Is general relativity ‘essentially understood’? (English) Zbl 1098.83005
This is a broad and very readable survey of known results on the structure of solutions to Einstein’s equations. The main topics are general properties of solutions (constraint equations, gauge system, evolution system, subsidiary system, global behavior of solutions), as well as gravitational radiation, black hole formation and cosmic censorship. The author’s conclusion is that despite huge advances in the field over the past decades, three main areas stand out as not sufficiently understood: highly dynamical processes, strong field situations, and evolution over long time scales under general assumptions (lack of symmetry). The article includes a number of open problems.
Reviewer: Michael Kunzinger (Wien)
MSC:
| 83C05 | Einstein’s equations (general structure, canonical formalism, Cauchy problems) |
| 83C35 | Gravitational waves |
| 83C57 | Black holes |
| 83C75 | Space-time singularities, cosmic censorship, etc. |
| 83C15 | Exact solutions to problems in general relativity and gravitational theory |
Keywords:
classical general relativity; singularities and cosmic censorship; initial value problem for Einstein’s field equations; asymptotic structure; numerical relativityReferences:
| [1] | Alcubierre, Phys. Rev. D 60 pp 064017– (1999) |
| [2] | Allen, Phys. Rev. D 70 pp 044038– (2004) |
| [3] | Existence and stability of even dimensional asymptotically de Sitter spaces. http://xxx.lanl.gov/abs/gr-qc/0408072. · Zbl 1100.83004 |
| [4] | and , Asymptotically simple solutions of the vacuum Einstein equations in even dimensions. http://xxx.lanl.gov/abs/gr-qc/0412020. |
| [5] | The global existence problem in general relativity, in: The Einstein equations and the large scale behaviour of gravitational fields, edited by P. T. Chruściel and H. Friedrich (Birkhäuser, Basel, 2004). |
| [6] | Andersson, Commun. Math. Phys. 161 pp 533– (1994) |
| [7] | , Solutions of the constraint equations in general relativity satisfying hyperboloidal boundary conditions. Dissertationes Mathematicae Polska Akademia Nauk, Inst. Matem., Warszawa, 1996. · Zbl 0873.35101 |
| [8] | Andersson, Commun. Math. Phys. 149 pp 587– (1992) |
| [9] | Andersson, Class. Quantum. Gravity 21 (2004) |
| [10] | Andersson, Phys. Rev. Lett. 94 pp 051101– (2005) |
| [11] | Andersson, Ann. Henri Poincaré 4 pp 1– (2003) |
| [12] | , Future complete vacuum space-times, in: The Einstein equations and the large scale behaviour of gravitational fields, edited by P. T. Chruściel and H. Friedrich (Birkhäuser, Basel, 2004). |
| [13] | Bartnik, Commun. Pure Appl. Math. 39 pp 661– (1986) |
| [14] | Bartnik, Phys. Rev. Lett. 61 pp 141– (1988) |
| [15] | and , The constraint equations, in: The Einstein equations and the large scale behaviour of gravitational fields, edited by P. T. Chruściel and H. Friedrich (Birkhäuser, Basel, 2004). |
| [16] | Baumgarte, Phys. Rev. D 59 pp 024007– (1999) |
| [17] | Beig, Phys. Rev. Lett. 66 pp 2421– (1991) |
| [18] | Beig, Class. Quantum Gravity 11 pp 419– (1994) |
| [19] | Belinskii, Adv. in Phys. 19 pp 525– (1970) |
| [20] | Belinskii, Adv. in Phys. 31 pp 639– (1982) |
| [21] | Numerical Approaches to Space-Time Singularities. Living Reviews in Relativity 2002. |
| [22] | Berger, Class. Quantum Gravity 14 (1997) |
| [23] | Berger, Phys. Rev. D 48 pp 4676– (1993) |
| [24] | Bizoń, Acta Phys. Polonica B 33 pp 1893– (2002) |
| [25] | Bizoń, Phys. Rev. D 64 pp 121701– (2001) |
| [26] | Bondi, Proc. Roy. Soc. A 269 pp 21– (1962) |
| [27] | and (eds.), Gravitational Lensing: Recent Progress and Future Go. ASP Conference Series, Vol. CS-237, 2001. |
| [28] | Bray, J. Diff. Geom. 59 pp 177– (2001) |
