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 p p是4维时空 ( M , g a b ) (M, g_{ab}) (M,gab)的一点， { ( e μ ) a } \{(e_\mu)^a\} {(eμ)a}是 p p p点的一个正交归一标架，定义 p p p点的4个特殊矢量如下：
l a = 1 2 [ ( e 0 ) a + ( e 1 ) a ] , n a = 1 2 [ ( e 0 ) a − ( e 1 ) a ] m a = 1 2 [ ( e 2 ) a + i ( e 3 ) a ] , m ˉ a = 1 2 [ ( e 2 ) a − i ( e 3 ) a ] 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] la=2 1[(e0)a+(e1)a], na=2 1[(e0)a−(e1)a]ma=2 1[(e2)a+i(e3)a], mˉa=2 1[(e2)a−i(e3)a]
容易验证 l a l a = n a n a = m a m a = m ˉ a m ˉ a = 0 l_al^a = n_an^a = m_am^a = \bar m_a\bar m^a = 0 lala=nana=mama=mˉamˉa=0，即它们都是类光矢量。这4个矢量构成 p p p点的一个基底，称为 p p p点的一个类光标架。以下以 { ( ε μ ) a } \{(\varepsilon_\mu)^a\} {(εμ)a}代表类光标架，并规定其编号为
( ε 1 ) a = l a , ( ε 2 ) a = n a , ( ε 3 ) a = m a , ( ε 4 ) a = m ˉ a (\varepsilon_1)^a = l^a, ~ (\varepsilon_2)^a = n^a, ~ (\varepsilon_3)^a = m^a, ~ (\varepsilon_4)^a = \bar m^a (ε1)a=la, (ε2)a=na, (ε3)a=ma, (ε4)a=mˉa
相应的对偶基矢为
( ε 1 ) a = n a , ( ε 2 ) a = l a , ( ε 3 ) a = − m ˉ a , ( ε 4 ) a = − m a (\varepsilon^1)_a = n_a, ~ (\varepsilon^2)_a = l_a, ~ (\varepsilon^3)_a = -\bar m_a, ~ (\varepsilon^4)_a = -m^a (ε1)a=na, (ε2)a=la, (ε3)a=−mˉa, (ε4)a=−ma
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 m ˉ a = − 1 l^an_a = 1, ~ m^a\bar m_a = -1 lana=1, mamˉa=−1
因此度规 g a b g_{ab} gab及其逆 g a b g^{ab} gab在该标架的分量 g μ ν g_{\mu\nu} gμν和 g μ ν g^{\mu\nu} gμν组成的矩阵为
( g μ ν ) = [ 0 1 0 0 1 0 0 0 0 0 0 − 1 0 0 − 1 0 ] = ( g μ ν ) (g_{\mu\nu}) = \begin{bmatrix} 0 &1 &0 &0 \\ 1 &0 &0 &0 \\ 0 &0 &0 &-1 \\ 0 &0 &-1 &0 \end{bmatrix} = (g^{\mu\nu}) (gμν)=⎣⎢⎢⎡01001000000−100−10⎦⎥⎥⎤=(gμν)
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
( ω μ ν ) a = ( ε μ ) c ∇ a ( ε ν ) c (\omega_\mu{}^\nu)_a = (\varepsilon_\mu)^c\nabla_a(\varepsilon^\nu)_c (ωμν)a=(εμ)c∇a(εν)c
同样地
( ω μ ν ) a = ( ε μ ) b ∇ a ( ε ν ) b = − ( ω ν μ ) a (\omega_{\mu\nu})_a = (\varepsilon_\mu)^b\nabla_a(\varepsilon_\nu)_b = -(\omega_{\nu\mu})_a (ωμν)a=(εμ)b∇a(εν)b=−(ωνμ)a
相应的里奇旋转系数为 ω μ ν ρ = ( ω μ ν ) a ( ε ρ ) a \omega_{\mu\nu\rho} = (\omega_{\mu\nu})_a(\varepsilon_\rho)^a ωμνρ=(ωμν)a(ερ)a. 黎曼曲率张量的计算过程与上节一样，只是其中的 ( e μ ) a (e_\mu)^a (eμ)a现在应理解为 ( ε μ ) a (\varepsilon_\mu)^a (εμ)a.

