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I want to convert a gravity action in terms of differential forms to tensorial expressions. A procedure known as tetrad formalism/Cartan formalism like Palatini action in this page: CARTAN FORMALISM

Is there any package in this way in/for Mathematica?

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I used CARTAN for this purpose. It's a package for Wolfram Mathematica by Harald Soleng. It was allowing me to use tetrad formalism in space-time with curvature and torsion. Anyway, it was more than ten years ago so it would be better to ask Harald if this package still maintained. http://library.wolfram.com/infocenter/Articles/2280/

I know that Olga Babourova and Boris Frolov made an extension for CARTAN package to add non-metricity object to it. I believe you can ask them to clarify if this project still alive. In this article, you can find their email https://arxiv.org/abs/1611.08251

By the way, you can try some free alternatives. For example GRG for Reduce. It allows you to make calculation in exterior forms formalism. Please take a look at http://www.reduce-algebra.com/grg32/grg32.php and installation instructions https://mrfroll.com/post/reduce-and-grg-installations-on-ubuntu/

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Alternatively, you might want to explore the xAct package in Mathematica, which is updated relatively recently: http://www.xact.es/

This is so far the best package I have seen involving tensors (though I have mostly worked with standard tensor formalism), but I believe what you want can be found here --- since they include spinor formalism in which (to my knowledge) is naturally implemented via tetrad formalism. This is also very well-documented.

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