黎曼曲率张量漫谈(续)

黎曼曲率张量

计算黎曼曲率张量

用Newman-Penrose形式计算

本节我们使用$(+,-,-,-)$号差。设$p$是4维时空$(M,g_{ab})$的一点，$\{(e_{\mu }{)}^a\}$是$p$点的一个正交归一标架，定义$p$点的4个特殊矢量如下：
$$\begin{gathered} l^a=\frac{1}{\sqrt{2}}[(e_0{)}^a+(e_1{)}^a],n^a=\frac{1}{\sqrt{2}}[(e_0{)}^a-(e_1{)}^a] \\ {m}^a=\frac{1}{\sqrt{2}}[(e_2{)}^a+i(e_3{)}^a],{\bar{m}}^a=\frac{1}{\sqrt{2}}[(e_2{)}^a-i(e_3{)}^a] \end{gathered}$$
容易验证$l_a{l}^a=n_a{n}^a=m_a{m}^a={\bar{m}}_a{\bar{m}}^a=0$，即它们都是类光矢量。这4个矢量构成$p$点的一个基底，称为$p$点的一个类光标架。以下以$\{({\epsilon }_{\mu }{)}^a\}$代表类光标架，并规定其编号为
$$({\epsilon }_1{)}^a=l^a,({\epsilon }_2{)}^a=n^a,({\epsilon }_3{)}^a=m^a,({\epsilon }_4{)}^a={\bar{m}}^a$$
相应的对偶基矢为
$$({\epsilon }^1{)}_a=n_a,({\epsilon }^2{)}_a=l_a,({\epsilon }^3{)}_a=-{\bar{m}}_a,({\epsilon }^4{)}_a=-m^a$$
不难看出类光标架中任意两个基矢的内积只有以下两对非零：
$$l^a{n}_a=1,m^a{\bar{m}}_a=-1$$
因此度规$g_{ab}$及其逆$g^{ab}$在该标架的分量$g_{\mu \nu }$和$g^{\mu \nu }$组成的矩阵为
$$(g_{\mu \nu })=\left[\begin{array}{cccc}0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& -1\\ 0& 0& -1& 0\end{array}\right]=(g^{\mu \nu })$$
这表明类光标架是刚性标架。与上节类似，联络1形式可以表示为
$$({\omega }_{\mu }{}^{\nu }{)}_a=({\epsilon }_{\mu }{)}^c{\nabla }_a({\epsilon }^{\nu }{)}_c$$
同样地
$$({\omega }_{\mu \nu }{)}_a=({\epsilon }_{\mu }{)}^b{\nabla }_a({\epsilon }_{\nu }{)}_b=-({\omega }_{\nu \mu }{)}_a$$
相应的里奇旋转系数为${\omega }_{\mu \nu \rho }=({\omega }_{\mu \nu }{)}_a({\epsilon }_{\rho }{)}^a$. 黎曼曲率张量的计算过程与上节一样，只是其中的$(e_{\mu }{)}^a$现在应理解为$({\epsilon }_{\mu }{)}^a$.

由于${\epsilon }_3$和${\epsilon }_4$的共轭性，容易证明交换3、4下标就得到对应量的共轭，例如${\omega }_{421}={\bar{\omega }}_{321},{\omega }_{14}={\bar{\omega }}_{13},R_{24}={\bar{R}}_{23}$等。这导致24个复${\omega }_{\mu \nu \rho }$中只有12个是独立的，以12个不带指标的希腊字母表示它们为
$$\begin{array}{rlr}& \kappa =-{\omega }_{311},\rho =-{\omega }_{314},& \epsilon =\frac{1}{2}({\omega }_{211}-{\omega }_{341})\\ & \sigma =-{\omega }_{313},\mu ={\omega }_{243},& \gamma =\frac{1}{2}({\omega }_{212}-{\omega }_{342})\\ & \lambda ={\omega }_{244},\tau =-{\omega }_{312},& \alpha =\frac{1}{2}({\omega }_{214}-{\omega }_{344})\\ & \nu ={\omega }_{242},\pi ={\omega }_{241},& \beta =\frac{1}{2}({\omega }_{213}-{\omega }_{343})\end{array}$$
这12个希腊字母称为自旋系数。又引入4个求导符号$$D=l^a{\nabla }_a,\Delta =n^a{\nabla }_a,\delta =m^a{\nabla }_a,\bar{\delta }={\bar{m}}^a{\nabla }_a$$它们之间满足如下的对易关系：
