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[t_]->cd[t][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]