Local and global existence theorems for the Einstein equations. (English) Zbl 1316.83007
Summary: This article is a guide to the literature on existence theorems for the Einstein equations which also draws attention to open problems in the field. The local in time Cauchy problem, which is relatively well understood, is treated first. Next global results for solutions with symmetry are discussed. This is followed by a presentation of global results in the case of small data, and some miscellaneous topics connected with the main theme.
MSC:
| 83-02 | Research exposition (monographs, survey articles) pertaining to relativity and gravitational theory |
| 83C15 | Exact solutions to problems in general relativity and gravitational theory |
| 83C05 | Einstein’s equations (general structure, canonical formalism, Cauchy problems) |
| 83A05 | Special relativity |
| 83C75 | Space-time singularities, cosmic censorship, etc. |
References:
| [1] | Andersson, L., Chruściel, P. T. and Friedrich, H., “On the regularity of solutions to the Yamabe equation and the existence of smooth hyperboloidal initial data for Einstein’s field equations”, Commun. Math. Phys., 149, 587-612 (1992). · Zbl 0764.53027 · doi:10.1007/BF02096944 |
| [2] | Arnold, VI; Ilyashenko, YS; Anosov, DV (ed.); Arnold, VI (ed.), Ordinary differential equations, No. 1, 1-148 (1988), Berlin; New York |
| [3] | Aubin, T., Nonlinear Analysis on Manifolds. Monge-Ampère equations, Grundlehren der mathematischen Wissenschaften, 252, (Springer, Berlin; New York, 1982). · Zbl 0512.53044 · doi:10.1007/978-1-4612-5734-9 |
| [4] | Bartnik, R., “Remarks on cosmological spacetimes and constant mean curvature hypersurfaces”, Commun. Math. Phys., 117, 615-624 (1988). · Zbl 0647.53044 · doi:10.1007/BF01218388 |
| [5] | Bartnik, R. and McKinnon, J., “Particlelike Solutions of the Einstein-Yang-Mills Equations”, Phys. Rev. Lett., 61, 141-143 (1988). · doi:10.1103/PhysRevLett.61.141 |
| [6] | Berger, B. K., Chruściel, P. T., Isenberg, J. A. and Moncrief, V., “Global Foliations of Vacuum Spacetimes with <Emphasis Type=”Italic“>T2 Isometry”, Ann. Phys. (N.Y.), 260, 117-148 (1997). · Zbl 0929.58013 · doi:10.1006/aphy.1997.5707 |
| [7] | Berger, B. K., Chruściel, P. T. and Moncrief, V., “On “Asymptotically Flat” Space-Times with G2-Invariant Cauchy Surfaces“, <Emphasis Type=”Italic“>Ann. Phys. (N.Y.), <Emphasis Type=”Bold”>237, 322-354 (1995). · Zbl 0967.83507 · doi:10.1006/aphy.1995.1012 |
| [8] | Bourguignon, J.-P., “Stabilité par déformation non-linéaire de la métrique de Minkowski (d’après D. Christodoulou et S. Klainerman)”, Asterisque, 201-203, 321-358 (1991). · Zbl 0754.53060 |
| [9] | Brauer, U., Rendall, A. D. and Reula, O. A., “The cosmic no-hair theorem and the nonlinear stability of homogeneous Newtonian cosmological models”, Class. Quantum Grav., 11, 2283-2296 (1994). · Zbl 0815.53092 · doi:10.1088/0264-9381/11/9/010 |
| [10] | Brodbeck, O., Heusler, M., Straumann, N. and Volkov, M., “Rotating solitons and non-rotating non-static black holes”, Phys. Rev. Lett., 79, 4310-4313 (1997). · doi:10.1103/PhysRevLett.79.4310 |
| [11] | Burnett, G. A. and Rendall, A. D., “Existence of maximal hypersurfaces in some spherically symmetric spacetimes”, Class. Quantum Grav., 13, 111-123 (1996). · Zbl 0843.53047 · doi:10.1088/0264-9381/13/1/010 |
| [12] | Cantor, M., “A necessary and sufficient condition for York data to specify an asymptotically flat spacetime”, J. Math. Phys., 20, 1741-1744 (1979). · Zbl 0427.35072 · doi:10.1063/1.524259 |
