Gravity actions, boundary terms and second-order field equations

Abstract

The Lovelock action is a natural generalization of the Einstein-Hilbert gravity action in higher (more than four) dimensions. Certain boundary terms have to be added to this action because of various reasons which we briefly review. Explicit expressions of these boundary terms can already be found in Chern's famous paper [6] on the generalization of the Gauss-Bonnet theorem. Dimensional reduction of the Lovelock action leads to a generalized Einstein-Yang-Mills (+ scalar field or σ-models) action in lower dimensions. Since the field equations derived from the Lovelock action are of second order only, the same holds for the dimensionally reduced action. The necessary boundary terms for the latter action are then obtained by dimensional reduction of the boundary terms in the Lovelock action. We consider reduction from n to n −1 dimensions in some detail and obtain generalizations of Horndeski's non-minimally coupled Einstein-Maxwell action with second-order field equations in more than four dimensions.