【相対論】Newman-Penrose formalism 3

相対論∩スピン幾何

　Newman-Penrose formalismはWeylスピノルに関する共変微分の係数、あるいはnull tetradに関する共変微分の係数を$\alpha ,\beta ,...$のように12個の関数(スピン係数なとど呼ばれる)で置いて、色々な量を計算する手法ですが、スピン係数の微分とWeylスカラーの微分に関するNewman-Penrose方程式（恒等式？）がわんさかあります。Newman-Penrose formalismを勉強するとこのNewman-Penrose方程式に面食らって面倒そうに見えるのですが、よく考えるとどんなformalismだろうと同等の情報量はあるわけなので、どこかの段階では必要なら計算しないといけないわけです。Newman-Penrose方程式は主要な関係式をはじめに全部計算して列挙しておいて応用する時に見ながら使うためのものなので、人生で一回計算する経験をしておいて残りの人生では辞書的に使いのがよいです。

　4次元時空$(M,g)$のnull tetradを$\{m,\overline{m},l,k\}$とします。

スピン係数

　$e_1=m,e_2=\overline{m},e_3=l,e_4=k$とし、${\mathrm{\Gamma }}_{ij}=g(e_i,\mathrm{\nabla }{e}_j)$と置きます。スピノル束を$S=S^{+}\oplus S^{-}$とし、o.n.f.$\{{\overline{e}}_1,{\overline{e}}_2,{\overline{e}}_3,{\overline{e}}_4\}$を
$$\begin{array}{rl}& e_1=\frac{1}{\sqrt{2}}({\overline{e}}_1-i{\overline{e}}_2),e_2=\frac{1}{\sqrt{2}}({\overline{e}}_1+i{\overline{e}}_2)\\ & e_3=-\frac{1}{\sqrt{2}}({\overline{e}}_0+{\overline{e}}_3),e_4=\frac{1}{\sqrt{2}}({\overline{e}}_0-{\overline{e}}_3)\end{array}$$
で定義するとき、このo.n.f.に関する$S^{+}$(right-handed Weyl spinor束)のスピン接続の接続形式は
$$\begin{array}{r}{\mathrm{\Gamma }}^{S^{+}}=\left(\begin{array}{cc}-\frac{{\mathrm{\Gamma }}_{12}}{2}+\frac{{\mathrm{\Gamma }}_{34}}{2}& -{\mathrm{\Gamma }}_{14}\\ \overline{{\mathrm{\Gamma }}_{13}}& \frac{{\mathrm{\Gamma }}_{12}}{2}-\frac{{\mathrm{\Gamma }}_{34}}{2}\end{array}\right)\end{array}$$
で与えられるので、スピン係数などと呼ばれる12個の複素関数$\alpha ,\beta ,...$を
$$\begin{array}{rl}\frac{1}{2}(-{\mathrm{\Gamma }}_{12}+{\mathrm{\Gamma }}_{34})=\frac{1}{2}(g(\mathrm{\nabla }k,\ell )-g(\mathrm{\nabla }\overline{m},m))& =\beta {\theta }^1+\alpha {\theta }^2+\gamma {\theta }^3+\epsilon {\theta }^4\\ {\mathrm{\Gamma }}_{23}=g(\mathrm{\nabla }\ell ,\overline{m})& =\mu {\theta }^1+\lambda {\theta }^2+\nu {\theta }^3+\pi {\theta }^4\\ {\mathrm{\Gamma }}_{14}=g(\mathrm{\nabla }k,m)& =\sigma {\theta }^1+\rho {\theta }^2+\tau {\theta }^3+\kappa {\theta }^4\end{array}$$
と定義します。

