On the Jacobi field equations

Abstract

A noncommutative geometry is studied in the quasi-classical, linear field approximation. Integrability conditions for the associative structure of the algebra, which we refer to as the Jacobi equations, force restrictions on the metric which can be expressed as an effective additional source term for the Einstein tensor. All calculations are carried out in the 'quasi-classical approximation', by which we mean that the noncommutativity is considered as a first-order perturbation of the limiting classical configuration.