I would like to attempt to calculate the laplacian operator using index notation and shoow that it gives the usually expected laplacian in spherical coordinates. I am able to do most of the steps however, I cannot get xCoba to reduce the final form out of index notation. Consider the following minimal example:
Assuming the metric has the form of:
$$ \text{d} s^2 = \text{d}r^2 + r^2 \text{d}\Omega^2$$
and that the laplacian may be calculated using:
$$ \Delta Y(r,\theta,\phi) = g^{mn}(\partial_m\partial_n - \Gamma^l_{\ mn} \partial_l) Y $$
Clear["Global`*"]
<< xAct`xCoba`
DefManifold[M, 3, {a, b, c, d, e, f}]
DefChart[coords, M, {0, 1, 2}, {r[], \[Theta][], \[Phi][]}]
metric = {{1, 0, 0}, {0, r[]^2, 0}, {0, 0, Sin[\[Theta[]]]^2*r[]^2}}
g = CTensor[metric, {-coords, -coords}]
SetCMetric[g, coords, SignatureOfMetric -> {3, 0, 0}]
CD = CovDOfMetric[g]
DefTensor[Chs[a, -b, -c], M]
DefScalarFunction[{Y}]
IndexSet[ChS[c_, -a_, -b_],
ChristoffelFromMetric[g[-d, -e], coords][c, -a, -b]];
laplacianY =
g[a, b][PDcoords[-a][PDcoords[-b][Y[r[], \[Theta][], \[Phi][]]]] -
ChS[c, -a, -b][PDcoords[-c][Y[r[], \[Theta][], \[Phi][] ]]]]
At this point as far as I can tell laplacianY
has the correct form in index notation, but I cannot find the required xCoba function that expands all of the indexes. I've tried: ToCanonical
, ContractBasis
,TraceDummy
but all of these either throw an error or return the same function. I can't currently see any other function in the xCoba package that sounds like it would reduce index notation.
Any help is greatly appreciated.
ContractBasis
to work as expected. $\endgroup$CD[-a][CD[a][Y]]
, or something like that? $\endgroup$