【广义相对论速成版】1. Riemann几何 1.5 Riemann曲率张量

1. Riemann曲率张量的定义

当曲率张量

$$R_{~\sigma\mu\nu}^{\lambda}=\partial_{\mu}\Gamma_{\nu\sigma}^{\lambda}-\partial_{\nu}\Gamma_{\mu\sigma}^{\lambda}+\Gamma_{\mu\alpha}^{\lambda}\Gamma_{\nu\sigma}^{\alpha}-\Gamma_{\nu\alpha}^{\lambda}\Gamma_{\mu\sigma}^{\alpha}$$

中的联络$\Gamma_{\mu\nu}^{\lambda}$为Riemann联络(Christoffel符号)时$R_{~\sigma\mu\nu}^{\lambda}$称为\textbf{Riemann曲率张量(Riemannian Curvature Tensor)}。即$R_{~\sigma\mu\nu}^{\lambda}$中的

$$\Gamma_{\mu\nu}^{\lambda}=\frac{1}{2}g^{\lambda\sigma}\left(\partial_{\mu}g_{\sigma\nu}+\partial_{\nu}g_{\sigma\mu}-\partial_{\sigma}g_{\mu\nu}\right)$$

且满足

$$\Gamma_{\mu\nu}^{\lambda}=\Gamma_{\nu\mu}^{\lambda}$$

也即$R_{~\sigma\mu\nu}^{\lambda}$能完全用$g_{\mu\nu}$表征时，称其为Riemann曲率张量。

Riemann曲率张量$R_{~\sigma\mu\nu}^{\lambda}$显然对$\mu,\nu$有下面的反对称性

$$R_{~\sigma\mu\nu}^{\lambda}=-R_{~\sigma\nu\mu}^{\lambda}$$

还可定义指标全部协变的Riemann曲率张量

$$\boxed{R_{\tau\sigma\mu\nu}=g_{\tau\lambda}R_{~\sigma\mu\nu}^{\lambda}}\tag{1.5.1}$$

2. $R_{\tau\sigma\mu\nu}$的重要特征(对称性)

把(1.5.1)写成Christoffel符号构成的形式

$$R_{\tau\sigma\mu\nu}=g_{\tau\alpha}R_{~\sigma\mu\nu}^{\alpha}=g_{\tau\alpha}\left(\partial_{\mu}\Gamma_{\nu\sigma}^{\alpha}-\partial_{\nu}\Gamma_{\mu\sigma}^{\alpha}+\Gamma_{\mu\beta}^{\alpha}\Gamma_{\nu\sigma}^{\beta}-\Gamma_{\nu\beta}^{\alpha}\Gamma_{\mu\sigma}^{\beta}\right)$$

将Christoffel符号用度规代入，第一项为

$$\begin{aligned}g_{\tau\alpha}\partial_{\mu}\Gamma_{\nu\sigma}^{\alpha}&=g_{\tau\alpha}\partial_{\mu}\left(\frac{1}{2}g^{\alpha\beta}\left(\partial_{\nu}g_{\sigma\beta}+\partial_{\sigma}g_{\nu\beta}-\partial_{\beta}g_{\nu\sigma}\right)\right)\\&=\frac{1}{2}g_{\tau\alpha}g^{\alpha\beta}\partial_{\mu}\left(\partial_{\nu}g_{\sigma\beta}+\partial_{\sigma}g_{\nu\beta}-\partial_{\beta}g_{\nu\sigma}\right)+\frac{1}{2}g_{\tau\alpha}\left(\partial_{\mu}g^{\alpha\beta}\right)\left(\partial_{\nu}g_{\sigma\beta}+\partial_{\sigma}g_{\nu\beta}-\partial_{\beta}g_{\nu\sigma}\right)\\&=\frac{1}{2}\delta_{\tau}^{\beta}\left(\partial_{\mu}\partial_{\nu}g_{\sigma\beta}+\partial_{\mu}\partial_{\sigma}g_{\nu\beta}-\partial_{\mu}\partial_{\beta}g_{\nu\sigma}\right)\\&=\frac{1}{2}\left(\partial_{\mu}\partial_{\nu}g_{\sigma\tau}+\partial_{\mu}\partial_{\sigma}g_{\nu\tau}-\partial_{\mu}\partial_{\tau}g_{\nu\sigma}\right)\end{aligned}$$

