Execute the code of PublicNPGHP.nb which is before the section “Bianchi identities”. Use the convention $RiemannSign=-1

Bianchi identities

Second Bianchi identity:

3*Antisymmetrize[CD[-a]@RiemannCD[-b,-c,-d,-f],{-a,-b,-c}]==0

ToCanonical/@%

BianchiIdentity=%;

ChangeCovD[#,CD,PDNP]&/@BianchiIdentity

%/.ChristoffelCDPDNP->ChristoffelCDePDNP

ToBasis[NP]/@ToBasis[NP]/@%

ExpandedBianchiIdentity=%;

Computation of Chandrasekar’s equation (321)-(a)

ExpandedBianchiIdentity/.{-{c,NP}->{1,-NP},-{a,NP}->{3,-NP},-{b,NP}->{4,-NP},-{d,NP}->{1,-NP},-{f,NP}->{3,-NP}}

TraceBasisDummy/@%

First@%-Last@%==0

FullSimplify/@%

Contracted Bianchi identity:

ContractedBianchiIdentity=g[a,b]CD[-a][RicciCD[-b,-c]-1/2 RicciScalarCD[]g[-b,-c]]==0

ChangeCovD[#,CD,PDNP]&/@ContractedBianchiIdentity

%/.ChristoffelCDPDNP->ChristoffelCDePDNP

ToBasis[NP]/@ToBasis[NP]/@%

ExpandedContractedBianchiIdentity=%;

Computation of Chandrasekar’s (322)-(i)

ExpandedContractedBianchiIdentity/.{-{c,NP}->{1,-NP}}

TraceBasisDummy/@%;

Flatten@ComponentArray@%

FullSimplify/@%