Noncommutative Spaces and the Theory of Geometrodynamics

This article is concerned with a generalisation of Connes' noncommutative framework. This is achieved by a general study of spectral triples, in particular through an analysis of the role played by the Dirac operator. The Dirac operator is used to construct a generalised covariant derivative, which is then employed to define a generalised spacetime. The spectral action is then modified to accommodate this generalisation and it is observed that in the appropriate limit, Connes' spectral action is recovered.