View source for Spin-coefficient formalism/The non-vacuum spin coefficient equations

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===The metric equations===

:<math>\label{metric eqsApp}
\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},
</math>

:<math>
\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},
</math>
:<math>
\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},
</math>
:<math>
\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}.
</math>

===The spin-coefficient equations===

:<math>\label{SC1App}
\Delta \lambda -\overline{\delta }\nu =-(\mu +\overline{\mu } + 3\gamma -\overline{\gamma }) \lambda + (3\alpha +\overline{\beta }+\pi -\overline{
\tau })\nu -\Psi _{4}
</math>

:<math>
\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}
</math>
:<math>
\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}
</math>
:<math>
\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}
</math>
:<math>
\delta \nu -\Delta \mu =\mu ^{2}+\lambda \overline{\lambda }+\mu (\gamma +
\overline{\gamma })-\overline{\nu }\pi +\nu (\tau -3\beta -\overline{\alpha })+\Phi_{22}
</math>
:<math>
\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}
</math>
:<math>
\delta \tau -\Delta \sigma =\mu \sigma +\rho \overline{\lambda }+\tau
(\tau +\beta -\overline{\alpha })-\sigma (3\gamma -\overline{\gamma }
)-\kappa \overline{\nu } +\Phi_{02}
</math>
:<math>
\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
</math>
:<math>
\Delta \alpha -\overline{\delta }\gamma =\nu (\rho +\epsilon )-\lambda
(\tau +\beta )+\alpha (\overline{\gamma }-\overline{\mu })+\gamma (\overline{
\beta }-\overline{\tau })-\Psi _{3}
</math>


:<math>\label{SC2App}
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}
</math>

:<math>
D\sigma -\delta \kappa =(\rho +\overline{\rho })\sigma +(3\epsilon -
\overline{\epsilon })\sigma -(\tau -\overline{\pi }+\overline{\alpha }
+3\beta )\kappa +\Psi _{0}
</math>
:<math>
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}
</math>
:<math>
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}
</math>
:<math>
D\beta -\delta \epsilon =(\alpha +\pi )\sigma +(\overline{\rho }-\overline{
\epsilon })\beta -(\mu +\gamma )\kappa -(\overline{\alpha }-\overline{\pi }
)\epsilon +\Psi _{1}
</math>
:<math>
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}
</math>
:<math>
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}
</math>
:<math>
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
</math>
:<math>
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}
</math>

===The Bianchi identities===

:<math>\label{BI1App}
\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}, 
</math>

:<math>\label{BI2App}
\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}, 
</math>

:<math>\label{BI3App}
\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}, 
</math>

:<math>\label{BI4App}
\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},
</math>



:<math>\label{BI5App}
\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}, 
</math>

:<math>\label{BI6App}
\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}, 
</math>

:<math>\label{BI7App}
\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}, 
</math>

:<math>\label{BI8App}
\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}.
</math>



:<math>\label{BI9App}
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} 
</math>

:<math>\label{BI10App}
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}, 
</math>

:<math>\label{BI11App}
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},
</math>

where <math>\Phi _{ij}</math> represents the trace-free Ricci tensor while the Ricci scalar is given by <math>R=24\Lambda:</math> 
:<math>\label{BI12App}
\Phi _{00} =-\frac{1}{2}R_{ab}l^{a}l^{b} </math>

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