あるいは同値ですが、スピン接続を持ち出さなくてもnull tetradに対して
$$\begin{array}{rl}& \alpha =\frac{1}{2}[g({\mathrm{\nabla }}_{\overline{m}}k,\ell )-g({\mathrm{\nabla }}_{\overline{m}}\overline{m},m)]\\ & \beta =\frac{1}{2}[g({\mathrm{\nabla }}_{m}k,\ell )-g({\mathrm{\nabla }}_m\overline{m},m)]\\ & \gamma =\frac{1}{2}[g({\mathrm{\nabla }}_{\ell }k,\ell )-g({\mathrm{\nabla }}_{\ell }\overline{m},m)]\\ & \epsilon =\frac{1}{2}[g({\mathrm{\nabla }}_{k}k,\ell )-g({\mathrm{\nabla }}_k\overline{m},m)]\\ & \mu =g({\mathrm{\nabla }}_m\ell ,\overline{m}),\lambda =g({\mathrm{\nabla }}_{\overline{m}}\ell ,\overline{m})\\ & \nu =g({\mathrm{\nabla }}_{\ell }\ell ,\overline{m}),\pi =g({\mathrm{\nabla }}_k\ell ,\overline{m})\\ & \sigma =g({\mathrm{\nabla }}_{m}k,m),\rho =g({\mathrm{\nabla }}_{\overline{m}}k,m)\\ & \tau =g({\mathrm{\nabla }}_{\ell }k,m),\kappa =g({\mathrm{\nabla }}_{k}k,m)\end{array}$$
あるいは同値ですが
$$\begin{array}{rl}& {\mathrm{\nabla }}_{k}k=(\epsilon +\overline{\epsilon })k+\overline{\kappa }m+\kappa \overline{m}\\ & {\mathrm{\nabla }}_{\ell }\ell =-(\gamma +\overline{\gamma })\ell +\nu m+\overline{\nu }\overline{m}\\ & {\mathrm{\nabla }}_{m}m=\overline{\lambda }k-\sigma \ell -(\overline{\alpha }-\beta )m\\ & {\mathrm{\nabla }}_m\overline{m}=-\mu k-\overline{\rho }\ell +(\overline{\alpha }-\beta )\overline{m}\\ & {\mathrm{\nabla }}_m\ell =-(\overline{\alpha }+\beta )\ell +\mu m+\overline{\lambda }\overline{m}\\ & {\mathrm{\nabla }}_{\ell }m=-\overline{\nu }k-\tau \ell +(\gamma -\overline{\gamma })m\\ & {\mathrm{\nabla }}_{m}k=(\overline{\alpha }+\beta )k+\overline{\rho }m+\sigma \overline{m}\\ & {\mathrm{\nabla }}_{k}m=-\overline{\pi }k-\kappa \ell +(\epsilon -\overline{\epsilon })m\\ & {\mathrm{\nabla }}_{\ell }k=(\gamma +\overline{\gamma })k+\overline{\tau }m+\tau \overline{m}\\ & {\mathrm{\nabla }}_k\ell =-(\epsilon +\overline{\epsilon })\ell +\pi m+\overline{\pi }\overline{m}\end{array}$$
あるいは同値ですが
$$\begin{array}{rl}& [m,\overline{m}]=(\overline{\beta }-\alpha )m+(\overline{\alpha }-\beta )\overline{m}+(\rho -\overline{\rho })l+(\overline{\mu }-\mu )k\\ & [m,l]=(\mu +\overline{\gamma }-\gamma )m+\overline{\lambda }\overline{m}+(\tau -\overline{\alpha }-\beta )l+\overline{\nu }k\\ & [m,k]=(\overline{\rho }+\overline{\epsilon }-\epsilon )m+\sigma \overline{m}+\kappa l+(\overline{\alpha }+\beta +\overline{\pi })k\\ & [\ell ,k]=(\overline{\tau }-\pi )m+(\tau -\overline{\pi })\overline{m}+(\overline{\epsilon }+\epsilon )l+(\overline{\gamma }+\gamma )k\end{array}$$
で定義します。

