Intrinsic General Relativity and Clifford Algebra

Abstract

This paper proposes to go beyond the Einstein General Relativity theory in a noncommutative geometric framework. As a first step, we rewrite the General Relativity theory intrinsically (coordinate-free formulation). As a second step, we rewrite the first and the second Bianchi identities, including torsion, within minimal algebraic hypotheses. Then, in order to extend the General Relativity theory for dealing with noncommutative scalar fields on a manifold \(\mathcal {M}\), we are forced to adapt the definition of vector fields and connections. It leads to the consideration of one associative product p of vector fields and four derivations (\(\nabla \): (vector, vector) to vector, \(\partial \): (vector, scalar) to scalar, \(\delta \): (scalar, vector) to scalar, \(\mathcal {C}\): (scalar, scalar) to scalar). At last, the particular case where scalars are a Clifford numbers motivates future investigations towards a common writing of the Einstein field equations and the Dirac equation.