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On the occurrence of gauge-dependent secularities in nonlinear gravitational waves. (English) Zbl 1373.83030

Summary: We study the plane (not necessarily monochromatic) gravitational waves at nonlinear quadratic order on a flat background in vacuum. We show that, in the harmonic gauge, the nonlinear waves are unstable. We argue that, at this order, this instability can not be eliminated by means of a multiscale approach, i.e. introducing suitable long variables, as is often the case when secularities appear in a perturbative scheme. However, this is a non-physical and gauge-dependent effect that disappears in a suitable system of coordinates. In facts, we show that in a specific gauge such instability does not occur, and that it is possible to solve exactly the second order nonlinear equations of gravitational waves. Incidentally, we note that this gauge coincides with the one used by Belinski and Zakharov to find exact solitonic solutions of Einstein’s equations, that is to an exactly integrable case, and this fact makes our second order nonlinear solutions less interesting. However, the important warning is that one must be aware of the existence of the instability reported in this paper, when studying nonlinear gravitational waves in the harmonic gauge.

MSC:

83C35 Gravitational waves
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
83C25 Approximation procedures, weak fields in general relativity and gravitational theory
83C75 Space-time singularities, cosmic censorship, etc.
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[1] Abbott B P et al and LIGO Scientific Collaboration and Virgo Collaboration 2016 Phys. Rev. Lett.116 061102 · doi:10.1103/PhysRevLett.116.061102
[2] Abbott B P et al and LIGO Scientific Collaboration and Virgo Collaboration 2016 Phys. Rev. Lett.116 241103 · doi:10.1103/PhysRevLett.116.241103
[3] Einstein A 1918 Naherungsweise Integration der Feldgleichungen der Gravitation(Sitzungsberichte der Koniglich Preussischen Akademie der Wissenschaften) p 688696
[4] Einstein A 1916 Uber Gravitationswellen(Sitzungsberichte der Koniglich Preussischen Akademie der Wissenschaften) p 154167
[5] Hulse R A and Taylor J H 1975 Discovery of a pulsar in a binary system Astrophys. J.195 L51 · doi:10.1086/181708
[6] Sathyaprakash B S and Schutz B F 2009 Physics, astrophysics and cosmology with gravitational waves Living Rev. Relativ.12 2 · Zbl 1166.85002 · doi:10.12942/lrr-2009-2
[7] Ade P A R et al and Planck Collaboration 2015 Planck 2015 results. XIII. Cosmological parameters Astron. Astrophys.594 A13 · doi:10.1051/0004-6361/201525830
[8] Ade P A R et al and BICEP2/Keck, Planck Collaborations 2015 A joint analysis of BICEP2/Keck array and planck data Phys. Rev. Lett.114 101301 · doi:10.1103/PhysRevLett.114.101301
[9] Pretorius F 2007 Binary black hole coalescence (arXiv:0710.1338)
[10] Cardoso V, Gualtieri L, Herdeiro C A R and Sperhake U 2015 Living Rev. Relativ.18 1 · doi:10.1007/lrr-2015-1
[11] Blanchet L 2014 Living Rev. Relativ.17 2 · doi:10.12942/lrr-2014-2
[12] Damour T 2008 Int. J. Mod. Phys. A 23 1130-48 · doi:10.1142/S0217751X08039992
[13] Belinskii V A and Zakharov V E 1978 Sov. Phys.—JETP48 985
[14] Belinski V and Verdaguer E 2001 Gravitational Solitons(Series: Cambridge Monographs on Mathematical Physics) (Cambridge: Cambridge University Press) · doi:10.1017/CBO9780511535253
[15] Amendola L et al 2015 Surfing gravitational waves: can bigravity survive growing tensor modes? J. Cosmol. Astropart. Phys.JCAP05(2015) 052 · doi:10.1088/1475-7516/2015/05/052
[16] Cusin G et al 2015 Gravitational waves in bigravity cosmology J. Cosmol. Astropart. Phys.JCAP05(2015) 030 · doi:10.1088/1475-7516/2015/05/030
[17] Fasiello M and Ribeiro R H 2015 Mild bounds on bigravity from primordial gravitational waves J. Cosmol. Astropart. Phys. JCAP07(2015)027 · doi:10.1088/1475-7516/2015/07/027
[18] Konnig F et al 2014 Stable and unstable cosmological models in bimetric massive gravity Phys. Rev. D 90 124014 · doi:10.1103/PhysRevD.90.124014
[19] Capozziello S and De Laurentis M 2011 Extended theories of gravity Phys. Rep.509 167-321 · doi:10.1016/j.physrep.2011.09.003
[20] Sotiriou T P and Faraoni V 2010 f(R) theories of gravity Rev. Mod. Phys.82 451-97 · Zbl 1205.83006 · doi:10.1103/RevModPhys.82.451
[21] Nojiri S and Odintsov S D 2011 Unified cosmic history in modified gravity: from F(R) theory to Lorentz non-invariant models Phys. Rep.505 59-144 · doi:10.1016/j.physrep.2011.04.001
[22] Nojiri S, Odintsov S D and Oikonomou V K 2017 Modified gravity theories on a nutshell:inflation, bounce and late-time evolution (arXiv:1705.11098) · Zbl 1370.83084
[23] Modesto L and Rachwal L 2015 Universally finite gravitational, gauge theories Nucl. Phys. B 900 147-69 · Zbl 1331.81201 · doi:10.1016/j.nuclphysb.2015.09.006
[24] Biswas T et al 2012 Towards singularity and ghost free theories of gravity Phys. Rev. Lett.108 031101 · doi:10.1103/PhysRevLett.108.031101
[25] Briscese F et al Super-renormalizable or finite completion of the Starobinsky theory Phys. Rev. D 89 024029 · doi:10.1103/PhysRevD.89.024029
[26] Briscese F et al 2013 Inflation in (super-) renormalizable gravity Phys. Rev. D 87 083507 · doi:10.1103/PhysRevD.87.083507
[27] Koshelev A S et al 2016 Occurrence of exact R2 inflation in non-local UV-complete gravity J. High Energy Phys.JHEP67(2016) 2016 · doi:10.1007/JHEP11(2016)067
[28] Nersisyan H et al 2017 Instabilities in tensorial nonlocal gravity Phys. Rev. D 95 043539 · doi:10.1103/PhysRevD.95.043539
[29] Capozziello S et al 2017 The gravitational energy-momentum pseudo-tensor of higher-order theories of gravity Ann. der Phys.529 1600376 · Zbl 1365.83026 · doi:10.1002/andp.201600376
[30] Briscese F and Pucheu M L 2016 Palatini formulation of non-local gravity Int. J. Geom. Methods Mod. Phys.14 1750019 · Zbl 1358.83070 · doi:10.1142/S0219887817500190
[31] Briscese F et al 2015 One-loop effective potential in nonlocal scalar field models Phys. Rev. D 92 104026 · doi:10.1103/PhysRevD.92.104026
[32] Aldrovandi R et al 2010 Nonlinear gravitational waves: their form and effects Int. J. Theor. Phys.49 549-63 · Zbl 1187.83027 · doi:10.1007/s10773-009-0236-2
[33] Weinberg S 1972 Gravitation and Cosmology (New York: Wiley)
[34] Landau L D and Lifshitz E M (ed) 1971 The Classical Theory of Fields(Course of Theoretical Physics Series vol 2) (Oxford: Butterworth-Heinemann)
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