Masud Chaichian, Josef Klusoň, Markku Oksanen, Anca Tureanu
In order to explore some general features of modified theories of gravity which involve higher derivatives and spontaneous Lorentz and/or diffeomorphism symmetry breaking, we study the recently proposed new version of Covariant Renormalizable Gravity (CRG). CRG is a higher derivative theory of gravity that aims to provide a generally covariant ultraviolet completion of general relativity. Its special features are the presence of projection operators constructed from a constrained scalar field in the action and the spontaneous breaking of Lorentz invariance, which enable the theory to be power-counting renormalizable. We obtain an Arnowitt-Deser-Misner (ADM) representation of the CRG action in 4-dimensional spacetime with respect to a foliation of spacetime adapted to the constrained scalar field. The resulting action is analyzed by using Hamiltonian formalism, which was originally developed for constrained systems by Dirac. It is found that the theory contains two extra propagating degrees of freedom. One is due to the presence of second order time derivatives in the ADM representation of the CRG action. The other is due to the lack of a local Hamiltonian constraint, similarly as in the projectable version of Ho\v{r}ava-Lifshitz gravity. It is argued that CRG contains a degree of freedom that carries negative energy (a ghost) which will destabilize the theory. Such a pathology jeopardizes all higher time derivative field theories where degrees of freedom with positive and negative energies interact with each other, unless the given Lagrangian is appropriately degenerate so that there exist constraints that protect the stability. We conjecture that generally covariant higher derivative theories of gravity which involve spontaneous (constraint induced) Lorentz and/or diffeomorphism symmetry breaking will in general share this problem with CRG.
View original:
http://arxiv.org/abs/1208.3990
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