Work in arbitrary dimension with a metric

[Alvaro Martinez]

Hi, I'm new to xAct

I want to know if there is any way to define a dimensional D metric as an anzats and then evaluate it in a Lagrangian, I was looking at xcoba but I would not know how to define a metric that only depends on a radial coordinate and the other dependency is given by the unit sphere D -2 dimensional.

Sorry if my question doesn't make a lot of sense, but I've never used xAct before

[Jose]

Hi,

There is no specific built-in support for something like this, but that doesn't mean it cannot be done.

If your metric has a symbolic D dimension, then xCoba will not be able to handle that. You need to do your computation in xTensor.

Imagine you want to work in D-dimensional spherical symmetry. The first steps would be something like this:

    << xAct`xTensor`

Define the symbolic dimension:

    DefConstantSymbol[dim, PrintAs -> "D"]

Define the radial manifold:

    DefManifold[M2, 2, {a, b, c, d}]

Define the D-dimensional spheres:

    DefManifold[MD, dim, {A, B, C, D}] // Quiet

Define the product manifold of those two manifolds:

    DefManifold[M, {M2, MD}, {\[Mu], \[Nu], \[Lambda], \[Sigma]}]

Now define the metrics of each manifold:

    DefMetric[-1, g[-a, -b], cd]

    DefMetric[1, G[-A, -B], CD]

And now you would use DefProductMetric to construct the metric (diagonal by blocks) of the product manifold. See section 7.6 of xTensorDoc.nb on how to work with product metrics. Here is a link to an old HTML dump of that section:

    http://xact.es/Documentation/HTML/HTMLLinks/xTensorDoc.nb_54.html

Then you would construct your Lagrangian from the object defined in those manifolds, and then I guess you would use VarD to compute the equations of motion.

Cheers,

Jose.