$$\begin{array}{rl}\Delta D-D\Delta & =(\gamma +\bar{\gamma })D+(\epsilon +\bar{\epsilon })\Delta -(\bar{\tau }+\pi )\delta -(\tau +\bar{\pi })\bar{\delta }\\ \delta D-D\delta & =(\bar{\alpha }+\beta -\bar{\pi })D+\kappa \Delta -(\bar{\rho }+\epsilon -\bar{\epsilon })\delta -\sigma \bar{\delta }\\ \delta \Delta -\Delta \delta & =-\bar{\nu }D+(\tau -\bar{\alpha }-\beta )\Delta +(\mu -\gamma +\bar{\gamma })\delta +\bar{\lambda }\bar{\delta }\\ \bar{\delta }\delta -\delta \bar{\delta }& =(\bar{\mu }-\mu )D+(\bar{\rho }-\rho )\Delta +(\alpha -\bar{\beta })\delta -(\bar{\alpha }-\beta )\bar{\delta }\end{array}$$
外尔张量有10个实的独立分量，用5个复数代表，定义为$${\Psi }_0=-C_{1313},{\Psi }_1=-C_{1213},{\Psi }_2=-C_{1342},{\Psi }_3=-C_{1242},{\Psi }_4=-C_{2424}$$里奇张量有10个，其中$R_{11},R_{12},R_{22}$显然为实数，$R_{34}=R_{43}={\bar{R}}_{34}$也是实数，由此定义以下四个实数$${\Phi }_{00}=-\frac{1}{2}{R}_{11},{\Phi }_{11}=-\frac{1}{4}(R_{12}+R_{34}),{\Phi }_{22}=-\frac{1}{2}{R}_{22},\Lambda =\frac{1}{12}(R_{12}-R_{34})$$其余6个分量为复数，分别为$${\Phi }_{01}=-\frac{1}{2}{R}_{13},{\Phi }_{10}=-\frac{1}{2}{R}_{14},{\Phi }_{02}=-\frac{1}{2}{R}_{33},{\Phi }_{20}=-\frac{1}{2}{R}_{44},{\Phi }_{12}=-\frac{1}{2}{R}_{23},{\Phi }_{21}=-\frac{1}{2}{R}_{24}$$易见$\Phi$构成一个厄米矩阵。由这些符号，我们可以写出黎曼曲率张量的分量表达式
$$\begin{array}{rl}D\rho -\bar{\delta }\kappa & =({\rho }^2+\sigma \bar{\sigma })+(\epsilon +\bar{\epsilon })\rho -\bar{\kappa }\tau -\kappa (3\alpha +\bar{\beta }-\pi )+{\Phi }_{00},[R_{1314}]\\ D\sigma -\delta \kappa & =(\rho +\bar{\rho })\sigma +(3\epsilon -\bar{\epsilon })\sigma -(\tau -\bar{\pi }+\bar{\alpha }+3\beta )\kappa +{\Psi }_0,[R_{1313}]\\ D\tau -\Delta \kappa & =(\tau +\bar{\pi })\rho +(\bar{\tau }+\pi )\sigma +(\epsilon -\bar{\epsilon })\tau -(3\gamma +\bar{\gamma })\kappa +{\Psi }_1+{\Phi }_{01},[R_{1312}]\\ D\alpha -\bar{\delta }\epsilon & =(\rho +\bar{\epsilon }-2\epsilon )\alpha +\beta \bar{\sigma }-\bar{\beta }\epsilon -\kappa \lambda -\bar{\kappa }\gamma +(\epsilon +\rho )\pi +{\Phi }_{10},[\frac{1}{2}(R_{3414}-R_{1214})]\\ D\beta -\delta \epsilon & =(\alpha +\pi )\sigma +(\bar{\rho }-\bar{\epsilon })\beta -(\mu +\gamma )\kappa -(\bar{\alpha }-\bar{\pi })\epsilon +{\Psi }_1,[\frac{1}{2}(R_{1213}-R_{3413})]\\ D\gamma -\Delta \epsilon & =(\tau +\bar{\pi })\alpha +(\bar{\tau }+\pi )\beta -(\epsilon +\bar{\epsilon })\gamma -(\gamma +\bar{\gamma })\epsilon +\tau \pi -\nu \kappa +{\Psi }_2+{\Phi }_{11}-\Lambda ,[\frac{1}{2}(R_{1212}-R_{3412})]\\ D\lambda -\bar{\delta }\pi & =(\rho \lambda +\bar{\sigma }\mu )+{\pi }^2+(\alpha -\bar{\beta })\pi -\nu \bar{\kappa }-(3\epsilon -\bar{\epsilon })\lambda +{\Phi }_{20},[R_{2441}]\\ D\mu -\delta \pi & =(\bar{\rho }\mu +\sigma \lambda )+\pi \bar{\pi }-(\epsilon +\bar{\epsilon })\mu -(\bar{\alpha }-\beta )\pi -\nu \kappa +{\Psi }_2+2\Lambda ,[R_{2431}]\\ D\nu -\Delta \pi & =(\pi +\bar{\tau })\mu +(\bar{\pi }+\tau )\lambda +(\gamma -\bar{\gamma })\pi -(3\epsilon +\bar{\epsilon })\nu +{\Psi }_3+{\Phi }_{21},[R_{2421}]\\ \Delta \lambda -\bar{\delta }\nu & =-(\mu +\bar{\mu })\lambda -(3\gamma -\bar{\gamma })\lambda +(3\alpha +\bar{\beta }+\pi -\bar{\tau })\nu -{\Psi }_4,[R_{2442}]\\ \delta \rho -\bar{\delta }\sigma & =\rho (\bar{\alpha }+\beta )-\sigma (3\alpha -\bar{\beta })+(\rho -\bar{\rho })\tau +(\mu -\bar{\mu })\kappa -{\Psi }_1+{\Phi }_{01},[R_{3143}]\\ \delta \alpha -\bar{\delta }\beta & =(\mu \rho -\lambda \sigma )+\alpha \bar{\alpha }+\beta \bar{\beta }-2\alpha \beta +\gamma (\rho -\bar{\rho })+\epsilon (\mu -\bar{\mu })-{\Psi }_2+{\Phi }_{11}+\Lambda ,[\frac{1}{2}(R_{1234}-R_{3434})]\\ \delta \lambda -\bar{\delta }\mu & =(\rho -\bar{\rho })\nu +(\mu -\bar{\mu })\pi +(\alpha +\bar{\beta })\mu +(\bar{\alpha }-3\beta )\lambda -{\Psi }_3+{\Phi }_{21},[R_{2443}]\\ \delta \nu -\Delta \mu & =({\mu }^2+\lambda \bar{\lambda })+(\gamma +\bar{\gamma })\mu -\bar{\nu }\pi +(\tau -3\beta -\bar{\alpha })\nu +{\Phi }_{22},[R_{2423}]\\ \delta \gamma -\Delta \beta & =(\tau -\bar{\alpha }-\beta )\gamma +\mu \tau -\sigma \nu -\epsilon \bar{\nu }-(\gamma -\bar{\gamma }-\mu )\beta +\alpha \bar{\lambda }+{\Phi }_{12},[\frac{1}{2}(R_{1232}-R_{3432})]\\ \delta \tau -\Delta \sigma & =(\mu \sigma +\bar{\lambda }\rho )+(\tau +\beta -\bar{\alpha })\tau -(3\gamma -\bar{\gamma })\sigma -\kappa \bar{\nu }+{\Phi }_{02},[R_{1332}]\\ \Delta \rho -\bar{\delta }\tau & =-(\rho \bar{\mu }+\sigma \lambda )+(\bar{\beta }-\alpha -\bar{\tau })\tau +(\gamma +\bar{\gamma })\rho +\nu \kappa -{\Psi }_2-2\Lambda ,[R_{1324}]\\ \Delta \alpha -\bar{\delta }\gamma & =(\rho +\epsilon )\nu -(\tau +\beta )\lambda +(\bar{\gamma }-\bar{\mu })\alpha +(\bar{\beta }-\bar{\tau })\gamma -{\Psi }_3,[\frac{1}{2}(R_{1242}-R_{3442})]\end{array}$$
以及比安基恒等式：