| [29] | and , The Penrose inequality, in: The Einstein equations and the large scale behaviour of gravitational fields, edited by P. T. Chruściel and H. Friedrich (Birkhäuser, Basel, 2004). |
| [30] | Brodbeck, J. Math. Phys. 40 pp 909– (1999) |
| [31] | Has the black hole equilibrium problem been solved?, in: The Eighth Marcel Grossmann Meeting, edited by T. Piran and R. Ruffini (World Scientific, Singapore, 1999). |
| [32] | Chae, Commun. Pure Appl. Math. 57 pp 1015– (2004) |
| [33] | Future complete U(1) symmetric Einsteinian space-times, the unpolarized case, in: The Einstein equations and the large scale behaviour of gravitational fields, edited by P. T. Chruściel and H. Friedrich (Birkhäuser, Basel, 2004). |
| [34] | Choquet-Bruhat, Commun. Math. Phys. 14 pp 329– (1969) |
| [35] | and , The Cauchy problem, in: General Relativity and Gravitation, edited by A. Held, Vol. 1 (New York, Plenum, 1980). |
| [36] | Choptuik, Phys. Rev. Lett. 70 pp 9– (1993) |
| [37] | Christodoulou, Ann. Math. 149 pp 183– (1999) |
| [38] | and , The Global Nonlinear Stability of the Minkowski Space (Princeton University Press, Princeton, 1993). |
| [39] | Black holes, in: The Conformal Structure of Space-Time, edited by J. Frauendiener and H. Friedrich (Springer, Berlin, 2002). |
| [40] | Chruściel, Class. Quantum Gravity 19 (2002) |
| [41] | Chruściel, Class. Quantum Gravity 19 pp 3389– (2002) |
| [42] | and , On mapping properties of the general relativistic constraints operator in weighted function spaces, with application. Mém. Soc. Math. France, submitted. http://xxx.lanl.gov/abs/gr-qc/0301073. |
| [43] | Chruściel, Phys. Rev. D 48 pp 1616– (1993) |
| [44] | Chruściel, Phys. Rev. Lett. 93 pp 081101– (2004) |
| [45] | Chruściel, Class. Quantum Gravity 21 (2004) |
| [46] | Chruściel, Ann. Phys. 242 pp 349– (1995) |
| [47] | , and , Singularities and Horizons - A Review Article, in: General Relativity and Gravitation, Vol. 2, edited by A. Held (Plenum, New York, 1980). |
| [48] | Corvino, Commun. Math. Phys. 214 pp 137– (2000) |
| [49] | and , On the Asymptotics for the Vacuum Einstein Constraint Equations. http://xxx.lanl.gov/abs/gr-qc/0301071. · Zbl 1122.58016 |
| [50] | Cutler, Class. Quantum Gravity 6 pp 453– (1989) |
| [51] | Black hole formation from a complete past. arXiv:gr-qc/0310040. · Zbl 1168.83010 |
| [52] | and , A Proof of Price’s law for the collapse of a self-gravitating scalar field. arXiv:gr-qc/0309115. · Zbl 1088.83008 |
| [53] | and , A note on boundary value problems for black hole evolutions. http://xxx.lanl.gov/abs/gr-qc/0403034. |
| [54] | Dain, Class. Quantum Gravity 21 pp 555– (2004) |
| [55] | Dain, Phys. Rev. Lett. 93 pp 231101– (2004) |
| [56] | Dain, Commun. Math. Phys. 222 pp 569– (2001) |
| [57] | , and , On the existence of initial data containing isolated black holes, gr-qc/0412061 (2004). |
| [58] | Dain, Phys. Rev. D 65 pp 084020– (2002) |
| [59] | Einstein, Ann. Phys. 49 pp 769– (1916) |
| [60] | Näherungsweise Integration der Feldgleichungen der Gravitation. Sitzungsberichte Königl. Preuss. Akademie der Wiss. (1916), pp. 688-696. |
| [61] | Über Gravitationswellen. Sitzungsberichte Königl. Preuss. Akademie der Wiss. (1918), pp. 154-167. |
| [62] | and , The initial value problem and the dynamical formulation of general relativity, in: General relativity. An Einstein centenary survey, edited by S. Hawking and W. Israel (Cambridge University Press, Cambridge, 1979). |
| [63] | Fourès-Bruhat, Acta Mathematica 88 pp 141– (1952) |
| [64] | Conformal infinity. Living reviews, 2002. http://www.livingreviews.org/lrr-2004-1. |
| [65] | Frauendiener, Class. Quantum Gravity 22 pp 1769– (2005) |
| [66] | Friedrich, Commun. Math. Phys. 100 pp 525– (1985) |
| [67] | Friedrich, Commun. Math. Phys. 103 pp 35– (1986) |
