Exit

<<xAct`xTensor`

DefManifold[M3, 3,{a,b,c,d,e,f,i,j,k,l,m}]

DefManifold[M1,1,{t,u,w,x,z}]

DefManifold[M4,{M3,M1},{A,B,F,G,J,L}]

DefMetric[-1,metricg[-a,-b],CD,{";","▽"},PrintAs->"g"];

DefTensor[n[a],M3]

AutomaticRules[n,MakeRule[{n[a]n[-a],-1}]]

AutomaticRules[n,MakeRule[{metricg[-a,-b]n[a]n[b],-1}]]

$Extrinsicksign=-1;

$Accelerationsign=-1;

DefMetric[1,metrich[-a,-b],cd,{"|","D"},InducedFrom->{metricg,n},PrintAs->"h"]

DefTensor[lapse[],M3,PrintAs->"α"]

LapseRule=Accelerationn[a_]->cd[a][lapse[]]/lapse[]

DefTensor[V[-a],M3]

DefTensor[W[-a],M3,OrthogonalTo->{n[a]},ProjectedWith->{metrich[a,-b]}]

InducedDecomposition[W[-a],{metrich,n}]

InducedDecomposition[V[-a],{metrich,n}]//NoScalar//ScreenDollarIndices

InducedDecomposition[7V[-a]V[-b]+W[-a]W[-b],{metrich,n}]//NoScalar//ScreenDollarIndices

InducedDecomposition[RiemannCD[-a,-b,-c,-d],{metrich,n}]//NoScalar//ScreenDollarIndices

ToCanonical[%]

n[d]%//Expand

n[c]%//Expand

n[b]%%//Expand

Projectormetrich[RicciCD[-a,-b]]

%/.Projectormetrich ->ProjectWith[metrich]//ScreenDollarIndices

%//ProjectorToMetric

%//Expand

InducedDecomposition[%,{metrich,n}]//ToCanonical

CD[-a][n[b]]

%//GradNormalToExtrinsicK

n[-a]CD[-b][W[a]]

%//GradNormalToExtrinsicK//ContractMetric

ExtrinsicKmetrich[-a,-b]

%//ExtrinsicKToGradNormal//ContractMetric

CD[-c][W[-b]]

%//ProjectWith[metrich]//ScreenDollarIndices

%//ProjectorToMetric

%//Expand

%//GradNormalToExtrinsicK

%//Simplification

cd[-c][W[-b]]

ProjectDerivative[%]

%/.Projectormetrich->ProjectWith[metrich]//ScreenDollarIndices

cd[-a][W[b]W[c]]

Catch@cd[-a][V[b]V[c]]

cd[-a][W[a]W[b]]

cd[-a][V[b]V[-b]]//ScreenDollarIndices

Catch@cd[-a][V[a]V[b]]

n[a]CD[-a][n[b]]//GradNormalToExtrinsicK//Expand

cd[-a][metrich[-b,-c]]

(cd[-a]@cd[-b][#]-cd[-b]@cd[-a][#])&@W[-c]

%//SortCovDs//ToCanonical//ScreenDollarIndices

Riemanncd[-a,-b,-c,d]{n[a],n[b],n[c],n[-d]}

Riemanncd[-a,-c,-b,c]

{Riccicd[-a,a],metrich[a,b]Riccicd[-a,-b],metricg[a,b]Riccicd[-a,-b]}

CD[-a][n[-b]]//GradNormalToExtrinsicK//ContractMetric

Antisymmetrize[ProjectWith[metrich][%],{-a,-b}]//ContractMetric//ToCanonical

ProjectWith[metrich][-1/2LieD[n[c],CD][metricg[-a,-b]]//GradNormalToExtrinsicK]//ContractMetric//ToCanonical

-1/2LieD[n[c],CD][metrich[-a,-b]]//ProjectorToMetric//GradNormalToExtrinsicK//ContractMetric//ToCanonical

