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<<xAct`xTerior`

$PrePrint=ScreenDollarIndices;

$DefInfoQ=False;
$UndefInfoQ=False;

<<xAct`ShowTime1`;

$ShowTimeThreshold=0.3;

DefManifold[AdS,4,IndexRange[a,h]];

DefChart[Global,AdS,{0,1,2,3},{t[],r[],θ[],ϕ[]}];

DefConstantSymbol[Mass,PrintAs->"M"];

DefConstantSymbol[Omega,PrintAs->"ω"];

$Assumptions=Mass>0 && r[]> 0 && r[]<π/2 && Element[Omega,Reals];

metric = CTensor[DiagonalMatrix[{-((Sec[r[]])^2),((Sec[r[]])^2),((Tan[r[]])^2),((Tan[r[]])^2)((Sin[θ[]])^2)}],{-Global,-Global}];

metric[-a,-b]

SetCMetric[metric,Global,SignatureOfMetric->{3,1,0}]

MetricCompute[metric,Global,All,CVSimplify-> Simplify];

CD = CovDOfMetric[metric]

Chr = Christoffel[CD,PDGlobal]

xAct`xCoba`Private`moveto[n_,n_]:={1}

Computation of the Laplacian acting on the spherical harmonics

DefConstantSymbol[l];

DefConstantSymbol[m];

We define a second metric which shall be used to define the Laplace operator on the sphere. We choose the flat Minkowski metric written in spherical coordinates:

metric2 = CTensor[DiagonalMatrix[{-1,1,r[]^2,r[]^2Sin[θ[]]^2}],{-Global,-Global}];

metric2[-a,-b]

We need to declare this metric as the new metric. To that end we remove first the metric declaration we did before:

UnsetCMetric@metric

SetCMetric[metric2,Global,SignatureOfMetric->{3,1,0}]

Now the Laplacian is just the Hodge Laplacian. Given that we used the flat metric in 4 dimensions the in this particular case the Laplacian will coincide with the Laplacian in the 2-sphere:

Codiff[metric2]@Diff@SphericalHarmonicY[l,m,θ[],ϕ[]]

We work out the Hodge Laplacian:

CodiffToDiff@%;

ExpandHodgeDual[%,dx[AdS],metric2];

ExpandHodgeDual[%,dx[AdS],metric2]

Simplify@%;

FullSimplify@%

Unfortunately this is not the simplest form for l,m natural numbers. We can, however, check that the result is indeed correct as follows:

%+l(l+1)/r[]^2SphericalHarmonicY[l,m,θ[],ϕ[]]

FunctionExpand@%

Simplify@%

FullSimplify@%