【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$\{m,\bar m,l,k\}$とします。

スピン係数

　$e_1=m,e_2=\bar m,e_3=l,e_4=k$とし、$\Gamma_{ij}=g(e_i,\nabla e_j)$と置きます。スピノル束を$S=S^+\oplus S^-$とし、o.n.f.$\{\bar e_1,\bar e_2,\bar e_3,\bar e_4\}$を
\begin{align} &e_1=\frac{1}{\sqrt{2}}(\bar{e}_1-i\bar{e}_2),\ e_2=\frac{1}{\sqrt{2}}(\bar{e}_1+i\bar{e}_2)\\ &e_3=-\frac{1}{\sqrt{2}}(\bar{e}_0+\bar{e}_3),\ e_4=\frac{1}{\sqrt{2}}(\bar{e}_0-\bar{e}_3) \end{align}
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{align} \Gamma^{S^+}=\begin{pmatrix}- \frac{\Gamma_{12}}{2} + \frac{\Gamma_{34}}{2} & - \Gamma_{14} \\ \overline{\Gamma_{13}} & \frac{\Gamma_{12}}{2} - \frac{\Gamma_{34}}{2} \end{pmatrix} \end{align}
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{align} \frac{1}{2}(-\Gamma_{12}+\Gamma_{34})=\frac{1}{2}(g(\nabla k,\ell)-g(\nabla\bar m,m))&=\beta\theta^1+\alpha\theta^2+\gamma\theta^3+\varepsilon\theta^4\\ \Gamma_{23}=g(\nabla \ell,\bar m)&=\mu\theta^1+\lambda\theta^2+\nu\theta^3+\pi\theta^4\\ \Gamma_{14}=g(\nabla k,m)&=\sigma\theta^1+\rho\theta^2+\tau\theta^3+\kappa\theta^4 \end{align}
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{align} &\alpha = \frac{1}{2}[g(\nabla_{\bar{m}}k,\ell) - g(\nabla_{\bar{m}}\bar m,m)] \\ &\beta = \frac{1}{2}[g(\nabla_mk,\ell) - g(\nabla_m\bar{m},m)] \\ &\gamma = \frac{1}{2}[g(\nabla_\ell k,\ell) - g(\nabla_\ell\bar{m},m)] \\ &\varepsilon = \frac{1}{2}[g(\nabla_kk,\ell) - g(\nabla_k\bar{m},m)] \\ &\mu = g(\nabla_m\ell,\bar m),\ \ \lambda = g(\nabla_{\bar m}\ell,\bar m) \\ &\nu = g(\nabla_\ell\ell,\bar m),\ \ \pi = g(\nabla_k\ell,\bar m) \\ &\sigma = g(\nabla_mk,m),\ \ \rho = g(\nabla_{\bar m}k,m) \\ &\tau = g(\nabla_\ell k,m),\ \ \kappa = g(\nabla_kk,m) \\ \end{align}
あるいは同値ですが
\begin{align} &\nabla_k k = (\varepsilon + \bar{\varepsilon})k + \bar{\kappa}m + \kappa\bar{m} \\ &\nabla_\ell\ell = -(\gamma + \bar{\gamma})\ell + \nu m + \bar{\nu}\bar{m} \\ &\nabla_m m = \bar{\lambda}k - \sigma\ell - (\bar{\alpha} - \beta)m\\ &\nabla_m\bar{m} = -\mu k - \bar{\rho}\ell + (\bar{\alpha} - \beta)\bar{m}\\ &\nabla_m\ell = -(\bar{\alpha} + \beta)\ell + \mu m + \bar{\lambda}\bar{m}\\ &\nabla_\ell m = -\bar{\nu}k - \tau\ell + (\gamma - \bar{\gamma})m \\ &\nabla_m k = (\bar{\alpha} + \beta)k + \bar{\rho}m + \sigma\bar{m}\\ &\nabla_k m = -\bar{\pi}k - \kappa\ell + (\varepsilon - \bar{\varepsilon})m \\ &\nabla_\ell k = (\gamma + \bar{\gamma})k + \bar{\tau}m + \tau\bar{m}\\ &\nabla_k\ell = -(\varepsilon + \bar{\varepsilon})\ell + \pi m + \bar{\pi}\bar{m}\\ \end{align}
