Unifying Einstein and Palatini gravities

TL;DR

This paper introduces C-theories, a new class of gravity models unifying Einstein and Palatini gravities, with potential to resolve their limitations and featuring a scalar-tensor formulation.

Contribution

It proposes C-theories, a novel framework linking Einstein and Palatini gravities through a conformally scaled metric dependent on scalar curvature.

Findings

C-theories include Einstein and Palatini as special cases

They can avoid some conceptual and observational issues of existing theories

C-theories have a scalar-tensor formulation similar to Brans-Dicke gravity

Abstract

We consider a novel class of $f(\R)$ gravity theories where the connection is related to the conformally scaled metric $\hat g_{\mu\nu}=C(\R)g_{\mu\nu}$ with a scaling that depends on the scalar curvature $\R$ only. We call them C-theories and show that the Einstein and Palatini gravities can be obtained as special limits. In addition, C-theories include completely new physically distinct gravity theories even when $f(\R)=\R$. With nonlinear $f(\R)$, C-theories interpolate and extrapolate the Einstein and Palatini cases and may avoid some of their conceptual and observational problems. We further show that C-theories have a scalar-tensor formulation, which in some special cases reduces to simple Brans-Dicke-type gravity. If matter fields couple to the connection, the conservation laws in C-theories are modified. The stability of perturbations about flat space is determined by a simple condition on the lagrangian.