# Expanding the Riemann tensor (perturbation)

I am trying to expand the following term (via the perturbation expansion $g_{\mu\nu} = \eta_{\mu\nu} + \kappa h_{\mu\nu}$):

$$\frac12 \epsilon^{abcd}R^e_{fcd}R^f_{eab}$$

Where $R$ is the Riemann Tensor and $\epsilon$ the Levi-Cevita Tensor (which depends on the determinant of the metric).

I have tried using the package xAct (specifically xPert), however I can't get a result from it. I also don't know how to input the Levi-Civita Symbol $\epsilon^{abcd}$.

A minimal code is below, which typically just crashes for me. If I set GR to be simpler (say, just GR = Sqrt[-Detg[]] RicciScalarCD[]) then it seems to work OK.

<< xActxPert;
\$PrePrint = ScreenDollarIndices;
(* Define a manifold *)
DefManifold[M4, 4, {a, b, c, d, e, m, n, p, q, r, s, t, l}];

(* Define a metric Subscript[g, ab] *)
DefMetric[{1, 3, 0}, g[-m, -n], CD, {";", "\[Del]"}, Torsion -> False, Curvature -> True, PrintAs -> "g"];

(* Define the metric perturbation *)
DefMetricPerturbation[g, H, \[Epsilon]];
DefConstantSymbol[k]

GR = 1/(2 k^2)Sqrt[-Detg[]].RiemannCD[s, -t, -a, -b].RiemannCD[t, -s, -m, -n];
Perturbation[GR] // ExpandPerturbation


## 1 Answer

For example,

<< xActxPert


Define your structures:

DefManifold[M4, 4, IndexRange[a, f]]

DefMetric[{1, 3, 0}, g[-a, -b], CD]

DefMetricPerturbation[g, H, \[Epsilon]]


Construct the object to perturb, as you give it above, including the epsilon tensor of the metric g:

GR = 1/2 epsilong[a, b, c, d] RiemannCD[e, -f, -c, -d] RiemannCD[f, -e, -a, -b]


Perturb to second order and manipulate:

Perturbation[GR, 2] // ExpandPerturbation // ContractMetric // ToCanonical


Particularize to flat background and simplify:

% /. RiemannCD -> Zero // Simplification

• Dear jose, thanks for your answer. if the background metric was not flat, how can plug it into the above code? I just want to expand Ricci tensor around a reference metric with known components. how to introduce it? I don't want a formal expansion. Commented Aug 4, 2021 at 10:38