Annales Henri Poincaré

, Volume 12, Issue 3, pp 419–482 | Cite as

The Cauchy Problem on a Characteristic Cone for the Einstein Equations in Arbitrary Dimensions

  • Yvonne Choquet-Bruhat
  • Piotr T. ChruścielEmail author
  • José M. Martín-García


We derive explicit formulae for a set of constraints for the Einstein equations on a null hypersurface, in arbitrary space–time dimensions n + 1 ≥ 3. We solve these constraints and show that they provide necessary and sufficient conditions so that a spacetime solution of the Cauchy problem on a characteristic cone for the hyperbolic system of the reduced Einstein equations in wave-map gauge also satisfies the full Einstein equations. We prove a geometric uniqueness theorem for this Cauchy problem in the vacuum case.


Cauchy Problem Einstein Equation Christoffel Symbol Null Hypersurface Null Cone 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bondi H., van der Burg M.G.J., Metzner A.W.K.: Gravitational waves in general relativity VII. Proc. Roy. Soc. Lond. A269, 21–51 (1962)ADSGoogle Scholar
  2. 2.
    Caciotta G., Nicolò F.: Global characteristic problem for Einstein vacuum equations with small initial data. I. The initial data constraints. J. Hyp. Differ. Equ. 2(1), 201–277 (2005) arXiv:gr-qc/0409028. MR MR2134959 (2006i:58042)CrossRefzbMATHGoogle Scholar
  3. 3.
    Caciotta G., Nicolò F.: On a class of global characteristic problems for the Einstein vacuum equations with small initial data. J. Math. Phys. 51, 102503 (2010) arXiv: gr-qc/0608038CrossRefADSMathSciNetGoogle Scholar
  4. 4.
    Cagnac F.: Problème de Cauchy sur les hypersurfaces caractéristiques des équations d’Einstein du vide. C. R. Acad. Sci. Paris Sér. A-B 262, A1488–A1491 (1966) MR MR0198931 (33 #7081)MathSciNetGoogle Scholar
  5. 5.
    Cagnac F.: Problème de Cauchy sur les hypersurfaces caractéristiques des équations d’Einstein du vide. C. R. Acad. Sci. Paris Sér. A-B 262, A1356–A1359 (1966) MR MR0198930 (33 #7080)MathSciNetGoogle Scholar
  6. 6.
    Cagnac, F.: Applications du problème de Cauchy caractéristique. C. R. Acad. Sci. Paris Sér. A-B 276, A195–A198 (1973); Problème de Cauchy caractéristique pour certains systèmes. C. R. Acad. Sci. Paris Sér. A-B 276, A133–A136 (1973). MR MR0320541 (47 #9078). MR MR0320542 (47 #9079)Google Scholar
  7. 7.
    Cagnac F.: Problème de Cauchy sur un conoï de caractéristique. Ann. Fac. Sci. Toulouse Math. (5) 2(1), 11–19 (1980) MR MR583901 (81m:35083)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Cagnac F.: Problème de Cauchy sur un conoï de caractéristique pour des équations quasi-linéaires. Ann. Mat. Pura Appl. 129(4), 13–41 (1981) MR MR648323 (84a:35185)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Choquet-Bruhat Y.: Problème des conditions initiales sur un conoïde caractéristique. C. R. Acad. Sci. Paris 256, 3971–3973 (1963)Google Scholar
  10. 10.
    Choquet-Bruhat, Y.: General relativity and the Einstein equations. In: Oxford Mathematical Monographs. Oxford University Press, Oxford (2009). MR MR2473363Google Scholar
  11. 11.
    Choquet-Bruhat Y., Chruściel P.T., Loizelet J.: Global solutions of the Einstein–Maxwell equations in higher dimension. Class. Quantum Gravity 24, 7383–7394 (2006) arXiv:gr-qc/0608108. MR 2279722 (2008i:83022)CrossRefADSGoogle Scholar
  12. 12.
    Choquet-Bruhat Y., Chruściel P.T., Martín-García J.M.: The light-cone theorem. Class. Quantum Gravity 26, 135011 (2009) arXiv:0905.2133 [gr-qc]CrossRefADSGoogle Scholar
  13. 13.
    Choquet-Bruhat, Y., Chruściel, P.T., Martín-García, J.M.: An existence theorem for the Cauchy problem on a characteristic cone for the Einstein equations. Cont. Math. (2010, in press). Proceedings of “Complex Analysis & Dynamical Systems IV”, Nahariya, May 2009. arXiv:1006.5558 [gr-qc]Google Scholar
  14. 14.
    Choquet-Bruhat, Y., DeWitt-Morette, C.: Analysis, Manifolds and Physics. Part II. North-Holland, Amsterdam (1989). MR MR1016603 (91e:58001)Google Scholar
