ADM perturbations in xPand

[kold...@gmail.com]

Hi, 

I am trying to perturb some expressions in the ADM formalism using xPand. What I want is: 1. Set flat FLRW. 2. Perturb to second order in tensor perturbations. 3. Plug back into the action and find equations of motion. In doing this I found some results which disagree with my own hand calculations. For example, the 3D Ricci scalar (derived with the induced metric) is zero in flat FLRW, but its perturbation should not be (I think?), but when I perturb it with xPand I get zero. I am getting similar results for the extrinsic curvature (ExtrinsicKh in xPand). It should be proportional to the derivative of the spatial metric, which is not zero in FLRW. 

Note that perturbing the 3D Ricci scalar with xPert directly gives a result which agrees with my hand calculations.

I am quite new to xPand and xPert, so I have probably misunderstood some trivial things here, but I would appreciate any input. I have attached an MWE (see the bottom of the file). 

Thanks!

[Benedict]

I ran your notebook and added at the very end the following line:

ToxPand[RicciScalarCD[], dg, u, du, h, "AnyGauge", 1]

This gives back the background + first order perturbation. The command is: ToxPand[expression, dg, u, du, h, gauge, order]

Furthermore, I saw a red back in your notebook asking for only tensor perturbations. The following commands might be useful to you as well:

$ConformalTime = True;

$FirstOrderVectorPerturbations =   True;(*Default=True*)

$FirstOrderTensorPerturbations =   True;(*Default=True*)

Good luck with everything!

[Benedict]

An addendum to my answer:

What ToxPand does is:

expr

Conformal[g, gah2][%]

ExpandPerturbation@Perturbed[%, 1] // org (*1 is the order*)

SplitPerturbations[%, SplitMetric[g, dg, h, "AnyGauge"], h] (*"AnyGauge" is the chosen gauge*)

I hope that helps.

kold...@gmail.com schrieb am Donnerstag, 11. Februar 2021 um 20:07:38 UTC+1:

[kold...@gmail.com]

Hi, and thanks for your reply, 

in particular, I am looking for perturbations of the 3D Ricci scalar, RicciScalarcd[] and the extrinsic curvature of the hypersurface defined in SetSlicing. For example, the extrinsic curvature is called by ExtrinsicCh[a,b], and in a homogenous and isotropic spacetime, this is proportional to the time derivative of the spatial metric. But, xPand is saying that the extrinsic curvature is zero, which is incorrect.

Also, the 3D Ricci Scalar, Ricciscalarcd[], is zero in flat FLRW, but it should still have non-zero perturbation contributions. I have calculated it by hand and using xPert where I get non-zero results, but xPand is again telling me that it is zero. 

Do you have any ideas? Maybe I have misunderstood something here?

Cheers

[kold...@gmail.com]

As an update to the above, I used the definition of extrinsic curvature (from Baumgarte & Shapiro) and perturbed it using xPand, rather than using xPand's automatically generated ExtrinsicCurvatureh, and I get the following result

and when forming the GR term K^ij K_ij - K^2 I get the expected result. The question is then what ExtrinsicCurvatureh in xPand actually is

[Benedict]

I used the commands from my previous post in the notebook you find attached. I hope this helps and explains a bit what they do and how you could use them.

[kold...@gmail.com]

Hi, 

I have had a look at your code, thanks for taking the time to do that. I still have a few questions: On the last line, under 'only tensor perturbations', you still get scalar perturbations \phi, which doesn't seem right? Also, you have looked at RicciScalarCD[] ,whereas I am interested in RicciScalarcd[], the (d-1) Ricci scalar. I have had a look, and RicciScalarcd[] is zero to all orders in perturbation, so that is not a big deal now. 

What I am more concerned about is the extrinsic curvature; Using the textbook definition of the extrinsic curvature tensor and perturbing it, I get a non-zero result. But when I use xPand's built-in ExtrinsicCh[a,b], this evaluates to zero, which is not correct. Do you know why this happens?

Thanks!