Towards a theory of macroscopic gravity

Abstract

By averaging out Cartan's structure equations for a four-dimensional Riemannian space over space regions, the structure equations for the averaged space have been derived with the procedure being valid on an arbitrary Riemannian space. The averaged space is characterized by a metric, Riemannian and non-Rimannian curvature 2-forms, and correlation 2-, 3- and 4-forms, an affine deformation 1-form being due to the non-metricity of one of two connection 1-forms. Using the procedure for the space-time averaging of the Einstein equations produces the averaged ones with the terms of geometric correction by the correlation tensors. The equations of motion for averaged energy momentum, obtained by averaging out the contracted Bianchi identities, also include such terms. Considering the gravitational induction tensor to be the Riemannian curvature tensor (the non-Riemannian one is then the field tensor), a theorem is proved which relates the algebraic structure of the averaged microscopic metric to that of the induction tensor. It is shown that the averaged Einstein equations can be put in the form of the Einstein equations with the conserved macroscopic energy-momentum tensor of a definite structure including the correlation functions. By using the high-frequency approximation of Isaacson with second-order correction to the microscopic metric, the self-consistency and compatibility of the equations and relations obtained are shown. Macrovacuum turns out to be Ricci non-flat, the macrovacuum source being defined in terms of the correlation functions. In the high-frequency limit the equations are shown to become Isaacson's ones with the macrovauum source becoming Isaacson's stress tensor for gravitational waves.