Global Behavior of Solutions to Einstein's Equations

doi:10.1017/CBO9781139583961.013

9 - Global Behavior of Solutions to Einstein's Equations

from Part Three - Gravity is Geometry, after all

Published online by Cambridge University Press: 05 June 2015

Igor Rodnianski and

Edited by

James Isenberg and

James Isenberg: Affiliation:
University of Oregon
Stefanos Aretakis: Affiliation:
Institute for Advanced Study and Princeton University
Igor Rodnianski: Affiliation:
Massachusetts Institute of Technology and Princeton University
Vincent Moncrief: Affiliation:
Yale University
Abhay Ashtekar: Affiliation:
Pennsylvania State University
Beverly K. Berger: Affiliation:
Formerly Program Director for Gravitational Physics, National Science Foundation
James Isenberg: Affiliation:
University of Oregon
Malcolm MacCallum: Affiliation:
University of Bristol

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Summary

Introduction

From a purely analytical perspective, Einstein's equations constitute a formidable PDE system. They mix constraint equations with evolution equations, their manifest character (hyperbolic or not) depends on the choice of coordinates, they are defined and studied on a spacetime manifold which is field-dependent and therefore not fixed, and the system is nonlinear in a serious way. These features make it challenging to study Einstein's equations using the analytical techniques and ideas which have been successfully applied to other nonlinear PDE systems. This is especially true of those analyses concerned with global, evolutionary aspects of solutions of Einstein's equations, which are the focus of interest in this chapter.

During the past thirty years, it has become apparent that the most successful way to meet these challenges and understand the behavior of solutions of Einstein's equations is to recognize the fundamental role played by spacetime geometry in general relativity and exploit some of its structures. Indeed, the Christodoulou–Klainerman proof of the stability of Minkowski spacetime [1] provides a good example of this: It relies strongly on the use of spacetime geometric structures such as null foliations, maximal hypersurfaces, “almost Killing fields”, and the Bel–Robinson tensor, combined with sophisticated use of standard analytical tools such as the control of energy functionals and hyperbolic radiation estimates. The more recent work of Christodoulou and others, which has discovered sufficient conditions for the formation of trapped surfaces and black holes, also relies on a strong alliance between geometric insight and the mastery of analytical technique.

As we saw in Chapters 3 and 4, gravitational effects play an important role both in astrophysics and cosmology. However, the two areas feature two fairly distinct branches of the mathematical analysis of solutions of Einstein's equations: One works with asymptotically flat solutions to Einstein's equations and often focuses on issues related to black holes, while the other works predominantly with spatially closed solutions and is more focused on the nature of cosmological singularities and possible mechanisms for isotropization and long-distance correlation.

Type: Chapter
Information: General Relativity and Gravitation
A Centennial Perspective
, pp. 449 - 498

DOI: https://doi.org/10.1017/CBO9781139583961.013 [Opens in a new window]

Publisher: Cambridge University Press

Print publication year: 2015

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References

[1] H., Nussbaumer and L., Bieri, Discovering the expanding universe (Cambridge University Press, Cambridge, 2009).Google Scholar

[2] G. F. R., Ellis, R., Maartens and M. A. H., MacCallum, Relativistic cosmology (Cambridge University Press, Cambridge, 2012).Google Scholar

[3] J., Wolf, Spaces of constant curvature (American Mathematical Society, Providence, 2010). See especially the discussion of homogeneous spaces in Chapter 8.Google Scholar

[4] Among numerous recent references on the Ricci flow and its implications for 3-manifold topology see, for example: J., Morgan and F., Fong, Ricci flow and geometrization of 3-manifolds, University Lecture Series (American Mathematical Society, Providence, 2010), H. D., Cao, B., Chow, S.C., Chu and S.T., Yau (editors), Collected papers on Ricci flow, Series in Geometry and Topology, vol. 37 (International Press, Boston, 2003), B., Chow and D., Knopf, The Ricci flow: an introduction, Mathematical Surveys and Monographs (American Mathematical Society, Providence, 2004).Google Scholar

[5] H., Kneser, Geschlossene Flächen in dreidimensionalen Mannigfaltigkeiten, Jahr. Deutschen Math. Vereinigung 38(1929) 248–260.Google Scholar