由于 ε 3 \varepsilon_3 ε3和 ε 4 \varepsilon_4 ε4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 ω 421 = ω ˉ 321 , ω 14 = ω ˉ 13 , R 24 = R ˉ 23 \omega_{421} = \bar\omega_{321}, ~ \omega_{14} = \bar\omega_{13}, ~ R_{24} = \bar R_{23} ω421=ωˉ321, ω14=ωˉ13, R24=Rˉ23等。这导致24个复 ω μ ν ρ \omega_{\mu\nu\rho} ωμνρ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
κ = − ω 311 , ρ = − ω 314 , ε = 1 2 ( ω 211 − ω 341 ) σ = − ω 313 , μ = ω 243 , γ = 1 2 ( ω 212 − ω 342 ) λ = ω 244 , τ = − ω 312 , α = 1 2 ( ω 214 − ω 344 ) ν = ω 242 , π = ω 241 , β = 1 2 ( ω 213 − ω 343 ) \begin{aligned} &\kappa = -\omega_{311}, ~ \rho = -\omega_{314}, ~ &\varepsilon = \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{aligned} κ=−ω311, ρ=−ω314, σ=−ω313, μ=ω243, λ=ω244, τ=−ω312, ν=ω242, π=ω241, ε=21(ω211−ω341)γ=21(ω212−ω342)α=21(ω214−ω344)β=21(ω213−ω343)
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 ∇ a , Δ = n a ∇ a , δ = m a ∇ a , δ ˉ = m ˉ a ∇ a D = l^a\nabla_a, ~ \Delta = n^a\nabla_a, ~ \delta = m^a\nabla_a, ~ \bar\delta = \bar m^a\nabla_a D=la∇a, Δ=na∇a, δ=ma∇a, δˉ=mˉa∇a它们之间满足如下的对易关系：
Δ D − D Δ = ( γ + γ ˉ ) D + ( ε + ε ˉ ) Δ − ( τ ˉ + π ) δ − ( τ + π ˉ ) δ ˉ δ D − D δ = ( α ˉ + β − π ˉ ) D + κ Δ − ( ρ ˉ + ε − ε ˉ ) δ − σ δ ˉ δ Δ − Δ δ = − ν ˉ D + ( τ − α ˉ − β ) Δ + ( μ − γ + γ ˉ ) δ + λ ˉ δ ˉ δ ˉ δ − δ δ ˉ = ( μ ˉ − μ ) D + ( ρ ˉ − ρ ) Δ + ( α − β ˉ ) δ − ( α ˉ − β ) δ ˉ \begin{aligned} \Delta D-D\Delta&=(\gamma+\bar{\gamma})D+(\varepsilon+\bar{\varepsilon})\Delta-(\bar{\tau}+\pi)\delta-(\tau+\bar{\pi})\bar{\delta} \\ \delta D-D\delta&=(\bar{\alpha}+\beta-\bar{\pi})D+\kappa\Delta-(\bar{\rho}+\varepsilon-\bar{\varepsilon})\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{aligned} ΔD−DΔδD−DδδΔ−Δδδˉδ−δδˉ=(γ+γˉ)D+(ε+εˉ)Δ−(τˉ+π)δ−(τ+πˉ)δˉ=(αˉ+β−πˉ)D+κΔ−(ρˉ+ε−εˉ)δ−σδˉ=−νˉD+(τ−αˉ−β)Δ+(μ−γ+γˉ)δ+λˉδˉ=(μˉ−μ)D+(ρˉ−ρ)Δ+(α−βˉ)δ−(αˉ−β)δˉ
