Tutorial

[Jolyon Bloomfield]

Hi folks,

I'm presenting a "tutorial" on various aspects of xTensor that I've found useful in my own research at a workshop tomorrow. I thought that the tutorial might be useful for new users of xTensor, and wanted to make it available to all. Please find it in the attachments. Note that I have deleted the output, which took an extra 2 Mb to store.

Best,
Jolyon

[Jose]

Many thanks Jolyon!

I've added the file to the page http://www.xact.es/documentation.html , after adding a title and reproducing the output.

Cheers,

Jose.

[Teake Nutma]

Hi Jolyon,

That looks very nice! I like how you tackle more advanced topics to explain basic features about xTensor; that way people will see how powerful it is. But allow me to promote some features of my xTras package that might come in handy for the cases you discuss :).

In section one you go through a bit of tensor manipulation to show that the 4D Lovelock tensor is zero. Alternatively, you one could have sufficed with the command

ConstructDDIs[RiemannCD[a, b, c, d] RiemannCD[e, f, g, h], IndexList[a, b]]

which constructs all dimensionally dependent identities that consist of contractions of two Riemanns and have two free indices. Of course, the whole point of your notebook is to go through the tensor manipulation business, but it might be mentioning as a shortcut.
Secondly, section three computes equations of motion of the metric, where you first need to perturb the action. xTras redefines the VarD command such that this isn't necessary; VarD[metric[-a,-b],CD] automatically gives the 'correct' equations of motion of the metric. Also, for the simplifying of the eom of the Gauss-Bonnet term xTras has the functions SortCovDsToDiv and CurvatureRelationsBianchi, which are used in the 'magic function' FullSimplification. Here's a 3 line derivation of the Gauss-Bonnet equations of motion:

GB = NoScalar@EulerDensity[CD]
eom = VarL[metric[-a, -b], CD][GB] // ContractMetric // ToCanonical


eom = FullSimplification[][eom]



Again, I guess the whole point of your tutorial is to go through all the smaller steps so people get a better feel of the various aspects of xTensor, but the lazy among us might welcome these shortcuts.

Though overall your tutorial looks very good! (And very long! How long did you take presenting all that material?)
Best,

Teake

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[Jolyon Bloomfield]

Hi Teake,

Thanks for the praise!

To be honest, I'm not all that familiar with your xTras package (although it has been on my list of things to investigate for a while now!). In the stuff that I present, I really wanted to demonstrate basic manipulations - using xTensor to reproduce basic manipulations that you would do by hand. Although, I must admit, I really should look into your FullSimplification function. Actually, I might even swap out my various rearrangement rules in favor of FullSimplification...

I haven't presented this yet. I want to use it more as a demonstration than an interactive tutorial. I expect it to take about an hour to present in total. Having the entire tutorial available for people to step through in detail in their own time afterwards will hopefully be helpful, too.

Cheers,
Jolyon

[Jolyon Bloomfield]

Hi all,

I just wanted to give you all an update on the tutorial. I presented it at a conference; it took about an hour to go through (although it was an interactive hour). The audience was incredibly appreciative! So much so, I was asked to present it again at the next university I visited. It kind of feels like I'm giving away trade secrets, but I'm sure you'll all tell me I'm being noble, right?

People were interested in seeing the TeX output stuff, so I inserted a brief segment for that. Jose will have the updated version in the documentation section of his website shortly.

Two things that I haven't played with but was asked about were the spinors handling and the harmonic decompositions. One person particularly wanted to know if spin 3/2 fields could be handled, and if spinors could be conformally transformed appropriately. If anybody wants to write a tutorial on those packages, I'm sure they'd be appreciated :-)

Cheers,
Jolyon