| [13] | Choquet-Bruhat, Y., “The Cauchy problem in classical supergravity”, Lett. Math. Phys., 7, 459-467 (1983). · Zbl 0529.58039 · doi:10.1007/BF00402245 |
| [14] | Choquet-Bruhat, Y. and Geroch, R., “Global aspects of the Cauchy problem in general relativity”, Commun. Math. Phys., 14, 329-335 (1969). · Zbl 0182.59901 · doi:10.1007/BF01645389 |
| [15] | Choquet-Bruhat, Y.; York, JW; Held, A. (ed.), The Cauchy problem, 99-172 (1980), New York |
| [16] | Christodoulou, D., “Global existence of generalized solutions of the spherically symmetric Einstein-scalar equations in the large”, Commun. Math. Phys., 106, 587-621 (1986). · Zbl 0613.53047 · doi:10.1007/BF01463398 |
| [17] | Christodoulou, D., “The problem of a self-gravitating scalar field”, Commun. Math. Phys., 105, 337-361 (1986). · Zbl 0608.35039 · doi:10.1007/BF01205930 |
| [18] | Christodoulou, D., “A mathematical theory of gravitational collapse”, Commun. Math. Phys., 109(4), 613-647 (1987). · Zbl 0613.53049 · doi:10.1007/BF01208960 |
| [19] | Christodoulou, D., “The structure and uniqueness of generalised solutions of the spherically symmetric Einstein-scalar equations”, Commun. Math. Phys., 109, 591-611 (1987). · Zbl 0613.53048 · doi:10.1007/BF01208959 |
| [20] | Christodoulou, D., “The formation of black holes and singularities in spherically symmetric gravitational collapse”, Commun. Pure Appl. Math., 44, 339-373 (1991). · Zbl 0728.53061 · doi:10.1002/cpa.3160440305 |
| [21] | Christodoulou, D., “Bounded variation solutions of the spherically symmetric Einstein-scalar field equations”, Commun. Pure Appl. Math., 46, 1131-1220 (1993). · Zbl 0853.35122 · doi:10.1002/cpa.3160460803 |
| [22] | Christodoulou, D., “Examples of naked singularity formation in the gravitational collapse of a scalar field”, Ann. Math., 140, 607-653 (1994). · Zbl 0822.53066 · doi:10.2307/2118619 |
| [23] | Christodoulou, D., “Self-Gravitating Fluids: A Two-Phase Model”, Arch. Ration. Mech. Anal., 130, 343-400 (1995). · Zbl 0841.76097 · doi:10.1007/BF00375144 |
| [24] | Christodoulou, D., “Self-Gravitating Fluids: The Continuation and Termination of a Free Phase Boundary”, Arch. Ration. Mech. Anal., 133, 333-398 (1996). · Zbl 1075.83517 · doi:10.1007/BF00375147 |
| [25] | Christodoulou, D., “Self-Gravitating Fluids: The Formation of a Free Phase Boundary in the Phase Transition from Soft to Hard”, Arch. Ration. Mech. Anal., 134, 97-154 (1996). · Zbl 0863.76098 · doi:10.1007/BF00379551 |
| [26] | Christodoulou, D., “The instability of naked singularities in the gravitational collapse of a scalar field”, Ann. Math. (2), 149, 183-217 (1999). · Zbl 1126.83305 · doi:10.2307/121023 |
| [27] | Christodoulou, D. and Klainerman, S., “Asymptotic properties of linear field equations in Minkowski space”, Commun. Pure Appl. Math., 43, 137-199 (1990). · Zbl 0715.35076 · doi:10.1002/cpa.3160430202 |
| [28] | Christodoulou, D. and Klainerman, S., The Global Nonlinear Stability of the Minkowski Space, Princeton Mathematical Series, 41, (Princeton University Press, Princeton, 1993). · Zbl 0827.53055 |
| [29] | Christodoulou, D. and Ó Murchadha, N., “The boost problem in general relativity”, Commun. Math. Phys., 80, 271-300 (1981). · Zbl 0477.35081 · doi:10.1007/BF01213014 |
| [30] | Christodoulou, D. and Tahvildar-Zadeh, A. S., “On the asymptotic behaviour of spherically symmetric wave maps”, Duke Math. J., 71, 31-69 (1993). · Zbl 0791.58105 · doi:10.1215/S0012-7094-93-07103-7 |
