dimension dependent identities

[Daniel Mahler]

There are identities that are true in only some dimensions.

For example in 2 dimensions $R_{ijkl} = R \varpsilon_{ij} \varepsilon{kl}$.

Can xAct  utilize this identity or do I need to add it?

If it has it, which function applies it?

thanks

Daniel

[Leo Stein]

Hi Daniel,

This will get you part of the way there:

Since this is a special case of the Weyl decomposition, you can enact this in two dimensions by using RiemannToWeyl. However, instead of being expressed in terms of \epsilon, it will be expressed in terms of the metric, but with the correct index symmetries. For example, in two dimensions:

  In[] := RiemannCD[-a, -b, -c, -d] // RiemannToWeyl

  Out[] = -(1/2)  met[-a, -d]   met[-b, -c]  RicciScalarCD[] + 1/2  met[-a, -c]   met[-b, -d]  RicciScalarCD[]

I know this is not exatly what you wanted, but I hope this helps.

Leo

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[Teake Nutma]

Hi Daniel,

I've written a bit of functionality in the xTras package that might come in handy for you. The main function you'll be interested in is ConstructDDIs (http://xact.es/xtras/documentation/ref/ConstructDDIs.html), which constructs all possible DDIs out of a given expression. How you might use these DDIs once constructed is explained in the tutorial for the Gauss-Bonnet term, http://xact.es/xtras/documentation/tutorial/GaussBonnet.html.
By the way, did you mean three dimensions instead of two in your example? In two (spacetime) dimensions not only the Weyl tensor but also the Einstein tensor namely vanishes.
Best,

Teake

On Monday, January 12, 2015 at 11:23 PM, Leo Stein wrote:

> Hi Daniel,
>
> This will get you part of the way there:
> Since this is a special case of the Weyl decomposition, you can enact this in two dimensions by using RiemannToWeyl. However, instead of being expressed in terms of \epsilon, it will be expressed in terms of the metric, but with the correct index symmetries. For example, in two dimensions:
> In[] := RiemannCD[-a, -b, -c, -d] // RiemannToWeyl
> Out[] = -(1/2) met[-a, -d] met[-b, -c] RicciScalarCD[] + 1/2 met[-a, -c] met[-b, -d] RicciScalarCD[]
>
> I know this is not exatly what you wanted, but I hope this helps.
> Leo
>

> On Mon, Jan 12, 2015 at 4:21 PM, Daniel Mahler <dma...@gmail.com (mailto:dma...@gmail.com)> wrote:
> > There are identities that are true in only some dimensions.For example in 2 dimensions $R_{ijkl} = R \varpsilon_{ij} \varepsilon{kl}$.

> > Can xAct utilize this identity or do I need to add it?
> > If it has it, which function applies it?
> >
> > thanks
> > Daniel
> >
> >
> > --
> > You received this message because you are subscribed to the Google Groups "xAct Tensor Computer Algebra" group.

> > To unsubscribe from this group and stop receiving emails from it, send an email to xact+uns...@googlegroups.com (mailto:xact+uns...@googlegroups.com).

> > For more options, visit https://groups.google.com/d/optout.
>
>

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