Abstract
We hereby study exact solutions in a wide range of local higher-derivative and weakly nonlocal gravitational theories. In particular, we give a list of exact classical solutions for two classes of gravitational theories both weakly nonlocal, unitary, and super-renormalizable (or finite) at quantum level. We prove that maximally symmetric spacetimes are exact solutions in both classes, while in dimension higher than four we can also have Anti-de Sitter solutions in the presence of positive cosmological constant. It is explicitly shown under which conditions flat and Ricci-flat spacetimes are exact solutions of the equation of motion (EOM) for the first class of theories not involving the Weyl tensor in the action. We find that the well-known physical spacetimes like Schwarzschild, Kerr, (Anti-) de Sitter serve as solutions for standard matter content, when the EOM does not contain the Riemann tensor alone (operators made out of only the Riemann tensor.) We pedagogically show how to obtain these exact solutions. Furthermore, for the second class of gravity theories, with terms in the Lagrangian written using Weyl tensors, the Friedmann-Robertson-Walker (FRW) spacetimes are also exact solutions (exactly in the same way like in Einstein theory), when the matter content is given by conformal matter (radiation). We also comment on rather inevitable presence and universality of singularities and possible resolution of them in finite and conformally invariant theories. “Delocalization” is proposed as a way to solve the black hole singularity problem in the first class. In order to solve the problem of cosmological singularities in the second class, it seems crucial to have a conformally invariant or asymptotically free quantum gravitational theory.
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References
S.W. Hawking and R. Penrose, The singularities of gravitational collapse and cosmology, Proc. Roy. Soc. Lond. A 314 (1970) 529 [INSPIRE].
S.W. Hawking and G.F.R. Ellis, The Large scale structure of space-time, Cambridge University Press, Cambridge, U.K. (1973).
V.P. Frolov, Mass-gap for black hole formation in higher derivative and ghost free gravity, Phys. Rev. Lett. 115 (2015) 051102 [arXiv:1505.00492] [INSPIRE].
V.P. Frolov, A. Zelnikov and T. de Paula Netto, Spherical collapse of small masses in the ghost-free gravity, JHEP 06 (2015) 107 [arXiv:1504.00412] [INSPIRE].
V.P. Frolov and G.A. Vilkovisky, Spherically Symmetric Collapse in Quantum Gravity, Phys. Lett. B 106 (1981) 307 [INSPIRE].
V.P. Frolov and G.A. Vilkovisky, Quantum Gravity Removes Classical Singularities And Shortens The Life Of Black Holes, IC-79-69 [INSPIRE].
E.T. Tomboulis, Superrenormalizable gauge and gravitational theories, hep-th/9702146 [INSPIRE].
E.T. Tomboulis, Renormalization and unitarity in higher derivative and nonlocal gravity theories, Mod. Phys. Lett. A 30 (2015) 1540005.
L. Modesto, Super-renormalizable Quantum Gravity, Phys. Rev. D 86 (2012) 044005 [arXiv:1107.2403] [INSPIRE].
L. Modesto, Super-renormalizable Multidimensional Quantum Gravity, arXiv:1202.3151 [INSPIRE].
L. Modesto, Multidimensional finite quantum gravity, arXiv:1402.6795 [INSPIRE].
L. Modesto, Super-renormalizable Higher-Derivative Quantum Gravity, arXiv:1202.0008 [INSPIRE].
L. Modesto, Towards a finite quantum supergravity, arXiv:1206.2648 [INSPIRE].
L. Modesto and L. Rachwał, Super-renormalizable and finite gravitational theories, Nucl. Phys. B 889 (2014) 228 [arXiv:1407.8036] [INSPIRE].
F. Briscese, L. Modesto and S. Tsujikawa, Super-renormalizable or finite completion of the Starobinsky theory, Phys. Rev. D 89 (2014) 024029 [arXiv:1308.1413] [INSPIRE].
N.V. Krasnikov, Nonlocal Gauge Theories, Theor. Math. Phys. 73 (1987) 1184 [INSPIRE].
S. Alexander, A. Marcianò and L. Modesto, The Hidden Quantum Groups Symmetry of Super-renormalizable Gravity, Phys. Rev. D 85 (2012) 124030 [arXiv:1202.1824] [INSPIRE].
F. Briscese, A. Marciano, L. Modesto and E.N. Saridakis, Inflation in (Super-)renormalizable Gravity, Phys. Rev. D 87 (2013) 083507 [arXiv:1212.3611] [INSPIRE].