| [31] | Christodoulou, D. and Tahvildar-Zadeh, A. S., “On the regularity of spherically symmetric wave maps”, Commun. Pure Appl. Math., 46, 1041-1091 (1993). · Zbl 0744.58071 · doi:10.1002/cpa.3160460705 |
| [32] | Chruściel, P. T., “On Space-Time with <Emphasis Type=”Italic“>U(1) × <Emphasis Type=”Italic“>U(1) Symmetric Compact Cauchy Surfaces”, Ann. Phys. (N.Y.), 202, 100-150 (1990). · Zbl 0727.53078 · doi:10.1016/0003-4916(90)90341-K |
| [33] | Chruściel, P. T., On Uniqueness in the Large of Solutions of Einstein’s Equations (Strong Cosmic Censorship), Proceedings of the Centre for Mathematics and its Applications, 27, (Australian National University Press, Canberra, Australia, 1991). · Zbl 0747.53050 |
| [34] | Chruściel, P. T., “Semi-global existence and convergence of solutions of the Robinson-Trautman (2-dimensional Calabi) equation”, Commun. Math. Phys., 137, 289-313 (1991). · Zbl 0729.53071 · doi:10.1007/BF02431882 |
| [35] | Chruściel, P. T., Isenberg, J. A. and Moncrief, V., “Strong cosmic censorship in polarised Gowdy spacetimes”, Class. Quantum Grav., 7, 1671-1680 (1990). · Zbl 0703.53081 · doi:10.1088/0264-9381/7/10/003 |
| [36] | Claudel, C. M. and Newman, K. P., “The Cauchy problem for quasi-linear hyperbolic evolution problems with a singularity in the time”, Proc. R. Soc. London, Ser. A, 454, 1073-1107 (1998). · Zbl 0916.35064 · doi:10.1098/rspa.1998.0197 |
| [37] | Dossa, M., “Espaces de Sobolev non isotropes, à poids et problèmes de Cauchy quasi-linéaires sur un conoïde caractéristique”, Ann. Inst. Henri Poincare A, 66, 37-107 (1997). · Zbl 0880.35074 |
| [38] | Eardley, D. M. and Moncrief, V., “The global existence of Yang-Mills fields in <Emphasis Type=”Italic“>M3+1”, Commun. Math. Phys., 83, 171-212 (1982). · Zbl 0496.35061 · doi:10.1007/BF01976040 |
| [39] | Friedrich, H., “Existence and structure of past asymptotically simple solutions of Einstein’s field equations with positive cosmological constant”, J. Geom. Phys., 3, 101-117 (1986). · Zbl 0592.53061 · doi:10.1016/0393-0440(86)90004-5 |
| [40] | Friedrich, H., “On the global existence and asymptotic behaviour of solutions to the Einstein-Yang-Mills equations”, J. Differ. Geom., 34, 275-345 (1991). · Zbl 0737.53070 |
| [41] | Friedrich, H., “Einstein equations and conformal structure: Existence of anti-de Sitter-type spacetimes”, J. Geom. Phys., 17, 125-184 (1995). · Zbl 0840.53055 · doi:10.1016/0393-0440(94)00042-3 |
| [42] | Friedrich, H., “Hyperbolic reductions of Einstein’s field equations”, Class. Quantum Grav., 13, 1451-1469 (1996). · Zbl 0851.58044 · doi:10.1088/0264-9381/13/6/014 |
| [43] | Friedrich, H., “Gravitational fields near spacelike and null infinity”, J. Geom. Phys., 24, 83-172 (1998). · Zbl 0896.53053 · doi:10.1016/S0393-0440(97)82168-7 |
| [44] | Fritelli, S. and Reula, O. A., “On the Newtonian limit of general relativity”, Commun. Math. Phys., 166, 221-235 (1994). · Zbl 0812.35147 · doi:10.1007/BF02112314 |
| [45] | Hartman, P., Ordinary Differential Equations, (Birkhäuser, Boston, 1982), 2nd edition. · Zbl 0476.34002 |
| [46] | Heusler, M., Black Hole Uniqueness Theorems, (Cambridge University Press, Cambridge; New York, 1996). · Zbl 0945.83001 · doi:10.1017/CBO9780511661396 |
| [47] | Heusler, M., “Stationary black holes: Uniqueness and beyond”, Living Rev. Relativity, 1 (1998). URL (accessed 3 January 1998): http://www.livingreviews.org/lrr-1998-6. · Zbl 1023.83006 |
| [48] | Hubbard, J. H. and West, B. H., Differential Equations: A Dynamical Systems Approach, Vol. 1: Ordinary Differential Equations, Texts in Applied Mathematics, 5, (Springer, Berlin; New York, 1991), 3rd edition. · Zbl 0987.34001 |