[6] J., Milnor, A unique decomposition theorem for 3-manifolds, Amer. J. Math. 84(1962) 1–7.Google Scholar

[7] P., Scott, The geometries of 3-manifolds, Bull. Lond. Math. Soc. 15 (1983) 401–487.Google Scholar

[8] From recent results in Ricci flow it is now known that no Whitehead manifold (i.e., open 3-manifold that is contractible but not homeomorphic to R3)can be the universal cover of a compact 3-manifold; cf., Theorem 10 appearing in J., Porti, Geometrization of three manifolds and Perelman's proof, Rev. R. Acad. Cien. Serie A. Mat. 102 (2008) 101–125.Google Scholar

[9] A., Fischer and V., Moncrief, Hamiltonian reduction of Einstein's equations, Encyclopedia of mathematical physics, Vol. 2, 607–623, J.-P., Françoise, G., Naber, Tsou, S. T. (editors), (Elsevier, Amsterdam, 2006).Google Scholar

[10] J., Lee and T., Parker, The Yamabe problem, Bull. Amer. Math. Soc. 17 (1987) 37–91. See also R., Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Diff. Geom. 20(1984) 479–495. For the case of metrics of negative Yamabe type of most interest herein, see the original proof by N. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact Manifolds, Ann. Scuola Norm. Sup. Pisa (3)22: (1968) 265–274.Google Scholar

[11] A., Fischer and J., Marsden, Deformations of the scalar curvature, Duke Math. J. 42(1975) 519–547.Google Scholar

[12] A., Tromba, Teichmüller theory in Riemannian geometry, Lectures in Mathematics, ETH Zürich (Birkhäuser, Zürich, 1992).

[13] A., Fischer and V., Moncrief, The structure of quantum conformal superspace, Global structure and evolution in general relativity, Lecture Notes in Physics vol. 460, S., Cotsakis and G., Gibbons (editors), (Springer-Verlag, New York, 1996) 111–173.Google Scholar

[14] V., Moncrief, Reduction of the Einstein equations in 2 + 1 dimensions to a Hamiltonian system over Teichmuller space, J. Math. Phys. 30(1989) 2907–2914.Google Scholar

[15] V., Moncrief, Relativistic Teichmüller theory–a Hamilton–Jacobi approach to 2 + 1 dimensional Einstein gravity, Surveys in Differential Geometry XII (International Press, Boston, 2008) 203–249.Google Scholar

[16] See especially Chapter 8 of the present volume.

[17] Y., Choquet-Bruhat, General relativity and the Einstein equations, Oxford Mathematical Monographs (Oxford University Press, Oxford, 2009). See especially Chapter VII.Google Scholar

[18] R., Bartnik and J., Isenberg, The constraint equations, in The Einstein equations and the large scale behavior of gravitational fields, edited by P.T., Chruściel and H., Friedrich (Birkhäuser-Verlag, Basel, 2004) 1–38.Google Scholar

[19] A., Fischer and V., Moncrief, Conformal volume collapse of 3-manifolds and the reduced Einstein flow, in the Springer volume Geometry dynamics and mechanics for the 60th birthday of Jerrold Marsden edited by D., HolmesP. K., Newton and A., Weinstein (Springer-Verlag, New York, 2002) 463–522.Google Scholar

[20] A., Fischer and V., Moncrief, Hamiltonian reduction and perturbations of continuously self-similar (n + 1)-dimensional Einstein vacuum spacetimes, Class. Quant. Grav. 19 (2002) 1–33.Google Scholar

[21] L., Andersson and V., Moncrief, Einstein spaces as attractors for the Einstein flow, J. Diff. Geom. 89(2011) 1–48.Google Scholar

[22] R., Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Topics in the calculus of variations, Lecture Notes in Mathematics vol. 1365 (Springer-Verlag, Berlin, 1989) 120–154.Google Scholar

[23] M., Anderson, Scalar curvature and geometrization conjectures for 3-manifolds, in Comparison geometry (MSRI Publications, Berkeley, 1997) 49–82.Google Scholar

[24] H., Bray and A., Neves, Classification of prime 3-manifolds with Yamabe invariant greater than RP3, Ann. Math. 159(2004) 407–424.Google Scholar

