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}-\cite{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 \cite{noakes}. The techniques employed here differ slightly from those in \cite{noakes}, but the main conclusions agree. Furthermore, the analysis of the initial value formulation in \cite{noakes} is enlarged and the use of harmonic coordinates is clarified. In particular, it is shown that the initial constraints found \cite{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 \cite{noakes} are derived explicitly. This procedure facilitates application of some of the mathematical theorems given in \cite{ringstrom}. As a consequence of these theorems, the existence of both $C^{\infty }$ 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.

34 pages. Expanded version, with more detailed steps. English typos corrected

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• Semiclassical gravity with a conformally covariant field in globally hyperbolic spacetimes
• Black Holes in Multi-Metric Gravity