Background field perturbation

[davw...@gmail.com]

Dear All,

I want to apply background field perturbation to the Einstein-Hilbert action around the Minkowski background up to (and including) cubic order, where the perturbation tensor $h$ is conformally flat. All raising and lowering of indices (apart from the inverse metric tensor) is done with respect to flat space. I have consulted the xPert documentation, but how do I indicate that $g$ is the Minkowski background and that $h$ is conformally flat? Any help welcome. I would appreciate it if the answer was in the form of .nb file.

 

Best,

David Woodward

[Teake Nutma]

Hi David,

Have a look at this tutorial for perturbations on AdS backgrounds:

http://xact.es/xtras/documentation/tutorial/AdSPerturbations.html

You’re interested in a flat background, so you can just set the cosmological constant to zero in the tutorial to zero.

As for h being conformally flat, do you mean that the perturbative expansion of the the Weyl tensor around flat space in terms of h should be zero order by order?

Best,

Teake

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[davw...@gmail.com]

Dear Dr Nutma,

Thanks very much for your reply. I am doing background field quantization where $g _ {\mu \nu} = \eta _ {\mu \nu} + h _ {\mu \nu}$ and $g ^ {\mu \nu} = \eta ^ {\mu \nu} - h ^ {\mu \nu} + h ^ {\mu \rho} h _ {\rho} ^ {\nu} - O(h ^ 3)$, essentially the same as the background field perturbation section of the xPertDoc.nb documentation. Moreover, $\eta _ {\mu \nu} = diag (-1,1,1,1)$ and $h _ {\mu \nu} = \Omega ^ {2} \eta _ {\mu \nu}$. All raising and lowering of indices is done with respect to flat space. The Weyl tensor, yes, is zero order by order. I would like to expand the Einstein-Hilbert action, that is $\sqrt{-g} R$, up to cubic order and I do not know how to indicate that the background metric is $diag (-1,1,1,1)$ and that $h _ {\mu \nu} =  \Omega ^ {2} \eta _ {\mu \nu} $, that is, $h _ {\mu \nu}$ is conformally flat. See Input[177],[178],[179] of the xPert documentation; in my case, $g _ {\mu \nu} = \eta _ {\mu \nu}$ and $h _ {\mu \nu} =  \Omega ^ {2} \eta _ {\mu \nu} $. I am completely new to xAct and even the simplest things seem like a mountain!

 

Best,

David

[Teake Nutma]

Hi David,

I’ve attached a notebook that hopefully does what you want. Oh, and please don’t attach xAct documentation notebooks next time; these come standard with xAct, so everyone on the mailing list should have them, and they’re pretty big.

Best,

Teake

Attachments:

- xPertDoc.nb

[davw...@gmail.com]

Dear Dr Nutma,

Thanks very much for your help. By the way, where do the $1/2 !$ and $1/3 !$ factors in the cubic expansion come from? Is it a Taylor series expansion?

 

Best,

David

[Teake Nutma]

Hi David,

Yes, the factorials come from the fact that it’s a Taylor-like power series expansion. These coefficients were discussed before here:

https://groups.google.com/d/msg/xact/jWhzQqNOAzw/RpWeOPIy3bsJ

Best,

Teake