What is the Correct Way to Perturb the Energy-Momentum Tensor and Express the Result in a Gauge?

[Benedict]

I am trying to derive the equations (30) from Ma-Bertschinger (1995) for a slightly different energy-momentum tensor. These equations describe the perturbed part of energy-momentum conservation in a Newtonian gauge.

My approach is to first define a tensor with two components using 'DefTensor[myt[-a, -b], M, PrintAs -> "T"]' and its perturbation with the corresponding 'DefTensorPerturbation' command, before I set the so created tensor equal to my expression for the energy-momentum tensor.

As a next step, I am interested in the perturbed tensor, which I derive using 'myt[-a, -b] //Perturbation //ExpandPerturbation //ContractMetric //ToCanonical //Simplification'.

Is this the correct way of getting the perturbation?

To get the conservation equations, I take the derivative and do some further manipulations with the result:

'cd[a][%] //Expand //ContractMetric //ToCanonical //Simplification

The result I get is of first order, which is what I want. I attached a notebook that contains all the steps in their full glory.

As a next step, I'd like to express this result in a flat FLRW metric with Newtonian gauge. How can this be done? My attempt using xPand was not very fruitful; I attached it anyways, maybe it is only a minor issue in using a wrong command in a single step or something alike - maybe not and there is a more fundamental issue...

Thank you very much for reading this. Have a great day.

[Obinna Umeh]

Hi Benedict

It is a little hard to understand exactly what you plan to accomplish. I have attached a minimal example that might help. 

Basically, you use xTras to obtain your EMT, then you xPand to perturb the result in any gauge of your choice. 

Hope it helps!

Obinna

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