Modified gravity: a unified approach

TL;DR

This paper introduces a new unified framework for modified gravity theories starting from Einstein's original action, establishing equivalences among various models like $f(Q)$, $f(R)$, and $f(T)$ gravity through boundary terms, and emphasizing the role of boundary terms in these theories.

Contribution

The paper proposes a novel formulation of modified gravity that unifies different models and highlights the significance of boundary terms in establishing their equivalences.

Findings

Unified framework for $f(Q)$, $f(R)$, and $f(T)$ gravity models.

Equivalence of these models at the action and field equation levels.

Boundary terms are crucial for establishing these equivalences.

Abstract

Starting from the original Einstein action, sometimes called the Gamma squared action, we propose a new setup to formulate modified theories of gravity. This can yield a theory with second order field equations similar to those found in other popular modified gravity models. Using a more general setting the theory gives fourth order equations. This model is based on the metric alone and does not require more general geometries. It is possible to show that our new theory and the recently proposed $f(Q)$ gravity models are equivalent at the level of the action and at the level of the field equations, provided that appropriate boundary terms are taken into account. Our theory can also match up with $f(R)$ gravity which is an expected result. Perhaps more surprisingly, we can also show that this equivalence extends to $f(T)$ gravity at the level of the action and its field equations, provided that appropriate boundary terms are taken in account. While these three theories are conceptually different and are based on different geometrical settings, we can establish the necessary conditions under which their field equations are indistinguishable. The final part requires matter to couple minimally to gravity. Through this work we emphasise the importance played by boundary terms which are at the heart of our approach.