Abstract
We discuss the presence of ghostly instabilities for metric-affine theories constructed with higher order curvature terms. We mainly focus on theories containing only the Ricci tensor and show the crucial role played by the projective symmetry. The pathological modes arise from the absence of a pure kinetic term for the projective mode and the non-minimal coupling of a 2-form field contained in the connection, and which can be related to the antisymmetric part of the metric in non-symmetric gravity theories. The couplings to matter are considered at length and cannot be used to render the theories stable. We discuss different procedures to avoid the ghosts by adding additional constraints. We finally argue how these pathologies are expected to be present in general metric-affine theories unless much care is taken in their construction.
Highlights
The intimate relation between GR and the geometry of the spacetime makes it natural to explore modifications to GR based on extending its geometrical framework, which in turn could be related to carefully accounting for the microscopic nature of spacetime itself
It is specially useful to check the breaking of gauge symmetries and the operators producing it1 as a manner to identify ghostly dof’s. This approach was used in [14] to show that generalized Ricci Based Gravity (RBG) where the projective symmetry is explicitly broken generally contain additional ghostly dof’s. These come in two forms: A spin-1 ghost arising from the absence of a pure kinetic term for the projective mode and Ostrogradski instabilities associated to non-minimal couplings of the additional 2-form field present in the theory
In the previous section we have shown how vacuum RBG without a projective symmetry are plagued by ghost-like instabilities arising from two sectors, namely: the dynamical projective mode whose mixing with the 2-form leads to the necessary presence of a spin-1 ghost and the non-minimal couplings of the 2-form field that gives rise to Ostrogradski instabilities
Summary
Since we are going to deal with metric-affine gravity theories, it will be convenient to introduce the corresponding geometrical framework. Let us note that projective transformations leave invariant the symmetric parts of the Ricci and co-Ricci tensors, but their antisymmetric part is not This fact is important because of what follows: It is well known that higher order curvature gravity theories in the metric formalism propagate ghostly degrees of freedom (except Lovelock theories), which can be traced back to the fact that their equations of motion for the metric are of fourth order and present Ostrogradski instabilities. This is not true in the metric-affine formalism, where the connection is no longer related to derivatives of the metric but a fundamental field, and arbitrary powers of the Riemann in the action do not render higher order equations of motion for the metric Holding on to this fact, as commented in the Introduction, it is sometimes argued that metric-affine higher order curvature gravity theories do not propagate ghost-like degrees of freedom.
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