×

The Newtonian limit on cosmological scales. (English) Zbl 1352.35120

Author’s abstract: We establish the existence of a wide class of inhomogeneous relativistic solutions to the Einstein-Euler equations that are well approximated on cosmological scales by solutions of Newtonian gravity. Error estimates measuring the difference between the Newtonian and relativistic solutions are provided.

MSC:

35Q35 PDEs in connection with fluid mechanics
35Q76 Einstein equations
76Y05 Quantum hydrodynamics and relativistic hydrodynamics
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Adams, R.A., Fournier, J.: Sobolev Spaces, 2nd edn. Academic Press, Amsterdam (2003) · Zbl 1098.46001
[2] Blanchet, L.; Faye, G.; Nissanke, S., On the structure of the post-Newtonian expansion in general relativity, Phys. Rev. D, 72, 44024, (2005)
[3] Blanchet, L.: Gravitational radiation from post-Newtonian sources and inspiralling compact binaries. Living Rev. Relativ. 17(2) (2014) · Zbl 1316.83003
[4] Browning, G.; Kreiss, H.O., Problems with different time scales for nonlinear partial differential equations, SIAM J. Appl. Math., 42, 704-718, (1982) · Zbl 0506.35006
[5] Buchert, T.; Räsänen, S., Backreaction in late-time cosmology, Ann. Rev. Nucl. Part. Sci., 62, 57-79, (2012)
[6] Chandrasekhar, S., The post-Newtonian equations of hydrodynamics in general relativity, Ap. J., 142, 1488-1512, (1965)
[7] Choquet-Bruhat Y.: General Relativity and the Einstein Equations. Oxford University Press, New York (2009) · Zbl 1157.83002
[8] Clarkson, C.; Ellis, G.; Larena, J.; Umeh, O., Does the growth of structure affect our dynamical models of the universe? the averaging, backreaction and Fitting problems in cosmology, Rept. Prog. Phys., 74, 112901, (2011)
[9] Dautcourt, G., Die newtonsche gravitationstheorie als strenger grenzfall der allgemeinen relativitätstheorie, Acta Phys. Polon., 25, 637-646, (1964)
[10] Ehlers, J.: On limit relations between, and approximative explanations of, physical theories. In: Marcus, B., Dorn, G.J.W., Weingartner, P. (eds.) Logic, methodology and philosophy of science VII, North-Holland, Amsterdam, pp. 387-403 (1986)
[11] Einstein, A.; Infeld, L.; Hoffmann, B., The gravitational equations and the problem of motion, Ann. Math., 39, 65-100, (1938) · Zbl 0018.28103
[12] Ellis, G.F.R., Inhomogeneity efffects in cosmology, Class. Qauntum Grav., 28, 164001, (2011) · Zbl 1225.83092
[13] Friedman A.: Partial Differential Equations. Krieger Publishing Company, Huntington (1976)
[14] Futamase, T., Itoh, Y.: The post-Newtonian approximation for relativistic compact binaries. Living Rev. Relativ. 10(2) (2007) · Zbl 1255.83005
[15] Grafakos L.: Classical Fourier Analysis, 2nd edn. Springer, Berlin (2008) · Zbl 1220.42001
[16] Grafakos L.: Modern Fourier Analysis, 2nd edn. Springer, Berlin (2009) · Zbl 1158.42001
[17] Green, S.R.; Wald, R.M., A new framework for analyzing the effects of small scale inhomogeneities in cosmology, Phys. Rev. D, 83, 084020, (2011)
[18] Green, S.R.; Wald, R.M., Newtonian and relativistic cosmologies, Phys. Rev. D, 85, 063512, (2012)
[19] Hwang, J.; Noh, H., Newtonian limit of fully nonlinear cosmological perturbations in einstein’s gravity, JCAP, 04, 035, (2013)
[20] Hwang, J.; Noh, H.; Puetzfeld, D., Cosmological non-linear hydrodynamics with post-Newtonian corrections, JCAP, 03, 010, (2008)
[21] Klainerman, S.; Majda, A., Singular perturbations of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Commun. Pure Appl. Math., 34, 481-524, (1981) · Zbl 0476.76068
[22] Klainerman, S.; Majda, A., Compressible and incompressible fluids, Commun. Pure Appl. Math., 35, 629-651, (1982) · Zbl 0478.76091
[23] Kopeikin, S.M.; Petrov, A.N., Post-Newtonian celestial dynamics in cosmology: field equations, Phys. Rev. D, 87, 044029, (2013)
[24] Kopeikin, S.M., Petrov, A.N.: Dynamic field theory and equations of motion in cosmology. Ann. Phys. 350, 379-440 (2015) · Zbl 1344.81122
[25] Kreiss, H.O., Problems with different time scales for partial differential equations, Commun. Pure Appl. Math., 33, 399-439, (1980) · Zbl 0439.35043
[26] Künzle, H.P., Galilei and Lorentz structures on space-time: comparison of the corresponding geometry and physics, Ann. Inst. Henri Poincaré, 17, 337-362, (1972)
[27] Künzle, H.P., Covariant Newtonian limit of Lorentz space-times, Gen. Relativ. Grav., 7, 445-457, (1976) · Zbl 0349.53025
[28] Künzle, H.P.; Duval, C., Relativistic and non-relativistic classical field theory on five-dimensional spacetime, Class. Quantum Grav., 3, 957-974, (1986) · Zbl 0604.70033
[29] Lottermoser, M., A convergent post-Newtonian approximation for the constraint equations in general relativity, Ann. l’inst. Henri Poincaré (A) Phys. Théor., 57, 279-317, (1992) · Zbl 0762.53053
[30] Majda A.: Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables. Springer, Berlin (1984) · Zbl 0537.76001
[31] Matarrese, S., Terranova, D.: Post-Newtonian cosmological dynamics in Lagrangian coordinates. Mon. Not. Roy. Astron. Soc. 283, 400-418 (1996)
[32] Oliynyk, T.A., The Newtonian limit for perfect fluids, Commun. Math. Phys., 276, 131-188, (2007) · Zbl 1194.83018
[33] Oliynyk, T.A., Post-Newtonian expansions for perfect fluids, Commun. Math. Phys., 288, 847-886, (2009) · Zbl 1175.83027
[34] Oliynyk, T.A., Cosmological post-Newtonian expansions to arbitrary order, Commun. Math. Phys., 295, 431-463, (2010) · Zbl 1195.35286
[35] Oliynyk, T.A., A rigorous formulation of the cosmological Newtonian limit without averaging, JHDE, 7, 405-431, (2010) · Zbl 1204.83030
[36] Oliynyk, T.A., Cosmological Newtonian limit, Phys. Rev. D, 89, 124002, (2014)
[37] Räsänen, S., Applicability of the linearly perturbed FRW metric and Newtonian cosmology, Phys. Rev. D, 81, 103512, (2010)
[38] Rendall, A.D., The initial value problem for a class of general relativistic fluid bodies, J. Math. Phys., 33, 1047-1053, (1992) · Zbl 0754.76098
[39] Rendall, A.D., The Newtonian limit for asymptotically flat solutions of the Vlasov-Einstein system, Commun. Math. Phys., 163, 89-112, (1994) · Zbl 0816.53058
[40] Taylor M.E.: Partial Differential Equations III: Nonlinear Equations. Springer, Berlin (1996)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.