The existence theorem for the general relativistic Cauchy problem on the light-cone. (English) Zbl 1297.83004
Summary: We prove existence of solutions of the vacuum Einstein equations with initial data induced by a smooth metric on a light-cone.
MSC:
| 83C05 | Einstein’s equations (general structure, canonical formalism, Cauchy problems) |
| 53Z05 | Applications of differential geometry to physics |
| 83C15 | Exact solutions to problems in general relativity and gravitational theory |
| 35L15 | Initial value problems for second-order hyperbolic equations |
References:
| [1] | L.Andersson and P. T.Chruściel, ‘On asymptotic behavior of solutions of the constraint equations in general relativity with hyperboloidal boundary conditions’, Dissert. Math.355 (1996), 1-100. · Zbl 0873.35101 |
| [2] | F.Cagnac, ‘Problème de Cauchy sur un conoïde caractéristique pour des équations quasi-linéaires’, Ann. Mat. Pura Appl. 4129 (1981), 13-41. · Zbl 0486.35023 |
| [3] | Y.Choquet-Bruhat, General Relativity and the Einstein Equations, Oxford Mathematical Monographs (Oxford University Press, Oxford, 2009). · Zbl 1157.83002 |
| [4] | Y.Choquet-Bruhat, P. T.Chruściel and J. M.Martín-García, ‘An existence theorem for the Cauchy problem on a characteristic cone for the Einstein equations’, Cont. Math.554 (2011), 73-81. arXiv:1006.5558 [gr-qc]. · Zbl 1235.83011 |
| [5] | Y.Choquet-Bruhat, P. T.Chruściel and J. M.Martín-García, ‘The Cauchy problem on a characteristic cone for the Einstein equations in arbitrary dimensions’, Ann. Henri Poincaré12 (2011), 419-482. arXiv:1006.4467 [gr-qc]. · Zbl 1215.83016 |
| [6] | Y.Choquet-Bruhat, P. T.Chruściel and J. M.Martín-García, ‘An existence theorem for the Cauchy problem on a characteristic cone for the Einstein equations with near-round analytic data’, Uchenye zapiski Kazanskogo universiteta3 (2012), arXiv:1012.0777 [gr-qc]. · Zbl 1344.83002 |
| [7] | P. T.Chruściel and J.Jezierski, ‘On free general relativistic initial data on the light cone’, J. Geom. Phys.62 (2012), 578-593. arXiv:1010.2098 [gr-qc]. · Zbl 1243.53111 |
| [8] | P. T.Chruściel and O.Lengard, ‘Solutions of wave equations in the radiating regime’, Bull. Soc. Math. France133 (2003), 1-72. arXiv:math.AP/0202015. |
| [9] | P. T.Chruściel and T.-T.Paetz, ‘The many ways of the characteristic Cauchy problem’, Classical Quantum Gravity29 (2012), 145006, 27. arXiv:1203.4534 [gr-qc]. · Zbl 1261.83003 |
| [10] | P. T.Chruściel and T.-T.Paetz, ‘Solutions of the vacuum Einstein equations with initial data on past null infinity’, Classical Quantum Gravity30 (2013), 235037. arXiv:1307.0321 [gr-qc]. · Zbl 1284.83020 |
| [11] | T.Damour and B.Schmidt, ‘Reliability of perturbation theory in general relativity’, J. Math. Phys.31 (1990), 2441-2453. · Zbl 0723.53050 |
| [12] | M.Dossa, ‘Espaces de Sobolev non isotropes, à poids et problèmes de Cauchy quasi-linéaires sur un conoïde caractéristique’, Ann. Inst. Henri Poincaré Phys. Théor.66 (1997), 37-107. · Zbl 0880.35074 |
| [13] | M.Dossa, ‘Problèmes de Cauchy sur un conoïde caractéristique pour les équations d’Einstein (conformes) du vide et pour les équations de Yang-Mills-Higgs’, Ann. Henri Poincaré4 (2003), 385-411. · Zbl 1026.81036 |
| [14] | H.Friedrich, ‘On purely radiative space – times’, Comm. Math. Phys.103 (1986), 35-65. · Zbl 0584.53038 |
| [15] | H.Friedrich, ‘The Taylor expansion at past time-like infinity’, Comm. Math. Phys.324 (2013), 263-300. arXiv:1306.5626 [gr-qc]. · Zbl 1277.83022 |
| [16] | G. J.Galloway, ‘Maximum principles for null hypersurfaces and null splitting theorems’, Ann. H. Poincaré1 (2000), 543-567. · Zbl 0965.53048 |
| [17] | D.Häfner and J.-P.Nicolas, ‘The characteristic Cauchy problem for Dirac fields on curved backgrounds’, J. Hyperbolic Differ. Equ.8 (2011), 437-483. arXiv:0903.0515 [math.AP]. · Zbl 1230.35136 |
| [18] | J.Luk, ‘On the local existence for the characteristic initial value problem in general relativity’, Int. Math. Res. Not. IMRN (2012), 4625-4678. · Zbl 1262.83011 |
| [19] | A. D.Rendall, ‘Reduction of the characteristic initial value problem to the Cauchy problem and its applications to the Einstein equations’, Proc. R. Soc. Lond. A427 (1990), 221-239. · Zbl 0701.35149 |
| [20] | T. Y.Thomas, The Differential Invariants of Generalized Spaces, (Cambridge University Press, 1934). · JFM 60.0363.02 |
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.
