Boundary Terms, Variational Principles and Higher Derivative Modified Gravity (0809.4033v2)

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.

Summary of "Boundary Terms, Variational Principles and Higher Derivative Modified Gravity"

The paper by Ethan Dyer and Kurt Hinterbichler presents an in-depth examination of boundary terms and variational principles in the context of higher derivative models of gravity, with a specific focus on $F(R)$ gravity. It provides critical insight into the selection and implications of boundary terms within the framework of higher derivative theories of gravity by addressing foundational questions about variational principles and their relation to boundary terms derived from both a geometrical and theoretical standpoint.

The authors begin by emphasizing the necessity of boundary terms for rendering a variational problem well-posed, which is critical for ensuring that the equations of motion are derived correctly. The traditional Einstein-Hilbert action of General Relativity necessitates the addition of the Gibbons-Hawking-York (GHY) boundary term to avoid fixing normal derivatives of the metric on the boundary, enhancing the well-posedness of the corresponding variational principle. This is applicable especially in the Hamiltonian formulation where it reproduces expected results for energy and entropy.

In the paper, the authors argue that higher derivative theories of gravity, such as those described by the $F(R)$ action, require boundary terms that align with their scalar-tensor theoretical equivalence. $F(R)$ gravity extends General Relativity by considering actions formulated as arbitrary functions of the Ricci scalar $R$, introducing higher-order derivatives and thereby modifying the criteria for boundary terms. The authors show that suitable boundary terms must be chosen to ensure the correspondence between these theories, where the variation of the action requires fixing both the metric and scalar field degrees of freedom at the boundary.

Key numerical findings include the derivation of the Hamiltonian formulation in such modified theories and calculations of the ADM energy and Schwartzschild black hole entropy for $F(R)$ gravity using the derived boundary terms. The paper confirms that the expected ADM energy form is achieved when these boundary terms are incorporated, aligning with classical General Relativity results when $F(R)$ reduces to linear functions. Similarly, the entropy calculations concur with previously established results from Wald's entropy formula for diffeomorphism invariant theories, suggesting that higher curvature corrections do not alter the entropy significantly.

In conclusion, the results presented by Dyer and Hinterbichler underscore the crucial role of well-defined boundary terms in formulating consistent variational principles for higher derivative theories of gravity. Such considerations provide a pathway for understanding modified gravity theories that attempt to extend General Relativity and offer insights into cosmic acceleration phenomena without invoking dark energy. The implications of this research are foundational for theoretical advancements in cosmology and quantum gravity, potentially influencing future developments in artificial gravitational theories within astrophysical observations and quantum simulations.