物理

【广义相对论速成版】1. Riemann几何 1.1 张量

1. 坐标变换

$n$维流形是二维曲面概念的推广，它的重要特点是局部与Euclid空间同胚，并以此局部Euclid空间的坐标来定义流形上的坐标。

设$n$维流形上$p$点的坐标为$x=(x^{1},x^{2},\ldots,x^{n})$。在另一坐标系中$p$点的坐标为$x^{\prime}=(x^{\prime 1},x^{\prime 2},\ldots,x^{\prime n})$。它们之间存在关系

$$x^{\prime \mu}=\overline{x}^{\mu}(x^{1},x^{2},\cdots,x^{n})~,\qquad \mu=1,2,\ldots,n$$

称为坐标变换(coordinate transformation)，它是一个多次可微函数。定义变换矩阵：

$$A=\left[A_{\nu}^{\mu}\right]~,\qquad A_{\nu}^{\mu}=\frac{\partial \overline{x}^{\mu}}{\partial x^{\nu}}~.$$

则$\det A\equiv J\left(\frac{\overline{x}}{x}\right)$称为Jacobi行列式(Jacobian)。

由于

$$\mathrm{d}\overline{x}^{\mu}=\frac{\partial \overline{x}^{\mu}}{\partial x^{\nu}}\mathrm{d}x^{\nu}$$

即

$$\mathrm{d}\overline{x}^{\mu}=A_{\nu}^{\mu}\mathrm{d}x^{\nu}$$

由线性方程理论可知，如果Jacobi行列式不为零$\det A\neq 0$，即$J\left(\frac{\overline{x}}{x}\right)\neq 0$，则$\mathrm{d}x^{\mu}$可用$\mathrm{d}\overline{x}^{\mu}$线性表示，即这时存在逆变换

$$x^{\mu}=x^{\mu}(x^{\prime 1},x^{\prime 2},\ldots,x^{\prime n})~,\qquad x^{\prime \mu}=\overline{x}^{\mu}$$

同理可定义

$$\overline{A}_{\nu}^{\mu}=\frac{\partial x^{\mu}}{\partial \overline{x}^{\nu}}$$

因为$\frac{\partial x^{\mu}}{\partial x^{\nu}}=\delta_{\nu}^{\mu}$，且$\frac{\partial x^{\mu}}{\partial x^{\nu}}=\frac{\partial x^{\mu}}{\partial \overline{x}^{\lambda}}\frac{\partial \overline{x}^{\lambda}}{\partial x^{\nu}}=\overline{A}_{\lambda}^{\mu}A_{\nu}^{\lambda}$，所以$\overline{A}_{\lambda}^{\mu}A_{\nu}^{\lambda}=\delta_{\nu}^{\mu}$，由此可知矩阵$\overline{A}=\left[\overline{A}_{\nu}^{\mu}\right]$为$A$的逆矩阵，即

$$\overline{A}=A^{-1}~,\qquad \left(A^{-1}A=AA^{-1}=I\right)$$

2. 标量$\phi(x)$

如果$x=(x^{1},x^{2},\ldots,x^{n})$的函数$\phi(x)$满足如下的变换规律

$$\phi^{\prime}(x^{\prime})=\phi(x)$$

即：$\phi(x)$在坐标变换下不变，则$\phi(x)$称为流形上的标量(scalar)。

3. 协变矢量$\phi_{\mu}(x)$

如果一个单一下指标的量$\phi_{\mu}(x)$满足下列变换规律

$$\phi^{\prime}_{\mu}(x^{\prime})=\overline{A}_{\mu}^{\nu}\phi_{\nu}(x)$$

则$\phi_{\mu}(x)$称为协变矢量(convariant vector)。

可证明标量$\phi(x)$的梯度是协变矢量：令$\phi_{\mu}(x)=\frac{\partial \phi(x)}{\partial x^{\mu}}$，$\phi(x)$为标量，则

$$\phi_{\mu}^{\prime}(x^{\prime})=\frac{\partial\phi^{\prime}(x^{\prime})}{\partial \overline{x}^{\mu}}=\frac{\partial \phi(x)}{\partial \overline{x}^{\mu}}=\frac{\partial \phi(x)}{\partial x^{\nu}}\frac{\partial x^{\nu}}{\partial \overline{x}^{\mu}}=\phi_{\nu}(x)\cdot \overline{A}_{\mu}^{\nu}$$

即：$\phi^{\prime}_{\mu}(x^{\prime})=\overline{A}_{\mu}^{\nu}\phi_{\nu}(x)$，故$\phi^{\prime}_{\mu}(x^{\prime})=\partial_{\mu}\phi(x)$为协变矢量。

4. 逆变矢量$\phi^{\mu}(x)$

如果一个单一上指标的量$\phi^{\mu}(x)$满足如下的变换规律

$$\phi^{\prime \mu}(x^{\prime})=A_{\nu}^{\mu}\phi^{\nu}(x)$$

则$\phi^{\mu}(x)$称为逆变矢量(contravariant vector)。

逆变矢量简单示例如下：因为

$$x^{\prime\mu}=\overline{x}^{\mu}(x^{1},x^{2},\ldots,x^{n})~,\qquad\mu=1,2,\ldots,n$$

则

$$\mathrm{d}\overline{x}^{\mu}=\frac{\partial \overline{x}^{\mu}}{\partial x^{\nu}}\mathrm{d}x^{\nu}=A_{\nu}^{\mu}\mathrm{d}x^{\nu}$$