外尔张量有10个实的独立分量，用5个复数代表，定义为 Ψ 0 = − C 1313 , Ψ 1 = − C 1213 , Ψ 2 = − C 1342 , Ψ 3 = − C 1242 , Ψ 4 = − C 2424 \Psi_0 = -C_{1313}, ~ \Psi_1 = -C_{1213}, ~ \Psi_2 = -C_{1342}, ~ \Psi_3 = -C_{1242}, ~ \Psi_4 = -C_{2424} Ψ0=−C1313, Ψ1=−C1213, Ψ2=−C1342, Ψ3=−C1242, Ψ4=−C2424里奇张量有10个，其中 R 11 , R 12 , R 22 R_{11}, ~ R_{12}, ~R_{22} R11, R12, R22显然为实数， R 34 = R 43 = R ˉ 34 R_{34} = R_{43} = \bar R_{34} R34=R43=Rˉ34也是实数，由此定义以下四个实数 Φ 00 = − 1 2 R 11 , Φ 11 = − 1 4 ( R 12 + R 34 ) , Φ 22 = − 1 2 R 22 , Λ = 1 12 ( R 12 − R 34 ) \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}) Φ00=−21R11, Φ11=−41(R12+R34), Φ22=−21R22, Λ=121(R12−R34)其余6个分量为复数，分别为 Φ 01 = − 1 2 R 13 , Φ 10 = − 1 2 R 14 , Φ 02 = − 1 2 R 33 , Φ 20 = − 1 2 R 44 , Φ 12 = − 1 2 R 23 , Φ 21 = − 1 2 R 24 \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} Φ01=−21R13, Φ10=−21R14, Φ02=−21R33, Φ20=−21R44, Φ12=−21R23, Φ21=−21R24易见 Φ \Phi Φ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
D ρ − δ ˉ κ = ( ρ 2 + σ σ ˉ ) + ( ε + ε ˉ ) ρ − κ ˉ τ − κ ( 3 α + β ˉ − π ) + Φ 00 , [ R 1314 ] D σ − δ κ = ( ρ + ρ ˉ ) σ + ( 3 ε − ε ˉ ) σ − ( τ − π ˉ + α ˉ + 3 β ) κ + Ψ 0 , [ R 1313 ] D τ − Δ κ = ( τ + π ˉ ) ρ + ( τ ˉ + π ) σ + ( ε − ε ˉ ) τ − ( 3 γ + γ ˉ ) κ + Ψ 1 + Φ 01 , [ R 1312 ] D α − δ ˉ ε = ( ρ + ε ˉ − 2 ε ) α + β σ ˉ − β ˉ ε − κ λ − κ ˉ γ + ( ε + ρ ) π + Φ 10 , [ 1 2 ( R 3414 − R 1214 ) ] D β − δ ε = ( α + π ) σ + ( ρ ˉ − ε ˉ ) β − ( μ + γ ) κ − ( α ˉ − π ˉ ) ε + Ψ 1 , [ 1 2 ( R 1213 − R 3413 ) ] D γ − Δ ε = ( τ + π ˉ ) α + ( τ ˉ + π ) β − ( ε + ε ˉ ) γ − ( γ + γ ˉ ) ε + τ π − ν κ + Ψ 2 + Φ 11 − Λ , [ 1 2 ( R 1212 − R 3412 ) ] D λ − δ ˉ π = ( ρ λ + σ ˉ μ ) + π 2 + ( α − β ˉ ) π − ν κ ˉ − ( 3 ε − ε ˉ ) λ + Φ 20 , [ R 2441 ] D μ − δ π = ( ρ ˉ μ + σ λ ) + π π ˉ − ( ε + ε ˉ ) μ − ( α ˉ − β ) π − ν κ + Ψ 2 + 2 Λ , [ R 2431 ] D ν − Δ π = ( π + τ ˉ ) μ + ( π ˉ + τ ) λ + ( γ − γ ˉ ) π − ( 3 ε + ε ˉ ) ν + Ψ 3 + Φ 21 , [ R 2421 ] Δ λ − δ ˉ ν = − ( μ + μ ˉ ) λ − ( 3 γ − γ ˉ ) λ + ( 3 α + β ˉ + π − τ ˉ ) ν − Ψ 4 , [ R 2442 ] δ ρ − δ ˉ σ = ρ ( α ˉ + β ) − σ ( 3 α − β ˉ ) + ( ρ − ρ ˉ ) τ + ( μ − μ ˉ ) κ − Ψ 1 + Φ 01 , [ R 3143 ] δ α − δ ˉ β = ( μ ρ − λ σ ) + α α ˉ + β β ˉ − 2 α β + γ ( ρ − ρ ˉ ) + ε ( μ − μ ˉ ) − Ψ 2 + Φ 11 + Λ , [ 1 2 ( R 1234 − R 3434 ) ] δ λ − δ ˉ μ = ( ρ − ρ ˉ ) ν + ( μ − μ ˉ ) π + ( α + β ˉ ) μ + ( α ˉ − 3 β ) λ − Ψ 3 + Φ 21 , [ R 2443 ] δ ν − Δ μ = ( μ 2 + λ λ ˉ ) + ( γ + γ ˉ ) μ − ν ˉ π + ( τ − 3 β − α ˉ ) ν + Φ 22 , [ R 2423 ] δ γ − Δ β = ( τ − α ˉ − β ) γ + μ τ − σ ν − ε ν ˉ − ( γ − γ ˉ − μ ) β + α λ ˉ + Φ 12 , [ 1 2 ( R 1232 − R 3432 ) ] δ τ − Δ σ = ( μ σ + λ ˉ ρ ) + ( τ + β − α ˉ ) τ − ( 3 γ − γ ˉ ) σ − κ ν ˉ + Φ 02 , [ R 1332 ] Δ ρ − δ ˉ τ = − ( ρ μ ˉ + σ λ ) + ( β ˉ − α − τ ˉ ) τ + ( γ + γ ˉ ) ρ + ν κ − Ψ 2 − 2 Λ , [ R 1324 ] Δ α − δ ˉ γ = ( ρ + ε ) ν − ( τ + β ) λ + ( γ ˉ − μ ˉ ) α + ( β ˉ − τ ˉ ) γ − Ψ 3 , [ 1 2 ( R 1242 − R 3442 ) ] \begin{aligned} D\rho -\bar{\delta}\kappa&=(\rho^2+\sigma\bar{\sigma})+(\varepsilon+\bar{\varepsilon})\rho-\bar{\kappa}\tau-\kappa(3\alpha+\bar{\beta}-\pi)+\Phi_{00}, ~ [R_{1314}] \\ D\sigma-\delta\kappa&=(\rho+\bar{\rho})\sigma+(3\varepsilon-\bar{\varepsilon})\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+(\varepsilon-\bar{\varepsilon})\tau-(3\gamma+\bar{\gamma})\kappa+\Psi_1+\Phi_{01}, ~ [R_{1312}] \\ D\alpha-\bar{\delta}\varepsilon&=(\rho+\bar{\varepsilon}-2\varepsilon)\alpha+\beta\bar{\sigma}-\bar{\beta}\varepsilon-\kappa\lambda-\bar{\kappa}\gamma+(\varepsilon+\rho)\pi+\Phi_{10}, ~ [\frac{1}{2}(R_{3414} - R_{1214})] \\ D\beta-\delta\varepsilon&=(\alpha+\pi)\sigma+(\bar{\rho}-\bar{\varepsilon})\beta-(\mu+\gamma)\kappa-(\bar{\alpha}-\bar{\pi})\varepsilon+\Psi_1, ~ [\frac{1}{2}(R_{1213} - R_{3413})] \\ D\gamma-\Delta\varepsilon&=(\tau+\bar{\pi})\alpha+(\bar{\tau}+\pi)\beta-(\varepsilon+\bar{\varepsilon})\gamma-(\gamma+\bar{\gamma})\varepsilon+\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\varepsilon-\bar{\varepsilon})\lambda+\Phi_{20}, ~ [R_{2441}] \\ D\mu-\delta\pi&=(\bar{\rho}\mu+\sigma\lambda)+\pi\bar{\pi}-(\varepsilon+\bar{\varepsilon})\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\varepsilon+\bar{\varepsilon})\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})+\varepsilon(\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-\varepsilon\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+\varepsilon)\nu-(\tau+\beta)\lambda+(\bar{\gamma}-\bar{\mu})\alpha+(\bar{\beta}-\bar{\tau})\gamma-\Psi_3, ~ [\frac{1}{2}(R_{1242} - R_{3442})] \end{aligned} Dρ−δˉκDσ−δκDτ−ΔκDα−δˉεDβ−δεDγ−ΔεDλ−δˉπDμ−δπDν−ΔπΔλ−δˉνδρ−δˉσδα−δˉβδλ−δˉμδν−Δμδγ−Δβδτ−ΔσΔρ−δˉτΔα−δˉγ=(ρ2+σσˉ)+(ε+εˉ)ρ−κˉτ−κ(3α+βˉ−π)+Φ00, [R1314]=(ρ+ρˉ)σ+(3ε−εˉ)σ−(τ−πˉ+αˉ+3β)κ+Ψ0, [R1313]=(τ+πˉ)ρ+(τˉ+π)σ+(ε−εˉ)τ−(3γ+γˉ)κ+Ψ1+Φ01, [R1312]=(ρ+εˉ−2ε)α+βσˉ−βˉε−κλ−κˉγ+(ε+ρ)π+Φ10, [21(R3414−R1214)]=(α+π)σ+(ρˉ−εˉ)β−(μ+γ)κ−(αˉ−πˉ)ε+Ψ1, [21(R1213−R3413)]=(τ+πˉ)α+(τˉ+π)β−(ε+εˉ)γ−(γ+γˉ)ε+τπ−νκ+Ψ2+Φ11−Λ, [21(R1212−R3412)]=(ρλ+σˉμ)+π2+(α−βˉ)π−νκˉ−(3ε−εˉ)λ+Φ20, [R2441]=(ρˉμ+σλ)+ππˉ−(ε+εˉ)μ−(αˉ−β)π−νκ+Ψ2+2Λ, [R2431]=(π+τˉ)μ+(πˉ+τ)λ+(γ−γˉ)π−(3ε+εˉ)ν+Ψ3+Φ21, [R2421]=−(μ+μˉ)λ−(3γ−γˉ)λ+(3α+βˉ+π−τˉ)ν−Ψ4, [R2442]=ρ(αˉ+β)−σ(3α−βˉ)+(ρ−ρˉ)τ+(μ−μˉ)κ−Ψ1+Φ01, [R3143]=(μρ−λσ)+ααˉ+ββˉ−2αβ+γ(ρ−ρˉ)+ε(μ−μˉ)−Ψ2+Φ11+Λ, [21(R1234−R