Levi-Civita tensor in tensor definitions

[Zach Elgood]

Hello,

I have recently started using xAct and am trying to find a way to implement the Levi-Civita tensor in order to do calculations.  For example, I wish to calculate the curl, so I want to use MakeRule to define a new tensor somewhat like below

MakeRule[{w[-a],epsilon[-a,-b,-c,-d]*u[b]*CD[c][u[d]]},MetricOn->All]

where the epsilon term is the Levi-Civita tensor. I can't seem to find any way to do this, though I may have missed something in the documentation.

[Thomas Bäckdahl]

Hi!

You made it almost right.
epsilon is not the Levi-Civita tensor itself, it is a function that takes the metric and gives the Levi-Civita tensor for that metric.
If your metric is called g, then epsilon[g] gives the correct name of the Levi-Civita tensor, which is epsilong in that case.
Hence, change epsilon to epsilon[g] or epsilong in your code, unless your metric is called something else.

Regards
Thomas

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[Leo Stein]

Hi Zach,

Every metric defines its own volume form (and they are all proportional to each other; see e.g. MTW Eq. (8.10).

If you named your metric met[-a,-b], then the associated volume form will be named epsilonmet[-a,-b,-c,-d]. You should have seen this name printed out as one of the auxiliary tensors defined by DefMetric[].

You can also access this tensor via epsilon[met][-a,-b,-c,-d] if you want to be more programmatic in some places.

L

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[Zach Elgood]

Thank you very much for the reply; it works perfectly. One small question. In my example above, I had a separate expression for the tensor u. Let say I have AB = MakeRule[{w[-a], epsilon[met][-a, -b, -c, -d] u[b] CD[c][u[d]]}, 

  MetricOn -> All], so w[-a]/.AB gives the result I wish. If I also had an expression for u, say AA=MakeRule[{u[-a],PD[-a]x},MetricOn->All], how would I be able to substitute that expression into my final tensor. In other words, is it possible to make nested substitutions for the tensors?