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What is the best software/package to calculate

$$2R_{\alpha\mu\beta\nu}R^{\mu\nu}-\nabla_\alpha\nabla_\beta R + \Box R_{\alpha\beta}-\frac12g_{\alpha\beta}\Big(R_{\mu\nu}R^{\mu\nu}-\Box R\Big)$$

for a non-diagonal metric?

I know the last term can be calculated by Maxima GPL, but what about the first three?

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  • $\begingroup$ I do this type of thing with standard Mathematica - not sure how useful that is for you though. In the end it's just Christoffel symbols and partial derivatives. $\endgroup$
    – Eletie
    Commented Apr 19, 2022 at 19:41
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    $\begingroup$ The diffgeo package from Matt Headrick (found here - people.brandeis.edu/~headrick/Mathematica/index.html) is absolutely amazing at doing tensor calculations. $\endgroup$
    – Prahar
    Commented Apr 19, 2022 at 20:03
  • $\begingroup$ @Prahar It seems an interesting one. Tanx. $\endgroup$
    – Astrolabe
    Commented Apr 19, 2022 at 20:08

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The free Mathematica package xTensor can perform calculations such as this. You will also need to companion package xCoba, assuming that your goal is to get the components of this expression in a component basis.

Mathematica itself is, of course, very much not free. But many institutions have a site license, and if you're at such an institution then this is not as much of an issue.

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  • $\begingroup$ You can also purchase a student edition of Mathematica for a lower price. (At least you could a few years ago.) That is what I purchased when I retired in 2019. $\endgroup$
    – John Darby
    Commented Apr 19, 2022 at 21:41

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