Re: [xAct: 1150] Conformal transformation

[Obinna Umeh]

Hi Max,

xAct has features/functions to help you solve your problems, however, the new package xPand (which may be downloaded from here: http://www2.iap.fr/users/pitrou/xpand.htm in case you don't have it yet) has an example notebook within the main xPand folder on the kind of problem you intend to solve.

Obinna

On Wed, Mar 20, 2013 at 6:57 PM, Max <max.g...@gmail.com> wrote:

Hello!

Sorry if this is a very stupid question...

I guess the (great) package xAct is too big for my little task, but after trying some other much smaller packages (GREAT,...) and beeing frustrated over some others with strange licencing issues, I played around with xAct for a day now - without much progress...

My problem is:

I want to conformally transform my metric via g_{mn} = exp(C(x))g2_{mn} where C(x) is an arbitrary function of the space time coordinates. Afterwards I want to show how the Ricci tensor and Ricci scalar tranform.

Is there a simple way to do it, because what I was doing now included a lot of "own" function definitions (e.g. my\[CapitalGamma][a_, b_, c_] :=  Christoffelcd[c, -a, -b] - delta[c , -a] PD[-b][cf[]] -  delta[c, -b] PD[-a][cf[]] + metric[-a, -b] PD[c][cf[]] ).

Thanks a lot for your help,

Max

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[Alex Vañó Viñuales]

You can first define a metric, for instance:
DefMetric[-1,metric1[-i,-j],cd1,PrintAs->"g1"]
Then define a scalar function, for example:
DefTensor[C[],M]
And finally define a metric conformally related to the previous one:
DefMetric[-1,metric2[-i,-j],cd2,ConformalTo->{metric1,Exp[-
C[]]},PrintAs->"g2"]
or
DefMetric[-1,metric2[-i,-j],cd2,ConformalTo->{metric1,1/C[]]},PrintAs-
>"g2"]
depending on how you want to relate both metrics.

With
ConformalFactor[metric1, metric2]
and
ConformalRules[metric1,metric2]
you can check their relation.

And then with the previous function and
ChristoffelToGradConformal[Christoffelcd2[i,-j,-k],metric2,metric1]
also using
Christoffelcd1cd2p[i,-j,-k]//BreakChristoffel
you will be able to establish the relation between the Christoffel
symbols of the two metrices.

Hope this little mess helps! Please, ask if more details needed.

Alex

On Mar 20, 6:42 pm, Obinna Umeh <umeobi...@gmail.com> wrote:
> Hi Max,
>
> xAct has features/functions to help you solve your problems, however, the

> new package xPand (which may be downloaded from here:http://www2.iap.fr/users/pitrou/xpand.htmin case you don't have it yet)

[Cyril Pitrou]

Dear Max,

xAct provides the tools to get the Christoffel which relates the covariant derivative associated with one metric to the covariant derivative associated with the conformally related metric. This is essentially what you tried to implement with

my\[CapitalGamma][a_, b_, c_] :=  Christoffelcd[c, -a, -b] - delta[c , -a] PD[-b][cf[]] -  delta[c, -b] PD[-a][cf[]] + metric[-a, -b] PD[c][cf[]]

Using a series of xTensor commands (I do not want to lose you in the details there), it is then possible to obtain the transformation of the Ricci tensor and Ricci scalar.

In the package xPand which I wrote for cosmological perturbations, I had to automatize all these tools because I was needing a conformal transformation. So what you can do is add this package to the xAct distribution (just go on the xAct webpage, find the xPand package and install it according to the installation notes). Then in the Example folder of xPand there is a notebook dedicated to conformal transformations (Named "0 Conformal TRansformations.nb"). In this notebook there are several examples, among which the conformal transformation of the Ricci tensor. 

If you want to know more about how it works (the part about conformal transformation), I have tried to explain it in the paper (http://arxiv.org/abs/1302.6174). Just read the section which deals with conformal transformations. If you want to reproduce the results of this paper for the conformal transformation of the Riemann tensor, you can use the notebook "8 Implementation of Paper.nb" which is also in the example folder. This should be enough.

I hope that this will help you,

Best,

Cyril

[Thomas Bäckdahl]

Hi Max!

You have already got some answers, but here is another one.
The attached notebook contains an example where I show that the
conformal Laplacian is conformally covariant only using xTensor. I hope
you can follow this.

Regards
Thomas