Weylテンソル

　4次元時空$(M,g)$において、ワイルテンソル$W$は
$$\begin{array}{rl}& W_{abcd}=R_{abcd}+g_{ad}{\text{P}}_{cb}-g_{ac}{\text{P}}_{db}+g_{bc}{\text{P}}_{da}-g_{bd}{\text{P}}_{ca}\\ & {\text{P}}_{ab}=\frac{1}{2}{R}_{ab}-\frac{1}{12}R{g}_{ab}\end{array}$$
で与えられます。

　またWeylスカラーは以下で定義されます。
$$\begin{array}{rl}& {\mathrm{\Psi }}_0=W(k,m,k,m),{\mathrm{\Psi }}_1=W(k,l,k,m),{\mathrm{\Psi }}_2=W(k,m,l,\overline{m})\\ & {\mathrm{\Psi }}_3=W(l,k,l,\overline{m}),{\mathrm{\Psi }}_4=W(l,\overline{m},l,\overline{m})\end{array}$$
（ Petrov分類1 のWeylスカラーと${\mathrm{\Psi }}_2$の符号が違うので注意）

スピン係数の微分

　スピン係数の定義式をnull基底で共変微分の公式などを使い微分していくと機械的な計算で以下が得られます。
$$\begin{array}{rl}(1)& m\kappa =k\sigma +\overline{\alpha }\kappa +3\beta \kappa +\kappa \overline{\pi }-3\epsilon \sigma +\overline{\epsilon }\sigma +\rho \sigma +\overline{\rho }\sigma +\kappa \tau +{\mathrm{\Psi }}_0\\ (2)& k\overline{\beta }=\overline{m}\overline{\epsilon }-\alpha \overline{\epsilon }-\overline{\beta }\epsilon -\overline{\gamma }\overline{\kappa }-\overline{\kappa }\overline{\mu }-\overline{\epsilon }\pi -\overline{\beta }\rho -\overline{\alpha }\overline{\sigma }+\overline{\pi }\overline{\sigma }-{\overline{\mathrm{\Psi }}}_1\\ (3)& m\rho =\overline{m}\sigma +\kappa \overline{\mu }-\kappa \mu +\overline{\alpha }\rho +\beta \rho -3\alpha \sigma +\overline{\beta }\sigma -\overline{\rho }\tau +\rho \tau -{\mathrm{\Psi }}_1-{\text{P}}_{14}\\ (4)& k\tau =l\kappa -\overline{\gamma }\kappa -3\gamma \kappa +\overline{\pi }\rho +\pi \sigma -\sigma \overline{\tau }-\overline{\epsilon }\tau +\epsilon \tau -\rho \tau -{\mathrm{\Psi }}_1+{\text{P}}_{14}\\ (5)& l\rho =\overline{m}\tau -\kappa \nu +\gamma \rho +\overline{\gamma }\rho -\overline{\mu }\rho -\lambda \sigma -\alpha \tau +\overline{\beta }\tau -\tau \overline{\tau }-{\mathrm{\Psi }}_2-{\text{P}}_{12}-{\text{P}}_{34}\\ (6)& l\alpha =\overline{m}\gamma +\overline{\beta }\gamma +\alpha \overline{\gamma }-\beta \lambda -\alpha \overline{\mu }-\epsilon \nu +\nu \rho -\lambda \tau -\gamma \overline{\tau }+{\mathrm{\Psi }}_3\\ (7)& l\lambda =\overline{m}\nu -3\gamma \lambda +\overline{\gamma }\lambda -\lambda \mu -\lambda \overline{\mu }+3\alpha \nu +\overline{\beta }\nu -\nu \pi -\nu \overline{\tau }-{\mathrm{\Psi }}_4\\ (8)& k\lambda =\overline{m}\pi -3\epsilon \lambda +\overline{\epsilon }\lambda -\overline{\kappa }\nu +\alpha \pi -\overline{\beta }\pi -{\pi }^2-\lambda \rho -\mu \overline{\sigma }-{\text{P}}_{22}\\ (9)& k\mu =m\pi -\epsilon \mu -\overline{\epsilon }\mu -\kappa \nu -\overline{\alpha }\pi +\beta \pi -\pi \overline{\pi }-\mu \overline{\rho }-\lambda \sigma -{\mathrm{\Psi }}_2-{\text{P}}_{12}-{\text{P}}_{34}\\ (10)& k\alpha =\overline{m}\epsilon +\alpha \overline{\epsilon }-2\alpha \epsilon -\overline{\beta }\epsilon -\gamma \overline{\kappa }-\kappa \lambda -\epsilon \pi -\alpha \rho +\pi \rho -\beta \overline{\sigma }+{\text{P}}_{24}\\ (11)& l\beta =m\gamma +\overline{\alpha }\gamma +2\beta \gamma -\beta \overline{\gamma }-\alpha \overline{\lambda }-\beta \mu -\epsilon \overline{\nu }+\nu \sigma -\gamma \tau -\mu \tau -{\text{P}}_{13}\\ (12)& k\rho =\overline{m}\kappa -3\alpha \kappa -\overline{\beta }\kappa -\kappa \pi +\epsilon \rho +\overline{\epsilon }\rho -{\rho }^2-\sigma \overline{\sigma }-\kappa \overline{\tau }-{\text{P}}_{44}\\ (13)& l\mu =m\nu -\lambda \overline{\lambda }-\gamma \mu -\overline{\gamma }\mu -{\mu }^2+\overline{\alpha }\nu +3\beta \nu -\nu \pi -\nu \tau -{\text{P}}_{33}\\ (14)& k\nu =l\pi -\overline{\epsilon }\nu -3\epsilon \nu +\lambda \overline{\pi }-\overline{\gamma }\pi +\gamma \pi +\mu \pi -\mu \overline{\tau }-\lambda \tau +{\mathrm{\Psi }}_3-{\text{P}}_{23}\\ (15)& k\gamma =l\epsilon -2\epsilon \gamma -\overline{\epsilon }\gamma -\epsilon \overline{\gamma }-\kappa \nu +\beta \pi +\alpha \overline{\pi }-\alpha \tau +\pi \tau -\beta \overline{\tau }-{\mathrm{\Psi }}_2+{\text{P}}_{34}\\ (16)& \overline{m}\mu =m\lambda -\overline{\alpha }\lambda +3\beta \lambda -\alpha \mu -\overline{\beta }\mu +\mu \pi -\overline{\mu }\pi -\nu \rho +\nu \overline{\rho }-{\mathrm{\Psi }}_3-{\text{P}}_{23}\\ (17)& m\tau =l\sigma +\kappa \overline{\nu }+\overline{\lambda }\rho -3\gamma \sigma +\overline{\gamma }\sigma +\mu \sigma -\overline{\alpha }\tau +\beta \tau +{\tau }^2+{\text{P}}_{11}\\ (18)& m\alpha =\overline{m}\beta +\alpha \overline{\alpha }-2\alpha \beta +\beta \overline{\beta }-\epsilon \mu +\epsilon \overline{\mu }+\gamma \rho +\mu \rho -\gamma \overline{\rho }-\lambda \sigma -{\mathrm{\Psi }}_2+{\text{P}}_{12}\end{array}$$

