相对论天体物理

从时间、空间到时空

用时空图画出洛伦兹变换

• 洛伦兹变换

  $$\begin{array}{rl}{t}^{\prime }& =\gamma (t-\frac{vx}{c^2}),\\ x^{\prime }& =\gamma (x-vt),\\ \gamma & \equiv \frac{1}{\sqrt{1-v^2/c^2}}.\end{array}$$

• 一般洛伦兹变化矩阵

  $$\left[\begin{array}{c}{t}^{\prime }\\ x^{\prime }\\ y^{\prime }\\ z^{\prime }\end{array}\right]=\left[\begin{array}{cccc}\gamma & -\gamma v^x& -\gamma v^y& -\gamma v^z\\ -\gamma v^x& 1+{\displaystyle \frac{(\gamma -1)(v^x{)}^2}{v^2}}& {\displaystyle \frac{(\gamma -1)v^x{v}^y}{v^2}}& {\displaystyle \frac{(\gamma -1)v^x{v}^z}{v^2}}\\ -\gamma v^y& {\displaystyle \frac{(\gamma -1)v^x{v}^y}{v^2}}& 1+{\displaystyle \frac{(\gamma -1)(v^y{)}^2}{v^2}}& {\displaystyle \frac{(\gamma -1)v^y{v}^z}{v^2}}\\ -\gamma v^z& {\displaystyle \frac{(\gamma -1)v^x{v}^z}{v^2}}& {\displaystyle \frac{(\gamma -1)v^y{v}^z}{v^2}}& 1+{\displaystyle \frac{(\gamma -1)(v^z{)}^2}{v^2}}\end{array}\right]\left[\begin{array}{c}t\\ x\\ y\\ z\end{array}\right]$$

  即时间分量

  $$t^{\prime }=\gamma (t-\overrightarrow{v}\cdot \overrightarrow{r})$$

  空间分量

  $${\overrightarrow{r}}^{\prime }=-\gamma \overrightarrow{v}t+\overrightarrow{r}+\frac{(\gamma -1)(\overrightarrow{v}\cdot \overrightarrow{r})}{v^2}\overrightarrow{v}$$

动钟变慢

双生子佯谬

闵氏几何

时空几何（闵氏几何）线元

$$d{s}^2=-d{t}^2+d{x}^2+d{y}^2+d{z}^2$$

从矢量到张量

相对论速度变换

$$\begin{array}{rl}{u}^{\prime x}& =\frac{u^x-v}{1-v{u}^x},\\ u^{\prime y}& =\frac{u^y}{\gamma (1-v{u}^x)},\\ u^{\prime z}& =\frac{u^z}{\gamma (1-v{u}^x)}.\end{array}$$

切失与偏导

坐标基底的变换关系

$${\left(\frac{\partial }{\partial {\tilde{x}}^{\mu }}\right)}^a={\left(\frac{\partial }{\partial x^{\nu }}\right)}^a\frac{\partial x^{\nu }}{\partial {\tilde{x}}^{\mu }}$$

坐标微分与对偶坐标基底

$$(d{\tilde{x}}^{\mu }{)}_a=\frac{\partial {\tilde{x}}^{\mu }}{\partial x^{\nu }}(d{x}^{\nu }{)}_a$$

从并矢到张量

并矢：把矢量放一块 $u^a{v}^b$

张量：

$${{T_a}^{bc}}_d={{T_{\mu }}^{\nu \sigma }}_{\tau }(d{x}^{\mu }{)}_a{\left(\frac{\partial }{\partial x^{\nu }}\right)}^b{\left(\frac{\partial }{\partial x^{\sigma }}\right)}^c(d{x}^{\tau }{)}_d$$

张量缩并：

$${\left(\frac{\partial }{\partial x^{\mu }}\right)}^a(d{x}^{\nu }{)}_a={{\delta }^{\nu }}_{\mu }$$

度规张量

度规系数 $g_{\mu \nu }$

不同坐标基底下的度规系数

$$g_{kl}={\tilde{g}}_{ij}\frac{d{\tilde{x}}^i}{d{x}^k}\frac{d{\tilde{x}}^j}{d{x}^l}$$

度规张量 $g_{ab}\equiv g_{\mu \nu }(d{x}^{\mu }{)}_a(d{x}^{\nu }{)}_b$

逆度规张量 ${\displaystyle g^{ab}\equiv g^{\mu \nu }{\left(\frac{\partial }{\partial x^{\mu }}\right)}^a{\left(\frac{\partial }{\partial x^{\nu }}\right)}^b}$ 其中 $g^{\mu \nu }\equiv (g_{\mu \nu }{)}^{-1}$

指标升降

$$T_a\equiv g_{ab}{T}^b$$$$W^a\equiv g^{ab}{W}_b$$

• 闵氏几何度规张量

$$g_{\mu \nu }=\left[\begin{array}{cccc}-1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\equiv {\eta }_{\mu \nu }$$

电磁场张量

电磁场张量

$$F_{\mu \nu }=\left[\begin{array}{cccc}0& -E_x& -E_y& -E_z\\ E_x& 0& B_z& -B_y\\ E_y& -B_z& 0& B_x\\ E_z& B_y& -B_x& 0\end{array}\right]$$$$F_{ab}\equiv F_{\mu \nu }(d{x}^{\mu }{)}_a(d{x}^{\nu }{)}_b$$

