Boundary terms, variational principles, and higher derivative modified gravity

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 through 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 Arnowitt-Deser-Misner energy in the Hamiltonian formalism, and the correct entropy for black holes in the semiclassical approximation.

Figure 1

Region over which a field theory is defined.

Figure 2

Foliation of spacetime.