3434)]=(ρ−ρˉ)ν+(μ−μˉ)π+(α+βˉ)μ+(αˉ−3β)λ−Ψ3+Φ21, [R2443]=(μ2+λλˉ)+(γ+γˉ)μ−νˉπ+(τ−3β−αˉ)ν+Φ22, [R2423]=(τ−αˉ−β)γ+μτ−σν−ενˉ−(γ−γˉ−μ)β+αλˉ+Φ12, [21(R1232−R3432)]=(μσ+λˉρ)+(τ+β−αˉ)τ−(3γ−γˉ)σ−κνˉ+Φ02, [R1332]=−(ρμˉ+σλ)+(βˉ−α−τˉ)τ+(γ+γˉ)ρ+νκ−Ψ2−2Λ, [R1324]=(ρ+ε)ν−(τ+β)λ+(γˉ−μˉ)α+(βˉ−τˉ)γ−Ψ3, [21(R1242−R3442)]
以及比安基恒等式：
− δ ˉ Ψ 0 + D Ψ 1 + ( 4 α − π ) Ψ 0 − 2 ( 2 ρ + ε ) Ψ 1 + 3 κ Ψ 2 + R a = 0 , R 13 [ 13 , 4 ] = 0 δ ˉ Ψ 1 − D Ψ 2 − λ Ψ 0 + 2 ( π − α ) Ψ 1 + 3 ρ Ψ 2 − 2 κ Ψ 3 + R b = 0 , R 13 [ 21 , 4 ] = 0 − δ ˉ Ψ 2 + D Ψ 3 + 2 λ Ψ 1 − 3 π Ψ 2 + 2 ( ϵ − ρ ) Ψ 3 + κ Ψ 4 + R c = 0 , R 42 [ 13 , 4 ] = 0 δ ˉ Ψ 3 − D Ψ 4 − 3 λ Ψ 2 + 2 ( 2 π + α ) Ψ 3 − ( 4 ϵ − ρ ) Ψ 4 + R d = 0 , R 42 [ 21 , 4 ] = 0 − Δ Ψ 0 + δ Ψ 1 + 4 ( γ − μ ) Ψ 0 − 2 ( 2 τ + β ) Ψ 1 + 3 σ Ψ 2 + R e = 0 , R 13 [ 13 , 2 ] = 0 − Δ Ψ 1 + δ Ψ 2 + ν Ψ 0 − 2 ( γ − μ ) Ψ 1 − 3 τ Ψ 2 + 2 σ Ψ 3 + R f = 0 , R 13 [ 43 , 2 ] = 0 − Δ Ψ 2 + δ Ψ 3 + 2 ν Ψ 1 − 3 μ Ψ 2 + 2 ( β − τ ) Ψ 3 + σ Ψ 4 + R g = 0 , R 42 [ 13 , 2 ] = 0 − Δ Ψ 3 + δ Ψ 4 + 3 ν Ψ 2 − 2 ( γ − 2 μ ) Ψ 3 − ( τ − 4 β ) Ψ 4 + R h = 0 , R 42 [ 43 , 2 ] = 0 \begin{aligned} -\bar\delta\Psi_0 + D \Psi_1 + (4\alpha - \pi)\Psi_0 - 2(2\rho + \varepsilon)\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(\epsilon - \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 - (4\epsilon- \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{aligned} −δˉΨ0+DΨ1+(4α−π)Ψ0−2(2ρ+ε)Ψ1+3κΨ2+Ra=0, R13[13,4]=0δˉΨ1−DΨ2−λΨ0+2(π−α)Ψ1+3ρΨ2−2κΨ3+Rb=0, R13[21,4]=0−δˉΨ2+DΨ3+2λΨ1−3πΨ2+2(ϵ−ρ)Ψ3+κΨ4+Rc=0, R42[13,4]=0δˉΨ3−DΨ4−3λΨ2+2(2π+α)Ψ3−(4ϵ−ρ)Ψ4+Rd=0, R42[21,4]=0−ΔΨ0+δΨ1+4(γ−μ)Ψ0−2(2τ+β)Ψ1+3σΨ2+Re=0, R13[13,2]=0−ΔΨ1+δΨ2+νΨ0−2(γ−μ)Ψ1−3τΨ2+2σΨ3+Rf=0, R13[43,2]=0−ΔΨ2+δΨ3+2νΨ1−3μΨ2+2(β−τ)Ψ3+σΨ4+Rg=0, R42[13,2]=0−ΔΨ3+δΨ4+3νΨ2−2(γ−2μ)Ψ3−(τ−4β)Ψ4+Rh=0, R42[43,2]=0
