Symmetric transverse traceless rank-2 tensor

[David Woodward]

Suppose we work in de Sitter space and we have an expression consisting of covariant derivatives, the metric tensor and a symmetric transverse traceless rank-2 tensor H_{\mu \nu}.

How can one tell Mathematica that H_{\mu \nu} is symmetric, transverse and traceless?

I mean, I use bgRules=SymmetricSpaceRules[CD, \Lambda], but what are the extraRules in SetOptions[ToBackground, BackgroundSolution -> bgRules, 

  ExtraRules -> extraRules]?

My goal is to use ToBackground and find the background value of the expression I mentioned at the beginning.

Best,

David