ExpandPerturbation not working properly + some strange index behaviour

[Johannes Pirsch]

Hey everyone,

I am currently trying to do some expansions around a flat background. //ExpandPerturbation works fine on the RiemannCD and RicciCD. But for RicciScalarCD and the tetrad fields of FieldsX it does not expand the expressions

RicciScalarCD+\kappa Delta RicciScalarCD

frame[-mu,a]+\kappa Delta frame[-mu,a]

SpinConnection[-mu,a,b]+\kappa Delta SpinConnection[-mu,a,b]

any further. Am I missing something?

Also on that note, when I put 

RiemannCD[-\mu, -\nu, -\varrho, -\sigma] // RiemannToChristoffel,

it writes the expression in terms of the framebundle metric eta[] with tangent space indices instead of flat ones. I thought that this maybe just refers to eta[-\mu,-\nu]frame[-\mu,a]frame[-\nu,b]eta[-a,-b]  but Mathematica does not put 

eta[-\mu,-\nu]frame[-\mu,a]frame[-\nu,b]eta[-a,-b]-g[-\mu,-\nu] to zero.

Johannes

[Markus B. Fröb]

Dear Johannes,

without a MWE it is difficult to see where the problem lies.

What I can say is that the non-expansion of the perturbed frame field
and the spin connection is intentional, since one might want to use
these variables as fundamental ones, like in the 1.5-order formalism
for supergravity.

You could use SpinConnectionToFrame (FieldsX manual Sec. 4.5.9) to
expand your spin connection in terms of derivatives of the frame field
and torsion. This should also work for perturbed spin connections.
Note that FieldsX uses the conventions of the Freedman/van Proeyen
Supergravity book, here in particular Eq. (7.91). If you don't have
torsion, you can simply set it to zero afterwards, or with an
automatic rule.

For the frame field, any perturbation can be decomposed into a
symmetric part (expressible using the metric perturbation) and an
antisymmetric part, which is a local Lorentz rotation. (See for
example Forger/Römer hep-th/0307199 Eq. 256.) If you want to ignore
the antisymmetric part, you could just add an automatic rule to
replace frame field perturbations with their symmetric part,
Forger/Römer Eq. 258. But again, this is not automatic because the
antisymmetric part is relevant in some applications.

Your last comment sounds very strange to me, and it seems that
something went wrong before. Certainly ToCanonical@ContractMetric[
frame[-\mu,a]frame[-\nu,b]eta[-a,-b] ] gives g[-\mu,-\nu], and FieldsX
doesn't touch RiemannToChristoffel. A MWE would be helpful to see what
the issue is.

Best, Markus



Zitat von Johannes Pirsch <acanto...@gmail.com>:

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