Symmetric gravitational closure

TL;DR

This paper introduces a method to derive symmetry-reduced gravitational field equations directly from matter equations, exemplified by obtaining Friedmann equations from a Klein-Gordon field without prior Einstein equations.

Contribution

It presents a general procedure to determine gravitational equations compatible with matter models using symmetry assumptions, without relying on Einstein's equations.

Findings

Successfully derives Friedmann equations from Klein-Gordon field using symmetry assumptions.

Method generalizes to any Killing symmetry algebra and matter models beyond the standard model.

Applicable to tensorial spacetime geometries beyond Lorentzian metrics.

Abstract

We show how to exploit symmetry assumptions to determine the dynamical equations for the particular geometry that underpins given matter field equations. The procedure builds on the gravitational closure equations for matter models without any a priori assumption of symmetry. It suffices to illustrate the symmetrization procedure for a Klein-Gordon field equation on a Lorentzian background, for which one obtains the Friedmann equations, without ever having known Einstein's equations, by careful imposition of maximal cosmological symmetry directly on the pertinent gravitational closure equations. This method of finding the family of symmetry-reduced gravitational field equations that are compatible with given matter dynamics directly generalizes to any Killing symmetry algebra, matter models beyond the standard model and indeed tensorial spacetime geometries beyond Lorentzian metrics.