FRW Suggestions

[Jolyon Bloomfield]

Hi folks,

I've been thinking a lot recently about how to implement a 3+1 split for an FRW (or FRW-like) system using xTensor. In the thread "using xACT/xCoba/xPert in particular basis", JMM suggested using a 3+1 decomposition for FRW stuffs, but I haven't had much success following this route. I've been trying to come up with a way of separating the spatial derivatives and the time derivatives, while keeping spatial contractions. The brute force way using xCoba to split everything means that you need to go back afterwards and piece everything back together, which is painful. Has anybody had success in implementing this kind of splitting? I know that the documentation specifically asks "If you have a good idea how to implement this, let me know!", but even bad ideas would be appreciated by this fellow traveller :-)

Best,
Jolyon

P.S. I'm sure I'm not alone in attempting to do these calculations; if I get something working, I'll be sure to post it here.

[Emilio Bellini]

Dear Jolyon,
what do you mean by 3+1 split? If you want to define constant time hypersurfaces to find the induced metric on these hypersurfaces, my suggestion is to follow the xTensor documentation on Induced Metrics.
If you simply want to decompose every dummy index in its spatial an temporal component (A[-\mu] A[\mu]=A[-0] A[0]+A[-i] A[i]) I think you may find interesting this discussion

http://groups.google.com/group/xact/browse_thread/thread/834e010975f8e19d/6480f41ec022820f?hl=en&lnk=gst&q=3%2B1#6480f41ec022820f

Here you can find a notebook. There is a function SplitDummySpaceTime that could be useful for your purposes. I think this is the right way, I don't like xCoba, because you are loosing many tensorial properties..

If you have any other question, please ask me!

Cheers,
Emilio

[Leo Stein]

Hi Jolyon,

I have had success doing a 3+1 split and separating space/time
derivatives, but this was for post-Newtonian work, so I had a flat
background, which made everything simpler. Can you elaborate on what
the difficulties are, so that they might be individually tackled?

I imagine that the following approaches might be useful:
1) The standard slicing for FRW probably has a simple form for the
acceleration vector and extrinsic curvature, so those quantities can
have automatic rules which are simple. Using GradNormalToExtrinsicK
will take care of derivatives of the foliation vector.
2) Similarly, the curvature quantities (Riemann, Ricci, etc.) have a
simple form for the 3-metric (being a maximally symmetric space) and a
more complicated form for the 4-metric. You can write rules to replace
curvature quantities with functions of the Hubble parameter and scale
factor. However, these can not be automatic rules if you wish to
perturb curvature quantities.
3) Make your own rule to decompose derivative indices only; something
along the lines of
CD[a_][expr_] :> Module[{b}, -t[a]t[b] CD[-b][expr] + gamma[a,b]
cd[-b][expr] ]
where CD is the 4-derivative and cd is the 3-derivative of the induced
3-metric gamma, and t the vector of foliation (is that decomposition
missing something related to the acceleration/extrinsic curvature?).
Note that you can also do something quite like this within xCoba --
use IndicesOf to tell SeparateBasis that you just want to split up the
derivatives into space/time derivatives.
4) The perturbation theory aspect is a bit more annoying. You'll have
to define an automatic rule for what Perturbation[t[i_],n_] is (that
is a 'gauge' choice); this implies rules for what Perturbation of the
acceleration vector and extrinsic curvature are, so these also need
automatic rules. You'll need to allow for perturbations of both the
4-metric and the 3-metric, but of course the two are not independent,
and xPert does not know that there is any relationship between the
full metric and the induced metric (is that correct?). Therefore there
needs to be a rule to, say, convert all perturbations of the 4-metric
into a decomposition which includes the 3-metric, something like
DecomposeHRule =
MakeRule[{metpert[LI[1], -b, -d],
t[-b] t[-d] Phi[] + 1/2 (t[-b] v[-d] + t[-d] v[-b]) +
smetpert[LI[1], -b, -d]}]
where metpert is the perturbation of the full metric, smetpert is the
perturbation of the induced metric, Phi is the perturbation to the
time-time part of the metric, and v is the perturbation to the
time-space part of the metric (which has been defined with
OrthogonalTo t and ProjectedWith gamma). Of course phi and v are
actually first order in epsilon, but xPert does not know that ...
perhaps one could instead have some "background" Phi which is
vanishing and define a tensor perturbation for it (similarly for v),
so that xPert can know that it has perturbative objects. I do not know
the best way to do this.

Have I missed any important points?

Good luck!
Leo

[junior...@gmail.com]

Dear Leo,

Maybe this is not the right place but I'm looking for tips on how to make post-Newtonian approximations using xTensor/xPert and I found your email above.

I don't know how to work with time and spatial components separately and also how to give a perturbative weight for every time derivative, as it must be in PN analysis.

Could you give me some advices on how to start?

Thanks in advance.

Júnior D. Toniato

[Leo Stein]

Dear Junior,

My first recommendation would be to mirror the 3+1 approach in xTensor for splitting spacetime in space and time. Are you familiar with the xTensor tools for the decomposition?

Now regarding time derivatives being 1/c relative to spatial derivatives. This comes from an approach that I implemented in a PN notebook, using xTensor, but my PN code is embarrassing (not very clean or well organized), so I am not going to make it public (and I do not have the time to re-write this notebook). I know that Guillaume Faye has very advanced PN notebooks, so he might have some other advice. Michele Levi has also implemented some PN tools for the EFT approach, that could be another useful source.

With your 3+1 splitting of Minkowski, you will have a (d/dt) vector field. To organize a calculation by powers of c would work as follows. Start with your calculation in its covariant form, then perform the 3+1 split, making sure that every index is either on (d/dt) or projected with the spatial project (just the spatial 3-metric, \delta_{ij} at 0PN order). You must make sure that all (d/dt)'s are explicit. Then, you can replace all (d/dt)'s by 1/c*(d/dt). It may be a bit dangerous to actually replace a symbol with a constant multiple of itself, because of how Mma will apply the replacements. To be cautious, you can introduce an auxiliary vector and proceed in two replacements: First, ReplaceRepeated so that all ddt's become ddtTEMP. Then, ReplaceRepeated so all ddtTEMP's become 1/c d/dt.

A similar approach would be to define the two timelike vector fields (d/dt) and (d/d(ct)). This way there are no TEMP quantities, and you won't be replacing a symbol by a multiple of itself, but instead replacing the well-defined d/d(ct) by 1/c * d/dt. However, you will have to teach Mma rules about both of these quantities, so there will be a lot more rules flying around.

Either way, once you get all of the c's explicit, you can use Series[] to truncate your calculation to whatever order you like, or just Collect[] to group terms by powers of c, etc.

Hope this was useful, and if I did a bad job of explaining anything, please feel free to ask for clarification!

Best

Leo

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