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THE EXISTENCE THEOREM FOR THE GENERAL RELATIVISTIC CAUCHY PROBLEM ON THE LIGHT-CONE

Part of: General relativity

Published online by Cambridge University Press:  22 May 2014

PIOTR T. CHRUŚCIEL
PIOTR T. CHRUŚCIEL*
Affiliation:
I.H.É.S., Bures sur Yvette, France University of Vienna, Austria; piotr.chrusciel@univie.ac.at

Abstract

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We prove existence of solutions of the vacuum Einstein equations with initial data induced by a smooth metric on a light-cone.

MSC classification

Secondary: 83C05: Einstein's equations (general structure, canonical formalism, Cauchy problems)

Information

Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 2 , 01 July 2014 , e10
Creative Commons
Creative Common License - CCCreative Common License - BY
The online version of this article is published within an Open Access environment subject to the conditions of the Creative Commons Attribution licence .
Copyright
© The Author 2014

References

Andersson, L. and Chruściel, P. T., ‘On asymptotic behavior of solutions of the constraint equations in general relativity with hyperboloidal boundary conditions’, Dissert. Math. 355 (1996), 1–100.Google Scholar
Cagnac, F., ‘Problème de Cauchy sur un conoïde caractéristique pour des équations quasi-linéaires’, Ann. Mat. Pura Appl. 4 129 (1981), 13–41.CrossRefGoogle Scholar
Choquet-Bruhat, Y., General Relativity and the Einstein Equations, Oxford Mathematical Monographs (Oxford University Press, Oxford, 2009).Google Scholar
Choquet-Bruhat, Y., Chruściel, P. T. and Martín-García, J. M., ‘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].CrossRefGoogle Scholar
Choquet-Bruhat, Y., Chruściel, P. T. and Martín-García, J. M., ‘The Cauchy problem on a characteristic cone for the Einstein equations in arbitrary dimensions’, Ann. Henri Poincaré 12 (2011), 419–482. arXiv:1006.4467 [gr-qc].CrossRefGoogle Scholar
Choquet-Bruhat, Y., Chruściel, P. T. and Martín-García, J. M., ‘An existence theorem for the Cauchy problem on a characteristic cone for the Einstein equations with near-round analytic data’, Uchenye zapiski Kazanskogo universiteta 3 (2012), arXiv:1012.0777 [gr-qc].Google Scholar
Chruściel, P. T. and Jezierski, J., ‘On free general relativistic initial data on the light cone’, J. Geom. Phys. 62 (2012), 578–593. arXiv:1010.2098 [gr-qc].CrossRefGoogle Scholar
Chruściel, P. T. and Lengard, O., ‘Solutions of wave equations in the radiating regime’, Bull. Soc. Math. France 133 (2003), 1–72. arXiv:math.AP/0202015.Google Scholar
Chruściel, P. T. and Paetz, T.-T., ‘The many ways of the characteristic Cauchy problem’, Classical Quantum Gravity 29 (2012), 145006, 27. arXiv:1203.4534 [gr-qc].CrossRefGoogle Scholar
Chruściel, P. T. and Paetz, T.-T., ‘Solutions of the vacuum Einstein equations with initial data on past null infinity’, Classical Quantum Gravity 30 (2013), 235037. arXiv:1307.0321 [gr-qc].CrossRefGoogle Scholar
Damour, T. and Schmidt, B., ‘Reliability of perturbation theory in general relativity’, J. Math. Phys. 31 (1990), 2441–2453.CrossRefGoogle Scholar
Dossa, M., ‘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.Google Scholar
Dossa, M., ‘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 Poincaré 4 (2003), 385–411.Google Scholar
Friedrich, H., ‘On purely radiative space–times’, Comm. Math. Phys. 103 (1986), 35–65.CrossRefGoogle Scholar
Friedrich, H., ‘The Taylor expansion at past time-like infinity’, Comm. Math. Phys. 324 (2013), 263–300. arXiv:1306.5626 [gr-qc].CrossRefGoogle Scholar
Galloway, G. J., ‘Maximum principles for null hypersurfaces and null splitting theorems’, Ann. H. Poincaré 1 (2000), 543–567.CrossRefGoogle Scholar
Häfner, D. and Nicolas, J.-P., ‘The characteristic Cauchy problem for Dirac fields on curved backgrounds’, J. Hyperbolic Differ. Equ. 8 (2011), 437–483. arXiv:0903.0515 [math.AP].CrossRefGoogle Scholar
Luk, J., ‘On the local existence for the characteristic initial value problem in general relativity’, Int. Math. Res. Not. IMRN (2012), 4625–4678.CrossRefGoogle Scholar
Rendall, A. D., ‘Reduction of the characteristic initial value problem to the Cauchy problem and its applications to the Einstein equations’, Proc. R. Soc. Lond. A 427 (1990), 221–239.Google Scholar
Thomas, T. Y., The Differential Invariants of Generalized Spaces, (Cambridge University Press, 1934).Google Scholar