How to define a minimally coupled gauge covariant derivative?

[Alejo Hernandez]

Hi Everybody,

I would like to know how to convert a covariant derivative into a minimally coupled gauge covariant derivative.

I have tried the following, but the behavior of the newly defined derivative is not what i expected it to be:

Block[{Print}, Needs["xAct`xCoba`"]];

$DefInfoQ = False;

$CVVerbose = False;

$PrePrint = ScreenDollarIndices;

DefManifold[M, 4, 

  Complement[IndexRange[{a, z}], {g, h, r, s, t, x, y, z}]];

DefChart[B, M, {0, 1, 2, 3}, {t[], r[], x[], y[]}, ChartColor -> blue];

DefScalarFunction[F, PrintAs -> "f"];

$Assumptions = {r[] > 0};

AdSMetric = DiagonalMatrix[{-F[r[]], F[r[]]^-1, r[]^2, r[]^2}];

g = CTensor[AdSMetric, {-B, -B}];

cd = CovDOfMetric@g;

SetCMetric[g, B, SignatureOfMetric -> {3, 1, 0}];

MetricCompute[g, B, All];

DefScalarFunction[at];

DefScalarFunction[ay];

A = CTensor[{at[r[]], 0, 0, ay[x[]]}, {-B}];

DD[a_] := cd[a] - I A[a];

Is this approach correct? I also intend to use this expression in the future to write an action for a massive scalar field and calculate the equations of motion.

Thank you very much in advance.

Regards