Spin-connection on xPert and xAct

[Hengameh Bagherian]

Hello, 

I am brand new to xAct and have been reading the documentation and forums for the past week. 

I am trying to derive the spin connection up to second order in perturbation for the quasi deSitter metric. 

I have a metric of the form: 

And my spin connection is:

 

, with 

 

 , and

 . 

I believe in xAct   is the RicciRotation[] with e^{\alpha}_{\beta} being the tetrads. 

What would be the best way I can do this with xAct and its sub packages?

[Thomas Bäckdahl]

Hi!

If I understand what you wrote correctly, you are using spinors with gamma matrix notation.
As far as I know no one has implemented this formalism in xAct.
The main spinorial packages Spinors, SymManipulator and SymSpin are all implemented in the abstract index 2-spinor formalism you can find in Penrose-Rindler.

Together with J Valente Kroon, I developed a formalism for calculus of variations for such abstract index 2-spinors.
https://arxiv.org/abs/1505.03770
I have a private implementation of that formalism in xAct that I could tidy up and release if it would be helpful.

If you want to stick to your notation, then one would need to spend some time telling xAct how to interact with the gamma matrices etc.

Regards
Thomas

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[Leo Stein]

Hi all,

In the xAct examples repo (https://github.com/xAct-contrib/examples), there are two notebooks relevant to the Dirac algebra,

- Clifford_Algebra_Traces.nb

- Gamma matrices (Dirac algebra).nb

However I suspect that it would be easier to use 2-component Weyl spinors with indices, and write the spinorial connection in terms of \sigma (the "soldering form" or "Infeld–Van der Waerden symbols"), rather than having to teach xTensor the rules of the capital \Sigma representation of the Lorentz algebra on 4-component spinors. A call to DefSpinStructure creates \sigma and its conjugate, and a covariant derivative that acts on the spin bundle you are declaring.

The usual xTensor difficulty in the type of calculation you want is to take the results of abstract perturbation theory calculations (coming from e.g. xPert) and evaluate their coordinate components (in your case, after specifying a metric and spin frame) in xCoba. I do not have an example notebook doing this, but maybe somebody else does?

Best

Leo

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[Hengameh Bagherian]

hmm, it would be nice if I could stick to my notation. Is there a sample anywhere that does anything remotely similar? So that I can then change to get what I want? 

If you think that is much harder, then I would be grateful if I could see your implementation as a starting point. 

Thank you so much, 

Hengameh

[Markus B. Fröb]

Dear all,

in principle FieldsX should work well for these types of computations,
section 4.5 in the manual https://arxiv.org/abs/2008.12422 deals with
frame bundles and spin connections. I attach a notebook with the
computation to first order. At second order one has to determine
manually the perturbation of the frame field in terms of the metric
perturbation (at first order I chose symmetric gauge), but otherwise
everything should generalize quite easily.
The not so easy part is the conformal transformation, since then xAct
complains about metric-incompatible derivatives sometimes :) Also, I
haven't tested FieldsX with conformal transformations apart from the
very basics.
The newest version of FieldsX can be found at
https://github.com/mfroeb/FieldsX. It uses by default the Freedman/van
Proeyen supergravity conventions, but those should be immaterial for
the present computation (and if not, can be changed).

Best, Markus

(BTW, when defining a conformally related metric xAct complains that
there is already a metric in the tangent bundle, but it seems to work
fine. Any idea why this happens?)


Zitat von Leo Stein <leo....@gmail.com>:

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