On xPert, TexAct

[user657]

Dear all,

I have 2 questions -

1) When doing perturbation theory, the (first order) perturbation of the metric tensor g_{\mu \nu} appears written as \Delta g^{1}_{\mu \nu} and that of g^{\mu \nu} as \Delta[g^{\mu \nu}], and so on. I want the first order perturbation of g_{\mu \nu} to be printed as h_{\mu \nu}. How can I do that, without spoiling further contractions that I might need to do with the new h_{\mu \nu}? Similarly, I would want the first order perturbation of g^{\mu \nu} to be printed as -h^{\mu \nu}, and so on for higher order perturbations.

2) I want to keep perturbations in the metric tensor g_{\mu \nu} only up to first order, so that, for example, implementing ExpandPerturbation@Perturbation[g[-mu,-nu], 3] prints out g_{\mu \nu} + \epsilon h_{\mu \nu}. How can I set all orders of perturbation in g_{\mu \nu} higher than 1 to zero? Note that I do want to keep all orders of perturbation for the inverse metric tensor, however.

Thank you for your help.

[zhanfe...@gmail.com]

Hi! I think you don't need to do it!

After you define the perturbation of metric tensor. like

DefMetricPerturbation[g,deltag , e] (where e is the perturbationparameter)

You can extract the perturbation tensor of metric like deltag[LI[n],-a,-b], where n is the perturbation order. 

and you can do 

PrintAs[deltag]^:="h"; 

I think it is ok and you can try by yourself.

I am still working on your second problem.

Best regards,

Zhan-Feng Mai

在 2018年8月29日星期三 UTC+8上午1:49:31，user657写道：

[user657]

Dear Zhan-Feng,

Thank you very much for your reply. It indeed does work. There is one more recurring issue, though - \Delta g^{1}_{\mu \nu} is replaced by h^{1}_{\mu \nu}. Is there a way I can get rid of the order of perturbation "1" in the superscript of h^{1}_{\mu \nu} to get h_{\mu \nu}? This also connects me to the second question I asked in my original post - basically, I want to express perturbations of any quantity (scalar, tensor, scalar density, etc.) in terms of only h_{\mu \nu}, which is the first order perturbation to the metric tensor g_{\mu \nu}.

Thanks again for working on my questions.

[Leo Stein]

Dear Shubham,

Please look in the following file in your xAct installation: xAct/Documentation/English/xPertDoc.nb . In that file, look in Section 6: Background field perturbations. That shows an example using the very simple rule:

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

This rule annihilates all instances of h^2, h^3, etc. (anything beyond h^1). Using this rule, you can use xPert's general perturbation theory to implement the "background field" formalism.

As for your earlier question, I often do something as follows. I let deltag[LI[order], -a, -b] be the perturbation to the metric, but since it's often more convenient to use the trace-reversed metric perturbation, I define another tensor field barh[-a, -b] and make rules to go back and forth between deltag[LI[1], -a, -b] and barh[-a,-b]. Then I just use PrintAs[barh] to print it as an h with a bar on top.

But for your purposes, if you just want h instead of the trace reverse, it's simple to just introduce another tensor field without the LI label index, so there's no superscript 1 in all your expressions.

Good luck

Leo

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[user657]

Dear Prof. Stein,

Thank you very much for your reply. This answers my questions.