UndefTensor[V]

DefTensor[V[a],M3,OrthogonalTo->{n[-a]},ProjectedWith->{metrich[-a,b]}]

CD[-c][V[-b]]

%//ProjectWith[metrich]//ScreenDollarIndices

%//ProjectorToMetric

%//Expand

%//GradNormalToExtrinsicK

%//Simplification

-%+CD[-c][V[-b]]-n[-b]V[-f]CD[-c][n[f]]+n[-c]n[e]CD[-e][V[-b]]-n[-c]n[-b]V[-f]Accelerationn[f]

%//GradNormalToExtrinsicK

%//Simplification

ProjectWith[metrich][CD[-d][CD[-c][V[-b]]]]-cd[-d][cd[-c][V[-b]]]//ScreenDollarIndices

%//ProjectDerivative//ProjectDerivative

%/.Projectormetrich->ProjectWith[metrich]//ScreenDollarIndices

%//ProjectorToMetric//Expand

%//GradNormalToExtrinsicK//Expand

%//ContractMetric

%//Simplification

expr1=Projectormetrich[CD[-d][CD[-c][V[-b]]]]-CD[-c][CD[-d][V[-b]]]

expr1A=expr1//SortCovDs//ScreenDollarIndices

expr1/.Projectormetrich->ProjectWith[metrich]//ProjectorToMetric//Expand//ScreenDollarIndices

expr1B=%//GradNormalToExtrinsicK//Simplification

expr2=cd[-d][cd[-c][V[-b]]]-cd[-c][cd[-d][V[-b]]]

expr2A=expr2//SortCovDs//ScreenDollarIndices

expr2//ProjectDerivative//ProjectDerivative

%/.Projectormetrich->ProjectWith[metrich]//ProjectorToMetric//Expand//ScreenDollarIndices

%//GradNormalToExtrinsicK//ContractMetric

expr2B=%//Simplification

(expr1A-expr1B)-(expr2A-expr2B)//Simplification

IndexSet[GCRiemann1[-a_,-b_,-c_,-d_],Simplification[(%/.V[_]->1//ScreenDollarIndices)]]

inducedRiemann[-a_,-b_,-c_,-d_]=InducedDecomposition[RiemannCD[-a,-b,-c,-d],{metrich,n}]//ToCanonical//ScreenDollarIndices

inducedRiemann[-a_,-b_,-c_,-d_]=inducedRiemann[-a,-b,-c,-d]-GCRiemann1[-a,-b,-c,-d]//ToCanonical//ScreenDollarIndices

expr1=Projectormetrich[CD[-d][CD[-c][n[-b]]]-CD[-c][CD[-d][n[-b]]]]

expr1A=expr1//SortCovDs//ScreenDollarIndices

expr1/.Projectormetrich->ProjectWith[metrich]//ProjectorToMetric//Expand//ScreenDollarIndices

expr1B=%//GradNormalToExtrinsicK//ContractMetric//GradNormalToExtrinsicK//Simplification

expr2A=cd[-c][ExtrinsicKmetrich[-b,-d]]-cd[-d][ExtrinsicKmetrich[-b,-c]]

expr2A//ProjectDerivative

%/.Projectormetrich->ProjectWith[metrich]//ProjectorToMetric//Expand//ScreenDollarIndices

expr2B=%//GradNormalToExtrinsicK//ContractMetric//Simplification

IndexSet[GCRiemann2[-b_,-c_,-d_],(expr1A-expr1B)-(expr2A-expr2B)//Simplification]

inducedRiemann[-a_,-b_,-c_,-d_]=inducedRiemann[-a,-b,-c,-d]-n[-d]GCRiemann2[-c,-a,-b]+n[-c]GCRiemann2[-d,-a,-b]-n[-b]GCRiemann2[-a,-c,-d]+n[-a]GCRiemann2[-b,-c,-d]//ToCanonical//ScreenDollarIndices