我们对上式将指标交换$\mu\leftrightarrow\nu$，得到第二项用度规的表达

$$g_{\tau\alpha}\partial_{\nu}\Gamma_{\mu\sigma}^{\alpha}=\frac{1}{2}\left(\partial_{\nu}\partial_{\mu}g_{\sigma\tau}+\partial_{\nu}\partial_{\sigma}g_{\mu\tau}-\partial_{\nu}\partial_{\tau}g_{\mu\sigma}\right)$$

所以前两项的差为

$$g_{\tau\alpha}\left(\partial_{\mu}\Gamma_{\nu\sigma}^{\alpha}-\partial_{\nu}\Gamma_{\mu\sigma}^{\alpha}\right)=\frac{1}{2}\left(\partial_{\mu}\partial_{\sigma}g_{\nu\tau}-\partial_{\nu}\partial_{\sigma}g_{\mu\tau}-\partial_{\nu}\partial_{\tau}g_{\mu\sigma}+\partial_{\nu}\partial_{\tau}g_{\mu\sigma}\right)$$

后两项的差为

$$\begin{aligned}g_{\tau\alpha}\left(\Gamma_{\mu\beta}^{\alpha}\Gamma_{\nu\sigma}^{\beta}-\Gamma_{\nu\beta}^{\alpha}\Gamma_{\mu\sigma}^{\beta}\right)&=g_{\tau\alpha}\cdot 1\cdot\left(\Gamma_{\mu\beta}^{\alpha}\Gamma_{\nu\sigma}^{\beta}-\Gamma_{\nu\beta}^{\alpha}\Gamma_{\mu\sigma}^{\beta}\right)\\&=g_{\tau\alpha}\cdot\delta_{\beta}^{\beta}\cdot\left(\Gamma_{\mu\beta}^{\alpha}\Gamma_{\nu\sigma}^{\beta}-\Gamma_{\nu\beta}^{\alpha}\Gamma_{\mu\sigma}^{\beta}\right)\\&=g_{\tau\alpha}\cdot\delta_{\beta}^{\tau}\cdot\delta_{\tau}^{\beta}\cdot\left(\Gamma_{\mu\beta}^{\alpha}\Gamma_{\nu\sigma}^{\beta}-\Gamma_{\nu\beta}^{\alpha}\Gamma_{\mu\sigma}^{\beta}\right)\\&=g_{\beta\alpha}\left(\Gamma_{\mu\tau}^{\alpha}\Gamma_{\nu\sigma}^{\beta}-\Gamma_{\nu\tau}^{\alpha}\Gamma_{\mu\sigma}^{\beta}\right)\\&=g_{\alpha\beta}\left(\Gamma_{\mu\tau}^{\alpha}\Gamma_{\nu\sigma}^{\beta}-\Gamma_{\nu\tau}^{\alpha}\Gamma_{\mu\sigma}^{\beta}\right)\end{aligned}$$

可得

$$R_{\tau\sigma\mu\nu}=\frac{1}{2}\left(\partial_{\mu}\partial_{\sigma}g_{\nu\tau}-\partial_{\nu}\partial_{\sigma}g_{\mu\tau}-\partial_{\mu}\partial_{\tau}g_{\nu\sigma}+\partial_{\nu}\partial_{\tau}g_{\mu\sigma}\right)+g_{\alpha\beta}\left(\Gamma_{\mu\tau}^{\alpha}\Gamma_{\nu\sigma}^{\beta}-\Gamma_{\nu\tau}^{\alpha}\Gamma_{\mu\sigma}^{\beta}\right)$$

