Positive Energy Theorems in Fourth-Order Gravity

Abstract

AbstractIn this paper we will review some recent results concerning conservation principles associated to higher order gravitational theories. In particular, we introduce a notion of energy associated to quadratic Lagrangians arising as a conserved quantity canonically associated to perturbations of asymptotically Minkowskian space-time solutions which possess a time-like Killing field. We then present some intuitions about it and prove two positive energy theorems associated to asymptotically Euclidean initial data, containing their corresponding rigidity statements. The first of these theorems concerns 4-dimensional Einstein solutions to the fourth order equations, which arise from initial data sets which are, in an appropriate sense, asymptotically umbilical. The second positive energy theorem concerns stationary fourth order solutions, in which case the problem becomes properly Riemannian. In this context, the analysis becomes closely related to Q-curvature analysis, and, in particular, we comment on how this positive energy theorem implies the known positive mass theorems associated to the Paneitz operator, which are important tools in the Q-curvature conformal prescription problem.KeywordsPositive energy theoremsQ-curvature analysisPaneitz positive massFourth order gravity