あるいは同値ですが
\begin{align} &[m,\bar{m}] = (\bar{\beta} - \alpha)m + (\bar{\alpha} - \beta)\bar{m} + (\rho - \bar{\rho})l + (\bar{\mu} - \mu)k \\ &[m,l] = (\mu + \bar{\gamma} - \gamma)m + \bar{\lambda}\bar{m} + (\tau - \bar{\alpha} - \beta)l + \bar{\nu}k \\ &[m,k] = (\bar{\rho} + \bar{\varepsilon} - \varepsilon)m + \sigma\bar{m} + \kappa l + (\bar{\alpha} + \beta + \bar{\pi})k \\ &[\ell,k] = (\bar{\tau} - \pi)m + (\tau - \bar{\pi})\bar{m} + (\bar{\varepsilon} + \varepsilon)l + (\bar{\gamma} + \gamma)k \end{align}
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{align} & 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}Rg_{ab} \end{align}
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{align} &\Psi_0=W(k,m,k,m),\Psi_1=W(k,l,k,m),\Psi_2=W(k,m,l,\bar m)\\ &\Psi_3=W(l,k,l,\bar m),\Psi_4=W(l,\bar m,l,\bar m) \end{align}
（ Petrov分類1 のWeylスカラーと$\Psi_2$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{align} (1) & m\kappa = k\sigma + \bar{\alpha}\kappa + 3\beta\kappa + \kappa\bar{\pi} - 3\varepsilon\sigma + \bar{\varepsilon}\sigma + \rho\sigma + \bar{\rho}\sigma + \kappa\tau + \Psi_0 \\ (2) & k\bar{\beta} = \bar{m}\bar{\varepsilon} - \alpha\bar{\varepsilon} - \bar{\beta}\varepsilon - \bar{\gamma}\bar{\kappa} - \bar{\kappa}\bar{\mu} - \bar{\varepsilon}\pi - \bar{\beta}\rho - \bar{\alpha}\bar{\sigma} + \bar{\pi}\bar{\sigma} - \bar{\Psi}_1 \\ (3) & m\rho = \bar{m}\sigma + \kappa\bar{\mu} - \kappa\mu + \bar{\alpha}\rho + \beta\rho - 3\alpha\sigma + \bar{\beta}\sigma - \bar\rho\tau + \rho\tau - \Psi_1 - \text{P}_{14} \\ (4) & k\tau = l\kappa - \bar{\gamma}\kappa - 3\gamma\kappa + \bar{\pi}\rho + \pi\sigma - \sigma\bar{\tau} - \bar{\varepsilon}\tau + \varepsilon\tau - \rho\tau - \Psi_1 + \text{P}_{14} \\ (5) & l\rho = \bar{m}\tau - \kappa\nu + \gamma\rho + \bar{\gamma}\rho - \bar{\mu}\rho - \lambda\sigma - \alpha\tau + \bar{\beta}\tau - \tau\bar{\tau} - \Psi_2 - \text{P}_{12} - \text{P}_{34} \\ (6) & l\alpha = \bar{m}\gamma + \bar{\beta}\gamma + \alpha\bar{\gamma} - \beta\lambda - \alpha\bar{\mu} - \varepsilon\nu + \nu\rho - \lambda\tau - \gamma\bar{\tau} + \Psi_3 \\ (7) & l\lambda = \bar{m}\nu - 3\gamma\lambda + \bar{\gamma}\lambda - \lambda\mu - \lambda\bar{\mu} + 3\alpha\nu + \bar{\beta}\nu - \nu\pi - \nu\bar{\tau} - \Psi_4 \\ (8) & k\lambda = \bar{m}\pi - 3\varepsilon\lambda + \bar{\varepsilon}\lambda - \bar{\kappa}\nu + \alpha\pi - \bar{\beta}\pi - \pi^2 - \lambda\rho - \mu\bar{\sigma} - \text{P}_{22} \\ (9) & k\mu = m\pi - \varepsilon\mu - \bar{\varepsilon}\mu - \kappa\nu - \bar{\alpha}\pi + \beta\pi - \pi\bar{\pi} - \mu\bar{\rho} - \lambda\sigma - \Psi_2 - \text{P}_{12} - \text{P}_{34} \\ (10) & k\alpha = \bar{m}\varepsilon + \alpha\bar{\varepsilon} - 2\alpha\varepsilon - \bar{\beta}\varepsilon - \gamma\bar{\kappa} - \kappa\lambda - \varepsilon\pi - \alpha\rho + \pi\rho - \beta\bar{\sigma} + \text{P}_{24} \\ (11) & l\beta = m\gamma + \bar{\alpha}\gamma + 2\beta\gamma - \beta\bar{\gamma} - \alpha\bar{\lambda} - \beta\mu - \varepsilon\bar{\nu} + \nu\sigma - \gamma\tau - \mu\tau - \text{P}_{13} \\ (12) & k\rho = \bar{m}\kappa - 3\alpha\kappa - \bar{\beta}\kappa - \kappa\pi + \varepsilon\rho + \bar{\varepsilon}\rho - \rho^2 - \sigma\bar{\sigma} - \kappa\bar{\tau} - \text{P}_{44} \\ (13) & l\mu = m\nu - \lambda\bar{\lambda} - \gamma\mu - \bar{\gamma}\mu - \mu^2 + \bar{\alpha}\nu + 3\beta\nu - \nu\pi - \nu\tau - \text{P}_{33} \\ (14) & k\nu = l\pi - \bar{\varepsilon}\nu - 3\varepsilon\nu + \lambda\bar{\pi} - \bar{\gamma}\pi + \gamma\pi + \mu\pi - \mu\bar{\tau} - \lambda\tau + \Psi_3 - \text{P}_{23} \\ (15) & k\gamma = l\varepsilon - 2\varepsilon\gamma - \bar{\varepsilon}\gamma - \varepsilon\bar{\gamma} - \kappa\nu + \beta\pi + \alpha\bar{\pi} - \alpha\tau + \pi\tau - \beta\bar{\tau} - \Psi_2 + \text{P}_{34} \\ (16) & \bar{m}\mu = m\lambda - \bar{\alpha}\lambda + 3\beta\lambda - \alpha\mu - \bar{\beta}\mu + \mu\pi - \bar{\mu}\pi - \nu\rho + \nu\bar{\rho} - \Psi_3 - \text{P}_{23} \\ (17) & m\tau = l\sigma + \kappa\bar{\nu} + \bar{\lambda}\rho - 3\gamma\sigma + \bar{\gamma}\sigma + \mu\sigma - \bar{\alpha}\tau + \beta\tau + \tau^2 + \text{P}_{11} \\ (18) & m\alpha = \bar{m}\beta + \alpha\bar{\alpha} - 2\alpha\beta + \beta\bar{\beta} - \varepsilon\mu + \varepsilon\bar{\mu} + \gamma\rho + \mu\rho - \gamma\bar{\rho} - \lambda\sigma - \Psi_2 + \text{P}_{12} \end{align}

Weylスカラーの微分

　Weyl曲率の成分の微分は
\begin{align} k[W(\ell, m, v, w)] = (\nabla_k W)(\ell, m, v, w) + W(\nabla_k\ell, m, v, w) + W(\ell, \nabla_km, v, w) \\ + W(\ell, m, \nabla_kv, w) + W(\ell, m, v, \nabla_kw) \end{align}
と表されますが、$\nabla W$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{align} &k[W(\ell, m, k, m)] + \ell[W(m, k, k, m)] + m[W(k, \ell, k, m)] = \\ &W(\nabla_k\ell, m, k, m) + W(\ell, \nabla_km, k, m) + W(\ell, m, \nabla_kk, m) + W(\ell, m, k, \nabla_km) \\ &+W(\nabla_\ell m, k, k, m) + W(m, \nabla_\ell k, k, m) + W(m, k, \nabla_\ell k, m) + W(m, k, k, \nabla_\ell m) \\ &+W(\nabla_mk, \ell, k, m) + W(k, \nabla_m\ell, k, m) + W(k, \ell, \nabla_mk, m) + W(k, \ell, k, \nabla_mm) \end{align}