  15. 15.
    Christodoulou, D.: The formation of black holes in general relativity. In: EMS Monographs in Mathematics. European Mathematical Society (2008)Google Scholar
  16. 16.
    Christodoulou D., Müllerzum Hagen H.: Problème de valeur initiale caractéristique pour des systèmes quasi linéaires du second ordre. C. R. Acad. Sci. Paris Sér. I Math. 293, 39–42 (1981) MR MR633558 (82i:35118)zbMATHGoogle Scholar
  17. 17.
    Damour T., Schmidt B.: Reliability of perturbation theory in general relativity. J. Math. Phys. 31, 2441–2453 (1990) MR MR1072957 (91m:83007)CrossRefzbMATHADSMathSciNetGoogle Scholar
  18. 18.
    Dautcourt G.: Zum charakteristischen Anfangswertproblem der Einsteinschen Feldgleichungen. Ann. Physik 12(7), 302–324 (1963) MR MR0165949 (29 #3229)CrossRefADSMathSciNetzbMATHGoogle Scholar
  19. 19.
    Dossa M.: Espaces de Sobolev non isotropes, à poids et problèmes de Cauchy quasi-linéaires sur un conoï de caractéristique. Ann. Inst. H. Poincaré Phys. Théor. 1, 37–107 (1997) MR MR1434115 (98b:35117)ADSMathSciNetGoogle Scholar
  20. 20.
    Dossa M.: Problèmes de Cauchy sur un conoï de caractéristique pour les équations d’Einstein (conformes) du vide et pour les équations de Yang-Mills-Higgs. Ann. Henri Poincaré 4, 385–411 (2003) MR MR1985778 (2004h:58041)CrossRefADSMathSciNetzbMATHGoogle Scholar
  21. 21.
    Fourès-Bruhat Y.: Théorème d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires. Acta Math. 88, 141–225 (1952)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Friedlander, F.G.: The wave equation on a curved space-time. In: Cambridge Monographs on Mathematical Physics, vol. 2. Cambridge University Press, Cambridge (1975). MR MR0460898 (57 #889)Google Scholar
  23. 23.
    Friedrich H.: The asymptotic characteristic initial value problem for Einstein’s vacuum field equations as an initial value problem for a first-order quasilinear symmetric hyperbolic system. Proc. Roy. Soc. Lond. Ser. A 378, 401–421 (1981) MR MR637872 (83a:83007)CrossRefzbMATHADSMathSciNetGoogle Scholar
  24. 24.
    Friedrich H.: On the regular and the asymptotic characteristic initial value problem for Einstein’s vacuum field equations. Proc. Roy. Soc. Lond. Ser. A 375, 169–184 (1981) MR MR618984 (82k:83002)CrossRefzbMATHADSMathSciNetGoogle Scholar
  25. 25.
    Friedrich H.: On the hyperbolicity of Einstein’s and other gauge field equations. Commun. Math. Phys. 100, 525–543 (1985) MR MR806251 (86m:83009)CrossRefzbMATHADSMathSciNetGoogle Scholar
  26. 26.
    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) MR MR855572 (88c:83006)CrossRefzbMATHADSMathSciNetGoogle Scholar
  27. 27.
    Friedrich H.: On purely radiative space-times. Commun. Math. Phys. 103, 35–65 (1986) MR MR826857 (87e:83029)CrossRefzbMATHADSMathSciNetGoogle Scholar
  28. 28.
    Galloway G.J.: Maximum principles for null hypersurfaces and null splitting theorems. Ann. H. Poincaré 1, 543–567 (2000) MR MR1777311 (2002b:53052)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Gourgoulhon E., Jaramillo J.L.: A 3 + 1 perspective on null hypersurfaces and isolated horizons. Phys. Rep. 423, 159–294 (2006) MR MR2195374 (2007f:83055)CrossRefADSMathSciNetGoogle Scholar
  30. 30.
    Gundlach C., Calabrese G., Hinder I., Martín-García J.M.: Constraint damping in the Z4 formulation and harmonic gauge. Class. Quantum Gravity 22, 3767–3773 (2005) MR MR2168553 (2006d:83012)CrossRefzbMATHGoogle Scholar
  31. 31.
    Hayward S.A.: The general solution to the Einstein equations on a null surface. Class. Quantum Gravity 10, 773–778 (1993) MR MR1214441 (94e:83004)CrossRefzbMATHADSMathSciNetGoogle Scholar
  32. 32.
