Boundary Terms, Variational Principles and Higher Derivative Modified Gravity

TL;DR

This work clarifies how to formulate a well-posed variational principle in higher-derivative gravity by identifying boundary terms that align with scalar-tensor duality. It argues that the natural boundary term for $F(R)$ gravity follows from the Einstein-frame GHY term and requires fixing appropriate boundary data, notably $ md R$ on the boundary. Through a Hamiltonian analysis, the authors derive the ADM energy expression $E=-2\oint\sqrt{\sigma}\left(fK+f' r^a\partial_a\phi\right)$ and show that the same boundary term yields the correct black-hole entropy in the Euclidean approach, in agreement with Wald's formula. Overall, the paper demonstrates that carefully chosen boundary terms, guided by scalar-tensor equivalence, render higher-derivative gravity theories consistent with both the Hamiltonian formalism and semiclassical thermodynamics.

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.

Figures (2)

• Figure 1: Region over which a field theory is defined.
• Figure 2: Foliation of spacetime