Solving equations of motion in xAct

Question

Varying with respect to metric (https://github.com/xAct-contrib/examples/blob/master/ActionVariation_Metric_Fields.nb) I obtained equations of motion for Einstein Gauss Bonnet gravity with two scalar fields.

$L=R+12+\lambda/2(R_{abcd}R^{abcd}+R^2-4R_{ab}R^{ab}+\phi^2+\chi^2-100\phi^2\chi^2-1/2(\nabla\phi)^2-1/2(\nabla\chi)^2 $

The code is as follows:

<< xAct`xTras`
dimension = 4; 
DefManifold[M4, 
  dimension, {a, b, c, d, e, i, j, k, l, p, q, r, s, t}];
DefMetric[-1, g[-a, -b], CD, {";", "\[Del]"}, PrintAs -> "g"];
DefTensor[\[Phi][], M4](* Scalar field *)
DefTensor[\[Chi][], M4](* Scalar field *)
DefConstantSymbol[{\[Alpha], gxx, G, \[Lambda]}]
Lm = -1/2 (CD[a]@[Phi][] CD[-a]@[Phi][]) + [Phi][]^2 + -1/
    2 (CD[a]@[Chi][] CD[-a]@[Chi][]) + [Chi][]^2 - 
  G [Phi][]^2 [Chi][]^2
L = RicciScalarCD[] + 
  12 + [Lambda]/
   2 (RiemannCD[a, b, c, d] RiemannCD[-a, -b, -c, -d] + 
     RicciScalarCD[] RicciScalarCD[] - 
     4 RicciCD[a, b] RicciCD[-a, -b]) + Lm
VarAction[L, g[a, b]]
VarAction[L, [Phi][]]
VarAction[L, [Chi][]]

I would like to first evaluate for a specific metric and coordinates

coords = {t[], x[], x1[], x2[]};
metric = {{-\[Alpha] a[x]^2 (2 x - x^2)^(2/3)*(1 - x)^2, 0, 0, 0}, {0,
gxx, 0, 0}, {0, 0, \[Alpha] a[x]^2/(2 x - x^2)^(2/3), 0}, {0, 0, 
0, \[Alpha] a[x]^2/(2 x - x^2)^(2/3)}}

Secondly, I would like to solve equations of motion with the ansatz $\phi(x), \chi(x)$.

Any help is more than welcome.

The problem is similar to Evaluating covariant derivative terms of a scalar function (xAct, xTras).

Solving equations of motion in xAct

0 Answers0