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

Abstract

In this short review, we discuss the approach of the commutator algebra of covariant derivative to analyze the gravitational theories, starting from the standard Einstein's general theory of relativity (GTR) 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 analyze the importance of the nascent gravitational wave (GW) astronomy as a tool to discriminate among the GTR and alternative theories of gravity.