A general scheme to average out an arbitrary 4-dimensional Riemannian space and to construct the geometry of the averaged space is proposed. It is shown that the averaged manifold has a metric and two equi-affine symmetric connections. The geometry of the space is characterized by the tensors of Riemannian and non-Riemannian curvatures, an affine deformation tensor being the result of non-metricity of one of the connections. To average out the differential Bianchi identities, correlation 2-form, 3-form and 4-form are introduced and the differential relations on these correlations tensors are derived, the relations being integrable on an arbitrary averaged manifold. Upon assuming a splitting rule for the average of the product including a covariantly constant tensor, an averaging out of the Einstein equations has been carried out which brings additional terms with the correlation tensors into them. As shown by averaging out the contracted Bianchi identities, the equations of motion for the averaged energy-momentum tensor do also include the geometric correction terms. Considering the gravitational induction tensor to be the Riemannian curvature tensor (then the non-Riemannian one is the macroscopic gravitational field), a theorem that relates the algebraic structure of the averaged microscopic metric with that of the induction tensor is proved. Due to the theorem the same field operator as in the Einstein equations is manifestly extracted from the averaged ones. Physical interpretation and application of the relations and equations obtained to treat macroscopic gravity are discussed.

On the Cosmological Equations in a Universe with Small Scale Condensations

Averaging out the Einstein equations

We introduce a simple and straightforward averaging procedure, which is a generalization of one which is commonly used in electrodynamics, and show that it possesses all the characteristics we require for linearized averaging in general relativity and cosmology — for weak-field and perturbed FLRW situations. In particular, we demonstrate that it yields quantities which are approximately tensorial in these situations, and that its application to an exact FLRW metric yields another FLRW metric, to first-order in integrals over the local coordinates. Finally, we indicate some important limits of any linearized averaging procedure with respect to cosmological perturbations which are the result of averages over large amplitude small and intermediate scale inhomogeneities, and show our averaging procedure can be approximately implemented by that of Zotov and Stoeger in these cases.

Averaging einstein's equations: the linearized case

The derivation of MacCallum-Taub's averaged high-frequency Lagrangian [1] is analysed with special attention paid to the assumptions made along the derivation It is shown that averaged high-frequency Lagrangians of the same form as MacCallum-Taub's Lagrangian can be derived by applying the Brill-Hartle and macroscopic gravity averaging schemes A procedure for the derivation of a Lagrangian of macroscopic gravity (an averaged Hilbert action) is proposed and its high-frequency limit (namely, its high-frequency perturbation expansion up to the second order terms in perturbations, which is referred to as MacCallum-Taub's limit) is calculated There is disagreement [2] in the expressions for MacCallum-Taub's averaged high-frequency Lagrangian and the high-frequency limit of the macroscopic gravity Lagrangian Possible reasons for such disagreement are analysed The origin of the difference is shown to consist in using the propagation equation for perturbations, ie the linearized Ricci tensor vanishes, during the derivation (averaging) carried out in [1] A new derivation of an averaged high-frequency Lagrangian without assuming the propagation equation to hold and by taking into account the proper correlation functions is given The newly derived expression is shown to coincide with MacCallum-Taub's limit of the macroscopic gravity Lagrangian, which resolves the disagreement

On the Cosmological Equations in a Universe with Small Scale Condensations

Citations

Averaging out the Einstein equations

Averaging einstein's equations: the linearized case

Averaged Lagrangians and MacCallum-Taub's limit in macroscopic gravity

Related Papers (5)

A Hypothesis, Unifying the Structure and the Entropy of the Universe

Cosmological models with variable gravitational and cosmological constants in r2 gravity

The cosmological constant problem

Scale Dependence of Nonlinear Gravitational Clustering in the Universe

Cosmology of the Very Early Universe