$$\begin{array}{r}-\bar{\delta }{\Psi }_0+D{\Psi }_1+(4\alpha -\pi ){\Psi }_0-2(2\rho +\epsilon ){\Psi }_1+3\kappa {\Psi }_2+R_a=0,R_{13[13,4]}=0\\ \bar{\delta }{\Psi }_1-D{\Psi }_2-\lambda {\Psi }_0+2(\pi -\alpha ){\Psi }_1+3\rho {\Psi }_2-2\kappa {\Psi }_3+R_b=0,R_{13[21,4]}=0\\ -\bar{\delta }{\Psi }_2+D{\Psi }_3+2\lambda {\Psi }_1-3\pi {\Psi }_2+2(ϵ-\rho ){\Psi }_3+\kappa {\Psi }_4+R_c=0,R_{42[13,4]}=0\\ \bar{\delta }{\Psi }_3-D{\Psi }_4-3\lambda {\Psi }_2+2(2\pi +\alpha ){\Psi }_3-(4ϵ-\rho ){\Psi }_4+R_d=0,R_{42[21,4]}=0\\ -\Delta {\Psi }_0+\delta {\Psi }_1+4(\gamma -\mu ){\Psi }_0-2(2\tau +\beta ){\Psi }_1+3\sigma {\Psi }_2+R_e=0,R_{13[13,2]}=0\\ -\Delta {\Psi }_1+\delta {\Psi }_2+\nu {\Psi }_0-2(\gamma -\mu ){\Psi }_1-3\tau {\Psi }_2+2\sigma {\Psi }_3+R_f=0,R_{13[43,2]}=0\\ -\Delta {\Psi }_2+\delta {\Psi }_3+2\nu {\Psi }_1-3\mu {\Psi }_2+2(\beta -\tau ){\Psi }_3+\sigma {\Psi }_4+R_g=0,R_{42[13,2]}=0\\ -\Delta {\Psi }_3+\delta {\Psi }_4+3\nu {\Psi }_2-2(\gamma -2\mu ){\Psi }_3-(\tau -4\beta ){\Psi }_4+R_h=0,R_{42[43,2]}=0\end{array}$$
其中
$$\begin{array}{rl}{R}_a& =-D{\Phi }_{01}+\delta {\Phi }_{00}+2(\bar{\rho }+ϵ){\Phi }_{01}+2\sigma {\Phi }_{10}-2\kappa {\Phi }_{11}-\bar{\kappa }{\Phi }_{02}+(\bar{\pi }-2\bar{\alpha }-2\beta ){\Phi }_{00}\\ R_b& =-\Delta {\Phi }_{00}+\bar{\delta }{\Phi }_{01}-2(\bar{\tau }+\alpha ){\Phi }_{01}+2\rho {\Phi }_{11}-2\tau {\Phi }_{10}+\bar{\sigma }{\Phi }_{02}-(\bar{\mu }-2\bar{\gamma }-2\gamma ){\Phi }_{00}-2D\Lambda \\ R_c& =-D{\Phi }_{21}+\delta {\Phi }_{20}+2(\bar{\rho }-ϵ){\Phi }_{21}+2\pi {\Phi }_{11}-2\mu {\Phi }_{10}-\bar{\kappa }{\Phi }_{22}+(\bar{\pi }-2\bar{\alpha }+2\beta ){\Phi }_{20}-2\bar{\delta }\Lambda \\ R_d& =-\Delta {\Phi }_{20}+\bar{\delta }{\Phi }_{21}-2(\bar{\tau }-\alpha ){\Phi }_{21}+2\nu {\Phi }_{10}-2\lambda {\Phi }_{11}+\bar{\sigma }{\Phi }_{22}-(\bar{\mu }-2\bar{\gamma }+2\gamma ){\Phi }_{20}\\ R_e& =-D{\Phi }_{02}+\delta {\Phi }_{01}+2(\bar{\pi }-\beta ){\Phi }_{01}+2\sigma {\Phi }_{11}-2\kappa {\Phi }_{12}-\bar{\lambda }{\Phi }_{00}+(\bar{\rho }-2\bar{\epsilon }+2\epsilon ){\Phi }_{02}\\ R_f& =\Delta {\Phi }_{01}-\bar{\delta }{\Phi }_{02}+2(\bar{\mu }-\gamma ){\Phi }_{01}+2\tau {\Phi }_{11}-2\rho {\Phi }_{12}-\bar{\nu }{\Phi }_{00}+(\bar{\tau }-2\bar{\beta }+2\alpha ){\Phi }_{02}+2\delta \Lambda \\ R_g& =-D{\Phi }_{22}+\delta {\Phi }_{21}+2(\bar{\pi }+\beta ){\Phi }_{21}+2\pi {\Phi }_{12}-2\mu {\Phi }_{11}-\bar{\lambda }{\Phi }_{20}+(\bar{\rho }-2\bar{\epsilon }-2\epsilon ){\Phi }_{22}-2\Delta \Lambda \\ R_h& =\Delta {\Phi }_{21}-\bar{\delta }{\Phi }_{22}+2(\bar{\mu }+\gamma ){\Phi }_{21}+2\lambda {\Phi }_{12}-2\nu {\Phi }_{11}-\bar{\nu }{\Phi }_{20}+(\bar{\tau }-2\bar{\beta }-2\alpha ){\Phi }_{22}\end{array}$$