J. Khoury, Fading gravity and self-inflation, Phys. Rev. D 76 (2007) 123513 [hep-th/0612052] [INSPIRE].
G. Calcagni and L. Modesto, Nonlocal quantum gravity and M-theory, Phys. Rev. D 91 (2015) 124059 [arXiv:1404.2137] [INSPIRE].
L. Modesto and S. Tsujikawa, Non-local massive gravity, Phys. Lett. B 727 (2013) 48 [arXiv:1307.6968] [INSPIRE].
M. Piva, Nonlocal theories of quantum gravity and gauge fields, MSc Thesis, Supervisor: Prof. D. Anselmi, Physics Department, Pisa University, etd-11192014-163737, (2014).
I.A. Batalin and G.A. Vilkovisky, Gauge Algebra and Quantization, Phys. Lett. B 102 (1981) 27 [INSPIRE].
I.A. Batalin and G.A. Vilkovisky, Quantization of Gauge Theories with Linearly Dependent Generators, Phys. Rev. D 28 (1983) 2567 [Erratum ibid. D 30 (1984) 508] [INSPIRE].
S. Weinberg, The quantum theory of fields, vol. II, Cambridge University Press, Cambridge U.K. (1995).
D. Anselmi, Weighted power counting and chiral dimensional regularization, Phys. Rev. D 89 (2014) 125024 [arXiv:1405.3110] [INSPIRE].
D. Anselmi, Functional integration measure in quantum gravity, Phys. Rev. D 45 (1992) 4473 [INSPIRE].
D. Anselmi, On δ(0) divergences and the functional integration measure, Phys. Rev. D 48 (1993) 680 [INSPIRE].
D. Anselmi, Covariant Pauli-Villars regularization of quantum gravity at the one loop order, Phys. Rev. D 48 (1993) 5751 [hep-th/9307014] [INSPIRE].
S. Talaganis, T. Biswas and A. Mazumdar, Towards understanding the ultraviolet behavior of quantum loops in infinite-derivative theories of gravity, Class. Quant. Grav. 32 (2015) 215017 [arXiv:1412.3467] [INSPIRE].
K.S. Stelle, Renormalization of Higher Derivative Quantum Gravity, Phys. Rev. D 16 (1977) 953 [INSPIRE].
I.L. Buchbinder, S.D. Odintsov and I.L. Shapiro, Effective action in quantum gravity, IOP Publishing Ltd (1992).
M. Asorey, J.L. Lopez and I.L. Shapiro, Some remarks on high derivative quantum gravity, Int. J. Mod. Phys. A 12 (1997) 5711 [hep-th/9610006] [INSPIRE].
F.d.O. Salles and I.L. Shapiro, Do we have unitary and (super)renormalizable quantum gravity below the Planck scale?, Phys. Rev. D 89 (2014) 084054 [Phys. Rev. D 90 (2014) 129903] [arXiv:1401.4583] [INSPIRE].
A. Accioly, A. Azeredo and H. Mukai, Propagator, tree-level unitarity and effective nonrelativistic potential for higher-derivative gravity theories in D dimensions, J. Math. Phys. 43 (2002) 473 [INSPIRE].
F.d.O. Salles and I.L. Shapiro, Do we have unitary and (super)renormalizable quantum gravity below the Planck scale?, Phys. Rev. D 89 (2014) 084054 [arXiv:1401.4583] [INSPIRE].
K.S. Stelle, Classical Gravity with Higher Derivatives, Gen. Rel. Grav. 9 (1978) 353.
V.P. Frolov, Do Black Holes Exist?, arXiv:1411.6981 [INSPIRE].
V.P. Frolov, Information loss problem and a ‘black hole’ model with a closed apparent horizon, JHEP 05 (2014) 049 [arXiv:1402.5446] [INSPIRE].
V.P. Frolov and I.L. Shapiro, Black Holes in Higher Dimensional Gravity Theory with Quadratic in Curvature Corrections, Phys. Rev. D 80 (2009) 044034 [arXiv:0907.1411] [INSPIRE].
L. Modesto, T. de Paula Netto and I.L. Shapiro, On Newtonian singularities in higher derivative gravity models, JHEP 04 (2015) 098 [arXiv:1412.0740] [INSPIRE].
T. Biswas, E. Gerwick, T. Koivisto and A. Mazumdar, Towards singularity and ghost free theories of gravity, Phys. Rev. Lett. 108 (2012) 031101 [arXiv:1110.5249] [INSPIRE].
L. Modesto, J.W. Moffat and P. Nicolini, Black holes in an ultraviolet complete quantum gravity, Phys. Lett. B 695 (2011) 397 [arXiv:1010.0680] [INSPIRE].
