CTensor contraction

[Maryna М]

Hi everyone,

I'm new to Mathematica, and I'm currently trying to define some tensors that I'll need in further calculations. However, Mathematica fails to contract the tensors I give it. Instead, it just lists all the tensors with respective indices. I tried to find some similar threads, but not seem to success. Could you may be point out, what I'm doing wrong? Thanks!

ClearAll["Global`*"]
<< xAct`xCoba`
Needs["xAct`xPert`"]
Quiet@Block[{Print}, << xAct`xTras`]
$DefInfoQ = False;

DefManifold[M,
  4, {\[Mu], \[Nu], \[Lambda], \[Sigma], \[Rho], i, j, k}];
DefChart[ch, M, {0, 1, 2, 3}, {t[], x[], y[], z[]}]
DefScalarFunction[a]
g = CTensor[{{-1, 0, 0, 0}, {0, a[t[]]^2, 0, 0}, {0, 0, a[t[]]^2,
     0}, {0, 0, 0, a[t[]]^2}}, {-ch, -ch}];
SetCMetric[g, ch]
DefScalarFunction[\[Phi]]
CD = CovDOfMetric[g]

A = CTensor[{\[Phi][t[]], 0, 0, 0}, {ch}]
X = -1/2 A[\[Mu]] A[-\[Mu]]
G = X^p
Fmn[-\[Mu], -\[Nu]] = CD[-\[Mu]]@A[-\[Nu]] - CD[-\[Nu]]@A[-\[Mu]]
F = -1/4 Fmn[-\[Mu], -\[Nu]] Fmn[\[Mu], \[Nu]]
Y = A[\[Mu]] A[\[Nu]] Fmn[-\[Mu], \[Alpha]] Fmn[\[Nu], \[Alpha]] 

[Jose]

Hi,

There are a few issues here:

1) With symbolic components, SetCMetric cannot compute the sign of the determinant of the metric. Therefore you should add the option SignatureOfMetric -> {3, 1, 0}, meaning three +1s, one -1 and zero 0s.

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

2) The vector A you have defined has zero antisymmetric part in its covariant derivative, so it's not a generic example here. Let me complicate it a bit so that the following computations are not trivial:

   A = CTensor[{\[Phi][t[]], t[]^2, 0, 0}, {ch}]

3) Once you do a computation with CTensor and you want to define a new tensor, this is what you need:

   Fmn = HeadOfTensor[CD[-\[Mu]]@A[-\[Nu]] - CD[-\[Nu]]@A[-\[Mu]] // Simplify, {-\[Mu], -\[Nu]}]

This returns a new CTensor object, which gets assigned to Fmn. Then the following computations will work as you expect.

4) Your final computation has index \[Alpha], but \[Alpha] was not in the list of indices of the manifold M (actually of its tangent bundle TangentM). You should define all the indices you use in inputs.

Ask again if you continue having problems.

Cheers,

Jose.