# Defining perturbed FLRW metric in Poisson gauge with xact

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### Mre

Dec 4, 2022, 5:52:53 AM12/4/22
to xAct Tensor Computer Algebra

Hi, I want to use the perturbed FLRW metric in the Poisson gauge (see the attached image) to expand the Einstein-Hilbert action and then take the variation of it to obtain the stress-energy tensor and perturbed Einstein equations to desired order.

the perturbed FLRW metric in the Poisson gauge:

$\mathrm{d} s^2=g_{\mu \nu} \mathrm{d} x^\mu \mathrm{d} x^\nu=a^2(\tau)\left[-(1+2 \Psi) \mathrm{d} \tau^2-2 B_i \mathrm{~d} x^i \mathrm{~d} \tau+(1-2 \Phi) \delta_{i j} \mathrm{~d} x^i \mathrm{~d} x^j+h_{i j} \mathrm{~d} x^i \mathrm{~d} x^j\right]$

My problem:

For this purpose, I used the method used in https://mathematica.stackexchange.com/a/67454/55218

But my problem is how to define the above metric with the method mentioned in the link above and which package is better to use.

More detail:

To be more precise, my problem is how to define the h_{i j} tensor and v_{i} vector because I don't know how to tell the xact that the i and j are the spatial indices.

should I define h_{i j} tensor and v_{i} vector on the original manifold (with dim=4) or should I define a manifold with dim=3 and then I will define them?

metric.PNG

### Cyril Pitrou

Dec 4, 2022, 6:11:51 AM12/4/22
to Mre, xAct Tensor Computer Algebra
Hi,

In that case I recommend to use the xPand package which goes on top of xTensor/xPert.
It allows for the 1+3 splitting of spacetime. In the example folder of the package you will find yo to perfrom such Poisson gauge perturbation and
how to perturb T and G.

Best,
Cyril Pitrou

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