xcoba: values to two connections

[Max Banados]

Hi, 

I need to work with two connections in xCoba. One is the standard Levi-Cevita associated to a given metric. I call it **cd[-m]**.

The second is the LeviCevita one plus a piece depending on a vector. (The vector, as the metric, have prescribed components on a chart already defined).  I call the second covariant derivative  **cd1[-m]**.  I have not managed to give values to **Christoffel[cd1,PDch]** which is necessary to compute covariant derivatives with **cd1[-m]**.      

I tried with *makeRule* and *IndexSetDelayed*, no luck...

See attached file for details. 

Any help?  Thanks!
Max

[Jose]

Hi,

This is a rather interesting question. I think the simplest thing to do is the following:

You have properly constructed the connection cd, as a CCovD object. This object has the structure CCovD[PDch, christoffel, metric], where christoffel is a rank 3 CTensor object and metric is a rank 2 CTensor object. We store the metric to indicate that this connection actually derives from a metric.

Now you want to have another connection that uses the same ordinary derivative PDch but a different Christoffel, and I assume it will not derive from a metric in general. Then what you need to do is to set

    cd1 = CCovD[PDch, newchristoffel, Null]

where the newchristoffel is again a rank 3 CTensor object, and Null means that this connection does not derive from a metric. You can use standard methods of computation with CTensor objects to construct the new Christoffel tensor.

Note that we don't need to go through a DefCovD[cd1[-a]] command. This command is needed when we will be using an abstract connection cd1. In your problem what you need is a concrete (given by components) connection, i.e. a CCovD object. It's exactly the same reason why you have a metric g but didn't need to evaluate a DefMetric command.

Cheers,

Jose.

[Max Bañados]

Dear José, 

Thank you very much for your reply (and for creating this amazing platform). 

I am sorry it took me so long to react to you email.  

I read you suggestion but I did not manage to put it to work. I tried 
various combinations but the command suggested finally 
did not give a component result.  

Would it be possible for you to produce a concrete example? 

Thank you very much, 

Max 

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[Jose]

Hi again,

I attach a suggestion.

Cheers,

Jose.

[Max Bañados]

Dear José, 
Fantastic.  I was missing the command HeadOfTensor. 

There is still a lot I need to learn about xTensor. 

Thanks a lot, this will make my calculation much easier.

Regards,

Max

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