Abstract
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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G. ’t Hooft and M.J.G. Veltman, One loop divergencies in the theory of gravitation, Ann. Inst. H. Poincare Phys. Theor. A 20 (1974) 69 [INSPIRE].
M.H. Goroff and A. Sagnotti, The Ultraviolet Behavior of Einstein Gravity, Nucl. Phys. B 266 (1986) 709 [INSPIRE].
K.S. Stelle, Renormalization of Higher Derivative Quantum Gravity, Phys. Rev. D 16 (1977) 953 [INSPIRE].
K.S. Stelle, Classical Gravity with Higher Derivatives, Gen. Rel. Grav. 9 (1978) 353 [INSPIRE].
D.G. Boulware, G.T. Horowitz and A. Strominger, Zero Energy Theorem for Scale Invariant Gravity, Phys. Rev. Lett. 50 (1983) 1726 [INSPIRE].
G.T. Horowitz, Quantum Cosmology With a Positive Definite Action, Phys. Rev. D 31 (1985) 1169 [INSPIRE].
F. David and A. Strominger, On the Calculability of Newton’s Constant and the Renormalizability of Scale Invariant Quantum Gravity, Phys. Lett. B 143 (1984) 125 [INSPIRE].
I.L. Buchbinder and S.L. Lyakhovich, Canonical Quantization and Local Measure of R2 Gravity, Class. Quant. Grav. 4 (1987) 1487 [INSPIRE].
S. Deser and B. Tekin, New energy definition for higher curvature gravities, Phys. Rev. D 75 (2007) 084032 [gr-qc/0701140] [INSPIRE].
G. ’t Hooft, A class of elementary particle models without any adjustable real parameters, Found. Phys. 41 (2011) 1829 [arXiv:1104.4543] [INSPIRE].
H. Lu, Y. Pang and C.N. Pope, Conformal Gravity and Extensions of Critical Gravity, Phys. Rev. D 84 (2011) 064001 [arXiv:1106.4657] [INSPIRE].
M. Park and L. Sorbo, Massive Gravity from Higher Derivative Gravity with Boundary Conditions, JHEP 01 (2013) 043 [arXiv:1210.7733] [INSPIRE].
L. Alvarez-Gaume et al., Aspects of Quadratic Gravity, Fortsch. Phys. 64 (2016) 176 [arXiv:1505.07657] [INSPIRE].
D.M. Capper and M.J. Duff, Conformal Anomalies and the Renormalizability Problem in Quantum Gravity, Phys. Lett. A 53 (1975) 361 [INSPIRE].
E.S. Fradkin and A.A. Tseytlin, Renormalizable asymptotically free quantum theory of gravity, Nucl. Phys. B 201 (1982) 469 [INSPIRE].
J. Julve and M. Tonin, Quantum Gravity with Higher Derivative Terms, Nuovo Cim. B 46 (1978) 137 [INSPIRE].
S.L. Adler, Einstein Gravity as a Symmetry-Breaking Effect in Quantum Field Theory, Rev. Mod. Phys. 54 (1982) 729 [Erratum ibid. 55 (1983) 837] [INSPIRE].
H. Weyl, Gravitation and electricity, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.) 1918 (1918) 465 [INSPIRE].
H. Weyl, A New Extension of Relativity Theory, Annalen Phys. 59 (1919) 101 [INSPIRE].
R. Bach, Zur Weylschen Relativitätstheorie und der Weylschen Erweiterung des Krümmungstensorbegriffs, Math. Z. 9 (1921) 110.
M. Kaku, P.K. Townsend and P. van Nieuwenhuizen, Gauge Theory of the Conformal and Superconformal Group, Phys. Lett. B 69 (1977) 304 [INSPIRE].
E.S. Fradkin and A.A. Tseytlin, Conformal supergravity, Phys. Rept. 119 (1985) 233 [INSPIRE].
