We study the metric perturbations around the de Sitter and Minkowski backgrounds in Conformal Gravity. We confirm the presence of ghosts in both cases. In the de Sitter case, by applying the Maldacena boundary conditions — the Neumann boundary condition and the positive-frequency mode condition — to the metric, we show that one cannot recover a general solution for the perturbations. In turn, alongside the Neumann boundary condition, we derive an additional condition with which the perturbations of conformal gravity and dS perturbations of Einstein gravity with cosmological constant coincide. We further show that the Neumann boundary condition does not lead to a general solution in Minkowski space. Conversely, we derive the alternative boundary conditions, with which we attain an agreement between the perturbations of conformal and Einstein gravity in full generality, thus removing the ghost of conformal gravity.
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A.H. would like to thank Misao Sasaki, for enlightening discussions, and for further inspiring the search for general perturbative solutions in the dS space. In addition, A.H. and G.Z. would like to thank Giorgos Anastasiou and George Manolakos for very useful correspondence, and the CERN theory department, where part of this work was completed, for hospitality. G.Z. would also like to thank MPP-Munich for hospitality, and MPP-Munich, CERN-TH and DFG Exzellenzcluster 2181:STRUCTURES of Heidelberg University for support. The work of A.H. is supported in part by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC-2111 – 390814868. The work of D.L. is supported by the Origins Excellence Cluster and by the German-Israel-Project (DIP) on Holography and the Swampland.
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Hell, A., Lüst, D. & Zoupanos, G. On the ghost problem of conformal gravity. J. High Energ. Phys. 2023, 168 (2023). https://doi.org/10.1007/JHEP08(2023)168