| [68] | Friedrich, Commun. Math. Phys. 107 pp 587– (1986) |
| [69] | Friedrich, J. Differ. Geom. 34 pp 275– (1991) |
| [70] | Friedrich, J. Geom. Phys. 17 pp 125– (1995) |
| [71] | Friedrich, J. Geom. Phys. 24 pp 83– (1998) |
| [72] | Conformal Einstein evolution, in: The Conformal Structure of Spacetime: Geometry, Analysis, Numerics, edited by J. Frauendiener and H. Friedrich (Springer, Berlin, 2002). |
| [73] | Friedrich, Class. Quantum. Gravity 20 pp 101– (2003) |
| [74] | Friedrich, Commun. Math. Phys. 235 pp 513– (2003) |
| [75] | Smoothness at null infinity and the structure of initial data, in: The Einstein equations and the large scale behaviour of gravitational fields, edited by P. T. Chruściel and H. Friedrich (Birkhäuser, Basel, 2004). |
| [76] | Friedrich, Class. Quantum Gravity 22 (2005) |
| [77] | Friedrich, Commun. Math. Phys. 201 pp 619– (1999) |
| [78] | and , The Cauchy Problem for the Einstein Equations, in: Einstein’s Field Equations and Their Physical Implications, edited by B. Schmidt (Lecture Notes in Physics, vol. 540, Springer, Berlin 2000). |
| [79] | Frittelli, J. Math. Phys. 40 pp 5143– (1999) |
| [80] | and , Black Hole Physics (Kluwer, Dordrecht, 1998). |
| [81] | Numerical simulations of generic singularities. arXiv: gr-qc/0312117 v2. |
| [82] | Gundlach, Phys. Rep. 367 pp 339– (2003) |
| [83] | , and , Constraint damping in the Z4 formulation and harmonic gauge. http://xxx.lanl.gov/abs/gr-qc/0504114. · Zbl 1154.83302 |
| [84] | Gundlach, Phys. Rev. D 49 pp 883– (1994) |
| [85] | Gundlach, Phys. Rev. D 49 pp 890– (1994) |
| [86] | and , The large scale structure of space-time (Cambridge University Press, Cambridge, 1973). |
| [87] | Hawking, Proc. Roy. Soc. A 314 pp 529– (1970) |
| [88] | and , Conformal infinity does not exist for radiating solutions in odd space-time dimensions. gr-qc/0407014. |
| [89] | Hübner, Class. Quantum Gravity 18 pp 1871– (2001) |
| [90] | Hughes, Arch. Ration. Mech. Anal. 63 pp 273– (1977) |
| [91] | Huisken, J. Diff. Geom. 59 pp 353– (2001) |
| [92] | Problems and successes in the numerical approach to the conformal field equations, in: The Conformal Structure of Spacetime: Geometry, Analysis, Numerics, edited by J. Frauendiener and H. Friedrich (Springer, Berlin, 2002). |
| [93] | Isenberg, Ann. Henri Pioncaré 4 pp 369– (2003) |
| [94] | Dark stars: the evolution of an idea, in: Three hundred years of gravitation, edited by S. Hawking and W. Israel (Cambridge University Press, Cambridge, 1989). |
| [95] | The internal structure of black holes, in: Black Holes and Relativistic Stars, edited by R. M. Wald (University of Chicago Press, Chicago, 1998). |
| [96] | and , The Evolution Problem in General Relativity. (Birkhäuser, Basel, 2003). |
| [97] | Klainerman, Class. Quantum Gravity 20 pp 3215– (2003) |
| [98] | and Causal geometry of Einstein vacuum space-times with finite curvature flux. math.ap/0308123 (2003). |
| [99] | et al., The double pulsar - A new testbed for relativistic gravity. Binary Radio Pulsars, ASP Conference Series, Vol. TBD, 2004, edited by F. A. Rasio and I. H. Stairs, arXiv: astro-ph/0405179. |
| [100] | Lee, Bull. Am. Math. Soc. 17 pp 37– (1987) |
| [101] | and , Status quo and open problems in the numerical construction of space-times, in: The Einstein equations and the large scale behaviour of gravitational fields, edited by P. T. Chruściel and H. Friedrich (Birkhäuser, Basel, 2004). |
| [102] | Lifshitz, Adv. in Phys. 12 pp 525– (1963) |
| [103] | and , Global existence for the Einstein vacuum equations in wave coordinates. arXiv:math.AP/0312479. · Zbl 1081.83003 |
| [104] | and , The global stability of Minkowski space-time in harmonic gauge. arXiv:math.AP/0411109. · Zbl 1192.53066 |
