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I am trying to use xAct package to calculate certain mixing terms of $\nabla_{\chi} b$ of a scalar function $b[ \tau, r, \theta, \phi]$ for a modified metric. Here is my Code:

<< xAct`xTras`
ddim = 4;
coords = {\[Tau][], r[], \[Theta][], \[Phi][]};

DefManifold[M4, 
  ddim, {[Alpha], [Beta], [Gamma], [Sigma], [Mu], [Nu]}];
DefMetric[-1, g[-[Alpha], -[Beta]], CD, PrintAs -> "g"];
DefScalarFunction[{a, [Eta], b}];
DefChart[ch, M4, {0, 1, 2, 3}, coords, ChartColor -> Red];
DefConstantSymbol[K]


FRWr = DiagonalMatrix[{
   -[Eta][[Tau][]]^2,
   b[[Tau][], r[], [Theta][], [Phi][]]^2/(1 - K r[]^2),
   b[[Tau][], r[], [Theta][], [Phi][]]^2 r[]^2,
   b[[Tau][], r[], [Theta][], [Phi][]]^2 r[]^2 Sin[[Theta][]]^2
   }
  ]


MatrixForm@MetricInBasis[g, -ch, FRWr]
MetricCompute[g, ch, Einstein, Verbose -> True]


CD[-[Alpha]]@CD[[Alpha]]@b


CD[-[Alpha]]@CD[[Alpha]]@b[[Tau][], r[], [Theta][], [Phi][]]


CD[-[Alpha]]@b CD[[Alpha]]@b


CD[-[Alpha]]@b CD[[Alpha]]@b g[-[Mu], -[Nu]]


(CD[-[Alpha]]@b CD[[Alpha]]@b) g[-[Mu], -[Nu]] g[[Gamma], [Sigma]]

The output of the last 5 lines is here:

enter image description here

How can I evaluate these abstract outputs? I have tried using //ToBasis[ch]// ToValues but didn't help either.

Any help would be greatly appreciated, thank you in advance.

Muktadir Rahman
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1 Answers1

3

I'd recommend to do the following. Keep this part of your code:

<< xAct`xTras`

ddim = 4;
coords = {[Tau][], r[], [Theta][], [Phi][]};


DefManifold[M4, ddim, {[Alpha], [Beta], [Gamma], [Sigma], [Mu], [Nu]}];
DefScalarFunction[{a, [Eta], b}];
DefChart[ch, M4, {0, 1, 2, 3}, coords, ChartColor -> Red];
DefConstantSymbol[K]


FRWr = DiagonalMatrix[{
    -[Eta][[Tau][]]^2, 
    b[[Tau][], r[], [Theta][], [Phi][]]^2/(1 - K r[]^2), 
    b[[Tau][], r[], [Theta][], [Phi][]]^2 r[]^2, 
    b[[Tau][], r[], [Theta][], [Phi][]]^2 r[]^2 Sin[[Theta][]]^2
}]

You don't need to use a symbolic metric g. We can just work with this explicit metric:

g = CTensor[FRWr, {-ch, -ch}]

Compute everything up to the Einstein tensor and set this metric as the one to be used to raise/lower indices automatically:

MetricCompute[g, ch, "Einstein", Verbose -> True]
SetCMetric[g, ch, SignatureOfMetric -> {3, 1, 0}]

Get the associated Levi-Civita connection:

CD = CovDOfMetric[g];

Now you can compute the Laplacian of a scalar:

bsc = b[\[Tau][], r[], \[Theta][], \[Phi][]]

CD[-[Alpha]]@CD[[Alpha]]@bsc

or the square of a gradient:

CD[-\[Alpha]]@bsc CD[\[Alpha]]@bsc

This case requires some further simplification:

BasisExpand[%, ch] // Simplify

Note that you cannot compute derivatives of b, because b is a function. You first need to specify the arguments of that scalar function (that's the scalar bsc).

jose
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