其中
R a = − D Φ 01 + δ Φ 00 + 2 ( ρ ˉ + ϵ ) Φ 01 + 2 σ Φ 10 − 2 κ Φ 11 − κ ˉ Φ 02 + ( π ˉ − 2 α ˉ − 2 β ) Φ 00 R b = − Δ Φ 00 + δ ˉ Φ 01 − 2 ( τ ˉ + α ) Φ 01 + 2 ρ Φ 11 − 2 τ Φ 10 + σ ˉ Φ 02 − ( μ ˉ − 2 γ ˉ − 2 γ ) Φ 00 − 2 D Λ R c = − D Φ 21 + δ Φ 20 + 2 ( ρ ˉ − ϵ ) Φ 21 + 2 π Φ 11 − 2 μ Φ 10 − κ ˉ Φ 22 + ( π ˉ − 2 α ˉ + 2 β ) Φ 20 − 2 δ ˉ Λ R d = − Δ Φ 20 + δ ˉ Φ 21 − 2 ( τ ˉ − α ) Φ 21 + 2 ν Φ 10 − 2 λ Φ 11 + σ ˉ Φ 22 − ( μ ˉ − 2 γ ˉ + 2 γ ) Φ 20 R e = − D Φ 02 + δ Φ 01 + 2 ( π ˉ − β ) Φ 01 + 2 σ Φ 11 − 2 κ Φ 12 − λ ˉ Φ 00 + ( ρ ˉ − 2 ε ˉ + 2 ε ) Φ 02 R f = Δ Φ 01 − δ ˉ Φ 02 + 2 ( μ ˉ − γ ) Φ 01 + 2 τ Φ 11 − 2 ρ Φ 12 − ν ˉ Φ 00 + ( τ ˉ − 2 β ˉ + 2 α ) Φ 02 + 2 δ Λ R g = − D Φ 22 + δ Φ 21 + 2 ( π ˉ + β ) Φ 21 + 2 π Φ 12 − 2 μ Φ 11 − λ ˉ Φ 20 + ( ρ ˉ − 2 ε ˉ − 2 ε ) Φ 22 − 2 Δ Λ R h = Δ Φ 21 − δ ˉ Φ 22 + 2 ( μ ˉ + γ ) Φ 21 + 2 λ Φ 12 − 2 ν Φ 11 − ν ˉ Φ 20 + ( τ ˉ − 2 β ˉ − 2 α ) Φ 22 \begin{aligned} R_a &= -D\Phi_{01} + \delta\Phi_{00} + 2(\bar\rho + \epsilon)\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 - \epsilon)\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\varepsilon + 2\varepsilon)\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\varepsilon - 2\varepsilon)\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{aligned} RaRbRcRdReRfRgRh=−DΦ01+δΦ00+2(ρˉ+ϵ)Φ01+2σΦ10−2κΦ11−κˉΦ02+(πˉ−2αˉ−2β)Φ00=−ΔΦ00+δˉΦ01−2(τˉ+α)Φ01+2ρΦ11−2τΦ10+σˉΦ02−(μˉ−2γˉ−2γ)Φ00−2DΛ=−DΦ21+δΦ20+2(ρˉ−ϵ)Φ21+2πΦ11−2μΦ10−κˉΦ22+(πˉ−2αˉ+2β)Φ20−2δˉΛ=−ΔΦ20+δˉΦ21−2(τˉ−α)Φ21+2νΦ10−2λΦ11+σˉΦ22−(μˉ−2γˉ+2γ)Φ20=−DΦ02+δΦ01+2(πˉ−β)Φ01+2σΦ11−2κΦ12−λˉΦ00+(ρˉ−2εˉ+2ε)Φ02=ΔΦ01−δˉΦ02+2(μˉ−γ)Φ01+2τΦ11−2ρΦ12−νˉΦ00+(τˉ−2βˉ+2α)Φ02+2δΛ=−DΦ22+δΦ21+2(πˉ+β)Φ21+2πΦ12−2μΦ11−λˉΦ20+(ρˉ−2εˉ−2ε)Φ22−2ΔΛ=ΔΦ21−δˉΦ22+2(μˉ+γ)Φ21+2λΦ12−2νΦ11−νˉΦ20+(τˉ−2βˉ−2α)Φ22