全对称/全反称张量

全对称操作

$$T^{(i_1{i}_2...i_m)}=\frac{1}{m!}\sum _{\mathrm{\Pi }}{T}^{\mathrm{\Pi }(i_1{i}_2...i_m)}$$

全反称操作

$$T^{[i_1{i}_2...i_m]}=\frac{1}{m!}\sum _{\mathrm{\Pi }}G(\mathrm{\Pi })T^{\mathrm{\Pi }[i_1{i}_2...i_m]}$$

体元张量

$${\epsilon }_{ab...c}=n!\sqrt{|\text{det}{g}_{\mu \nu }|}(d{x}^1{)}_{[a}(d{x}^2{)}_b\cdots (d{x}^n{)}_{c]}$$

四维力

$$F^{\mu }=\left[\begin{array}{c}{F}^0\\ F^1\\ F^2\\ F^3\end{array}\right]\equiv \left[\begin{array}{c}\gamma \overrightarrow{f}\cdot \overrightarrow{u}\\ \gamma f_1\\ \gamma f_2\\ \gamma f_3\end{array}\right]$$

四维速度

$$U^{\mu }=\left[\begin{array}{c}{U}^0\\ U^1\\ U^2\\ U^3\end{array}\right]\equiv \left[\begin{array}{c}\gamma \\ \gamma u_1\\ \gamma u_2\\ \gamma u_3\end{array}\right]$$$$\gamma =\frac{1}{\sqrt{1-(u^x{)}^2-(u^y{)}^2-(u^z{)}^2}}$$

• 正交性：

$$g_{ab}{F}^a{U}^b=0$$

四维动量

$$P^{\mu }=\left[\begin{array}{c}{P}^0\\ P^1\\ P^2\\ P^3\end{array}\right]\equiv \left[\begin{array}{c}\gamma m\\ \gamma m{u}^1\\ \gamma m{u}^2\\ \gamma m{u}^3\end{array}\right]$$

这里 $m$ 代表对象的静质量。

四维加速度与动力学方程

• 四维加速度矢量

$$A^a\equiv U^b{\partial }_b{U}^a$$

• 动力学方程

$$F^a=m{U}^b{\partial }_b{U}^a$$$$F^a={\left(\frac{\mathrm{d}}{\mathrm{d}\tau }\right)}^b{\mathrm{\nabla }}_b{P}^a$$

矢量的类时、类空、类光性

对于 $g_{ab}{A}^a{A}^b$ ：

1. 正：类空 ${\left(\frac{\partial }{\partial t}\right)}^a$
2. 负：类时 ${\left(\frac{\partial }{\partial x}\right)}^a$
3. $0$ ：类光 ${\left(\frac{\partial }{\partial t}\right)}^a+{\left(\frac{\partial }{\partial x}\right)}^a$

• 相对论火箭方程

与通常的齐奥尔科夫斯基火箭方程一致

$$\mathrm{\Delta }{v}_{\text{roket}}=v\ln \frac{m_0}{m_1}$$

曲线参数化及其物理意义

线长

$$L={\int }_{s_1}^{s_2}\sqrt{\left|g_{ab}{\left(\frac{\mathrm{d}}{\mathrm{d}s}\right)}^a{\left(\frac{\mathrm{d}}{\mathrm{d}s}\right)}^b\right|}\mathrm{d}s$$

类时曲线参数化

类时曲线对应某观者世界线，四速 $U^a$, 固有时 $\tau$

$${\left(\frac{\mathrm{d}}{\mathrm{d}\tau }\right)}^a=U^a$$$$g_{ab}{U}^a{U}^b=-1$$

类空曲线参数化

$$g_{ab}{\left(\frac{\mathrm{d}}{\mathrm{d}s}\right)}^a{\left(\frac{\mathrm{d}}{\mathrm{d}s}\right)}^b=1$$

类光曲线参数化

参数化 （光源四速 $U^a$ ，发光频率 $f$ ）

$$g_{ab}{\left(\frac{\mathrm{d}}{\mathrm{d}\lambda }\right)}^a{U}^b=-f$$

光子频率 $f$ ，光的空间波矢 $\overrightarrow{k}$

$${\left(\frac{\mathrm{d}}{\mathrm{d}\lambda }\right)}^a=f{\left(\frac{\partial }{\partial t}\right)}^a+{\overrightarrow{k}}^a$$

光子能量为 $hf$ ，动量为 $h\overrightarrow{k}$ ，故光子的四动量

$$P^a=h{\left(\frac{\mathrm{d}}{\mathrm{d}\lambda }\right)}^a$$

广义四加速及其与四速正交性

广义四加速

$$A^a\equiv U^b{\partial }_b{U}^a$$

这里 $U^a$ 对于类时、类空、类光曲线分别为 ${\left(\frac{\mathrm{d}}{\mathrm{d}\tau }\right)}^a,{\left(\frac{\mathrm{d}}{\mathrm{d}s}\right)}^a,{\left(\frac{\mathrm{d}}{\mathrm{d}\lambda }\right)}^a$

正交性

$$g_{ac}{U}^c{A}^a=0$$