M. Kaku and P.K. Townsend, Poincaré supergravity as broken superconformal gravity, Phys. Lett. B 76 (1978) 54 [INSPIRE].
M. Kaku, P.K. Townsend and P. van Nieuwenhuizen, Properties of Conformal Supergravity, Phys. Rev. D 17 (1978) 3179 [INSPIRE].
B. de Wit, J.W. van Holten and A. Van Proeyen, Structure of N = 2 Supergravity, Nucl. Phys. B 184 (1981) 77 [Erratum ibid. 222 (1983) 516] [INSPIRE].
E. Bergshoeff, M. de Roo and B. de Wit, Extended Conformal Supergravity, Nucl. Phys. B 182 (1981) 173 [INSPIRE].
H. Liu and A.A. Tseytlin, D = 4 superYang-Mills, D = 5 gauged supergravity, and D = 4 conformal supergravity, Nucl. Phys. B 533 (1998) 88 [hep-th/9804083] [INSPIRE].
S. Ferrara, A. Kehagias and D. Lüst, Aspects of Weyl Supergravity, JHEP 08 (2018) 197 [arXiv:1806.10016] [INSPIRE].
L. Andrianopoli and R. D’Auria, N = 1 and N = 2 pure supergravities on a manifold with boundary, JHEP 08 (2014) 012 [arXiv:1405.2010] [INSPIRE].
R. D’Auria and L. Ravera, Conformal gravity with totally antisymmetric torsion, Phys. Rev. D 104 (2021) 084034 [arXiv:2101.10978] [INSPIRE].
M. Dunajski and P. Tod, Self-Dual Conformal Gravity, Commun. Math. Phys. 331 (2014) 351 [arXiv:1304.7772] [INSPIRE].
S. Ferrara, A. Kehagias and D. Lüst, Aspects of Conformal Supergravity, in the proceedings of the 57th International School of Subnuclear Physics: In Search for the Unexpected, Erice Italy, June 21–30 (2019) [arXiv:2001.04998] [INSPIRE].
A.H. Chamseddine and A. Connes, The Spectral action principle, Commun. Math. Phys. 186 (1997) 731 [hep-th/9606001] [INSPIRE].
G. Manolakos, P. Manousselis and G. Zoupanos, Four-Dimensional Gravity on a Covariant Noncommutative Space (II), Fortsch. Phys. 69 (2021) 2100085 [arXiv:2104.13746] [INSPIRE].
G. Manolakos, P. Manousselis and G. Zoupanos, Four-dimensional Gravity on a Covariant Noncommutative Space, JHEP 08 (2020) 001 [arXiv:1902.10922] [INSPIRE].
N. Berkovits and E. Witten, Conformal supergravity in twistor-string theory, JHEP 08 (2004) 009 [hep-th/0406051] [INSPIRE].
K. Sen, A. Sinha and N.V. Suryanarayana, Counterterms, critical gravity and holography, Phys. Rev. D 85 (2012) 124017 [arXiv:1201.1288] [INSPIRE].
D. Grumiller, M. Irakleidou, I. Lovrekovic and R. McNees, Conformal gravity holography in four dimensions, Phys. Rev. Lett. 112 (2014) 111102 [arXiv:1310.0819] [INSPIRE].
O. Miskovic and R. Olea, Topological regularization and self-duality in four-dimensional anti-de Sitter gravity, Phys. Rev. D 79 (2009) 124020 [arXiv:0902.2082] [INSPIRE].
G. Anastasiou et al., Conformal renormalization of scalar-tensor theories, Phys. Rev. D 107 (2023) 104049 [arXiv:2212.04364] [INSPIRE].
G. Anastasiou, O. Miskovic, R. Olea and I. Papadimitriou, Counterterms, Kounterterms, and the variational problem in AdS gravity, JHEP 08 (2020) 061 [arXiv:2003.06425] [INSPIRE].
P.D. Mannheim and J.G. O’Brien, Galactic rotation curves in conformal gravity, J. Phys. Conf. Ser. 437 (2013) 012002 [arXiv:1211.0188] [INSPIRE].
