Generalized GHY term for f(R) Gravity

TL;DR

This paper introduces a generalized Gibbons-Hawking-York boundary term for metric f(R) gravity theories, enabling consistent field equations without boundary restrictions.

Contribution

It proposes a new definition of the Gibbons-Hawking-York term for f(R) gravity, ensuring boundary variations are properly canceled and extending the formalism.

Findings

Derived a consistent set of field equations for f(R) gravity

Proposed a boundary term that cancels surface variations

Ensured no boundary restrictions on metric variations

Abstract

A generalization to the Gibbons-Hawking-York boundary term for metric $f(R)$ gravity theories is introduced. A redefinition of the Gibbons-Hawking-York term is proposed. The proposed new definition is used to derive a consistent set of field equations and is extended to metric $f(R)$ gravity theories. The surface terms in the action are gathered into a total variation of some quantity. A total divergence term is added to the action to cancel these terms. Finally, the new definition is proven to demand no restrictions on the value of ${\delta g}_{ab}$ or ${\partial}_{c}{\delta g}_{ab}$ on the boundary.