$$

 

Field Strength Tensor

$$F_{\mu\nu}=\begin{bmatrix}0 & B_3 & -B_2 & -iE_1 \newline-B_3 & 0 & B_1 & -iE_2 \newline B_2 & -B_1 & 0 & -iE_3 \newline iE_1 & iE_2 & iE_3 & 0\newline\end{bmatrix} $$

 

Einstein's Convention : repeated index variable with $\sum$ -> skip $\sum$ symbol

 

Let $A_{\mu\nu}$ be the matrix of Lorenz Transform.

Then $x_{\mu}=A_{\mu\nu}x_{\nu}$, $F_{\mu\nu}=A_{\mu\lambda}A_{\nu\delta}F_{\lambda\delta}$