Space / time decomposition and epsilon tensor

[Sjoerd]

Hello

In the attached notebook I try to manipulate expressions with epsilon tensors after making a split between space and time. To that end I introduce a product manifold, which means there are two metrics and two epsilon tensors. My issue with that approach is that not all four combinations of a derivative acting on an epsilon tensor vanish automatically.

There was a related discussion in this group,  called "No ProductMetricRules for epsilon tensor". What I understood from this discussion is that in that particular case a construction with induced metric has advantages. I looked at the documentation of induced metric, but it is not clear to me how base my calculation with induced metrics. In particular I can't see whether it would resolve my problem.

My question is: which approach do you advise? A product manifold or an induced metric?

Best regards,
Sjoerd

[Teake Nutma]

Hi Sjoerd,

While I can't directly answer your last question, I can try to solve your derivative problem. If you want to simply set all covariant derivatives on the two epsilon tensors to zero, you can use

CDs[_][(epsilong | epsilongs)[__]] = 0;
CD[_][(epsilong | epsilongs)[__]] = 0;



But you might not want to do that. Because the covariant derivative in the time submanifold is in general not equal to the the time component of the covariant derivative of the whole manifold. It might be that for the epsilon tensors they actually are the same (I forgot the details), but it's definitely safer to work with epsilon symbols and partial derivatives.

By the way, I noticed that you don't define a metric on the two-dimensional spatial submanifold, and that the whole manifold has Euclidean signature. Is that on purpose?

Best,


Teake

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> Attachments:
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[Jose]

Hi Sjoerd,

I´m afraid there is no definite answer to your question, because
xTensor is not yet prepared to handle general decompositions. In
principle, for n+1 decompositions it is better to use induced metrics,
but if you need to use epsilon tensors then things get more
complicated. If your computations will need the use of extrinsic
curvatures or the Gauss-Codazzi equations then I would certainly
recommend the induced metric method. But if your computations are of a
more general type, then perhaps constructing your own framework by
writing your own definitions, as you do in your notebook, is not a bad
idea at all. The main problem is the odd treatment of the 1-manifold
using indices.

Best,
Jose.

[Sjoerd]

Dear Teake and Jose,

Thank you for your responses to my question.
I am not using the extrinsic curvatures or the Gauss-Codazzi equations.
Now I am writing my own definitions in the way Teake suggested. Thanks for notify me about the odd (or actually even) metric signature I used. It is a mistake.

All the best,
Sjoerd