DifferentialGeometry/Tensor/NPRicciBianchiIdentitiesDetails - Maple Help

Details for NPRicciIdentities and NPBianchiIdentities

Description

 • Let $\mathrm{κ},\mathrm{ρ},\mathrm{σ},\mathrm{τ},\mathrm{π},\mathrm{λ},\mathrm{μ},\mathrm{ν},\mathrm{α},\mathrm{β},\mathrm{γ},\mathrm{ε}$ be the Newman-Penrose spin coefficients. Let $\mathrm{D},\mathrm{Δ},\mathrm{δ},\stackrel{‾}{\mathrm{δ}}$ be the Newman-Penrose directional derivatives.
 • Here is the list of Newman-Penrose Ricci idenitites, taken from the paper of Newman and Penrose .

[a]

[b]

[c]

[d]

[e]

[f]  $\mathrm{Dγ}-\mathrm{Δε}=\left(\mathrm{τ}+\stackrel{‾}{\mathrm{π}}\right)\mathrm{α}+\left(\stackrel{‾}{\mathrm{τ}}+\mathrm{π}\right)\mathrm{β}-\left(\mathrm{ε}+\stackrel{‾}{\mathrm{ε}}\right)\mathrm{γ}-\left(\mathrm{γ}+\stackrel{‾}{\mathrm{γ}}\right)\mathrm{ε}+\mathrm{τπ}-\mathrm{νκ}+{\mathrm{Ψ}}_{2}-\mathrm{Λ}+{\mathrm{Φ}}_{11}$

[g]

[h]

[i]

[j]

[k]

[l]

[m]

[n]

[o]  $\mathrm{δγ}-\mathrm{Δβ}=\left(\mathrm{τ}-\stackrel{‾}{\mathrm{α}}-\mathrm{β}\right)\mathrm{γ}+\mathrm{μτ}-\mathrm{σν}-\mathrm{ε}\stackrel{‾}{\mathrm{ν}}-\left(\mathrm{γ}-\stackrel{‾}{\mathrm{γ}}-\mathrm{μ}\right)\mathrm{β}+\mathrm{α}\stackrel{‾}{\mathrm{λ}}+{\mathrm{Φ}}_{12}$

[p]

[q]

[r]  $\mathrm{Δα}-\stackrel{‾}{\mathrm{δ}}\mathrm{γ}=\left(\mathrm{ρ}+\mathrm{ε}\right)\mathrm{ν}-\left(\mathrm{τ}+\mathrm{β}\right)\mathrm{λ}+\left(\stackrel{‾}{\mathrm{γ}}-\stackrel{‾}{\mathrm{μ}}\right)\mathrm{α}+\left(\stackrel{‾}{\mathrm{β}}-\stackrel{‾}{\mathrm{τ}}\right)\mathrm{γ}-{\mathrm{Ψ}}_{3}$

 • Here the list of Newman-Penrose Bianchi idenitites, taken from the book of Stewart.

[a]

[b]

[c]

[d]

[e]

[f]

[g]

[h]

[i]  [j]

[k]