| [49] | Isenberg, J. A., “Constant mean curvature solutions of the Einstein constraint equations on closed manifolds”, Class. Quantum Grav., 12, 2249-2274 (1995). · Zbl 0840.53056 · doi:10.1088/0264-9381/12/9/013 |
| [50] | Isenberg, J. A. and Moncrief, V., “Asymptotic Behaviour of the Gravitational Field and the Nature of Singularities in Gowdy Spacetimes”, Ann. Phys. (N.Y.), 199, 84-122 (1990). · Zbl 0723.53061 · doi:10.1016/0003-4916(90)90369-Y |
| [51] | Isenberg, J. A. and Moncrief, V., “A set of nonconstant mean curvature solutions of the Einstein constraint equations on closed manifolds”, Class. Quantum Grav., 13, 1819-1847 (1996). · Zbl 0860.53056 · doi:10.1088/0264-9381/13/7/015 |
| [52] | John, F., Partial Differential Equations, Applied Mathematical Sciences, 1, (Springer, Berlin; New York, 1982), 4th edition. · Zbl 0472.35001 · doi:10.1007/978-1-4684-9333-7 |
| [53] | John, F., Nonlinear Wave Equations, Formation of Singularities, University Lecture Series, 2, (American Mathematical Society, Providence, 1990). · Zbl 0716.35043 |
| [54] | Kichenassamy, S., Nonlinear Wave Equations, Monographs and Textbooks in Pure and Applied Mathematics, 194, (Marcel Dekker, New York, 1996). · Zbl 0845.35001 |
| [55] | Klainerman, S. and Machedon, M., “Finite energy solutions of the Yang-Mills equations in <Emphasis Type=”Bold“>R3+1”, Ann. Math., 142, 39-119 (1995). · Zbl 0827.53056 · doi:10.2307/2118611 |
| [56] | Kleihaus, B. and Kunz, J., “Static axially symmetric Einstein-Yang-Mills-dilaton solutions: Regular solutions”, Phys. Rev. D, 57, 834-856 (1997). Online version (accessed 7 January 1998): http://arXiv.org/abs/gr-qc/9707045. · Zbl 1042.83513 · doi:10.1103/PhysRevD.57.834 |
| [57] | Majda, A., Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, 53, (Springer, Berlin; New York, 1984). · Zbl 0537.76001 |
| [58] | Moncrief, V., “Global Properties of Gowdy Spacetimes with <Emphasis Type=”Italic“>T3 × <Emphasis Type=”Italic“>R Topology”, Ann. Phys. (N.Y.), 132, 87-107 (1981). · doi:10.1016/0003-4916(81)90270-0 |
| [59] | Moncrief, V., “Neighbourhoods of Cauchy horizons in cosmological spacetimes with one Killing field”, Ann. Phys. (N.Y.), 141, 83-103 (1982). · Zbl 0539.53046 · doi:10.1016/0003-4916(82)90273-1 |
| [60] | Moncrief, V. and Eardley, D. M., “The global existence problem and cosmic censorship in general relativity”, Gen. Relativ. Gravit., 13, 887-892 (1981). · doi:10.1007/BF00764275 |
| [61] | Newman, R. P. A. C., “On the Structure of Conformal Singularities in Classical General Relativity”, Proc. R. Soc. London, 443, 473-492 (1993). see also Pt. II: Evolution Equations and a Conjecture, pp. 493-515. · Zbl 0810.53085 · doi:10.1098/rspa.1993.0158 |
| [62] | Racke, R., Lectures on Nonlinear Evolution Equations: Initial Value Problems, Aspects of Mathematics, 19, (Vieweg, Wiesbaden, 1992). · Zbl 0811.35002 |
| [63] | Rein, G., “Cosmological solutions of the Vlasov-Einstein system with spherical, plane and hyperbolic symmetry”, Math. Proc. Camb. Phil. Soc., 119, 739-762 (1996). · Zbl 0851.53058 · doi:10.1017/S0305004100074569 |
| [64] | Rein, G., “Nonlinear Stability of Homogeneous Models in Newtonian Cosmology”, Arch. Ration. Mech. Anal., 140, 335-351 (1997). Online version (accessed 7 January 1998): http://arXiv.org/abs/gr-qc/9603033. · Zbl 0910.35134 · doi:10.1007/s002050050070 |
| [65] | Rein, G. and Rendall, A. D., “Global Existence of Classical Solutions to the Vlasov-Poisson System in a Three Dimensional, Cosmological Setting”, Arch. Ration. Mech. Anal., 126, 183-201 (1994). · Zbl 0808.35109 · doi:10.1007/BF00391558 |
