Riemann tensor first order perturbation

[Petr Tretyakov]

Dear All. (But maybe rather Developers.)

I'm interested in perturbation in GR. So I start from the simplest formula for first order perturbation for Riemann tensor. By using xPert I wrote Perturbation[ RiemannCD[\[Alpha], -\[Beta], -\[Mu], -\[Nu]] ] after putting additional options // ExpandPerturbation // ContractMetric // ToCanonical

I obtain some answer.

 In principle the result is very well known and may found in almost any classical book. For instance in Landau-Lifshitz(LL) it contain 6 terms like \nabla\nabla h_{ik} -- where    h_{ik} -- first order perturbation of metric tensor. In "Introduction to Quantum Field Theory with Applications to Quantum Gravity", Buchbinder, Shapiro, 2021 (BS) p.435 this result contain 4 terms with second cov. derivatives and two like h_{ik}Rmn, where Rmn--Riemann tensor (due to commutation relation apply). I recalculate that results by my own hands, and know that it is correct. 

The result which is produced by the Program is very different. Ok -- I try to apply some additional specifications like  // ExpandPerturbation // SortCovDs // Simplification //
    ScreenDollarIndices // ContractMetric // Simplification // ToCanonical

The result is the next: four terms with second derivatives totally  coincide with four terms from BS, but instead of two terms  like h_{ik}Rmn Program generate 11(!)

Any my attempt to simplify its to known form was unsuccessfull. This very strange  simplification. I trust that Program works correctly, but how it is possible to obtain results in more convenient form?

Reguards,
Petr

[Juan Margalef]

In order to help you, we would need a minimal working example with your code, your result and the expected result. Otherwise it is very hard to give you useful feedback.

[Petr Tretyakov]

Ok. The code is next

Needs["Xact`xTensor`"]

Needs["xAct`xPert`"]

DefManifold[M4, 4, {l, m, n, r, s, a, b, c, d, i, k}]

DefMetric[-1, g[-m, -n], CD];

DefMetricPerturbation[g, h1, \[Epsilon]]

Perturbation[ RiemannCD[a, -b, -m, -n] ] // ExpandPerturbation // ContractMetric // ToCanonical

%%%%%%%%%%%%

The result is:

%%

h1[ -n, c]   RiemannCD[a, -b, -m, -c] + h1[ a, c]   RiemannCD[-b, -c, -m, -n] - 1/2  CD[a]@CD[-b]@h1[ -m, -n] - 1/2  CD[a]@CD[-m]@h1[ -b, -n] + 1/2  CD[a]@CD[-n]@h1[ -b, -m] + 1/2  CD[-b]@CD[a]@h1[ -m, -n] + 1/2  CD[-b]@CD[-m]@h1[ a, -n] - 1/2  CD[-b]@CD[-n]@h1[ a, -m]

%%%%%%%%%%%%

After putting 

%%%%%%%%%%%%

Perturbation[ RiemannCD[a, -b, -m, -n] ] // ExpandPerturbation // SortCovDs // Simplification // ScreenDollarIndices // ContractMetric // Simplification // ToCanonical

%%%%%%%%%%%%

The result is: A+B, where

%%

A= -(1/2)  CD[-m]@CD[a]@h1[ -b, -n] + 1/2  CD[-m]@CD[-b]@h1[ a, -n] + 1/2  CD[-n]@CD[a]@h1[ -b, -m] - 1/2  CD[-n]@CD[-b]@h1[ a, -m]

%%

B=1/2  h1[ -n, c]   RiemannCD[a, -b, -m, -c] - 1/2  h1[-m, c]   RiemannCD[a, -b, -n, -c] + 1/2  h1[ -n, c]   RiemannCD[a, -c, -b, -m] - 1/2  h1[ -m, c]   RiemannCD[a, -c, -b, -n] - 1/2  h1[ -n, c]   RiemannCD[a, -m, -b, -c] - 1/2  h1[ -b, c]   RiemannCD[a, -m, -n, -c] + 1/2  h1[ -m, c]   RiemannCD[a, -n, -b, -c] + 1/2  h1[ -b, c]   RiemannCD[a, -n, -m, -c] + h1[ a, c]   RiemannCD[-b, -c, -m, -n] + 1/2  h1[a, c]   RiemannCD[-b, -m, -n, -c] - 1/2  h1[a, c]   RiemannCD[-b, -n, -m, -c]

%%

If we take this result from the book we find that A -- is correct, whereas instead of B must be just B*= 1/2  h1[r, -b]   RiemannCD[a, -r, -m, -n]  - 1/2  h1[-r, a]   RiemannCD[r, -b, -m, -n] -- two terms instead 11.