| [105] | Commun. Math. Phys. (2004). |
| [106] | Maxwell, J. Hyp. Diff. Equ. 2 pp 521– (2005) |
| [107] | Misner, Phys. Rev. Lett. 29 pp 1071– (1969) |
| [108] | Moncrief, Ann. Phys. 132 pp 87– (1981) |
| [109] | Moncrief, Gen. Rel. Grav. 13 pp 887– (1981) |
| [110] | Ori, Phys. Rev. D 61 pp 024001– (1999) |
| [111] | Subtle is the Lord (Oxford University Press, 1982). |
| [112] | Penrose, Phys. Rev. Lett. 10 pp 66– (1963) |
| [113] | Penrose, Phys. Rev. Lett. 14 pp 57– (1965) |
| [114] | Penrose, Rev. Nuovo Cimento 1 pp 252– (1969) |
| [115] | Penrose, Ann. N.Y. Acad. Sci. 224 pp 125– (1973) |
| [116] | Singularities and time-asymmetry, in: General relativity. An Einstein centenary survey, edited by S. Hawking and W. Israel (Cambridge University Press, Cambridge, 1979). |
| [117] | Poisson, Phys. Rev. D 41 pp 1796– (1990) |
| [118] | Pound, Phys. Rev. Lett. 4 pp 337– (1960) |
| [119] | Pretorius, Class. Quantum Gravity 22 pp 425– (2005) |
| [120] | Price, Phys. Rev. D 5 pp 2419– (1972) |
| [121] | Black holes in the real universe and their prospects as probes of relativistic gravity, in: The Future of Theoretical Physics and Cosmology, edited by G. Gibbons, E. Shellard, and S. Rankin (Cambridge University Press, Cambridge, 2003). |
| [122] | Rendall, Class. Quantum Gravity 14 pp 2341– (1997) |
| [123] | Theorems on Existence and Global Dynamics for the Einstein Equations. Living Reviews in Relativity (2002). http://www.livingreviews.org/lrr-2002-6. |
| [124] | Asymptotics of solutions of the Einstein equations with positive cosmological constant. gr-qc/0312020. |
| [125] | The Einstein-Vlasov system, in: The Einstein equations and the large scale behaviour of gravitational fields, edited by P. T. Chruściel and H. Friedrich (Birkhäuser, Basel, 2004). |
| [126] | Rendall, Class. Quantum Gravity 21 (2004) |
| [127] | Rendall, Class. Quantum Gravity 18 pp 2959– (2001) |
| [128] | Reula, J. Hyp. Diff. Equ. 2 pp 397– (2005) |
| [129] | Ringström, Class. Quantum Gravity 17 pp 713– (2000) |
| [130] | Ringström, Ann. Inst. Henri Poincare 2 pp 405– (2000) |
| [131] | Ringström, Class. Quantum Gravity 20 pp 1943– (2003) |
| [132] | Ringström, J. Hyp. Diff. Equ. 2 pp 547– (2005) |
| [133] | and , J. Hyp. Diff. Equ. (to appear). |
| [134] | Schoen, J. Diff. Geom. 20 pp 479– (1984) |
| [135] | Schoen, Commun. Math. Phys. 79 pp 231– (1981) |
| [136] | Smoller, Commun. Math. Phys. 143 pp 115– (1991) |
| [137] | Stellmacher, Math. Ann. 115 pp 136– (1938) |
| [138] | Quasi-Local Energy-Momentum and Angular Momentum in GR: A Review Article. Living Reviews in Relativity (2004). http://www.livingreviews.org/lrr-2004-4. · Zbl 1068.83506 |
| [139] | Nonlinear wave equations, in: Proceedings of the International Congress of Mathematicians (Beijing, 2002), Vol. III, pp. 209-220. (Higher Ed. Press, Beijing, 2002). |
| [140] | Black Holes and Time Warps: Einstein’s Outrageous Legacy (Norton, New York, 1994). · Zbl 0949.83500 |
| [141] | Valiente Kroon, Commun. Math. Phys. 251 pp 211– (2004) |
| [142] | Valiente Kroon, Class. Quantum Gravity 22 pp 1683– (2005) |
| [143] | Wainwright, Class. Quantum Gravity 6 pp 1409– (1989) |
| [144] | Gravitational collapse and cosmic censorship, in: Black Holes, Gravitational Radiation and the Universe, edited by B. R. Iyer and B. Bhawal (Dordrecht, Kluwer Academic Publishers, 1999). |
| [145] | and , Relativistic Binary Pulsar B1913+16: Thirty Years of Observation and Analysis. Binary Radio Pulsars. ASP Conference Series, Vol. TBD, 2004, edited by F. A. Rasio and I. H. Stairs (arXiv: astro-ph/0407149). |
| [146] | Witten, Commun. Math. Phys. 80 pp 381– (1981) |
| [147] | Yoneda, Class. Quantum Gravity 20 (2003) |
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.