expr1=Projectormetrich[n[d](CD[-d][CD[-c][n[-b]]]-CD[-c][CD[-d][n[-b]]])]

expr1A=expr1//SortCovDs//ScreenDollarIndices

expr1/.Projectormetrich->ProjectWith[metrich]//ProjectorToMetric//Expand//ScreenDollarIndices

expr1B=%//GradNormalToExtrinsicK//ContractMetric//GradNormalToExtrinsicK//Simplification

expr2A=-cd[-c][Accelerationn[-b]]-Projectormetrich[n[d]CD[-d][ExtrinsicKmetrich[-b,-c]]]

expr2A//ProjectDerivative

%/.Projectormetrich->ProjectWith[metrich]//ProjectorToMetric//Expand//ScreenDollarIndices

expr2B=%//GradNormalToExtrinsicK//ContractMetric//Simplification

IndexSet[GCRiemann3[-b_,-c_],(expr1A-expr1B)-(expr2A-expr2B)//Simplification]

ProjectWith[metrich][LieD[n[a],CD][ExtrinsicKmetrich[-b,-c]]-n[a]CD[-a][ExtrinsicKmetrich[-b,-c]]//GradNormalToExtrinsicK]//Simplification//ContractMetric

inducedRiemann[-a_,-b_,-c_,-d_]=inducedRiemann[-a,-b,-c,-d]+n[-b]n[-d]GCRiemann3[-a,-c]-n[-b]n[-c]GCRiemann3[-a,-d]-n[-a]n[-d]GCRiemann3[-b,-c]+n[-a]n[-c]GCRiemann3[-b,-d]//ToCanonical//ScreenDollarIndices

GausCodazziRiemannRule=IndexRule[RiemannCD[-a,-b,-c,-d],inducedRiemann[-a,-b,-c,-d]]

inducedRicci[-a_,-b_]=InducedDecomposition[RicciCD[-a,-b],{metrich,n}]//NoScalar//ScreenDollarIndices

zero=RicciCD[-a,-b]-inducedRiemann[-a,-c,-b,-d]metricg[c,d]//ContractMetric//NoScalar//ScreenDollarIndices

zeroA=ProjectWith[metrich][%]//ContractMetric//ToCanonical

zeroB=Projectormetrich[RicciCD[-a,-b]]-ProjectWith[metrich][RicciCD[-a,-b]]//ScreenDollarIndices

IndexSet[GCRicci1[-a_,-b_],zeroA+zeroB//ToCanonical]

inducedRicci[-a_,-b_]=inducedRicci[-a,-b]-GCRicci1[-a,-b]//ToCanonical//ScreenDollarIndices

zero=(RicciCD[-a,-c]-inducedRiemann[-a,-b,-c,-d]metricg[b,d])n[c]//ContractMetric//NoScalar//ScreenDollarIndices

zeroA=ProjectWith[metrich][%]//ContractMetric//ToCanonical

zeroB=Projectormetrich[RicciCD[-a,-b]n[b]]-ProjectWith[metrich][RicciCD[-a,-b]n[b]]//ScreenDollarIndices

IndexSet[GCRicci2[-a_],zeroA+zeroB//ToCanonical]

inducedRicci[-a_,-b_]=inducedRicci[-a,-b]+GCRicci2[-a]n[-b]+GCRicci2[-b]n[-a]//ToCanonical//ScreenDollarIndices

zero=(RicciCD[-a,-c]-inducedRiemann[-a,-b,-c,-d]metricg[b,d])n[a]n[c]//Expand//ReplaceDummies//ScreenDollarIndices

inducedRiemann[-a,-b,-c,-d]metricg[b,d]metricg[a,c]//ContractMetric//NoScalar//ScreenDollarIndices

%/.Projectormetrich->ProjectWith[metrich]//ContractMetric//ToCanonical

%//GradNormalToExtrinsicK

inducedRicciScalar[]=%/.LapseRule//Expand