Space-time Averages in Macroscopic Gravity and Volume-preserving Coordinates

The definition of the covariant space-time averaging scheme for the objects (tensors, geometric objects, etc.) on differentiable metric manifolds with a volume n-form, which has been proposed for the formulation of macroscopic gravity, is analyzed. An overview of the space-time averaging procedure in Minkowski spacetime is given and comparison between this averaging scheme and that adopted in macroscopic gravity is carried out throughout the paper. Some new results concerning the algebraic structure of the averaging operator are precisely formulated and proved, the main one being that the averaging bilocal operator is idempotent iff it is factorized into a bilocal product of a matrix-valued function on the manifold, taken at a point, by its inverse at another point. The previously proved existence theorems for the averaging and coordination bilocal operators are revisited with more detailed proofs of related results. A number of new results concerning the structure of the volume-preserving averaging operators and the class of proper coordinate systems are given. It is shown, in particular, that such operators are defined on an arbitrary n-dimensional differentiable metric manifold with a volume n-form up to the freedom of (n-1) arbitrary functions of n arguments and 1 arbitrary function of (n-1) arguments. All the results given in this paper are also valid whenever appropriate for affine connection manifolds including (pseudo)-Riemannian manifolds.