Weylスカラーの微分

　Weyl曲率の成分の微分は
$$\begin{array}{r}k[W(\ell ,m,v,w)]=({\mathrm{\nabla }}_{k}W)(\ell ,m,v,w)+W({\mathrm{\nabla }}_k\ell ,m,v,w)+W(\ell ,{\mathrm{\nabla }}_{k}m,v,w)\\ +W(\ell ,m,{\mathrm{\nabla }}_{k}v,w)+W(\ell ,m,v,{\mathrm{\nabla }}_{k}w)\end{array}$$
と表されますが、$\mathrm{\nabla }W$はリーマンテンソルと同様にBianchi恒等式を満たすので、例えば$v=k,w=m$などとすると
$$\begin{array}{rl}& k[W(\ell ,m,k,m)]+\ell [W(m,k,k,m)]+m[W(k,\ell ,k,m)]=\\ & W({\mathrm{\nabla }}_k\ell ,m,k,m)+W(\ell ,{\mathrm{\nabla }}_{k}m,k,m)+W(\ell ,m,{\mathrm{\nabla }}_{k}k,m)+W(\ell ,m,k,{\mathrm{\nabla }}_{k}m)\\ & +W({\mathrm{\nabla }}_{\ell }m,k,k,m)+W(m,{\mathrm{\nabla }}_{\ell }k,k,m)+W(m,k,{\mathrm{\nabla }}_{\ell }k,m)+W(m,k,k,{\mathrm{\nabla }}_{\ell }m)\\ & +W({\mathrm{\nabla }}_{m}k,\ell ,k,m)+W(k,{\mathrm{\nabla }}_m\ell ,k,m)+W(k,\ell ,{\mathrm{\nabla }}_{m}k,m)+W(k,\ell ,k,{\mathrm{\nabla }}_{m}m)\end{array}$$
が成り立ちます。（ Petrov分類1 で見たようにWeylテンソルの０になる成分などを考慮すると(32)が得られます。
同様に以下の式が得られます。