C. Bambi, D. Malafarina and L. Modesto, Non-singular quantum-inspired gravitational collapse, Phys. Rev. D 88 (2013) 044009 [arXiv:1305.4790] [INSPIRE].
C. Bambi, D. Malafarina and L. Modesto, Terminating black holes in asymptotically free quantum gravity, Eur. Phys. J. C 74 (2014) 2767 [arXiv:1306.1668] [INSPIRE].
G. Calcagni, L. Modesto and P. Nicolini, Super-accelerating bouncing cosmology in asymptotically-free non-local gravity, Eur. Phys. J. C 74 (2014) 2999 [arXiv:1306.5332] [INSPIRE].
Y. Zhang, Y. Zhu, L. Modesto and C. Bambi, Can static regular black holes form from gravitational collapse?, Eur. Phys. J. C 75 (2015) 96 [arXiv:1404.4770] [INSPIRE].
B. Craps, T. De Jonckheere and A.S. Koshelev, Cosmological perturbations in non-local higher-derivative gravity, JCAP 11 (2014) 022 [arXiv:1407.4982] [INSPIRE].
A.S. Koshelev and S. Yu. Vernov, Cosmological Solutions in Nonlocal Models, Phys. Part. Nucl. Lett. 11 (2014) 960 [arXiv:1406.5887] [INSPIRE].
A.S. Koshelev, Stable analytic bounce in non-local Einstein-Gauss-Bonnet cosmology, Class. Quant. Grav. 30 (2013) 155001 [arXiv:1302.2140] [INSPIRE].
T. Biswas, A.S. Koshelev, A. Mazumdar and S. Yu. Vernov, Stable bounce and inflation in non-local higher derivative cosmology, JCAP 08 (2012) 024 [arXiv:1206.6374] [INSPIRE].
A.S. Koshelev and S. Yu. Vernov, On bouncing solutions in non-local gravity, Phys. Part. Nucl. 43 (2012) 666 [arXiv:1202.1289] [INSPIRE].
A.S. Koshelev, Modified non-local gravity, Rom. J. Phys. 57 (2012) 894 [arXiv:1112.6410] [INSPIRE].
S. Yu. Vernov, Nonlocal Gravitational Models and Exact Solutions, Phys. Part. Nucl. 43 (2012) 694 [arXiv:1202.1172] [INSPIRE].
A.S. Koshelev and S. Yu. Vernov, Cosmological perturbations in SFT inspired non-local scalar field models, Eur. Phys. J. C 72 (2012) 2198 [arXiv:0903.5176] [INSPIRE].
A.S. Koshelev, Non-local SFT Tachyon and Cosmology, JHEP 04 (2007) 029 [hep-th/0701103] [INSPIRE].
L. Modesto and L. Rachwał, Universally finite gravitational and gauge theories, Nucl. Phys. B 900 (2015) 147 [arXiv:1503.00261] [INSPIRE].
L. Modesto, M. Piva and L. Rachwał, Finite quantum gauge theories, arXiv:1506.06227 [INSPIRE].
A. Conroy, T. Koivisto, A. Mazumdar and A. Teimouri, Generalized quadratic curvature, non-local infrared modifications of gravity and Newtonian potentials, Class. Quant. Grav. 32 (2015) 015024 [arXiv:1406.4998] [INSPIRE].
G. ’t Hooft and M.J.G. Veltman, One loop divergencies in the theory of gravitation, Annales Poincare Phys. Theor. A 20 (1974) 69 [INSPIRE].
V. Iyer and R.M. Wald, Some properties of Noether charge and a proposal for dynamical black hole entropy, Phys. Rev. D 50 (1994) 846 [gr-qc/9403028] [INSPIRE].
P. Donà, S. Giaccari, L. Modesto, L. Rachwal and Y. Zhu, Scattering amplitudes in super-renormalizable gravity, JHEP 08 (2015) 038 [arXiv:1506.04589] [INSPIRE].
D. Anselmi, Renormalization and causality violations in classical gravity coupled with quantum matter, JHEP 01 (2007) 062 [hep-th/0605205] [INSPIRE].
D. Anselmi, Absence of higher derivatives in the renormalization of propagators in quantum field theories with infinitely many couplings, Class. Quant. Grav. 20 (2003) 2355 [hep-th/0212013] [INSPIRE].
M.H. Goroff and A. Sagnotti, The Ultraviolet Behavior of Einstein Gravity, Nucl. Phys. B 266 (1986) 709 [INSPIRE].
G. Calcagni, M. Montobbio and G. Nardelli, Localization of nonlocal theories, Phys. Lett. B 662 (2008) 285 [arXiv:0712.2237] [INSPIRE].