[25] M., Anderson, Geometrization of 3-manifolds via the Ricci flow, Notices Amer. Math. Soc. 51 (2004) 184–2193.Google Scholar

[26] M., Anderson, Canonical metrics on 3-manifolds and 4-manifolds, Asian J. Math. 10(2006) 127–164.Google Scholar

[27] A., Fischer and V., Moncrief, The reduced Hamiltonian of general relativity and the σ-constant of conformal geometry, Proceedings of the 2nd Samos Meeting on Cosmology, Geometry and Relativity, Mathematical and Quantum Aspects of Relativity and Cosmology, Lecture Notes in Physics, vol. 537 S., Cotsakis and G., Gibbons (editors) (Springer-Verlag, Berlin, 2000) 70–101.Google Scholar

[28] M., Anderson, On long-time evolution in general relativity and geometrization of 3-manifolds, Commun. Math. Phys. 222(2001) 533–567.Google Scholar

[29] M., Anderson, Asymptotic behavior of future complete cosmological spacetimes, appearing in the Spacetime Safari issue of Classical and Quantum Gravity in honor of the 60th birthday of Vincent Moncrief, Class. Quant. Grav. 21(2004) S11–S27.Google Scholar

[30] M., Reiris, The ground state and the long-time evolution in the CMC Einstein flow, Ann. Inst. Henri Poincaré 10 (2010) 1559–1604.Google Scholar

[31] M., Reiris, On the asymptotic spectrum of the reduced volume in cosmological solutions of the Einstein equations, Gen. Rel. Grav. 41 (2009) 1083–1106.Google Scholar

[32] L., Andersson, V., Moncrief and A., Tromba, On the global evolution problem in 2 + 1 gravity, J. Geom. Phys. 23 (1997) 191–205.Google Scholar

[33] R., Benedetti and E., Guadagnini, Cosmological time in (2 + 1)-gravity, Nucl. Phys. B613 (2001) 330–352.Google Scholar

[34] L., Andersson, Constant mean curvature foliations of simplicial flat spacetimes, Commun. Anal. Geom. 13(2005) 963–979. See also: T., Barbot, F., Béguin, A., Zeghib, Feuilletages des espaces temps globalement hyperboliques par des hypersurfaces à courbure moyenne constante, C. R. Math. Acad. Sci. Paris 336(2003) 245–250.Google Scholar

[35] Y., Fourès-Bruhat, Théorème d'existence pour certains systèmes d'équations aux derivèes partielles non linéeaires, Acta Math. 88(1) (1952) 141–225. See also: Y., Bruhat, The Cauchy problem, in Gravitation: an introduction to current research, L., Witten (editor) (John Wiley & Sons, San Francisco, 1962) 130–168.

[36] L., Andersson and V., Moncrief, Elliptic-hyperbolic systems and the Einstein equations, Ann. Inst. Henri Poincaré 4(2003) 1–34.Google Scholar

[37] L., Andersson and V., Moncrief, Future complete vacuum spacetimes, appearing in the Proceedings of the Cargèse Summer SchoolFifty years of the Cauchy problem in general relativity, edited by P., Chruściel and H., Friedrich (Birkhauser, Basel, 2004) 299–330.Google Scholar

[38] Y., Choquet-Bruhat and V., Moncrief, Future global-in-time Einsteinian spacetimes with U(1) isometry group, Ann. Inst. Henri Poincaré 2(2001) 1007–1064.Google Scholar

[39] Y., Choquet-Bruhat and V., Moncrief, Future complete Einsteinian spacetimes with U(1) isometry group, C. R. Acad. Sci. Paris, 332, Série I, (2001) 137–144.Google Scholar

[40] Y., Choquet-Bruhat, Future complete U(1) symmetric Einsteinian spacetimes, the unpolarized case, in The Einstein equations and the large scale behavior of gravitational fields, edited by P. T., Chruściel and H., Friedrich (Birkhäuser-Verlag, Basel, 2004). See also Chapter XVI of Ref. [17].Google Scholar

[41] H., Ringström, Cosmic censorship in Gowdy spacetimes, Living Rev. Rel. 13 (2010).Google Scholar

[42] For a discussion of the asymptotic behavior of the reduced Hamiltonian for spatially compact-ifiable, vacuum Bianchi models of negative Yamabe type see Refs. [9] and [19].

[43] W., Schlag and J., Krieger, Concentration compactness for critical wave maps, EMS Monographs in Mathematics (European Mathematical Society, Zürich, 2012).Google Scholar

[44] D., Tataru and J., Sterbenz, Regularity of wave-maps in dimension 2 + 1, Commun. Math. Phys. 298(2010) 231–264.Google Scholar

[45] F. G., Friedlander, The wave equation on a curved space-time (Cambridge University Press, Cambridge, 1975). See also C., Bär, N., Ginoux, F., Pfäffle, Wave equations on Lorentzian manifolds and quantization (European Mathematical Society, Zürich, 2007).Google Scholar

[46] S., Klainerman, I., Rodnianski and J., Szeftel, The bounded L2 curvature conjecture, arXiv:1204.1767v2 [math.AP]. See also by S., Klainerman and I., Rodnianski, Rough solutions of the Einstein-vacuum equations, Ann. Math. 161(2005) 1143–1193.Google Scholar

[47] Q., Wang, Rough solutions of Einstein vacuum equations in CMCSH gauge, arXiv:1201.0049v1 [math.AP]. This article improves the result of Ref. [36] by proving local existence for the Einstein equations in CMCSH (constant-mean-curvature-spatial-harmonic) gauge for Cauchy data g, k in the space Hs x Hs−1 (M) for s > 2.

[48] D., Eardley and V., Moncrief, The global existence of Yang–Mills–Higgs fields in 4-dimensional Minkowski space I. Local existence and smoothness properties, Commun. Math. Phys. 83(1982) 171–191.Google Scholar

[49] D., Eardley and V., Moncrief, The global existence of Yang–Mills–Higgs fields in 4-dimensional Minkowski space II. Completion of proof, Commun. Math. Phys. 83(1982) 193–212.Google Scholar

[50] P., Chruściel and J., Shatah, Solutions of the Yang–Mills equations on globally hyperbolic four dimensional Lorentzian manifolds, Asian J. Math. 1(1997) 530–548.Google Scholar

[51] S., Klainerman and I., Rodnianski, A Kirchhoff–Sobolev parametrix for wave equations in a Curved Spacetime, J. Hyperbolic Diff. Eqs. 4 (2007) 401–433.Google Scholar

[52] J., Hadamard, Lectures on Cauchy's problem in linear partial differential equations (Yale University Press, New Haven, 1923).Google Scholar

[53] V., Moncrief, An integral equation for spacetime curvature in general relativity, Isaac Newton Institute preprint NI05086 (2005). Published in the Journal of Differential Geometry volume Surveys in differential geometry X honoring the memory of S. S. Chern (International Press, Boston, 2006).Google Scholar

[54] V., Moncrief, Curvature propagation in general relativity, in preparation for Living Reviews in Relativity (Albert Einstein Institute, Potsdam).

[55] P., LeFloch, Injectivity radius and optimal regularity for Lorentzian manifolds with bounded curvature, Actes Semin. Theor. Spectr. Geom. 26(2007) 77–90.Google Scholar

[56] For the Yang–Mills equations in Minkowski spacetime the argument to bound the divergence of the connection given in [48, 49] depends upon the translational invariance of the metric and thus fails to apply in curved backgrounds.

[1] D., Christodoulou and S., Klainerman, The global nonlinear stability of Minkowski space (Princeton University Press, Princeton, 1994)Google Scholar

[2] P., Chruściel, J., Isenberg and V., Moncrief, Class. Quant. Grav. 7 1671–1679 (1990).

[3] V., Belinskii, I., Khalatnikov and E., Lifshitz, Adv. Phys. 19 525–573 (1970).

[4] V., Belinskii, I., Khalatnikov and E., Lifshitz, Adv. Phys. 31 639–667 (1982).

[5] B., Berger and V., Moncrief, Phys. Rev. D 48 4676 (1993).

[6] B., Berger and V., Moncrief, Phys. Rev. D 57 7235 (1998).

[7] P., Chruściel, Ann. Phys. 202 100–150 (1990).

[8] S., Kichenassamy and A., Rendall, Class. Quant. Grav. 15 1339–1335 (1999).

[9] A., Rendall, Class. Quant. Grav. 17 3305–3316 (2000).

[10] J., Isenberg and V., Moncrief, Class. Quant. Grav. 19, 5361 (2002).

[11] J., Isenberg and V., Moncrief, Ann. Phys. 199 84–122 (1990).

[12] H., Ringström, Ann. Math. 170 1181–1240 (2009).

[13] B., Berger, J., Isenberg and M., Weaver, Phys. Rev. D 64 084006 (2001).

[14] W., Lim, L., Andersso., D., Garfinkle and F., Pretorius, Phys.Rev. D 79 123526 (2009).

[1] A., Einstein, Sitzungsber. Preuss. Akad. Wiss. 50 118–186 (1915).

[2] A., Einstein, Sitzungsber. Preuss. Akad. Wiss. 50844–841 (1915).

[3] R., Schoen and S.T., Yau, Commun. Math. Phys. 65 45–16 (1919).

[4] R., Schoen and S.T., Yau, Commun. Math. Phys. 79 231–260 (1981).

[5] E., Witten, Commun. Math. Phys. 80, 381–402 (1981).

[6] D., Christodoulou and S., Klainerman, The global nonlinear stability of Minkowski space, Princeton University Press (1994).Google Scholar

[7] D., Christodoulou, The formation of black holes in general relativity, EMS (2009).

[8] Y., Choquet-Bruhat, Acta Math. 88141–225 (1952).

[9] Y., Choquet-Bruhat and R., Geroch, Commun. Math. Phys. 14 329–335 (1969).

[10] J., Sbierski, unpublished, arXiv:1309.1591.

[11] A., Rendall, Proc. R. Soc. Lond. A 427 221–239 (1990).

[12] J., Luk, Int. Math. Res. Not. 4625–4618 (2012).

[13] P., Dionne, J. Analyze Math. 10 1–90 (1962).

[14] A., Fischer and J., Marsden, Commun. Math. Phys. 28 1–38 (1912).

[15] T., Hughes, R., Kato and J., Marsden, Arch. Ration. Mech. Anal. 63 213–394 (1911).

[16] F., Planchon and I., Rodnianski, unpublished.

[17] G. Ponce, G., and T., Sideris, Commun. Partial Diff. Eq. 18 2419–2438 (1993).CrossRef

[18] H., Smith, Ann. Inst. Fourier 48 191–835 (1998).

[19] H., Bahouri and J., Chemin, Amer. J. Math. 21 1331–1311 (1999).

[20] H., Bahouri and J., Chemin, Int. Math. Res. Notices 21 1141–1118 (1999).

[21] D., Tataru, Amer. J. Math. 1122 349–316 (2000).

[22] D., Tataru, Amer. J. Math. 123 385–423 (2001).

[23] H., Smith and D., Tataru, Math.Res.Lett. 9 199–204 (2002).

[24] S., Klainerman and I., Rodnianski, Duke Math. J. 117 1–124 (2003).

[25] S., Klainerman and I., Rodnianski, Ann. Math. 161 1143–1193 (2005).

[26] H., Smith and D., Tataru, Ann. Math. 162 291–366 (2005).

[27] M., Anderson, Commun. Math. Phys. 222 533–561 (2001).

[28] S., Klainerman and I., Rodnianski, J. AMS23345–382 (2010).

[29] T., Beale, T., Kato and A., Majda, Commun. Math. Phys. 94 61–66 (1984).

[30] D., Eardley and V., Moncrief, Commun. Math. Phys. 83 111–191 (1982).

[31] D., Eardley and V., Moncrief, Commun. Math. Phys. 83 193–212 (1982).

[32] S., Klainerman and I., Rodnianski, J. Hyper. Diff. Eq. 4 401–403 (2001).

[33] V., Moncrief, J. DiffGeom: Surveys in Diff. Geom. X (2006).

[34] S., Klainerman and I., Rodnianski, Invent. Math. 159 431–529 (2005).

[35] S., Klainerman and I., Rodnianski, Geom. Funct. Anal. 16 126–163 (2006).

[36] S., Klainerman and I., Rodnianski, Geom. Funct. Anal. 16 164–229 (2006).

[37] S., Klainerman and I., Rodnianski, J. AMS 21 115–195 (2008).

[38] Q., Wang, Commun. Pure Appl. Math. 65 21–16 (2012).

[39] D., Christodoulou, Commun. Pure Appl. Math. 46, 1093–1220 (1993).

[40] S., Klainerman, Proceedings of Visions in Mathematics, Geom Funct. Anal. Special Vol. 219–315 (2000).

[41] S., Klainerman, I., Rodnianski and J., Szeftel, unpublished, arXiv:1204.1161.

[42] J., Szeftel, unpublished, arXiv:1204.1168.

[43] J., Szeftel, unpublished, arXiv:1204.1169.

[44] J., Szeftel, unpublished, arXiv:1204.1110.

[45] J., Szeftel, unpublished, arXiv:1204.1111.

[46] J., Szeftel, unpublished, arXiv:1301.0112.

[47] S., Klainerman and M., Machedon, Commun. Pure Appl. Math. 46 1221–1268 (1993).

[48] S., Klainerman and M., Machedon, Ann. Math. 142 39–119 (1995).

[49] S., Klainerman, Proceedings of the International Congress on Mathematics (Warsaw) 1209–1215 (1983).

[50] S., Klainerman and I., Rodnianski, J. Hyper. Diff. Eq. 161 219–291 (2005).

[51] H., Friedrich, Commun. Math. Phys. 107581–609 (1986).

[52] S., Klainerman and F., Nicolo, Evolution problem in general relativityBirkhäuser (2002).Google Scholar

[53] L., Bieri and N., Zipser, Extensions of the stability theorem of the Minkowski space in general relativityAMS/IP (2009).Google Scholar

[54] H., Friedrich, J. Geom. Phys. 3 101–111 (1986).

[55] M., Anderson, Ann. Inst. H. Poincaré 6 801–820 (2005).

[56] H., Ringström, Invent. Math. 173 123–208 (2008).

[51] L., Andersson and V., Moncrief, The Einstein equations and the large scale behavior of gravitational fields, in 50 years of the Cauchy problem in general relativity (ed. by P. T., hrusciel and H., Friedrich) Birkhäuser, 299–330 (2004).Google Scholar

[58] M., Reiris, Ph.D. Thesis, Stony Brook, (2005).

[59] D., Christodoulou, Ninth Marcel Grossmann Meeting (Rome 2000), ed. by V. G., Gurzadyan, World Scientific, 44–54 (2002).

[60] S., Klainerman and F., Nicolo, Class. Quant. Grav. 20 3215–3251 (2003).

[61] D., Christodoulou, Phys. Rev. Lett. 67 1486–1489 (1991).

[62] Y. Zel'dovich and A., Polnarev, Astron. Zh. 51 30 (1914).

[63] L., Bieri and D., Garfinkle, Phys. Rev. D 89 084039 (2014).

[64] H., Lindblad and I., Rodnianski, Ann. Math. 171 401–1411 (2010).

[65] D., Christodoulou, Commun. Pure Appl. Math. 39 261–282 (1986).

[66] S., Klainerman, Lect. Appl. Math. 23 293–326 (1986).

[67] Y., Choquet-Bruhat, C. R. Acad. Sci. Paris, Ser. A 276 281–284 (1913).

[68] H., Lindblad and I., Rodnianski, C. R. Acad. Sci. Paris, Ser. A 335 901–906 (2003).

[69] L., Hörmander, Pseudo-differential operators, Lecture Notes in Mathematics, vol. 1256 Springer-Verlag, Berlin214–280 (1981).Google Scholar

[70] H., Lindblad, Commun. PureAppl. Math. 45 1063–1096 (1992).

[71] S., Alinhac, Astérisque 284 (2003) 1–91 (2003).

[72] I., Rodnianski and J., Speck, unpublished, arXiv:0911.5501.

[73] J., Speck, to be published in Analysis and PDE.

[74] M., Dafermos, G., Holzegel and I., Rodnianski, unpublished, arXiv:1306.5364.

[75] S., Aretakis, unpublished, arXiv:1206.6598.

[76] L., Andersson and P., Blue, unpublished, arXiv:0908.2265.

