xCoba and covariant derivatives

[pelayo]

Using xCoba,  once i define a manifold, a chart, a metric and its covariant derivative:

DefManifold[M, 4, Complement[IndexRange[a, q], {g, n, h}]]
DefChart[coor, M, {0, 1, 2, 3}, {t[], r[], \[Theta][], \[Phi][]}] 

metric = CTensor[metricarray, {-coor, -coor}] ;
SetCMetric[metric, -coor, SignatureOfMetric -> {3, 1, 0}];
cd = CovDOfMetric[metric];

 

for a CTensor, call it 'n', I don't get an equivalent result computing

cd[-a][n[-b]]//ToBasis[coor]

or doing

cd[{-a,-coor}][n[{-b,-coor}]]

Do anyone know why? 

Thank you very much

[Jose]

Hi,

We need to distinguish n[b], which is an abstract vector field, and n[{b, coor}], which is a component in the coor frame of that abstract vector field. In particular, n[b] is a vector but n[{b, coor}] is a scalar. More explicitly we have:

    n[{b, coor}] == n[a] Basis[-a, {b, coor}]

Therefore, cd[-a][n[b]] is a covariant derivative of a vector field, a tensor with two free abstract indices. ToBasis[coor] basically contracts basis vectors and expands:

   ToBasis[coor][ cd[-a][n[b]] ]   ->   cd[-c][n[d]] Basis[{-a, coor}, c] Basis[-d, {b, coor}]

This is a useful operation, but not meaningful from a tensorial point of view because it converts abstract indices into basis indices with the same name.

Then cd[{-a, coor}][ n[{b, coor}] ] is a component of the covariant derivative of a scalar field. Hence the result does not contain any information of the connection:

   cd[{-a, coor}][ n[{b, coor}] ] == Basis[{-a, coor}, c] cd[-c][ n[d] Basis[-d, {b, coor}] ]

Hope that helps.

Cheers,

Jose.