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{align} (19) & m\Psi_1 = l\Psi_0 - k\text{P}_{11} + m\text{P}_{14} - 4\gamma\Psi_0 + \mu\Psi_0 + 2\beta\Psi_1 - 3\sigma\Psi_2 + 4\tau\Psi_1 - 2\kappa\text{P}_{13} + 2\varepsilon\text{P}_{11} - 2\bar{\varepsilon}\text{P}_{11} - 2\beta\text{P}_{14} - 2\bar{\pi}\text{P}_{14} + \bar{\lambda}\text{P}_{44} - \bar{\rho}\text{P}_{11} - \sigma\text{P}_{12} + \sigma\text{P}_{34} \\ (20) & k\Psi_1 = -\bar{m}\Psi_0 - k\text{P}_{14} + m\text{P}_{44} + 4\alpha\Psi_0 + \pi\Psi_0 + 2\varepsilon\Psi_1 - 3\kappa\Psi_2 - 4\Psi_1\rho + \bar{\kappa}\text{P}_{11} + \kappa\text{P}_{12} + 2\varepsilon\text{P}_{14} - \kappa\text{P}_{34} - 2\bar{\alpha}\text{P}_{44} - 2\beta\text{P}_{44} - \bar{\pi}\text{P}_{44} - 2\rho\text{P}_{14} - 2\sigma\text{P}_{24} \\ (21) & l\Psi_1 = m\Psi_2 + k\text{P}_{13} - m\text{P}_{34} + \nu\Psi_0 + 2\gamma\Psi_1 - 2\mu\Psi_1 - 2\sigma\Psi_3 - 3\tau\Psi_2 - \pi\text{P}_{11} - \bar{\pi}\text{P}_{12} + 2\bar{\varepsilon}\text{P}_{13} + \mu\text{P}_{14} + \bar{\lambda}\text{P}_{24} + \kappa\text{P}_{33} + \bar{\pi}\text{P}_{34} + \bar{\rho}\text{P}_{13} + \sigma\text{P}_{23} \\ (23) & m\bar{\Psi}_1 = -k\bar{\Psi}_2 + k\text{P}_{12} - \bar{m}\text{P}_{14} + \bar{\lambda}\bar{\Psi}_0 + 2\bar{\alpha}\bar{\Psi}_1 + 2\bar{\pi}\bar{\Psi}_1 + 2\bar{\kappa}\bar{\Psi}_3 - 3\bar{\rho}\bar{\Psi}_2 + \kappa\text{P}_{23} + \bar{\pi}\text{P}_{24} + \bar{\kappa}\text{P}_{13} + 2\alpha\text{P}_{14} + \pi\text{P}_{14} - \bar{\mu}\text{P}_{44} + \rho\text{P}_{12} - \rho\text{P}_{34} + \bar{\sigma}\text{P}_{11} \\ (24) & l\Psi_2 = -m\Psi_3 + l\text{P}_{12} - \bar{m}\text{P}_{13} + 2\nu\Psi_1 - 3\mu\Psi_2 - 2\beta\Psi_3 + 2\tau\Psi_3 + \sigma\Psi_4 + \lambda\text{P}_{11} + \bar\mu\text{P}_{12} - 2\bar{\beta}\text{P}_{13} + \nu\text{P}_{14} + \bar{\nu}\text{P}_{24} - \rho\text{P}_{33} - \bar{\mu}\text{P}_{34} + \tau\text{P}_{23} + \bar{\tau}\text{P}_{13} \\ (25) & k\Psi_3 = \bar{m}\Psi_2 + l\text{P}_{24} - \bar{m}\text{P}_{34} - 2\lambda\Psi_1 - 3\pi\Psi_2 - 2\varepsilon\Psi_3 + \kappa\Psi_4 - 2\rho\Psi_3 + \lambda\text{P}_{14} + \rho\text{P}_{23} - 2\bar{\gamma}\text{P}_{24} + \bar{\mu}\text{P}_{24} + \nu\text{P}_{44} + \bar{\sigma}\text{P}_{13} - \tau\text{P}_{22} - \bar{\tau}\text{P}_{12} + \bar{\tau}\text{P}_{34} \end{align}