[77] P., Blue and A., Soffer, Adv. Diff. Eqs. 8 595–614 (2003).

[78] P., Blue and J., Sterbenz, Commun. Math. Phys. 268 481–504 (2006).

[79] F., Finster, N., Kamra., J., Smoller and S., Yau, Commun Math. Phys. 264 465–503 (2006).

[80] M., Dafermos and I., Rodnianski, Commun Pure Appl. Math. 62 859–919 (2009).

[81] M., Dafermos and I., Rodnianski, Invent. Math. 185 467–559 (2011).

[82] M., Dafermos and I., Rodnianski, unpublished, arXiv:1010.5132.

[83] M., Dafermos and I., Rodnianski, Proc. 12th Marcel Grossmann Meeting, ed. by T., Damouret al., World Scientific132–189 (2011).

[84] B., Kay and R., Wald, Class. Quant. Grav. 4 893–898 (1987).

[85] J., Marzuola, J., Metcalf., D., Tataru and M., Tohaneanu, Commun. Math. Phys. 293 37–83 (2010).

[86] J., Metcalfe, D., Tataru and M., Tohaneanu, Adv. Math. 230 995–1028 (2012).

[87] Y., Shlapentokh-Rothman, unpublished, arXiv:1302.6902.

[88] D., Tataru and M., Tohaneanu, Int. Math. Res. Not. 2011 248–292 (2008).

[89] R., Wald, J. Math. Phys. 20 1056–1058 (1979).

[90] B., Whiting, J. Math. Phys. 30 (1989), 1301 (1989).

[91] G., Holzegel, Commun. Math. Phys. 294 169–197 (2010).

[92] G., Holzegel and J., Smulevici, Commun. PureAppl. Math. 66 1751–1802 (2013).

[93] J., Bony, and D., Hafner, Commun. Math. Phys. 282 697–719 (2008).

[94] M., Dafermos and I., Rodnianski, unpublished, arXiv:0709.2766.

[95] R., Melrose, A., Barreto and A., Vasy, unpublished, arXiv:0811.2229.

[96] A., Vasy and S., Dyatlov, unpublished, arXiv:1012.4391.

[97] S., Dyatlov, Commun. Math. Phys. 306 119–163 (2011).

[98] A., Barreto and M., Zworski, Math. Res. Lett. 4 103–121 (1997).

[99] D., Christodoulou, Commun. Math. Phys. 109 613–647 (1987).

[100] D., Christodoulou, Commun. PureAppl. Math. 44, 339–373 (1991).

[101] D., Christodoulou, Ann. Math. 149 183–217 (1999).

[102] D., Christodoulou, Ann. Math. 140 607–653 (1994).

[103] M., Dafermos, Ann. Math. 158 875–928 (2003).

[104] S., Klainerman and I., Rodnianski, Act a Math. 208 211–333 (2012).

[105] S., Klainerman and I., Rodnianski, unpublished, arXiv:1002.2656.

[106] M., Reiterer and E., Trubowitz, unpublished, arXiv:0906.3812.

[107] J., Wang and P., Yu, unpublished, arXiv:1207.3164.

[108] P., Yu, unpublished, arXiv:1105.5898.

[109] S., Klainerman, J., Luk and I., Rodnianski, I., unpublished, arXiv:1302.5951 (2013).

[110] R., Penrose, General relativity (papers in honour of J. L. Synge)101–115 (1972).

[111] M., Brinkmann, Proc. Natl. Acad. Sci. U.S.A. 9 1–3 (1923).

[112] V., Lanczos, Z. Phys. 21.273–110 (1924).

[113] J., Luk and I., Rodnianski, unpublished, arXiv:1209.1130.

[114] A., Majda, Mem. Amer. Math. Soc. 43 (1983).

[115] A., Majda, Mem. Amer. Math. Soc. 41 (1983).

[116] M., Dafermos and I., Rodnianski, XVIth International Congress on Mathematics and Physics,421–433 (2009).

[117] G., Holzegel, unpublished, arXiv:1010.3216.

[118] J., Luk and I., Rodnianski, unpublished, arXiv:1301.1072.

[119] K., Khan and R., Penrose, Nature 229185–186 (1971).

[120] P., Szekeres, Nature 2281183–1184 (1970).