About the Cauchy problem in Stelle's quadratic gravity

Abstract

The focus of the present work is on the Cauchy problem for the quadratic gravity models introduced in \cite{stelle,stelle2}. These are renormalizable higher order derivative models of gravity, but at cost of ghostly states propagating in the phase space. A previous work on the subject is [1]. The techniques employed here differ slightly from those in [2], but the main conclusions agree. Furthermore, the analysis of the initial value formulation in [3] is enlarged and the use of harmonic coordinates is clarified. In particular, it is shown that the initial constraints found [4] include a redundant one. In other words, this constraint is satisfied when the equations of motion are taken into account. In addition, some terms that are not specified in [5] are derived explicitly. This procedure facilitates application of some of the mathematical theorems given in [6]. As a consequence of these theorems, the existence of both C∞ solutions and maximal globally hyperbolic developments is proved. The obtained equations may be relevant for the stability analysis of the solutions under small perturbations of the initial data.