Spin-coefficient formalism/The non-vacuum spin coefficient equations

The metric equations

\[\tag{1} \Delta l^{a}-Dn^{a} =(\gamma +\overline{\gamma })l^{a}+(\epsilon + \overline{\epsilon })n^{a}-(\tau +\overline{\pi })\overline{m}^{a}-( \overline{\tau }+\pi )m^{a}, \]

\[ \delta l^{a}-Dm^{a} =(\overline{\alpha }+\beta -\overline{\pi } )l^{a}+\kappa n^{a}-\sigma \overline{m}^{a}-(\overline{\rho }+\epsilon - \overline{\epsilon })m^{a}, \] \[ \delta n^{a}-\Delta m^{a} =-{\overline\nu} l^{a}+(\tau -\overline{\alpha }-\beta )n^{a}+\overline{\lambda }\overline{m}^{a}+(\mu -\gamma +\overline{\gamma } )m^{a}, \] \[ \overline{\delta }m^{a}-\delta \overline{m}^{a} =(\overline{\mu }-\mu )l^{a}+(\overline{\rho }-\rho )n^{a}-(\overline{\alpha }-\beta )\overline{m} ^{a}+(\alpha -\overline{\beta })m^{a}. \]

The spin-coefficient equations

\[\tag{2} \Delta \lambda -\overline{\delta }\nu =-(\mu +\overline{\mu } + 3\gamma -\overline{\gamma }) \lambda + (3\alpha +\overline{\beta }+\pi -\overline{ \tau })\nu -\Psi _{4} \]

\[ \delta \rho -\overline{\delta }\sigma =\rho (\overline{\alpha }+\beta )-\sigma (3\alpha -\overline{\beta })+(\rho -\overline{\rho })\tau +(\mu - \overline{\mu })\kappa -\Psi _{1} +\Phi_{01} \] \[ \delta \alpha -\overline{\delta }\beta =\mu \rho -\lambda \sigma +\alpha \overline{\alpha }+\beta \overline{\beta }-2\alpha \beta +\gamma (\rho - \overline{\rho })+\epsilon (\mu -\overline{\mu })-\Psi _{2}+\Lambda+\Phi_{11} \] \[ \delta \lambda -\overline{\delta }\mu =(\rho -\overline{\rho })\nu +(\mu - \overline{\mu })\pi +\mu (\alpha +\overline{\beta })+\lambda (\overline{ \alpha }-3\beta )-\Psi _{3}+\Phi_{21} \] \[ \delta \nu -\Delta \mu =\mu ^{2}+\lambda \overline{\lambda }+\mu (\gamma + \overline{\gamma })-\overline{\nu }\pi +\nu (\tau -3\beta -\overline{\alpha })+\Phi_{22} \] \[ \delta \gamma -\Delta \beta =\gamma (\tau -\overline{\alpha }-\beta )+\mu \tau -\sigma \nu -\epsilon \overline{\nu }-\beta (\gamma -\overline{\gamma } -\mu )+\alpha \overline{\lambda } +\Phi_{12} \] \[ \delta \tau -\Delta \sigma =\mu \sigma +\rho \overline{\lambda }+\tau (\tau +\beta -\overline{\alpha })-\sigma (3\gamma -\overline{\gamma } )-\kappa \overline{\nu } +\Phi_{02} \] \[ \Delta \rho -\overline{\delta }\tau =-(\rho \overline{\mu }+\sigma \lambda )+\tau (\overline{\beta }-\alpha -\overline{\tau })+(\gamma +\overline{ \gamma })\rho +\kappa \nu -\Psi _{2} - 2\Lambda \] \[ \Delta \alpha -\overline{\delta }\gamma =\nu (\rho +\epsilon )-\lambda (\tau +\beta )+\alpha (\overline{\gamma }-\overline{\mu })+\gamma (\overline{ \beta }-\overline{\tau })-\Psi _{3} \]

\[\tag{3} D\rho -\overline{\delta }\kappa =\rho ^{2}+\sigma \overline{\sigma } +(\epsilon +\overline{\epsilon })\rho -\overline{\kappa }\tau -\kappa (3\alpha +\overline{\beta }-\pi ) +\Phi_{00} \]