G. Calcagni and G. Nardelli, Non-local gravity and the diffusion equation, Phys. Rev. D 82 (2010) 123518 [arXiv:1004.5144] [INSPIRE].
T. Biswas, A. Mazumdar and W. Siegel, Bouncing universes in string-inspired gravity, JCAP 03 (2006) 009 [hep-th/0508194] [INSPIRE].
A.G. Mirzabekian and G.A. Vilkovisky, The one loop form-factors in the effective action and production of coherent gravitons from the vacuum, Class. Quant. Grav. 12 (1995) 2173 [hep-th/9504028] [INSPIRE].
P.D. Mannheim and D. Kazanas, Exact Vacuum Solution to Conformal Weyl Gravity and Galactic Rotation Curves, Astrophys. J. 342 (1989) 635 [INSPIRE].
H. Lü, A. Perkins, C.N. Pope and K.S. Stelle, Black Holes in Higher-Derivative Gravity, Phys. Rev. Lett. 114 (2015) 171601 [arXiv:1502.01028] [INSPIRE].
Planck collaboration, P.A.R. Ade et al., Planck 2015 results. XX. Constraints on inflation, arXiv:1502.02114 [INSPIRE].
WMAP collaboration, E. Komatsu et al., Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Interpretation, Astrophys. J. Suppl. 180 (2009) 330 [arXiv:0803.0547] [INSPIRE].
K. Godel, An example of a new type of cosmological solutions of Einstein’s field equations of graviation, Rev. Mod. Phys. 21 (1949) 447 [INSPIRE].
P. Vaidya, The Gravitational Field of a Radiating Star, Proc. Indian Acad. Sci. A 33 (1951) 264.
R. Penrose, The basic ideas of conformal cyclic cosmology, AIP Conf. Proc. 1446 (2012) 233 [INSPIRE].
J.v. narlikar and A.k. kembhavi, Space-Time Singularities and Conformal Gravity, Lett. Nuovo Cim. 19 (1977) 517 [INSPIRE].
H. Rahman and S. Banerji, Can the big-bang singularity be avoided in the scale-covariant theory?, Astrophys. Space Sci. 113 (1985) 405.
A. Beesham, Power law singularities in the scale covariant theory, J. Math. Phys. 27 (1986) 2995.
A. Beesham, Comment on the Big-Bang singularity in the scale-covariant theory, Astrophys. Space Sci. 123 (1986) 405.
I. Bars, P. Steinhardt and N. Turok, Local Conformal Symmetry in Physics and Cosmology, Phys. Rev. D 89 (2014) 043515 [arXiv:1307.1848] [INSPIRE].
I.J. Araya, I. Bars and A. James, Journey Beyond the Schwarzschild Black Hole Singularity, arXiv:1510.03396 [INSPIRE].
D. Anselmi, Absence of higher derivatives in the renormalization of propagators in quantum field theories with infinitely many couplings, Class. Quant. Grav. 20 (2003) 2355 [hep-th/0212013] [INSPIRE].
D. Anselmi, Properties Of The Classical Action Of Quantum Gravity, JHEP 05 (2013) 028 [arXiv:1302.7100] [INSPIRE].
A.A. Starobinsky, A New Type of Isotropic Cosmological Models Without Singularity, Phys. Lett. B 91 (1980) 99 [INSPIRE].
S. Hervik, V. Pravda and A. Pravdova, Type III and N universal spacetimes, Class. Quant. Grav. 31 (2014) 215005 [arXiv:1311.0234] [INSPIRE].
S. Hervik, T. Malek, V. Pravda and A. Pravdova, Type II universal spacetimes, Class. Quant. Grav. 32 (2015) 245012 [arXiv:1503.08448] [INSPIRE].
T. Malek and V. Pravda, Type III and N solutions to quadratic gravity, Phys. Rev. D 84 (2011) 024047 [arXiv:1106.0331] [INSPIRE].
M. Ortaggio, V. Pravda and A. Pravdova, On higher dimensional Einstein spacetimes with a warped extra dimension, Class. Quant. Grav. 28 (2011) 105006 [arXiv:1011.3153] [INSPIRE].
M.A. Luty, J. Polchinski and R. Rattazzi, The a-theorem and the Asymptotics of 4D Quantum Field Theory, JHEP 01 (2013) 152 [arXiv:1204.5221] [INSPIRE].
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Li, YD., Modesto, L. & Rachwał, L. Exact solutions and spacetime singularities in nonlocal gravity. J. High Energ. Phys. 2015, 1–50 (2015). https://doi.org/10.1007/JHEP12(2015)173
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DOI: https://doi.org/10.1007/JHEP12(2015)173