Note About Born-Infeld Inspired Gravity Coupled Non-Minimally to Scalar Fields and Eddington Gravity with Matter

TL;DR

The paper addresses canonical formulations of gravity theories inspired by Born-Infeld and Eddington, focusing on coupling to scalar matter. It develops the Hamiltonian framework for a Born-Infeld gravity non-minimally coupled to scalars, revealing a $V^2$-dependent modification to the Hamiltonian constraint, and shows that a Vollick-type modification restores a GR-like constraint structure. It then proposes an Eddington-type action for gravity coupled to scalar fields without a metric, $S_{EG\phi}=\int d^4x \frac{1}{V(\phi)} \sqrt{-\det \mathbf{A}}$ with $\mathbf{A}_{\mu\nu}=R_{(\mu\nu)}+\partial_\mu \phi^A \partial_\nu \phi^B K_{AB}(\phi)$, and demonstrates that its canonical form reproduces GR with scalar matter, with the dynamical content carried by momenta and the metric emerging as an auxiliary field. Overall, the work shows how affine-like formulations can yield GR-equivalent dynamics for scalar fields and provides a framework for exploring more general matter couplings within BI- and Eddington-inspired gravity. The results have implications for understanding the role of the metric and for constructing alternative gravity theories with controlled deviations from General Relativity.

Abstract

We study canonical formulation of Born-Infeld inspired gravity coupled non-minimally to scalar field. Then we propose form of Eddington Gravity coupled to collection of scalar fields whose canonical form is the same as Hamiltonian for General Relativity coupled minimally to scalar field.