Einstein's equations constrained by homogeneous and isotropic expansion: Initial value problems and applications

TL;DR

This paper develops a new framework for solving Einstein's equations under homogeneous and isotropic conditions, demonstrating well-posedness, stability, and novel energy transfer mechanisms in cosmological models.

Contribution

It introduces a general scheme for initial value problems in Einstein's equations constrained by homogeneous and isotropic expansion, with proofs of existence, uniqueness, and stability.

Findings

Proved well-posedness and stability of solutions.

Discovered a new energy transfer mechanism among matter components.

Derived exact solutions exemplifying the theoretical framework.

Abstract

In this manuscript, we put forth a general scheme for defining initial value problems from Einstein's equations of General Relativity constrained by homogeneous and isotropic expansion. The cosmological models arising as solutions are naturally interpreted as spatially homogeneous and isotropic on ``large scales". In order to show the well-posedness and applicability of such a scheme, we specialize in a class of spacetimes filled with the general homogeneous perfect fluid and inhomogeneous viscoelastic matter. We prove the existence, uniqueness, and relative stability of solutions, and an additional inequality for the energy density. As a consequence of our theorems, a new mechanism of energy transfer appears involving the different components of matter. A class of exact solutions is also obtained to exemplify the general results.