(In all expressions imply   h1[ -a, -b] = h1[LI[1], -a, -b], I delete LI[1] for more readability )

[Juan Margalef]

I don't have the book at hand so I cannot compare, but my I guess is that the difference is due to a problem with the conventions. If you define a (non-metric) covariant derivative:

DefCovD[der[-a], {";", "D"}]

The Riemann tensor Riemannder has the first three indices down and the last index up as you can check using:

SlotsOfTensor[Riemannder]

{-TangentM4, -TangentM4, -TangentM4, -TangentM4}

If the covariant derivative is the Levi-Civita one, all indices are down (to implement better some additional symmetries):

SlotsOfTensor[RiemannCD]

{-TangentM4, -TangentM4, -TangentM4, -TangentM4}

When you write Perturbation[ RiemannCD[a, -b, -m, -n] ] you are actually writing the same as Perturbation[g[a,c] g[-n,-k] RiemannCD[-c, -b, -m, k] ] so you have additional terms coming from the perturbations of the metric and its inverse. Check if

Perturbation[RiemannCD[-a, -b, -m, n]]

coincide with the formula of the book.

Best,

Juan Margalef

[Petr Tretyakov]

I think the problem is deeper. For 

Perturbation[RiemannCD[-a, -b, -m, n]]

It produce 10 terms instead of 2, and not coincide with the known answer 

[Juan Margalef]

Hi Petr,

Thanks for your message. After checking the formulas of the book, I can confirm that it is indeed a matter of conventions. Here you have the correct code to check that xPert indeed provides the same as formula 18.37:

Needs["Xact`xTensor`"]
Needs["xAct`xPert`"]

DefManifold[M4, 4, {\[Alpha], \[Beta], \[Gamma], \[Phi], \[Delta], \[Mu], \[Nu], \[Tau], \[Lambda]}] (* I changed to greek letter to simplify the comparison*)
DefMetric[-1, g[-\[Alpha], -\[Beta]], CD];
DefMetricPerturbation[g, h1, \[Epsilon]]

$CovDFormat = "Prefix";
$RiemannSign = -1; (*This includes an overall minus so that it agrees with 11.13 when the risen index is the last one instead of the first one, this is equivalent to exchange the first pair with the second pair, and then flip the last two indices, an overall minus appears *)
RiemannCD[-\[Alpha], -\[Beta], -\[Tau], \[Lambda]] == (RiemannCD[-\[Alpha], -\[Beta], -\[Tau], \[Lambda]] // RiemannToChristoffel)

equation18dot37 = (CD[-\[Mu]]@CD[-\[Beta]]@h1[LI[1], \[Alpha], -\[Nu]] - CD[-\[Nu]]@CD[-\[Beta]]@h1[LI[1], \[Alpha], -\[Mu]] + CD[-\[Nu]]@CD[\[Alpha]]@h1[LI[1], -\[Mu], -\[Beta]] - CD[-\[Mu]]@CD[\[Alpha]]@h1[LI[1], -\[Nu], -\[Beta]] + RiemannCD[-\[Mu], -\[Nu], -\[Delta], \[Alpha]] h1[LI[1], \[Delta], -\[Beta]] - RiemannCD[-\[Mu], -\[Nu], -\[Beta], \[Delta]] h1[LI[1], -\[Delta], \[Alpha]]);

2 Perturbation[RiemannCD[-\[Mu], -\[Nu], -\[Beta], \[Alpha]]] -equation18dot37 // ExpandPerturbation // SortCovDs // Simplification

If you run this code, the last expression returns zero.

Best and good luck,

Juan Margalef

[Petr Tretyakov]

Dear  Juan,

Thank you very much for your assistance!

Best

Petr