# The commutator algebra of covariant derivative as general framework for extended gravity. The Rastall theory case and the role of the torsion

## TL;DR

This paper explores the algebraic structure of covariant derivatives in extended gravity theories, emphasizing the role of torsion and using the Rastall theory as a case study, while also highlighting gravitational wave astronomy's potential to distinguish between theories.

## Contribution

It introduces a commutator algebra framework for analyzing extended gravity theories, including the impact of torsion and the application to Rastall theory.

## Key findings

- Torsion plays a significant role in the algebraic structure of extended gravity.
- The commutator algebra approach helps compare different gravitational theories.
- Gravitational wave astronomy can discriminate between general relativity and alternative theories.

## Abstract

In this short review, we discuss the approach of the commutator algebra of covariant derivative to analyse the gravitational theories, starting from the standard Einstein's general theory of relativity and focusing on the Rastall theory. After that, we discuss the important role of the torsion in this mathematical framework. In the Appendix of the paper we analyse the importance of the nascent gravitational wave astronomy as a tool to discriminate among the general theory of relativity and alternative theories of gravity.