Some basic questions

[David Pirtskhalava]

Hi,

Sorry for asking such basic questions:

1. How should I specify the background metric when working with
xPert? (Say if I want to work with flat background)

2. Can I somehow have only the first perturbation of g_{a,b} nonzero,
setting all higher order perturbations like h^(2,3,..)_{a,b} to zero?

Thank you very much,

david

[Ethan]

David,

Hi,

The second question is simpler so let's start with that.

2) You can do this by some command like:

bgfield = h[LI[order_], __] :> 0 /; order > 1


This is in the background field section of the xPert documentation.

1) This depends on what exactly you are looking for. Two possible
options are i) specify a flat background (this is a bit subtle if not
done in xCoba, because presumably what you want is a flat background
in a particular set of coordinates, however xPert and xTensor is
naturally covariant). ii) work with xCoba.

i) Here is an example with a flat background and no xCoba

One method is to define two metrics:

DefMetric[-1, g[-a, -b], CD, {";", "\[Del]"},
WeightedWithBasis -> AIndex]
DefMetric[-1, \[Eta][-a, -b], PD, PrintAs -> "\[Eta]",
SymbolOfCovD -> {",", "\[PartialD]"}, FlatMetric -> True]

Take your perturbed expression and convert everything to the metric,
and then substitute in the flat metric,

exp=perturbedexp// RiemannToChristoffel // CovDToChristoffel //
ChristoffelToMetric/.{g[a_, b_] -> \[Eta][a, b], Detg[] -> -1}

Then switch metrics

MetricsOfVBundle[TangentM] ^= {\[Eta], g}
Inv\[Eta] = \[Eta]

Then simplify

exp// ContractMetric // Simplification

Note, this only works for a flat background not an arbitrary
background.

Alternatively you can work with xCoba as well.

Here you only need to define 1 metric (which is more natural), and you
can have an arbitrary background. What i do is

a) define my background metric and have xCoba calculate all of the
relevant symbols
b) do background perturbation -> turn everything into christoffels
(this should not be necessary, but I haven't figured out a good way to
get xCoba to calculate the values of all the different index
structures for the curvature tensors. For instance it knows R[-a,-b]
and g[a,b], but then R[a,b] has to be put in by hand)
c) sum over dummies and evaluate everything in some basis

Hope this helps.

Ethan

On Aug 5, 11:36 pm, David Pirtskhalava <dato.pirtskhal...@gmail.com>
wrote: