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Editorial note to: On the Newtonian limit of Einstein’s theory of gravitation (by Jürgen Ehlers). (English) Zbl 1437.83009
Summary: We give an overview of literature related to J. Ehlers’s pioneering 1981 paper [“Über den Newtonschen Grenzwert der Einsteinschen Gravitationstheorie”, in: J. Nitsch (ed.) et al., Grundlagenprobleme der modernen Physik. Mannheim: Bibliographisches Institut. 65–84 (1981); Gen. Relativ. Gravitation 51, No. 12, Paper No. 163, 20 p. (2019; Zbl 1437.83013)] on Frame theory – a theoretical framework for the unification of general relativity and the equations of classical Newtonian gravitation. This unification encompasses the convergence of one-parametric families of four-dimensional solutions of Einstein’s equations of General relativity to a solution of equations of a Newtonian theory if the inverse of a causality constant goes to zero. As such, the corresponding light cones open up and become space-like hypersurfaces of constant absolute time on which Newtonian solutions are found as a limit of the Einsteinian ones. It is explained what it means to not consider the ‘standard-textbook’ Newtonian theory of gravitation as a complete theory unlike Einstein’s theory of gravitation. In fact, Ehlers’ Frame theory brings to light a modern viewpoint in which the ‘standard’ equations of a self-gravitating Newtonian fluid are Maxwell-type equations. The consequences of Frame theory are presented for Newtonian cosmological dust matter expressed via the spatially projected electric part of the Weyl tensor, and for the formulation of characteristic quasi-Newtonian initial data on the light cone of a Bondi-Sachs metric.

MSC:
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
83C25 Approximation procedures, weak fields in general relativity and gravitational theory
53Z05 Applications of differential geometry to physics
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[1] Ehlers, J.: Über den Newtonschen Grenzwert der Einsteinschen Gravitationstheorie. In: Nitsch, J., Pfarr, J., Stachow, E.-W. (eds.) Grundlagenprobleme der modernen Physik: Festschrift für Peter Mittelstaedt zum 50. Geburtstag, pp. 65-84. Bibliographisches Institut, Mannheim, Wien, Zürich (1981). Translation: Gen. Relativ. Gravit. 51 (2019)
[2] Buchert, T.; Ehlers, J., Lagrangian theory of gravitational instability of Friedmann-Lemaître cosmologies—second-order approach: an improved model for non-linear clustering, Mon. Not. R. Astron. Soc., 264, 375 (1993)
[3] Buchert, T.; Ehlers, J., Averaging inhomogeneous Newtonian cosmologies, Astron. Astrophys., 320, 1 (1997)
[4] Ehlers, J.; Buchert, T., Newtonian cosmology in Lagrangian formulation: foundations and perturbation theory, Gen. Relativ. Gravit., 29, 733 (1997) · Zbl 0915.70005
[5] Ehlers, J.; Buchert, T., On the Newtonian limit of the Weyl tensor, Gen. Relativ. Gravit., 41, 2153 (2009) · Zbl 1177.83059
[6] Ehlers, J.; Ferrarese, G., The Newtonian limit of general relativity, Classical Mechanics and Relativity: Relationship and Consistency, 95-106 (1991), Napoli: Bibliopolis, Napoli
[7] Ehlers, J., Examples of Newtonian limits of relativistic space-times, Class. Quantum Gravity, 14, A119-A126 (1997) · Zbl 0878.70013
[8] Oliynyk, Ta; Schmidt, B., Existence of families of space-times with a Newtonian limit, Gen. Relativ. Gravit., 41, 2093 (2009) · Zbl 1177.83060
[9] Levin, J.; Scannapieco, E.; Silk, J., The topology of the universe: the biggest manifold of them all, Class. Quantum Gravity, 15, 2689 (1998) · Zbl 0946.83075
[10] Steiner, F.: Do Black Holes exist in a finite universe having the topology of a flat 3-torus? In: Arendt, W. et al. (eds.) Ulmer Seminare 2016/2017, vol. 20, pp. 331-351. Universität Ulm (2018). arXiv:1608.03133
[11] Geroch, R., Limits of spacetimes, Commun. Math. Phys., 13, 3, 180 (1969)
[12] Heilig, U., On the existence of rotating stars in General Relativity, Commun. Math. Phys., 166, 457 (1995) · Zbl 0813.53058
[13] Jantzen, Rt; Carini, P.; Bini, D., The many faces of gravitoelectromagnetism, Ann. Phys., 215, 1 (1992)
[14] Heaviside, O.: A gravitational and electromagnetic analogy. Electr. vol. 31, 281 (part I), 359 (part II) (1893)
[15] Al Roumi, F.; Buchert, T.; Wiegand, A., Lagrangian theory of structure formation in relativistic cosmology. IV. Lagrangian approach to gravitational waves, Phys. Rev. D, 96, 123538 (2017)
[16] Rendall, Ad, The Newtonian limit for asymptotically flat solutions of the Vlasov-Einstein system, Commun. Math. Phys., 163, 89 (1994) · Zbl 0816.53058
[17] Buchert, T., An exact Lagrangian integral for the Newtonian gravitational field strength, Phys. Lett. A, 354, 8 (2006) · Zbl 1255.70009
[18] Buchert, T.; Götz, G., A class of solutions for self-gravitating dust in Newtonian gravity, J. Math. Phys., 28, 2714 (1987) · Zbl 0645.76133
[19] Buchert, T.: Lagrangian perturbation approach to the formation of large-scale structure, In: Bonometto, S., Primack, J., Provenzale, A. (eds.) International School of Physics Enrico Fermi, Course CXXXII: Dark Matter in the Universe, Varenna 1995, pp. 543-564. IOP Press, Amsterdam (1996). arXiv:astro-ph/9509005
[20] Buchert, T., Toward physical cosmology: focus on inhomogeneous geometry and its non-perturbative effects, Class. Quantum Gravity, 28, 164007 (2011) · Zbl 1225.83087
[21] Buchert, T.; Ostermann, M., Lagrangian theory of structure formation in relativistic cosmology I: Lagrangian framework and definition of a nonperturbative approximation, Phys. Rev. D, 86, 023520 (2012)
[22] Alles, A.; Buchert, T.; Al Roumi, F.; Wiegand, A., Lagrangian theory of structure formation in relativistic cosmology. III. Gravitoelectric perturbation and solution schemes at any order, Phys. Rev. D, 92, 023512 (2015)
[23] Lottermoser, M., A convergent post-Newtonian approximation for the constraints in General Relativity, Ann. Inst. Henri Poincaré, 57, 279 (1992) · Zbl 0762.53053
[24] Ehlers, J.: Akad. Wiss. Lit. (Mainz); Abh. Math.-Nat. Kl. No. 11, 793 (1961); translation: Contributions to the relativistic mechanics of continuous media, Gen. Relativ. Gravit. 25, 1225 (1993)
[25] Ellis, George F. R.; Elst, Henk, Cosmological Models, Theoretical and Observational Cosmology, 1-116 (1999), Dordrecht: Springer Netherlands, Dordrecht
[26] Van Elst, H.; Uggla, C.; Lesame, Wm; Ellis, Gfr; Maartens, R., Integrability of irrotational silent cosmological models, Class. Quantum Gravity, 14, 1151 (1997) · Zbl 0874.53075
[27] Kofman, L.; Pogosyan, D., Dynamics of gravitational instability is nonlocal, Astrophys. J., 442, 30 (1995)
[28] Winicour, Jh, Newtonian gravity on the null cone, J. Math. Phys., 24, 1193 (1983) · Zbl 0522.76128
[29] Winicour, Jh, Null infinity from a quasi-Newtonian view, J. Math. Phys., 25, 2506 (1984)
[30] Winicour, Jh, The quadrupole radiation formula, Gen. Relativ. Gravit., 19, 281 (1987)
[31] Bondi, H.; Van Der Burg, Mgj; Metzner, Awk, Gravitational waves in general relativity. VII. Waves from axi-symmetric isolated systems, Proc. R. Soc. Lond. Ser. A, 269, 21 (1962) · Zbl 0106.41903
[32] Sachs, Rk, Gravitational waves in general relativity. VIII. Waves in asymptotically flat space-time, Proc. R. Soc. Lond. Ser. A, 270, 103 (1962) · Zbl 0101.43605
[33] Mädler, T.; Winicour, Jh, Bondi-Sachs Formalism, Scholarpedia, 11, 12, 33528 (2016)
[34] Winicour, Jh, Characteristic evolution and matching, Living Rev. Relativ., 15, 2 (2012) · Zbl 1316.83009
[35] Schutz, B.: Symposium 14. Radiative Spacetimes and Approximation Methods, In: MacCallum, M.A.H. (ed.) General Relativity and Gravitation: Proceedings of the 11th International Conference on General Relativity and Gravitation, pp. 369-376. Cambridge University Press, Cambridge (1987)
[36] Penrose, R., Asymptotic properties of fields and space-times, Phys. Rev. Lett., 10, 66 (1963)
[37] Tamburino, La; Winicour, Jh, Gravitational fields in finite and conformal Bondi frames, Phys. Rev., 150, 1039 (1966)
[38] Müller, B., The status of multi-dimensional core-collapse supernova models, Publ. Astron. Soc. Aust., 33, e048 (2016)
[39] Cerdá-Durán, P., Elias-Rosa, N.: Neutron stars formation and core collapse supernovae. In: Rezzolla L., Pizzochero P., Jones D., Rea N., Vidaña I. (eds.) The Physics and Astrophysics of Neutron Stars. Astrophysics and Space Science Library, vol 457. Springer, Cham. arXiv:1806.07267
[40] Siebel, F.; Font, Ja; Müller, E.; Papadopoulos, P., Axisymmetric core collapse simulations using characteristic numerical relativity, Phys. Rev. D, 67, 124018 (2003)
[41] Cerdá-Durán, P.; Debrye, N.; Aloy, Ma; Font, Ja; Obergaulinger, M., Gravitational wave signatures in black hole forming core collapse, Astrophys. J. Lett., 779, L18 (2013)
[42] Manasse, Fk; Misner, Cw, Fermi normal coordinates and some basic concepts in differential geometry, J. Math. Phys., 4, 735 (1963) · Zbl 0118.22903
[43] Mädler, T.; Müller, E., The Bondi-Sachs metric at the vertex of a null cone: axially symmetric vacuum solutions, Class. Quantum Gravity, 30, 055019 (2013) · Zbl 1263.83049
[44] Thorne, K.S.: The theory of gravitational radiation-an introductory review, in Gravitational radiation. In: Deruelle, N., Piran, T. (eds.) Proceedings of the Advanced Study Institute, Les Houches, Haute-Savoie, France, June 2-21, 1982 (A84-35026 16-90), pp. 1-57. North-Holland Publishing, Amsterdam (1985)
[45] Mädler, T.; Winicour, Jh, Boosted Schwarzschild metrics from a Kerr-Schild perspective, Class. Quantum Gravity, 35, 035009 (2018) · Zbl 1382.83064
[46] Mädler, Thomas; Winicour, Jeffrey, Kerr black holes and nonlinear radiation memory, Classical and Quantum Gravity, 36, 9, 095009 (2019)
[47] Mädler, T.; Winicour, Jh, The sky pattern of the linearized gravitational memory effect, Class. Quantum Gravity, 33, 175006 (2016) · Zbl 1349.83024
[48] Mädler, T.; Winicour, Jh, Radiation memory, boosted Schwarzschild spacetimes and supertranslations, Class. Quantum Gravity, 34, 115009 (2017) · Zbl 1370.83018
[49] Isaacson, Ra; Welling, Js; Winicour, Jh, Gravitational radiation from dust, J. Math. Phys., 26, 2859 (1985) · Zbl 0624.76134
[50] Newman, Et; Penrose, R., An approach to gravitational radiation by a method of spin coefficients, J. Math. Phys., 3, 566 (2004)
[51] Newman, Et; Penrose, R., Spin-coefficient formalism, Scholarpedia, 4, 6, 7445 (2009)
[52] Isaacson, Ra; Welling, Js; Winicour, Jh, Extension of the Einstein quadrupole formula, Phys. Rev. Lett., 53, 1870 (1984)
[53] Winicour, Jh, Logarithmic asymptotic flatness, Found. Phys., 15, 605 (1985)
[54] Chruściel, P.T., MacCallum, M.A.H., Singleton, D.B.: Gravitational waves in general relativity. XIV: Bondi expansions and the “Polyhomogeneity” of Scri. Philos. Trans. R. Soc. Lond. A 350, 19950004 (1995). 10.1098/rsta.1995.0004, arXiv:gr-qc/9305021 · Zbl 0829.53065
[55] Christodoulou, Demetrios, THE GLOBAL INITIAL VALUE PROBLEM IN GENERAL RELATIVITY, The Ninth Marcel Grossmann Meeting, 44-54 (2002) · Zbl 1032.83014
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