\begin{align} (26) & l\Psi_3 = -m\Psi_4 - l\text{P}_{23} + \bar{m}\text{P}_{33} - 3\nu\Psi_2 - 2\gamma\Psi_3 - 4\mu\Psi_3 - 4\beta\Psi_4 + \tau\Psi_4 + \nu\text{P}_{12} - 2\lambda\text{P}_{13} + \bar{\nu}\text{P}_{22} - 2\gamma\text{P}_{23} - 2\bar{\mu}\text{P}_{23} + 2\alpha\text{P}_{33} + 2\bar{\beta}\text{P}_{33} - \nu\text{P}_{34} - \bar{\tau}\text{P}_{33} \\ (27) & \bar{m}\Psi_3 = k\Psi_4 - l\text{P}_{22} + \bar{m}\text{P}_{23} - 3\lambda\Psi_2 - 2\alpha\Psi_3 + 4\pi\Psi_3 + 4\varepsilon\Psi_4 + \rho\Psi_4 + 2\bar{\gamma}\text{P}_{22} - 2\gamma\text{P}_{22} - \bar{\mu}\text{P}_{22} - \lambda\text{P}_{12} + 2\alpha\text{P}_{23} - 2\nu\text{P}_{24} + \lambda\text{P}_{34} + \bar{\sigma}\text{P}_{33} - 2\bar{\tau}\text{P}_{23} \\ (28) & m\text{P}_{12} = k\text{P}_{13} + l\text{P}_{14} + \bar{m}\text{P}_{11} - 2m\text{P}_{34} - 2\alpha\text{P}_{11} + 2\bar{\beta}\text{P}_{11} - \pi\text{P}_{11} - \bar{\pi}\text{P}_{12} + 2\bar{\varepsilon}\text{P}_{13} + 2\rho\text{P}_{13} - 2\gamma\text{P}_{14} + \mu\text{P}_{14} + 2\bar{\mu}\text{P}_{14} + \bar{\lambda}\text{P}_{24} + \kappa\text{P}_{33} + \bar{\pi}\text{P}_{34} + \bar{\nu}\text{P}_{44} + \bar{\rho}\text{P}_{13} + \sigma\text{P}_{23} - \tau\text{P}_{12} + \tau\text{P}_{34} - \bar{\tau}\text{P}_{11} \\ (29) & k\text{P}_{34} = -2k\text{P}_{12} + l\text{P}_{44} + \bar{m}\text{P}_{14} + m\text{P}_{24} - \rho\text{P}_{12} - \bar{\kappa}\text{P}_{13} - 2\alpha\text{P}_{14} - \pi\text{P}_{14} - \kappa\text{P}_{23} - 2\bar{\alpha}\text{P}_{24} - \bar{\pi}\text{P}_{24} + \rho\text{P}_{34} - 2\gamma\text{P}_{44} - 2\bar{\gamma}\text{P}_{44} + \mu\text{P}_{44} + \bar{\mu}\text{P}_{44} - \bar{\rho}\text{P}_{12} + \bar{\rho}\text{P}_{34} - \sigma\text{P}_{22} - \bar{\sigma}\text{P}_{11} - 2\tau\text{P}_{24} - 2\bar{\tau}\text{P}_{14} \\ (30) & l\text{P}_{34} = k\text{P}_{33} - 2l\text{P}_{12} + \bar{m}\text{P}_{13} + m\text{P}_{23} - \lambda\text{P}_{11} - \mu\text{P}_{12} - \bar{\mu}\text{P}_{12} + 2\bar{\beta}\text{P}_{13} - 2\pi\text{P}_{13} - \nu\text{P}_{14} - \bar{\lambda}\text{P}_{22} + 2\beta\text{P}_{23} - 2\bar{\pi}\text{P}_{23} - \bar{\nu}\text{P}_{24} + 2\varepsilon\text{P}_{33} + 2\bar{\varepsilon}\text{P}_{33} + \rho\text{P}_{33} + \mu\text{P}_{34} + \bar{\mu}\text{P}_{34} + \bar{\rho}\text{P}_{33} - \tau\text{P}_{23} - \bar{\tau}\text{P}_{13} \end{align}