Christoffel of induced LC derivative to ambient quantities

Author: Leo C. Stein (leo.stein@gmail.com)
Date: 2021 Oct 9

Needs["xAct`xPert`"]
$DefInfoQ=False;

DefManifold[M,4,Complement[IndexRange[a,q],{n,h,g}]]

DefMetric[-1,g[-a,-b],CD,PrintAs->"g"]

DefTensor[n[a],M]
AutomaticRules[n,MakeRule[{n[a]n[-a],-1}]]
AutomaticRules[n,MakeRule[{g[-a,-b]n[a]n[b],-1}]]

$ExtrinsicKSign=-1;
$AccelerationSign=-1;

DefMetric[1,h[-a,-b],cd,SymbolOfCovD->{"|","D"},InducedFrom->{g,n}]

Our convention:

g[-a,-b]
%//MetricToProjector

PrintAs[ExtrinsicKh]^="K";

DefTensor[Lapse[],M,PrintAs->"α"]

LapseRule={Accelerationn[a_]:>cd[a][Lapse[]]/Lapse[]};

More convention:

CD[-a]@n[-b]
%//GradNormalToExtrinsicK
%/.LapseRule

Alex’s question:

Start from Γ[D], go to Γ[∇].

Christoffelcd[a,-b,-c]
%//ChristoffelToGradMetric
%//ProjectorToMetric//Expand
%//GradMetricToChristoffel//Expand
ChangeCovD[%,PD,CD]//ToCanonical
ToCanonical[%//GradNormalToExtrinsicK,UseMetricOnVBundle->None]
%/.LapseRule

Notice the first two terms can be combined:

ChristoffelCD[e,-b,-c]h[a,-e]//ProjectorToMetric//Expand