Index of a Tensor/Object in xCoba

[Prosenjit Paul]

Hiii, 

I am trying to compute specific components of the double covariant derivative of the weyl tensor in xCoba. Here is my code: 

<< xAct`xCoba

DefManifold[M, 4, {\[Alpha], \[Beta], \[Gamma], \[Delta], \[Mu], \[Nu], \[Lambda], \[Sigma]}]

 

DefChart[ch, M, {0, 1, 2, 3}, {t[], r[], \[Theta][], \[Phi][]}, ChartColor -> Blue]

DefScalarFunction[f]

DefScalarFunction[h]

met = CTensor[DiagonalMatrix[{-h[r[]], 1/f[r[]], r[]^2,r[]^2 Sin[\[Theta][]]^2}], {-ch, -ch}];

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

cd = CovDOfMetric[met];

Now, I want to compute the double derivative of weyl Tensor as:

expr = cd[\[Mu]] [cd[\[Nu]] [Weyl[cd][-\[Alpha], -\[Beta], -\[Gamma], -\[Delta]]]];

result = Head[expr];

Now I want the specific component of the expr Tensor/Object.

Let's say I want \[Del]^2 [ \[Del]^3 [Weyl[-1, -2, -1, -3] ]], (where  \[Del] is the covariant derivative) Then which one is the correct :

result[{1, -ch}, {2, -ch}, {1, -ch}, {3, -ch}, {3, ch}, {2, ch}];

                                      or

result[{3, ch}, {2, ch}, {1, -ch}, {2, -ch}, {1, -ch}, {3, -ch}]  

I also attached the nb.

Thank You.

 

[Jose]

Hi,

The correct answer is your first line. In xAct cd[-a][v[-b]] gives a tensor with index structure cdv[-b, -a]. This follows Wolfram Language, which also "adds indices" at the end:

In[1]:= D[{f1[x, y], f2[x, y], f3[x, y]}, {{x, y}}] // Dimensions
Out[1]= {3, 2}

This differentiates a 3-vector wrt coordinates {x, y}, giving a matrix of dimensions {3, 2}.

You can also check the list of bases in your result (recall the structure CTensor[array, bases, weight]):

In[14]:= result[[2]]
Out[14]= {-ch, -ch, -ch, -ch, ch, ch}

This is because you started with your Weyl tensor with four down-indices and then differentiated it with two covariant derivatives with up-indices.

Cheers,

Jose.