Load Packages

Needs["xAct`xCoba`"]

Needs["xAct`xTensor`"];

Define manifold, chart and metric

DefManifold[M,2,{a,b,c}]

DefChart[ads,M,{1,2},{t[],z[]},ChartColor->Brown,FormatBasis->{"Partials","Differentials"}]

DefScalarFunction[Φ]

met=CTensor[DiagonalMatrix[{-1/((z[])^2),1/((z[])^2)}],{-ads,-ads}];

SetCMetric[met,ads,SignatureOfMetric->{1,1,0}]

Evaluate covariant derivatives and Ricci scalar

cd=CovDOfMetric[met]

RicciScalar[cd][]

Evaluate covariant derivative of a tensor

cd[-a][cd[a][Φ[t[],z[]]]]//Simplification

cd[-a][cd[-b][Φ[t[],z[]]]]//Simplification