DifferentialGeometry/Tensor/NPRicciBianchiIdentitiesDetails - Maple Help
For the best experience, we recommend viewing online help using Google Chrome or Microsoft Edge.

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : DifferentialGeometry/Tensor/NPRicciBianchiIdentitiesDetails

Details for NPRicciIdentities and NPBianchiIdentities

Description

• 

Let κ,ρ,σ,τ,π,λ,μ,ν,α,β,γ,ε be the Newman-Penrose spin coefficients. Let D,Δ,δ,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]  Dρ−δ‾κ=ρ2+σσ‾+ε+ε‾ρ −κ‾τ−κ3 α+β‾−π+ Φ00

[b]  Dσ−δκ=ρ+ρ‾σ+3 ε−ε‾σ−τ −π‾+α‾+3 βκ+Ψ0

[c]  Dτ−Δκ=τ+π‾ρ+τ‾+πσ+ε −ε‾τ−3 γ+γ‾κ+Ψ1+Φ01

[d]  Dα−δ‾ε=ρ+ε‾−2 εα+βσ‾−β‾ε−κλ −κ‾γ  +ε+ϱπ+ Φ10

[e]  Dβ−δε=α+πσ+ρ‾−ε‾β−μ+γκ−α‾−π‾ε+Ψ1

[f]  Dγ−Δε=τ+π‾α+τ‾+πβ−ε+ε‾γ−γ+γ‾ε+τπ−νκ+Ψ2−Λ+Φ11

[g]  Dλ−δ‾π=ρλ+σ‾μ+π2+α −β‾π−νκ‾−3 ε −ε‾λ+Φ20

[h]  Dμ−δπ=ρ‾μ+σλ+ππ‾−ε+ε‾μ−α‾−βπ−νκ +Ψ2+2 Λ

[i]  Dν−Δπ=π+τ‾μ+π‾+τλ+γ−γ‾π−3 ε+ε‾ν+Ψ3+Φ21

[j]  Δλ−δ‾ν=−μ+μ‾λ−3 γ−γ‾λ+3 α+β‾+π −τ‾ν−Ψ4

[k]  δρ−δ‾σ=α‾+βρ−3 α − β‾σ+ρ −ρ‾τ+μ −μ‾κ−Ψ1+Φ01

[l]  δα−δ‾β=μρ−λσ+αα‾+ββ‾−2 αβ+ρ−ρ‾γ+μ−μ‾ε−Ψ2+Λ+Φ11

[m]  δλ−δ‾μ=ρ−ρ‾ν+μ−μ‾π+α+β‾μ+α‾−3 βλ−Ψ3+Φ21

[n]  δν−Δμ=μ2+λλ‾+γ+γ‾μ−ν‾π+τ−3 β−α‾ν+Φ22

[o]  δγ−Δβ=τ−α‾−βγ+μτ−σν−εν‾−γ−γ‾−μβ+αλ‾+Φ12

[p]  δτ−Δσ=μσ+λ‾ρ+τ+β−α‾τ−3 γ−γ‾σ−κν‾+Φ02

[q]  Δρ−δ‾τ=−ρμ‾−σλ+β‾−α−τ‾τ+γ+γ‾ρ+νκ−Ψ2+2 Λ

[r]  Δα−δ‾γ=ρ+εν−τ+βλ+γ‾−μ‾α+β‾−τ‾γ−Ψ3

 

• 

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

[a]  DΨ1−δ‾Ψ0−DΦ01+δΦ00=π−4 αΨ0+22 ρ+εΨ1−3 κΨ2−π‾−2α‾−2 βΦ00−2ρ‾+εΦ01−2 σ Φ10+2 κ Φ11+κ‾ Φ02

[b]  ΔΨ0−δΨ1+DΦ02−δΦ01=4 γ−μΨ0−22 τ+βΨ1+3 σ Ψ2−λ‾Φ00+2π‾−βΦ01+2 σ Φ11+ρ‾+2 ε−2ε‾Φ02−2 κ Φ12

[c]  DΨ2−δ‾Ψ1+ΔΦ00−δ‾Φ01+2 DΛ=−λΨ0+2π−αΨ1+3 ρΨ2−2 κΨ3+2 γ+2γ‾−μ‾Φ00−2α+τ‾Φ01−2 τ Φ10+2 ρΦ11+σ‾Φ02

[d]  DΨ1−δΨ2−ΔΦ01+δ‾Φ02−2 δΛ=νΨ0+2γ−μΨ1−3 τ Ψ2+2 σ Ψ3−ν‾Φ00+2μ‾−γΦ01+2 α+τ‾−2β‾Φ02+2 τ Φ11−2 ρΦ12

[e]  DΨ3−δ‾Ψ2−DΦ21+δΦ20−2δ‾Λ=−2 λΨ1+3 π Ψ2+2 ρ−εΨ3−κ Ψ4+2 μΦ10−2 πΦ11−2 β+π‾−2α‾Φ20−2ρ‾−εΦ21+κ‾Φ22

[f]  ΔΨ2−δΨ3+DΦ22−δΦ21+2 ΔΛ=2 νΨ1−3 μΨ2+2β−τΨ3+σΨ4−2 μΦ11−λ‾Φ20+2 π Φ12+2β+π‾Φ21+ρ‾−2 ε−2ε‾Φ22

[g]  DΨ4−δ‾Ψ3+ΔΦ20−δ‾Φ21=−3 λΨ2+2α+2 πΨ3+ρ−4 εΨ4+2 ν Φ10−2 λΦ11−2 γ−2γ‾+μ‾Φ20−2τ‾−αΦ21+σ‾Φ22

[h]  ΔΨ3−δΨ4−ΔΦ21+δ‾Φ22=3 νΨ2−2γ+2 μΨ3+4 β−τΨ4−2 ν Φ11−ν‾Φ20+2 λΦ12+2γ+μ‾Φ21+τ‾−2β‾−2 αΦ22

[i]  DΦ11−δΦ10+ΔΦ00−δ‾Φ01+3 DΛ=2 γ+2γ‾−μ−μ‾Φ00+π−2 α−2τ‾Φ01+π‾−2α‾−2 τΦ10+2ρ+ρ‾Φ02+σ‾Φ02+σ Φ20−κ‾Φ12−κΦ21[j]  DΦ12−δΦ11+ΔΦ01−δ‾Φ02+3 δΛ=2 γ−μ−2μ‾Φ01+ν‾Φ00−λ‾Φ10+2π‾−τΦ11+π+2β‾−2 α−τ‾Φ02+2 ρ+ρ‾−2ε‾Φ12+σΦ21−κΦ22

[k]  DΦ22−δΦ21+ΔΦ11−δ‾Φ12+3 ΔΛ=νΦ01+ν‾Φ10−2μ+μ‾Φ11−λΦ02−λ‾Φ20+2 π−τ‾+2β‾Φ12+2 β−τ+2π‾Φ21+ρ+ρ‾−2 ε−2ε‾Φ22