ADM equations of motion (and xTensor bug?)

[kold...@gmail.com]

Hi, 

I have working on a problem where I need the (rather complicated) equations of motion in a higher-order gravity model expressed in the ADM formalism. 

To figure out this problem, I am trying to derive the equations of motion for GR in the same way, but I am encountering a lot of trouble when doing the 3+1 decomposition. xAct can do that just fine (with one exception), but I have not had any luck trying to apply VarD or VarL to the result. 

First of all, following the decompositions in xTensorDoc.nb gives the wrong expression for the (N-1)D Ricci scalar (unless there is a sign convention here which I am not aware of). In my attached MWE notebook, just above the section "GR ADM Lagrangian", the expression for the 'induced' Ricci scalar is given. I assume this refers to the 4D Ricci scalar? The expression is:

ExtrinsicKmetrich[-a, -b] ExtrinsicKmetrich[a, b] + ExtrinsicKmetrich[a, -a]ExtrinsicKmetrich[b, -b] + RicciScalarcd[] - (2 (cd[-a][
cd[a][lapse[]]]))/lapse[] - 2 n[a] CD[-a][ExtrinsicKmetrich[-b, b]]

but in the above, the term ExtrinsicKmetrich[a, -a]ExtrinsicKmetrich[b, -b]  should have the opposite sign to the other terms, I think. Am I just confused here?

As to how to get the equations of motion, I assume I have to replace Extrinsic.. etc with the ADM expressions, add a parametric time dependence, and then vary w.r.t the lapse function and shift vectors, but I can't seem to figure out how to do that. 

If anyone has any input, I would be very grateful!

Thanks!

[Jose]

Hi,

It seems you did not attach any notebook.

Regarding the sign of the Ricci scalar formula, can you point out a reference where the same formula for the 4D Ricci scalar appears with the opposite sign in that term? Note that the standard formula to quote (the "scalar Gauss relation", say 2.95 of Gourgoulhon's https://arxiv.org/pdf/gr-qc/0703035.pdf) does show mixed signs, but it is a different formula because the LHS is  RicciScalarCD[] + 2 RicciCD[-a, -b] n[a]n [b] and not just RicciScalarCD[], which is what xTensorDoc.nb shows at the very end of section 7.8. The RHS is also different. With the additional RicciCD term, xTensor reproduces the scalar Gauss relation.

Finally, I'm not sure VarD can be used in combination with induced derivatives in xTensor. Perhaps it can, but I'd be extra careful. Note that induced covariant derivatives only obey the Leibniz rule on tangent tensors. VarD (and as far as I know VarL) is not yet aware of induced derivatives.

Cheers,

Jose.

[kold...@gmail.com]

Hi, 

apologies, please find it attached now. 

I am likely confused about the sign, as the wording 'induced' normally refers to objects on the hypersurface. A simple reference is for example Eq.14 in http://www.math.toronto.edu/mccann/assignments/426/Tong.pdf or (my own paper) Eq.95 in https://arxiv.org/pdf/2009.00949.pdf. But it seems that these are different formulae, in which there is no issue. Thanks for clearing that up!

About getting the ADM equations of motion: the standard way to do this with xTensor seems to be to derive the covariant field equations first, and then project and decompose the result. In the model I am considering, there is no 4D spacetime analogue, and everything is expressed in ADM form from the start. Plugging in a metric to reduce the symmetry and then varying w.r.t the lapse seems to work, and gives the correct expressions. 

In the attached notebook (now actually attached!) I have tried another approach, which so far is quick and dirty; I consider a 3D spatial slice and use time as a parameter. This should work eventually, I think. I have used this technique before to get Hamilton's equations in GR. Nevertheless, I would very much like an easier way :)

[Jose]

Yes, xTensorDoc uses the terminology inducedRiemann to express the full 4D Riemann, but decomposed in many projected objects. I agree it is confusing. 

With respect to variation, I tried your method and VarD got stuck in the commutation of parametric and covariant derivatives. I'm not sure what's the right thing to do in this variational context.

I reattach your notebook with a few comments. Look for the word Jose.

Cheers,

Jose.

[kold...@gmail.com]

Hi, 

thanks for the comments, I have fixed the things you found. 

I am trying yet another way to get the ADM equations of motion: In the attached notebook (last section, called "Another try with SplitExpression") I try to use the xAct package SplitExpression, with some luck! It seems that it is possible to get the eoms w.r.t the lapse function, but not without some strange things which have to be done by hand, like swapping between covariant derivatives. 

I have commented that section a bit more, let me know if you have any thoughts, thanks!

Cheers, 

Albin