Emergent metric and geodesic analysis in cosmological solutions of (torsion-free) polynomial affine gravity

Abstract

Starting from an affinely connected space, we consider a model of gravity whose fundamental field is the connection. We build up the action using as sole premise the invariance under diffeomorphisms, and study the consequences of a cosmological ansatz for the affine connection in the torsion-free sector. Although the model is built without requiring a metric, we show that the nondegenerated Ricci curvature of the affine connection can be interpreted as an emergent metric on the manifold. We show that there exists a parametrization in which the (r, ϕ)-restriction of the geodesics coincides with that of the Friedman-Robertson-Walker model. Additionally, for connections with nondegenerated Ricci we are able to distinguish between space-, time-and null-like self-parallel curves, providing a way to differentiate trajectories of massive and massless particles.

Aspects of the polynomial affine model of gravity in three dimensions

“…A nice feature of this torsion-descendent metric is that, unlike the emergent metric from Ref. [49], it might be welldefined even when the space is Ricci-flat.…”

“…[44]), and used recently in Ref. [49]. A notable disadvantage is that interesting cases, such as Minkowski and Schwarzschild, cannot be described using this notion of metric.…”

“…Further studies, which take these tools into account, will allow us to complete the classification of the solutions in four dimensions started in Refs. [48,49]. which can be written as a total derivative, 1 X ∂ t (X ∂ t t) = 0, for j = 1 X ∂ t X , or equivalently X = e dt j(t) = e J (t) .…”

“…A dimensional analysis similar to the one presented in Refs. [21,22,[47][48][49] shows that the most general action (up to boundary terms) is given by 3…”

“…Unlike the analysis of the truncations in the four-dimensional polynomial affine model of gravity [22,48,49], the presence of the Chern-Simons terms in Eq. ( 3) allows to switch-off certain fields at the level of the action, and not just at the level of the field equations.…”