先将前两个指标和后两个指标交换($\tau\leftrightarrow\mu$和$\sigma\leftrightarrow\nu$)得

$$\begin{aligned}R_{\mu\nu\tau\sigma}&=\frac{1}{2}\left(\partial_{\tau}\partial_{\nu}g_{\sigma\mu}-\partial_{\sigma}\partial_{\nu}g_{\tau\mu}-\partial_{\tau}\partial_{\mu}g_{\sigma\nu}+\partial_{\sigma}\partial_{\mu}g_{\tau\nu}\right)+g_{\alpha\beta}\left(\Gamma_{\tau\mu}^{\alpha}\Gamma_{\sigma\nu}^{\beta}-\Gamma_{\sigma\mu}^{\alpha}\Gamma_{\tau\nu}^{\beta}\right)\\&=\frac{1}{2}\left(\partial_{\mu}\partial_{\sigma}g_{\nu\tau}-\partial_{\nu}\partial_{\sigma}g_{\mu\tau}-\partial_{\mu}\partial_{\tau}g_{\nu\sigma}+\partial_{\nu}\partial_{\tau}g_{\mu\sigma}\right)+g_{\alpha\beta}\left(\Gamma_{\mu\tau}^{\alpha}\Gamma_{\nu\sigma}^{\beta}-\Gamma_{\nu\tau}^{\alpha}\Gamma_{\mu\sigma}^{\beta}\right)=R_{\tau\sigma\mu\nu}\end{aligned}$$

将前两个指标交换($\sigma\leftrightarrow\tau$)

$$\begin{aligned}R_{\sigma\tau\mu\nu}&=~~~\frac{1}{2}\left(\partial_{\mu}\partial_{\tau}g_{\nu\sigma}-\partial_{\nu}\partial_{\tau}g_{\mu\sigma}-\partial_{\mu}\partial_{\sigma}g_{\nu\tau}+\partial_{\nu}\partial_{\sigma}g_{\mu\tau}\right)+g_{\alpha\beta}\left(\Gamma_{\mu\tau}^{\alpha}\Gamma_{\nu\sigma}^{\beta}-\Gamma_{\nu\tau}^{\alpha}\Gamma_{\mu\tau}^{\beta}\right)\\&=-\frac{1}{2}\left(\partial_{\mu}\partial_{\sigma}g_{\nu\tau}-\partial_{\nu}\partial_{\sigma}g_{\mu\tau}-\partial_{\mu}\partial_{\tau}g_{\nu\sigma}+\partial_{\nu}\partial_{\tau}g_{\mu\sigma}\right)-g_{\beta\alpha}\left(\Gamma_{\mu\tau}^{\beta}\Gamma_{\nu\tau}^{\alpha}-\Gamma_{\nu\sigma}^{\beta}\Gamma_{\mu\tau}^{\alpha}\right)=-R_{\tau\sigma\mu\nu}\end{aligned}$$

将后两个指标交换($\mu\leftrightarrow\nu$)

$$\begin{aligned}R_{\tau\sigma\nu\mu}&=~~~\frac{1}{2}\left(\partial_{\nu}\partial_{\sigma}g_{\mu\tau}-\partial_{\mu}\partial_{\sigma}g_{\nu\tau}-\partial_{\nu}\partial_{\tau}g_{\mu\sigma}+\partial_{\mu}\partial_{\tau}g_{\nu\sigma}\right)+g_{\alpha\beta}\left(\Gamma_{\nu\tau}^{\alpha}\Gamma_{\mu\sigma}^{\beta}-\Gamma_{\mu\tau}^{\alpha}\Gamma_{\nu\sigma}^{\beta}\right)\\&=-\frac{1}{2}\left(\partial_{\mu}\partial_{\sigma}g_{\nu\tau}-\partial_{\nu}\partial_{\sigma}g_{\mu\tau}-\partial_{\mu}\partial_{\tau}g_{\nu\sigma}+\partial_{\nu}\partial_{\tau}g_{\mu\sigma}\right)+g_{\alpha\beta}\left(\Gamma_{\mu\tau}^{\alpha}\Gamma_{\nu\sigma}^{\beta}-\Gamma_{\nu\tau}^{\alpha}\Gamma_{\mu\sigma}^{\beta}\right)=-R_{\tau\sigma\mu\nu}\end{aligned}$$

