Defining a perturbation of metric which is not symmetric in lower indices in xAct

Question

I am doing variation and perturbation in xAct, when perturb the metric, can I set the perturbation of metric is not symmetric in lower indices?

DefConstantSymbol[dim, PrintAs -> "D"]
DefManifold[M4, dim, {α, β, γ, μ, ν, ρ, σ, κ, ξ, ζ, δ}];
DefMetric[-1, Met[-μ, -ν], 
  CD, {";", "\!\(\*OverscriptBox[\(∇\), \(-\)]\)"}, 
  PrintAs -> "\!\(\*OverscriptBox[\(g\), \(_\)]\)"];
DefMetricPerturbation[Met, δMet, τ, PrintAs -> "h"];
DefTensor[h1[-α, -β], M4];

Answer 1

The metric perturbation is the difference between two metrics, the perturbed metric and the unperturbed metric. Both metrics are symmetric, and hence its difference is also symmetric.

The symmetry of the background metric (Met in your case) and of the metric perturbation (δMet in your case) is used in many places in xPert, and trying to remove this condition would probably lead to incorrect results.

If you want to try to remove the symmetry of the perturbation, do this:

SymmetryGroupOfTensor[\[Delta]Met] ^= StrongGenSet[{}, GenSet[]]

but, again, this will probably lead to problems if you want to use xPert.