Laplacian in coordinate chart

[Robert Maxton]

As part of a general attempt to understand xCoba, I'm trying to get xCoba to calculate the vector Laplacian in a given coordinate chart for me.

Abstractly, I can certainly get roughly the right answer with something like

PDcyl[-b][PDcyl[b][X[a]]] // CovDToChristoffel

but I can't seem to get xCoba to actually expand the Christoffel symbols in the particular chart I'm using.

For example, in cylindrical coordinates, I can write

<< xAct`xTensor`
<< xAct`xCoba`
<< xAct`SymManipulator`

DefManifold[M, 3, {a, b, c, d, e, f, g, h, i, j, k}];
DefBasis[frame, TangentM, {1, 2, 3}, BasisColor -> Blue];
DefChart[cyl, M, {1, 2, 3}, {\[Rho][], \[Phi][], z[]},
 ChartColor -> Red]

cylg = CTensor[( {
    {1, 0, 0},
    {0, \[Rho][]^2, 0},
    {0, 0, 1}
   } ), {-cyl, -cyl}]

SetCMetric[cylg, cyl]

However, this doesn't seem to update PDcyl or ChristoffelPDcyl to match the new metric; at any rate ChristoffelPDcyl doesn't have any values associated with it. If I explicitly use the derivative associated with my CMetric, xCoba just refuses to actually expand the expression:

(Indeed, I can't get xAct to expand the covariant derivatives at all except by using PD with abstract indices.)

-- Unrelatedly, what exactly is the distinction between a "parallel derivative" and a "covariant derivative"? Is it just the possible existence of a spin connection in the latter case?

Any advice would be appreciated!

[Jose]

Hi,

Apologies again for the late reply.

Let me start with your last question: in xAct we have the general concept of covariant derivative, of the form cd[-a] for some symbol cd. Those covariant derivatives can have curvature and torsion, or just one, or none of them. The covariant derivatives may act on fields on the tangent bundle and/or on other bundles. We have the following particular cases:

1) If there is a frame field (as defined with DefBasis) such that cd[-a] gives zero on all vector fields of the frame, then we say that the covariant derivative is parallel to the frame. The parallel derivative of the frame F (as defined with DefBasis[F, ...]) is called PDF in xCoba. Parallel derivatives do not have curvature, but in general will have torsion. This torsion is essentially (modulo sign) the tensor containing the Lie brackets of the frame vectors.

2) If the covariant derivative cd does not have curvature and does not have torsion, then we say it is an "ordinary derivative" (name taken from Wald's book).

3) If we have a coordinate chart S (as defined with DefChart[S, ...]), whose associated coordinated frame will also be called S, then the parallel derivative PDS does not have torsion, and therefore it is an ordinary derivative. (We know that the vectors of a coordinated frame commute.) The object PDS[-a] is a covariant derivative like any other, but it is special because it does not have curvature, nor torsion, and gives zero on the vector fields of the frame S. Then the derivatives PDS[{1, -S}], PDS[{2, -S}], ... can be understood as the "partial derivatives" with respect to the scalar fields of the chart S.

4) Any ordinary derivative derives locally from some coordinate chart. Given that coordinate charts are handled in xCoba, it might seem that we cannot have ordinary derivatives in xTensor. We use a special (sometimes called "fiducial") derivative PD in xTensor without specifying the chart to which it is associated. Therefore, once you have loaded xCoba and you are working with particular frames, charts or coordinate fields, it is better to forget about PD and use the actual parallel derivatives of the frames you have constructed.

Having said all that, let's go back to your example. Load xCoba (which loads xTensor) and define a manifold and a chart (with its coordinated frame with the same name):

    << xAct`xCoba`

    DefManifold[M, 3, {a, b, c, d, e, f, g, h, i, j, k}];

    DefChart[cyl, M, {1, 2, 3}, {\[Rho][], \[Phi][], z[]}, ChartColor -> Red]

A vector field in that chart could be:

   DefScalarFunction /@ {v1, v2, v3};

   v = CTensor[{

       v1[\[Rho][], \[Phi][], z[]],

       v2[\[Rho][], \[Phi][], z[]],

       v3[\[Rho][], \[Phi][], z[]]

   }, {cyl}];

So you can now compute PDcyl derivatives (to any order):

    PDcyl[-a][v[-b]]

You cannot yet compute PDcyl "Laplacians" because there is no metric to contract indices. So we now introduce your cylg metric, set it as the first-metric (so that indices are automatically raised and lowered with this metric), and compute its Levi-Civita covariant derivative cd:

   cylg = CTensor[({{1, 0, 0}, {0, \[Rho][]^2, 0}, {0, 0, 1}}), {-cyl, -cyl}];

   SetCMetric[cylg, cyl, SignatureOfMetric -> {3, 0, 0}]

   cd = LC[cylg]

Now you can compute the Laplacians:

   PDcyl[a]@PDcyl[-a]@v[b]

   cd[a]@cd[-a]@v[b]

Recall that Christoffels always take two covariant derivatives in xAct (to make them true tensors), so the Christoffel you need is

   Christoffel[cd, PDcyl]

Hope that clarifies things.

Cheers,

Jose.