我们将协变曲率张量的对称性总结，即

$$\boxed{\begin{aligned}R_{\tau\sigma\mu\nu}&=~~~R_{\mu\nu\tau\sigma}\\R_{\tau\sigma\mu\nu}&=-R_{\sigma\tau\mu\nu}\\R_{\tau\sigma\mu\nu}&=-R_{\tau\sigma\nu\mu}\end{aligned}}$$

为了讨论循环等式我们写出关于后三个指标循环的曲率张量

$$\begin{aligned}R_{\tau\sigma\mu\nu}=\frac{1}{2}\left(\underset{\substack{\textcircled{\footnotesize{1}}}}{\underline{\partial_{\mu}\partial_{\sigma}g_{\nu\tau}}}-\underset{\substack{\textcircled{\footnotesize{2}}}}{\underline{\partial_{\nu}\partial_{\sigma}g_{\mu\tau}}}-\underset{\substack{\textcircled{\footnotesize{3}}}}{\underline{\partial_{\mu}\partial_{\tau}g_{\nu\sigma}}}+\underset{\substack{\textcircled{\footnotesize{4}}}}{\underline{\partial_{\nu}\partial_{\tau}g_{\mu\sigma}}}\right)+g_{\alpha\beta}\left(\underline{\Gamma_{\mu\tau}^{\alpha}\Gamma_{\nu\sigma}^{\beta}}-\underline{\underline{\Gamma_{\nu\tau}^{\alpha}\Gamma_{\mu\sigma}^{\beta}}}\right)\\R_{\tau\mu\nu\sigma}=\frac{1}{2}\left(\underset{\substack{\textcircled{\footnotesize{5}}}}{\underline{\partial_{\nu}\partial_{\mu}g_{\sigma\tau}}}-\underset{\substack{\textcircled{\footnotesize{1}}}}{\underline{\partial_{\sigma}\partial_{\mu}g_{\nu\tau}}}-\underset{\substack{\textcircled{\footnotesize{4}}}}{\underline{\partial_{\nu}\partial_{\tau}g_{\sigma\mu}}}+\underset{\substack{\textcircled{\footnotesize{6}}}}{\underline{\partial_{\sigma}\partial_{\tau}g_{\nu\mu}}}\right)+g_{\alpha\beta}\left(\underline{\underline{\Gamma_{\nu\tau}^{\alpha}\Gamma_{\sigma\mu}^{\beta}}}-\underline{\underline{\underline{\Gamma_{\sigma\tau}^{\alpha}\Gamma_{\nu\mu}^{\beta}}}}\right)\\R_{\tau\nu\sigma\mu}=\frac{1}{2}\left(\underset{\substack{\textcircled{\footnotesize{2}}}}{\underline{\partial_{\sigma}\partial_{\nu}g_{\mu\tau}}}-\underset{\substack{\textcircled{\footnotesize{5}}}}{\underline{\partial_{\mu}\partial_{\nu}g_{\sigma\tau}}}-\underset{\substack{\textcircled{\footnotesize{6}}}}{\underline{\partial_{\sigma}\partial_{\tau}g_{\mu\nu}}}+\underset{\substack{\textcircled{\footnotesize{3}}}}{\underline{\partial_{\mu}\partial_{\tau}g_{\sigma\nu}}}\right)+g_{\alpha\beta}\left(\underline{\underline{\underline{\Gamma_{\sigma\tau}^{\alpha}\Gamma_{\mu\nu}^{\beta}}}}-\underline{\Gamma_{\mu\tau}^{\alpha}\Gamma_{\sigma\nu}^{\beta}}\right)\end{aligned}$$

注意到求偏导顺序可以交换且度规是对称张量，分别消去对应项，可证明循环等式

$$R_{\tau\sigma\mu\nu}+R_{\tau\mu\nu\sigma}+R_{\tau\nu\sigma\mu}=0$$

上式乘以$g^{\rho\tau}$可得

$$R_{~\sigma\mu\nu}^{\rho}+R_{~\mu\nu\sigma}^{\rho}+R_{~\nu\sigma\mu}^{\rho}=0$$