Special Relativity

Lorentz 变换

正变换

$$\left(\begin{array}{c}ct\\ x\\ y\\ x\end{array}\right)=\left(\begin{array}{cccc}\gamma & \gamma \beta & 0& 0\\ \gamma \beta & \gamma & 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)\left(\begin{array}{c}c{t}^{\prime }\\ x^{\prime }\\ y^{\prime }\\ z^{\prime }\end{array}\right)$$

负变换

$$\left(\begin{array}{c}c{t}^{\prime }\\ x^{\prime }\\ y^{\prime }\\ x^{\prime }\end{array}\right)=\left(\begin{array}{cccc}\gamma & -\gamma \beta & 0& 0\\ -\gamma \beta & \gamma & 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)\left(\begin{array}{c}ct\\ x\\ y\\ z\end{array}\right)$$

几何表示

速度变换

$$\begin{array}{rl}{u}_x& =\frac{u_x^{\prime }+v}{1+\frac{u_x^{\prime }v}{c^2}}\\ u_y& =\frac{\sqrt{1-{\beta }^2}{u}_y^{\prime }}{1+\frac{u_x^{\prime }v}{c^2}}\\ u_z& =\frac{\sqrt{1-{\beta }^2}{u}_z^{\prime }}{1+\frac{u_x^{\prime }v}{c^2}}\end{array}$$

相对论Doppler 效应

$$\nu =\frac{\sqrt{1-{\beta }^2}}{1-\beta \cos \varphi }{\nu }_0$$

其中$\varphi$是光源-观察者连线和光源速度之间的夹角，$\beta =\frac{v_{\text{source}}}{c}$（在观察者的参考系中）

4-动量

$$\mathbf{P}=\left(\begin{array}{c}\frac{E}{c}\\ p_x\\ p_y\\ p_z\end{array}\right)$$

是Lorentz协变量

能量-动量关系

$$E^2=(m_0{c}^2{)}^2+(pc{)}^2$$