设$C$为流形上的曲线，其方程为

$$x^{\mu}=x^{\mu}(s)$$

其中，$s$为曲线$C$上从某点起的弧长。在另一个坐标系中$C$的方程为

$$\overline{x}^{\mu}=\overline{x}^{\mu}(s^{\prime})$$

若规定

$$\mathrm{d}s^{\prime}=\mathrm{d}s$$

则

$$\frac{\mathrm{d}\overline{x}^{\mu}}{\mathrm{d}s^{\prime}}=A_{\nu}^{\mu}\cdot\frac{\mathrm{d}x^{\nu}}{\mathrm{d}s}$$

定义$u^{\mu}(x)=\frac{\mathrm{d}x^{\mu}}{\mathrm{d}s}$为$C$上某点的切矢量，则

$$u^{\prime\mu}(x^{\prime})=A_{\nu}^{\mu}u^{\nu}(x)$$

故$u^{\mu}(x)$为逆变矢量。

注：

$$\left.\begin{aligned}\phi_{\mu}^{\prime}(x^{\prime})&=\overline{A}_{\mu}^{\nu}\phi_{\nu}(x)\qquad &\text{协有横}\\\phi^{\prime\mu}(x^{\prime})&=A_{\nu}^{\mu}\phi^{\nu}(x)\qquad &\text{逆无横}\end{aligned}\right\}\text{上逆下协}$$

5. 二阶张量

流形上二指标的量如果满足一定的变换规律，称为二阶张量(second-order tensor)。二阶张量有三种

$$\begin{aligned}\phi_{\mu\nu}(x)&\text{——二阶协变张量}\\\phi^{\mu\nu}(x)&\text{——二阶逆变张量}\\\phi_{\nu}^{\mu}(x)&\text{——二阶混合张量}\end{aligned}$$

它们的变换规律分别为

$$\begin{aligned}\phi_{\mu\nu}^{\prime}(x^{\prime})&=\overline{A}_{\mu}^{\alpha}\overline{A}_{\nu}^{\beta}\phi_{\alpha\beta}(x)\\\phi^{\prime\mu\nu}(x^{\prime})&=\overline{A}_{\alpha}^{\mu}\overline{A}_{\beta}^{\nu}\phi^{\alpha\beta}(x)\\\phi_{~\nu}^{\prime\mu}(x^{\prime})&=\overline{A}_{\alpha}^{\mu}\overline{A}_{\nu}^{\beta}\phi_{\beta}^{\alpha}(x)\end{aligned}$$

很容易推广到高阶张量$\phi_{\nu_{1}\cdots\nu_{l}}^{\mu_{1}\cdots\mu_{k}}$

$$\phi_{~\nu_{1}\nu_{2}\cdots\nu_{l}}^{\prime\mu_{1}\mu_{2}\cdots\mu_{k}}(x^{\prime})=A_{\alpha_{1}}^{\mu_{1}}A_{\alpha_{2}}^{\mu_{2}}\cdots A_{\alpha_{k}}^{\mu_{k}}\overline{A}_{\nu_{1}}^{\beta_{1}}\overline{A}_{\nu_{2}}^{\beta_{2}}\cdots\overline{A}_{\nu_{l}}^{\beta_{l}}\phi_{~\beta_{1}\beta_{2}\cdots\beta_{l}}^{\alpha_{1}\alpha_{2}\cdots\alpha_{k}}(x)$$

练习：证明：$\phi_{\mu\nu}(x)\phi^{\mu\nu}(x)$和$\phi_{\nu}^{\mu}(x)\phi_{\mu}^{\nu}(x)$是标量。

证明：

$$\begin{aligned}\phi_{\mu\nu}^{\prime}(x^{\prime})\phi^{\prime\mu\nu}(x^{\prime})&=\overline{A}_{\mu}^{\lambda}\overline{A}_{\nu}^{\sigma}\phi_{\lambda\sigma}(x)\cdot A_{\alpha}^{\mu}A_{\beta}^{\nu}\phi^{\alpha\beta}(x)\\&=\left(\overline{A}_{\mu}^{\lambda}A_{\alpha}^{\mu}\right)\left(\overline{A}_{\nu}^{\sigma}A_{\beta}^{\nu}\right)\phi_{\lambda\sigma}(x)\phi^{\alpha\beta}(x)\\&=\delta_{\alpha}^{\lambda}\delta_{\beta}^{\sigma}\phi_{\lambda\sigma}(x)\phi^{\alpha\beta}(x)\\&=\phi_{\alpha\beta}(x)\phi^{\alpha\beta}(x)\end{aligned}$$

故$\phi_{\mu\nu}(x)\phi^{\mu\nu}(x)$是标量。

$$\begin{aligned}\phi_{~\nu}^{\prime\mu}(x^{\prime})\phi_{~\mu}^{\prime\nu}(x^{\prime})&=A_{\lambda}^{\mu}\overline{A}_{\nu}^{\sigma}\phi_{\sigma}^{\lambda}(x)\cdot A_{\alpha}^{\nu}\overline{A}_{\mu}^{\beta}\phi_{\beta}^{\alpha}(x)\\&=\left(A_{\lambda}^{\mu}\overline{A}_{\mu}^{\beta}\right)\left(\overline{A}_{\nu}^{\sigma}A_{\alpha}^{\nu}\right)\phi_{\sigma}^{\lambda}(x)\phi_{\beta}^{\alpha}(x)\\&=\delta_{\lambda}^{\beta}\delta_{\alpha}^{\sigma}\phi_{\sigma}^{\lambda}(x)\phi_{\beta}^{\alpha}(x)\\&=\phi_{\alpha}^{\beta}(x)\phi_{\beta}^{\alpha}(x)\end{aligned}$$

故$\phi_{\nu}^{\mu}(x)\phi_{\mu}^{\nu}(x)$是标量。