M. Hobson and A. Lasenby, Conformal gravity does not predict flat galaxy rotation curves, Phys. Rev. D 104 (2021) 064014 [arXiv:2103.13451] [INSPIRE].
Y. Meng, X.-M. Kuang and Z.-Y. Tang, Photon regions, shadow observables, and constraints from M87* of a charged rotating black hole, Phys. Rev. D 106 (2022) 064006 [arXiv:2204.00897] [INSPIRE].
M. Momennia and S.H. Hendi, Quasinormal Modes of Black Holes in Weyl Gravity: Electromagnetic and Gravitational Perturbations, Eur. Phys. J. C 80 (2020) 505 [arXiv:1910.00428] [INSPIRE].
R.J. Riegert, The particle content of linearized conformal gravity, Phys. Lett. A 105 (1984) 110 [INSPIRE].
M. Ostrogradsky, Mémoires sur les équations différentielles, relatives au problème des isopérimètres, Mem. Acad. St. Petersbourg 6 (1850) 385 [INSPIRE].
J. Maldacena, Einstein Gravity from Conformal Gravity, arXiv:1105.5632 [INSPIRE].
G. Anastasiou and R. Olea, From conformal to Einstein Gravity, Phys. Rev. D 94 (2016) 086008 [arXiv:1608.07826] [INSPIRE].
G. Anastasiou, I.J. Araya and R. Olea, Einstein Gravity from Conformal Gravity in 6D, JHEP 01 (2021) 134 [arXiv:2010.15146] [INSPIRE].
T. Wang, Z. Zhang, X. Kong and L. Zhao, Topological black holes in Einstein-Maxwell and 4D conformal gravities revisited, arXiv:2211.16904 [INSPIRE].
V. Dzhunushaliev and V. Folomeev, Masking singularities in Weyl gravity and Ricci flows, Eur. Phys. J. C 81 (2021) 387 [arXiv:2102.07494] [INSPIRE].
H. Lu, Y. Pang, C.N. Pope and J.F. Vazquez-Poritz, AdS and Lifshitz Black Holes in Conformal and Einstein-Weyl Gravities, Phys. Rev. D 86 (2012) 044011 [arXiv:1204.1062] [INSPIRE].
P.D. Mannheim and D. Kazanas, Exact Vacuum Solution to Conformal Weyl Gravity and Galactic Rotation Curves, Astrophys. J. 342 (1989) 635 [INSPIRE].
R.J. Riegert, Birkhoff’s Theorem in Conformal Gravity, Phys. Rev. Lett. 53 (1984) 315 [INSPIRE].
C. Corral, G. Giribet and R. Olea, Self-dual gravitational instantons in conformal gravity: Conserved charges and thermodynamics, Phys. Rev. D 104 (2021) 064026 [arXiv:2105.10574] [INSPIRE].
H.-S. Liu and H. Lu, Charged Rotating AdS Black Hole and Its Thermodynamics in Conformal Gravity, JHEP 02 (2013) 139 [arXiv:1212.6264] [INSPIRE].
V. Mukhanov, Physical Foundations of Cosmology, Cambridge University Press, Oxford (2005) [https://doi.org/10.1017/CBO9780511790553] [INSPIRE].
A.A. Starobinsky, Isotropization of arbitrary cosmological expansion given an effective cosmological constant, JETP Lett. 37 (1983) 66 [INSPIRE].
C. Fefferman and C.R. Graham, Conformal Invariants, in Elie Cartan et les Mathématiques d’aujourd’hui, Lyon France, June 25–29 (1984) [Astérisque S131 (1985) 95].
C.R. Graham, Volume and area renormalizations for conformally compact Einstein metrics, Rend. Circ. Mat. Palermo S 63 (2000) 31 [math/9909042] [INSPIRE].
Acknowledgments
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
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DOI: https://doi.org/10.1007/JHEP08(2023)168