Boundary Term in Metric f(R) Gravity: Field Equations in the Metric Formalism

TL;DR

This paper derives the field equations for metric f(R) gravity using elementary variational principles, emphasizing the importance of boundary terms for a well-defined action principle, and compares with the Gibbons-York-Hawking term in GR.

Contribution

It provides a straightforward derivation of metric f(R) gravity field equations including boundary terms, avoiding scalar-tensor reformulations.

Findings

Boundary term is essential for a well-defined variational principle.

Derived explicit field equations in metric f(R) gravity.

Compared boundary terms with those in General Relativity.

Abstract

The main goal of this paper is to get in a straightforward form the field equations in metric f(R) gravity, using elementary variational principles and adding a boundary term in the action, instead of the usual treatment in an equivalent scalar-tensor approach. We start with a brief review of the Einstein-Hilbert action, together with the Gibbons-York-Hawking boundary term, which is mentioned in some literature, but is generally missing. Next we present in detail the field equations in metric f(R) gravity, including the discussion about boundaries, and we compare with the Gibbons-York-Hawking term in General Relativity. We notice that this boundary term is necessary in order to have a well defined extremal action principle under metric variation.