Einstein eqautions in f(R) gravity

[Albert escrivà]

Hi all !,

 I would like to ask you some problem that I have with xAct and I don't get any solution. Let me explain:

I am working in f(R) gravity (is a generalization of general relativity, in the sense that you have more curvature terms than the Einstein-Hilbert action given only by the Ricci scalar) under a FRW background and I want to get the "new" Einstein equations, which can be got doing a variation of the action in terms of the metric of the space-time g_{\mu\nu}. xAct can get the typical Einstein equations when you consider only the Einstein-Hilbert term in a very easy way. For instance, you can get the Friedman equation (given the inflationary action) that claims: H^2=(1/3Mpl^2)*(V+\phi'^2/2), where H is the Hubble constant, V is the potential of inflation, Mpl is the Planck mass and phi' the time-derivative of the inflaton.

In particular: I don't have the idea how to get this Friedman equation in modified f(R) in xAct, see the notebook that I post. With my code you can get easily the Frieddman-equations when you have only the Einstein-Hilbert action, but for instance if you take a term like the contraction of the Riemann by itself

L = Sqrt[-Detmetricg[]] RiemannCD[-\[Alpha], -\[Beta], -\[Eta], -\[Lambda]] RiemannCD[\[Alpha], \[Beta], \[Eta], \[Lambda]]

then you get a lot of errors, which doesn't make sense, since the you are able to compute the Riemann, Ricci and Christoffel components for the FRW metric, and then xAct only has to substitute this on the expression that he is able to get when he performs a variation of the action with the metric:

metricg[-\[Mu], -\[Nu]]  
  RiemannCD[-\[Alpha], -\[Beta], -\[Eta], -\[Lambda]]  
  RiemannCD[\[Alpha], \[Beta], \[Eta], \[Lambda]] -
 4  RiemannCD[-\[Mu], \[Alpha], \[Beta], \[Eta]]  
  RiemannCD[-\[Nu], -\[Alpha], -\[Beta], -\[Eta]] - 4  CD[-\[Alpha]][
CD[-\[Beta]][
RiemannCD[-\[Mu], \[Alpha], -\[Nu], \[Beta]]]] - 4  CD[-\[Beta]][
CD[-\[Alpha]][
RiemannCD[-\[Mu], \[Alpha], -\[Nu], \[Beta]]]]

This problem also happens when you take the Ricci contraction for instance:

L = Sqrt[-Detmetricg[]] RicciCD[-\[Alpha], -\[Beta]]  RicciCD[\[Alpha], \[Beta]]

Anyone knows how to solve that? See my notebook please, and comment something if you have some question or already the answer :) ! Thanks.

[Armando Roque]

Hi,

I checked your notebook, and I think that the problem is because xCoba not work with contravariant index, you need to change the expression to covariant index, in my case i used '' ImplodedTensorValues '' and ''ChangeComponents'' when i had the same problem (many message error and red contravariant index) but with contravariant derivate of Ricci scalar.

I recomen that check this links,

https://groups.google.com/forum/?hl=en#!searchin/xAct/ChangeComponents|sort:date/xact/TwjiUheOtzM/RCERK1bVBQAJ

https://groups.google.com/forum/?hl=en#!searchin/xAct/ChangeComponents|sort:date/xact/Yi08vSOxxMs/ez9nq9fsAQAJ

https://groups.google.com/forum/?hl=en#!searchin/xAct/ChangeComponents|sort:date/xact/hFVZs7WRHmA/JrHaSyW4k08J 

Good luck

[Albert escrivà]

Than you very much for your answer Armando. I have been trying to solve the problems that I was using '' ImplodedTensorValues '' and ''ChangeComponents'' but still there are problems. Of course the situation has improved, a lot of uncontrated Rieman tensors has been solved thanks to ''ChangeComponents''. But there are 2-covariant derivatives of Riemann tensors and Christoffel symbols that are uncontracted and I don't have idea why:

1- The contraction of the covariant derivatives of Rieman tensor would be possible to solve using '' ImplodedTensorValues '', I tried but it doesn't work at all.

2- This is curios, why Christoffel symbols appear uncontested? This is already computed with the metric that you give to xAct, and as I understand, there is not a problem of covariant/contravariant problems as happened in the Rieman tensor.

I send the modified code if someone wants to check, it takes 10 min to do computation, take care!

Thanks.