$$\begin{array}{rl}(19)& m{\mathrm{\Psi }}_1=l{\mathrm{\Psi }}_0-k{\text{P}}_{11}+m{\text{P}}_{14}-4\gamma {\mathrm{\Psi }}_0+\mu {\mathrm{\Psi }}_0+2\beta {\mathrm{\Psi }}_1-3\sigma {\mathrm{\Psi }}_2+4\tau {\mathrm{\Psi }}_1-2\kappa {\text{P}}_{13}+2\epsilon {\text{P}}_{11}-2\overline{\epsilon }{\text{P}}_{11}-2\beta {\text{P}}_{14}-2\overline{\pi }{\text{P}}_{14}+\overline{\lambda }{\text{P}}_{44}-\overline{\rho }{\text{P}}_{11}-\sigma {\text{P}}_{12}+\sigma {\text{P}}_{34}\\ (20)& k{\mathrm{\Psi }}_1=-\overline{m}{\mathrm{\Psi }}_0-k{\text{P}}_{14}+m{\text{P}}_{44}+4\alpha {\mathrm{\Psi }}_0+\pi {\mathrm{\Psi }}_0+2\epsilon {\mathrm{\Psi }}_1-3\kappa {\mathrm{\Psi }}_2-4{\mathrm{\Psi }}_1\rho +\overline{\kappa }{\text{P}}_{11}+\kappa {\text{P}}_{12}+2\epsilon {\text{P}}_{14}-\kappa {\text{P}}_{34}-2\overline{\alpha }{\text{P}}_{44}-2\beta {\text{P}}_{44}-\overline{\pi }{\text{P}}_{44}-2\rho {\text{P}}_{14}-2\sigma {\text{P}}_{24}\\ (21)& l{\mathrm{\Psi }}_1=m{\mathrm{\Psi }}_2+k{\text{P}}_{13}-m{\text{P}}_{34}+\nu {\mathrm{\Psi }}_0+2\gamma {\mathrm{\Psi }}_1-2\mu {\mathrm{\Psi }}_1-2\sigma {\mathrm{\Psi }}_3-3\tau {\mathrm{\Psi }}_2-\pi {\text{P}}_{11}-\overline{\pi }{\text{P}}_{12}+2\overline{\epsilon }{\text{P}}_{13}+\mu {\text{P}}_{14}+\overline{\lambda }{\text{P}}_{24}+\kappa {\text{P}}_{33}+\overline{\pi }{\text{P}}_{34}+\overline{\rho }{\text{P}}_{13}+\sigma {\text{P}}_{23}\\ (23)& m{\overline{\mathrm{\Psi }}}_1=-k{\overline{\mathrm{\Psi }}}_2+k{\text{P}}_{12}-\overline{m}{\text{P}}_{14}+\overline{\lambda }{\overline{\mathrm{\Psi }}}_0+2\overline{\alpha }{\overline{\mathrm{\Psi }}}_1+2\overline{\pi }{\overline{\mathrm{\Psi }}}_1+2\overline{\kappa }{\overline{\mathrm{\Psi }}}_3-3\overline{\rho }{\overline{\mathrm{\Psi }}}_2+\kappa {\text{P}}_{23}+\overline{\pi }{\text{P}}_{24}+\overline{\kappa }{\text{P}}_{13}+2\alpha {\text{P}}_{14}+\pi {\text{P}}_{14}-\overline{\mu }{\text{P}}_{44}+\rho {\text{P}}_{12}-\rho {\text{P}}_{34}+\overline{\sigma }{\text{P}}_{11}\\ (24)& l{\mathrm{\Psi }}_2=-m{\mathrm{\Psi }}_3+l{\text{P}}_{12}-\overline{m}{\text{P}}_{13}+2\nu {\mathrm{\Psi }}_1-3\mu {\mathrm{\Psi }}_2-2\beta {\mathrm{\Psi }}_3+2\tau {\mathrm{\Psi }}_3+\sigma {\mathrm{\Psi }}_4+\lambda {\text{P}}_{11}+\overline{\mu }{\text{P}}_{12}-2\overline{\beta }{\text{P}}_{13}+\nu {\text{P}}_{14}+\overline{\nu }{\text{P}}_{24}-\rho {\text{P}}_{33}-\overline{\mu }{\text{P}}_{34}+\tau {\text{P}}_{23}+\overline{\tau }{\text{P}}_{13}\\ (25)& k{\mathrm{\Psi }}_3=\overline{m}{\mathrm{\Psi }}_2+l{\text{P}}_{24}-\overline{m}{\text{P}}_{34}-2\lambda {\mathrm{\Psi }}_1-3\pi {\mathrm{\Psi }}_2-2\epsilon {\mathrm{\Psi }}_3+\kappa {\mathrm{\Psi }}_4-2\rho {\mathrm{\Psi }}_3+\lambda {\text{P}}_{14}+\rho {\text{P}}_{23}-2\overline{\gamma }{\text{P}}_{24}+\overline{\mu }{\text{P}}_{24}+\nu {\text{P}}_{44}+\overline{\sigma }{\text{P}}_{13}-\tau {\text{P}}_{22}-\overline{\tau }{\text{P}}_{12}+\overline{\tau }{\text{P}}_{34}\end{array}$$