\[ D\sigma -\delta \kappa =(\rho +\overline{\rho })\sigma +(3\epsilon - \overline{\epsilon })\sigma -(\tau -\overline{\pi }+\overline{\alpha } +3\beta )\kappa +\Psi _{0} \] \[ D\tau -\Delta \kappa =(\tau +\overline{\pi })\rho +(\overline{\tau }+\pi )\sigma +(\epsilon -\overline{\epsilon })\tau -(3\gamma +\overline{\gamma } )\kappa +\Psi _{1} +\Phi_{01} \] \[ D\alpha -\overline{\delta }\epsilon =(\rho +\overline{\epsilon }-2\epsilon )\alpha +\beta \overline{\sigma }-\overline{\beta }\epsilon -\kappa \lambda - \overline{\kappa }\gamma +(\epsilon +\rho )\pi +\Phi_{10} \] \[ D\beta -\delta \epsilon =(\alpha +\pi )\sigma +(\overline{\rho }-\overline{ \epsilon })\beta -(\mu +\gamma )\kappa -(\overline{\alpha }-\overline{\pi } )\epsilon +\Psi _{1} \] \[ D\gamma -\Delta \epsilon =(\tau +\overline{\pi })\alpha +(\overline{\tau } +\pi )\beta -(\epsilon +\overline{\epsilon })\gamma -(\gamma +\overline{ \gamma })\epsilon +\tau \pi -\nu \kappa +\Psi _{2} - \Lambda +\Phi_{11} \] \[ D\lambda -\overline{\delta }\pi =\rho \lambda +\overline{\sigma }\mu +\pi ^{2}+(\alpha -\overline{\beta })\pi -\nu \overline{\kappa }-(3\epsilon -\overline{\epsilon })\lambda +\Phi_{20} \] \[ D\mu -\delta \pi =\overline{\rho }\mu +\sigma \lambda +\pi \overline{\pi }-(\epsilon +\overline{\epsilon })\mu -\pi (\overline{\alpha }-\beta )-\nu \kappa +\Psi _{2} + 2\Lambda \] \[ D\nu -\Delta \pi =(\overline{\tau }+\pi )\mu +(\tau +\overline{\pi } )\lambda +(\gamma -\overline{\gamma })\pi -(3\epsilon +\overline{\epsilon } )\nu +\Psi _{3} +\Phi_{21} \]

The Bianchi identities

\[\tag{4} \overline{\delta }\Psi _{0}-D\Psi _{1}+D\Phi _{01}-\delta \Phi _{00} =(4\alpha -\pi )\Psi _{0}-2(2\rho +\epsilon )\Psi _{1}+3\kappa \Psi _{2} +(\overline{\pi }-2\overline{\alpha }-2\beta )\Phi _{00}+2(\overline{\rho } +\epsilon )\Phi _{01}+2\sigma \Phi _{10}-2\kappa \Phi _{11}-\overline{\kappa }\Phi _{02}, \]

\[\tag{5} \overline{\delta }\Psi _{1}-D\Psi _{2}-\Delta \Phi _{00}+\overline{\delta } \Phi _{01}-2D\Lambda =\lambda \Psi _{0}+2(\alpha -\pi )\Psi _{1}-3\rho \Psi _{2}+2\kappa \Psi _{3} -(2\gamma +2\overline{\gamma }-\overline{\mu })\Phi _{00}+2(\overline{\tau }+\alpha )\Phi _{01}+2\tau \Phi _{10}-2\rho \Phi _{11}-\overline{\sigma } \Phi _{02}, \]

\[\tag{6} \overline{\delta }\Psi _{2}-D\Psi _{3}+D\Phi _{21}-\delta \Phi _{20}+2 \overline{\delta }\Lambda =2\lambda \Psi _{1}-3\pi \Psi _{2}+2(\epsilon -\rho )\Psi _{3}+\kappa \Psi _{4} -2\mu \Phi _{10}+2\pi \Phi _{11}+(2\beta +\overline{\pi }-2\overline{\alpha })\Phi _{20}+2(\overline{\rho }-\epsilon )\Phi _{21}-\overline{\kappa }\Phi _{22}, \]

\[\tag{7} \overline{\delta }\Psi _{3}-D\Psi _{4}-\Delta \Phi _{20}+\overline{\delta } \Phi _{21} =3\lambda \Psi _{2}-2(\alpha +2\pi )\Psi _{3}+(4\epsilon -\rho )\Psi _{4} -2\nu \Phi _{10}+2\lambda \Phi _{11}+(2\gamma -2\overline{\gamma }+ \overline{\mu })\Phi _{20}+2(\overline{\tau }-\alpha )\Phi _{21}-\overline{ \sigma }\Phi _{22}, \]

\[\tag{8} \Delta \Psi _{0}-\delta \Psi_1 +D\Phi _{02}-\delta \Phi _{01} =(4\gamma -\mu )\Psi _{0}-2(2\tau +\beta )\Psi _{1}+3\sigma \Psi _{2} -\overline{\lambda }\Phi _{00}+2(\overline{\pi }-\beta )\Phi _{01} +2\sigma\Phi _{11} +(\overline{\rho }+2\epsilon -2\overline{\epsilon })\Phi_{02} -2\kappa \Phi _{12}, \]

