Lie derivative with coordinates?

[Mattia Villani]

How do you calculate the Lie derivative with coordinates in xCoba? As far as I understand, LieD only works on abstract tensors.

[Jose]

Hi,

CTensor[...] can be used in all places in which you can use an abstract tensor T defined with DefTensor. For example, set up Schwarzschild with standard coordinates:

<< xAct`xCoba`

DefManifold[M4, 4, {a, b, c, d, e, f}]

DefChart[sch, M4, {0, 1, 2, 3}, {t[], r[], \[Theta][], \[Phi][]}]

DefConstantSymbol[M]

met = CTensor[DiagonalMatrix[{-1 + 2 M/r[], 1/(1 - 2 M/r[]), r[]^2, r[]^2 Sin[\[Theta][]]^2}], {-sch, -sch}]

Introduce a vector field with only radial component:

DefScalarFunction[vr]

v = CTensor[{0, vr[t[], r[], \[Theta][], \[Phi][]], 0, 0}, {sch}]

Now you can compute the Lie derivative of the metric along this vector field as

LieD[v[c]][met[-a, -b]] // Simplify

Recall that index c is not really an index of the expression. The result still has only indices -a, -b.

Check that you get the same result via symmetrization of Levi-Civita differentiation of the vector field:

cd = CovDOfMetric[met];

met[-b, -c] cd[-a]@v[c] + met[-a, -c] cd[-b]@v[c] // Simplify

To see all entries of the result use

$LargeComponentSize = 2000

Cheers,

Jose.