| [66] | Rendall, A. D., “Reduction of the characteristic initial value problem to the Cauchy problem and its applications to the Einstein equations”, Proc. R. Soc. London, Ser. A, 427, 221-239 (1990). · Zbl 0701.35149 · doi:10.1098/rspa.1990.0009 |
| [67] | Rendall, A. D., “On the definition of post-Newtonian approximations”, Proc. R. Soc. London, Ser. A, 438, 341-360 (1992). · Zbl 0754.53062 · doi:10.1098/rspa.1992.0111 |
| [68] | 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 · doi:10.1007/BF02101736 |
| [69] | Rendall, A. D., “Crushing singularities in spacetimes with spherical, plane and hyperbolic symmetry”, Class. Quantum Grav., 12, 1517-1533 (1995). · Zbl 0823.53072 · doi:10.1088/0264-9381/12/6/017 |
| [70] | Rendall, A. D., “Constant mean curvature foliations in cosmological spacetimes”, Helv. Phys. Acta, 69, 490-500 (1996). · Zbl 0867.53026 |
| [71] | Rendall, A. D., “The initial singularity in solutions of the Einstein-Vlasov system of Bianchi type I”, J. Math. Phys., 37, 438-451 (1996). · Zbl 0862.35129 · doi:10.1063/1.531400 |
| [72] | Rendall, A. D., “Existence and non-existence results for global constant mean curvature foliations”, Nonlinear Anal., 30, 3589-3598 (1997). · Zbl 0891.53067 · doi:10.1016/S0362-546X(96)00203-9 |
| [73] | Rendall, A. D., “Existence of Constant Mean Curvature Foliations in Spacetimes with Two-Dimensional Local Symmetry”, Commun. Math. Phys., 189, 145-164 (1997). · Zbl 0897.53049 · doi:10.1007/s002200050194 |
| [74] | Rendall, A. D., “Existence of constant mean curvature hypersurfaces in spacetimes with two-dimensional local symmetry”, Commun. Math. Phys., 189, 145-164 (1997). · Zbl 0897.53049 · doi:10.1007/s002200050194 |
| [75] | Smoller, J. A., Wasserman, A. G., Yau, S.-T. and McLeod, J. B., “Smooth static solutions of the Einstein-Yang-Mills equations”, Commun. Math. Phys., 143, 115-147 (1991). · Zbl 0755.53061 · doi:10.1007/BF02100288 |
| [76] | Strauss, W., Nonlinear Wave Equations, Regional Conference Series in Mathematics, 73, (American Mathematical Society, Providence, 1989). |
| [77] | Taylor, M. E., Pseudodifferential Operators and Nonlinear PDE, Progress in Mathematics, 100, (Birkhäuser, Boston, 1991). · Zbl 0746.35062 · doi:10.1007/978-1-4612-0431-2 |
| [78] | Taylor, M. E., Partial Differential Equations, Applied Mathematical Sciences, 115-117, (Springer, Berlin; New York, 1996). · Zbl 0869.35004 · doi:10.1007/978-1-4684-9320-7 |
| [79] | Tod, K. P., “Isotropic singularities”, Rend. Sem. Mat. Univ. Pol. Torino, 50, 69-93 (1992). · Zbl 0793.53096 |
| [80] | Tod, KP; Wit, D. (ed.); Bracken, AJ (ed.); Gould, MD (ed.); Pearce, PA (ed.), Isotropic cosmological singularities, Proceedings of the congress, University of Queensland, Brisbane, Australia, July 13-19 1997, Somerville · Zbl 1253.83031 |
| [81] | Wainwright, J. and Ellis, G. F. R., Dynamical Systems in Cosmology, (Cambridge University Press, Cambridge; New York, 1997). · Zbl 1072.83002 · doi:10.1017/CBO9780511524660 |
| [82] | Witt, D., “Vacuum spacetimes that admit no maximal slice”, Phys. Rev. Lett., 57, 1386-1389 (1986). · doi:10.1103/PhysRevLett.57.1386 |
| [83] | Woodhouse, NMJ; Chruściel, PT (ed.), Integrability and Einstein’s equations, Proceedings of the Workshop on Mathematical Aspects of Theories of Gravitation, Warsaw, Poland, February 29-March 30, 1996, Warsaw · Zbl 0891.35158 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