$$\begin{array}{rl}(26)& l{\mathrm{\Psi }}_3=-m{\mathrm{\Psi }}_4-l{\text{P}}_{23}+\overline{m}{\text{P}}_{33}-3\nu {\mathrm{\Psi }}_2-2\gamma {\mathrm{\Psi }}_3-4\mu {\mathrm{\Psi }}_3-4\beta {\mathrm{\Psi }}_4+\tau {\mathrm{\Psi }}_4+\nu {\text{P}}_{12}-2\lambda {\text{P}}_{13}+\overline{\nu }{\text{P}}_{22}-2\gamma {\text{P}}_{23}-2\overline{\mu }{\text{P}}_{23}+2\alpha {\text{P}}_{33}+2\overline{\beta }{\text{P}}_{33}-\nu {\text{P}}_{34}-\overline{\tau }{\text{P}}_{33}\\ (27)& \overline{m}{\mathrm{\Psi }}_3=k{\mathrm{\Psi }}_4-l{\text{P}}_{22}+\overline{m}{\text{P}}_{23}-3\lambda {\mathrm{\Psi }}_2-2\alpha {\mathrm{\Psi }}_3+4\pi {\mathrm{\Psi }}_3+4\epsilon {\mathrm{\Psi }}_4+\rho {\mathrm{\Psi }}_4+2\overline{\gamma }{\text{P}}_{22}-2\gamma {\text{P}}_{22}-\overline{\mu }{\text{P}}_{22}-\lambda {\text{P}}_{12}+2\alpha {\text{P}}_{23}-2\nu {\text{P}}_{24}+\lambda {\text{P}}_{34}+\overline{\sigma }{\text{P}}_{33}-2\overline{\tau }{\text{P}}_{23}\\ (28)& m{\text{P}}_{12}=k{\text{P}}_{13}+l{\text{P}}_{14}+\overline{m}{\text{P}}_{11}-2m{\text{P}}_{34}-2\alpha {\text{P}}_{11}+2\overline{\beta }{\text{P}}_{11}-\pi {\text{P}}_{11}-\overline{\pi }{\text{P}}_{12}+2\overline{\epsilon }{\text{P}}_{13}+2\rho {\text{P}}_{13}-2\gamma {\text{P}}_{14}+\mu {\text{P}}_{14}+2\overline{\mu }{\text{P}}_{14}+\overline{\lambda }{\text{P}}_{24}+\kappa {\text{P}}_{33}+\overline{\pi }{\text{P}}_{34}+\overline{\nu }{\text{P}}_{44}+\overline{\rho }{\text{P}}_{13}+\sigma {\text{P}}_{23}-\tau {\text{P}}_{12}+\tau {\text{P}}_{34}-\overline{\tau }{\text{P}}_{11}\\ (29)& k{\text{P}}_{34}=-2k{\text{P}}_{12}+l{\text{P}}_{44}+\overline{m}{\text{P}}_{14}+m{\text{P}}_{24}-\rho {\text{P}}_{12}-\overline{\kappa }{\text{P}}_{13}-2\alpha {\text{P}}_{14}-\pi {\text{P}}_{14}-\kappa {\text{P}}_{23}-2\overline{\alpha }{\text{P}}_{24}-\overline{\pi }{\text{P}}_{24}+\rho {\text{P}}_{34}-2\gamma {\text{P}}_{44}-2\overline{\gamma }{\text{P}}_{44}+\mu {\text{P}}_{44}+\overline{\mu }{\text{P}}_{44}-\overline{\rho }{\text{P}}_{12}+\overline{\rho }{\text{P}}_{34}-\sigma {\text{P}}_{22}-\overline{\sigma }{\text{P}}_{11}-2\tau {\text{P}}_{24}-2\overline{\tau }{\text{P}}_{14}\\ (30)& l{\text{P}}_{34}=k{\text{P}}_{33}-2l{\text{P}}_{12}+\overline{m}{\text{P}}_{13}+m{\text{P}}_{23}-\lambda {\text{P}}_{11}-\mu {\text{P}}_{12}-\overline{\mu }{\text{P}}_{12}+2\overline{\beta }{\text{P}}_{13}-2\pi {\text{P}}_{13}-\nu {\text{P}}_{14}-\overline{\lambda }{\text{P}}_{22}+2\beta {\text{P}}_{23}-2\overline{\pi }{\text{P}}_{23}-\overline{\nu }{\text{P}}_{24}+2\epsilon {\text{P}}_{33}+2\overline{\epsilon }{\text{P}}_{33}+\rho {\text{P}}_{33}+\mu {\text{P}}_{34}+\overline{\mu }{\text{P}}_{34}+\overline{\rho }{\text{P}}_{33}-\tau {\text{P}}_{23}-\overline{\tau }{\text{P}}_{13}\end{array}$$