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I want to use Mathematica to evaluate an expression like;

$h^{\alpha\beta}_{\,\,\,\,\,|\mu}h_{\alpha\beta\,|\nu} +\mbox{similar}$

where $h_{\alpha\beta}$ is the perturbation to a specified metric (FLRW: $ds^2=dt^2-a^2(t)\delta_{ij}dx^idx^j +h_{\alpha\beta}dx^\alpha dx^\beta$ ) and $|$ denotes a covariant derivative with respect to the background metric. To be precise I want to evaluate the covariant derivatives and write the expression solely in terms of partial derivatives and products of the tensor perturbation.

It looks like some of the required functionality is included in xact(/xpert) but I couldn't see how to input the background metric. I don't think it's actually necessary to know that $h$ is a perturbation to do this, it could be any general tensor.

Any help in explaining what would be the best packages to use and how to get them to do this would be greatly appreciated (my Mathematica knowledge is pretty limited...).

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