\[\tag{9} \Delta \Psi _{1}-\delta \Psi _{2}-\Delta \Phi _{01}+\overline{\delta }\Phi_{02}-2\delta \Lambda =\nu \Psi _{0}+2(\gamma -\mu )\Psi _{1}-3\tau \Psi _{2}+2\sigma \Psi _{3}-\overline{\upsilon }\Phi _{00}+2(\overline{\mu }-\gamma )\Phi_{01}+(2\alpha +\overline{\tau }-2\overline{\beta })\Phi _{02}+2\tau \Phi_{11}-2\rho \Phi _{12}, \]

\[\tag{10} \Delta \Psi _{2}-\delta \Psi _{3}+D\Phi _{22}-\delta \Phi _{21}+2\Delta\Lambda =2\nu \Psi _{1}-3\mu \Psi _{2}+2(\beta -\tau )\Psi _{3}+\sigma \Psi _{4} -2\mu \Phi _{11}-\overline{\lambda }\Phi _{20}+2\pi \Phi _{12}+2(\beta +\overline{\pi })\Phi _{21}+(\overline{\rho }-2\epsilon -2\overline{\epsilon } )\Phi _{22}, \]

\[\tag{11} \Delta \Psi _{3}-\delta \Psi _{4}-\Delta \Phi _{21}+\overline{\delta }\Phi_{22} =3\nu \Psi _{2}-2(\gamma +2\mu )\Psi _{3}+(4\beta -\tau )\Psi _{4} -2\upsilon \Phi _{11}-\overline{\nu }\Phi _{20}+2\lambda \Phi_{12}+2(\gamma +\overline{\mu })\Phi _{21}+(\overline{\tau }-2\overline{\beta }-2\alpha )\Phi _{22}. \]

\[\tag{12} D\Phi _{11}-\delta \Phi _{10}+\Delta \Phi _{00}-\overline{\delta }\Phi_{01}+3D\Lambda =(2\gamma +2\overline{\gamma }-\mu -\overline{\mu })\Phi _{00}+(\pi -2\alpha -2\overline{\tau })\Phi _{01}+(\overline{\pi }-2\overline{\alpha }-2\tau )\Phi _{10} +2(\rho +\overline{\rho })\Phi _{11}+\overline{\sigma }\Phi _{02}+\sigma\Phi _{20}-\overline{\kappa }\Phi _{12}-\kappa \Phi _{21} \]

\[\tag{13} D\Phi _{12}-\delta \Phi _{11}+\Delta \Phi _{01}-\overline{\delta }\Phi _{02}+3\delta \Lambda =(2\gamma -\mu -2\overline{\mu })\Phi _{01}+\overline{\nu }\Phi _{00}-\overline{\lambda }\Phi _{10}+2(\overline{\pi }-\tau )\Phi _{11} +(\pi +2\overline{\beta }-2\alpha -\overline{\tau })\Phi _{02}+(2\rho +\overline{\rho }-2\overline{\epsilon })\Phi _{12}+\sigma \Phi _{21}-\kappa\Phi _{22}, \]

\[\tag{14} D\Phi _{22}-\delta \Phi_{21}+\Delta \Phi _{11}-\overline{\delta }\Phi _{12}+3\Delta \Lambda =\upsilon \Phi _{01}+\overline{\nu }\Phi _{10}-2(\mu +\overline{\mu })\Phi _{11}-\lambda \Phi _{02}-\overline{\lambda }\Phi _{20} +(2\pi -\overline{\tau }+2\overline{\beta })\Phi _{12}+(2\beta -\tau +2 \overline{\pi })\Phi _{21}+(\rho +\overline{\rho }-2\epsilon -2\overline{\epsilon })\Phi _{22}, \]

where \(\Phi _{ij}\) represents the trace-free Ricci tensor while the Ricci scalar is given by \(R=24\Lambda:\) \[\tag{15} \Phi _{00} =-\frac{1}{2}R_{ab}l^{a}l^{b} \]

\[ \Phi _{01} =-\frac{1}{2}R_{ab}l^{a}m^{b}=\overline{\Phi }_{10} \] \[ \Phi _{02} =-\frac{1}{2}R_{ab}m^{a}m^{b}=\overline{\Phi }_{20} \] \[ \Phi _{10} =-\frac{1}{2}R_{ab}l^{a}\overline{m}^{b}=\overline{\Phi }_{01} \] \[ \Phi _{11} =-\frac{1}{4}R_{ab}(l^{a}m^{b}+m^{a}\overline{m}^{b}) \] \[ \Phi _{12} =-\frac{1}{2}R_{ab}n^{a}m^{b}=\overline{\Phi }_{21} \] \[ \Phi _{20} =-\frac{1}{2}R_{ab}\overline{m}^{a}\overline{m}^{b}=\overline{ \Phi }_{02} \] \[ \Phi _{21} =-\frac{1}{2}R_{ab}n^{a}\overline{m}^{b}=\overline{\Phi }_{12} \] \[ \Phi _{22} =-\frac{1}{2}R_{ab}n^{a}n^{b} \]