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 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 noakes. The techniques employed here differ slightly from those in noakes, but the main conclusions agree. Furthermore, the analysis of the initial value formulation in noakes is enlarged and the use of harmonic coordinates is clarified. In particular, it is shown that the initial constraints found noakes 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 noakes are derived explicitly. This procedure facilitates application of some of the mathematical theorems given in ringstrom. 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.
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