Boundary Terms, Variational Principles and Higher Derivative Modified Gravity

TL;DR

This paper clarifies the role of boundary terms in higher derivative gravity theories, ensuring well-posed variational principles and correct physical quantities like energy and entropy.

Contribution

It identifies the appropriate boundary terms for F(R) gravity, aligning variational principles with scalar-tensor correspondence and physical consistency.

Findings

Boundary terms reproduce correct ADM energy.

Boundary terms yield correct black hole entropy.

Clarifies variational principles in higher derivative gravity.

Abstract

We discuss the criteria that must be satisfied by a well-posed variational principle. We clarify the role of Gibbons-Hawking-York type boundary terms in the actions of higher derivative models of gravity, such as F(R) gravity, and argue that the correct boundary terms are the naive ones obtained though the correspondence with scalar-tensor theory, despite the fact that variations of normal derivatives of the metric must be fixed on the boundary. We show in the case of F(R) gravity that these boundary terms reproduce the correct ADM energy in the hamiltonian formalism, and the correct entropy for black holes in the semi-classical approximation.