Symmetric gravitational closure

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.