    Hörmander L.: A remark on the characteristic Cauchy problem. J. Funct. Anal. 93(2), 270–277 (1990) MR MR1073287 (91m:58154)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Jezierski, J., Kijowski, J., Czuchry, E.: Geometry of null-like surfaces in general relativity and its application to dynamics of gravitating matter. Rep. Math. Phys. 46, 399–418 (2000). Dedicated to Professor Roman S. Ingarden on the occasion of his 80th birthday. MR MR1811080 (2002c:83033)Google Scholar
  34. 34.
    Jezierski J., Kijowski J., Czuchry E.: Dynamics of a self-gravitating lightlike matter shell: a gauge-invariant Lagrangian and Hamiltonian description. Phys. Rev. D (3) 65, 064036 (2002) MR MR1918464 (2003f:83063)CrossRefADSMathSciNetGoogle Scholar
  35. 35.
    Klainerman, S., Rodnianski, I.: On emerging scarred surfaces for the Einstein vacuum equations. arXiv:1002.2656 [gr-qc] (2010)Google Scholar
  36. 36.
    Lee, J.M.: Riemannian manifolds. In: Graduate Texts in Mathematics, vol. 176. Springer-Verlag, New York (1997). MR MR1468735 (98d:53001)Google Scholar
  37. 37.
    Leray, J.: Hyperbolic differential equations. Mimeographed notes. Princeton (1953)Google Scholar
  38. 38.
    Lindblad H., Rodnianski I.: The global stability of the Minkowski space-time in harmonic gauge. Ann. Math. (2) 171, 1401–1477 (2004) arXiv:math.ap/0411109. MR 2680391CrossRefMathSciNetGoogle Scholar
  39. 39.
    Lindblad H., Rodnianski I.: Global existence for the Einstein vacuum equations in wave coordinates. Commun. Math. Phys. 256, 43–110 (2005) arXiv: math.ap/0312479. MR MR2134337 (2006b:83020)CrossRefzbMATHADSMathSciNetGoogle Scholar
  40. 40.
    Loizelet J.: Solutions globales d’équations Einstein Maxwell. Ann. Fac. Sci. Toulouse 18, 565–610 (2009) MR 2582443zbMATHMathSciNetCrossRefGoogle Scholar
  41. 41.
    Martín-García, J.M.: xAct: Efficient Tensor Computer Algebra.
  42. 42.
    Moncrief V., Isenberg J.: Symmetries of cosmological Cauchy horizons. Commun. Math. Phys. 89, 387–413 (1983)CrossRefzbMATHADSMathSciNetGoogle Scholar
  43. 43.
    Nicolas J.-P.: On Lars Hörmander’s remark on the characteristic Cauchy problem. C. R. Math. Acad. Sci. Paris 344(10), 621–626 (2007) MR MR2334072 (2008c:35165)zbMATHMathSciNetGoogle Scholar
  44. 44.
    Penrose R.: Null hypersurface initial data for classical fields of arbitrary spin and for general relativity. Gen. Rel. Grav. 12, 225–264 (1980) MR MR574333 (81d:83044)CrossRefzbMATHADSMathSciNetGoogle Scholar
  45. 45.
    Pretorius F.: Evolution of binary black hole space-times. Phys. Rev. Lett. 95, 121101 (2005) arXiv:gr-qc/0507014CrossRefADSMathSciNetGoogle Scholar
  46. 46.
    Reiterer, M., Trubowitz, E.: Strongly focused gravitational waves. arXiv: 0906.3812 [gr-qc] (2009)Google Scholar
  47. 47.
    Rendall A.D.: Reduction of the characteristic initial value problem to the Cauchy problem and its applications to the Einstein equations. Proc. Roy. Soc. Lond. A 427, 221–239 (1990) MR MR1032984 (91a:83004)CrossRefzbMATHADSMathSciNetGoogle Scholar
  48. 48.
    Rendall, A.D.: The characteristic initial value problem for the Einstein equations. In: Nonlinear Hyperbolic Equations and Field Theory (Lake Como, 1990). Pitman Res. Notes Math. Ser., vol. 253, pp. 154–163. Longman Sci. Tech., Harlow (1992). MR MR1175208 (93j:83010)Google Scholar
  49. 49.
    Sachs R.K.: On the characteristic initial value problem in gravitational theory. J. Math. Phys. 3, 908–914 (1962)CrossRefzbMATHADSMathSciNetGoogle Scholar
  50. 50.
    Thomas T.Y.: The Differential Invariants of Generalized Spaces. Cambridge University Press, Cambridge (1934)zbMATHGoogle Scholar
  51. 51.
    Müllerzum Hagen H., Seifert H.-J.: On characteristic initial-value and mixed problems. Gen. Rel. Grav. 8, 259–301 (1977) MR MR0606056 (58 #29307)CrossRefADSGoogle Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  • Yvonne Choquet-Bruhat
    • 1
  • Piotr T. Chruściel
    • 2
    Email author
  • José M. Martín-García
    • 3
  1. 1.Académie des SciencesParisFrance
  2. 2.Gravitational PhysicsUniversity of ViennaViennaAustria
  3. 3.Institut d’Astrophysique de Paris and Laboratoire Univers et Théories, UMR 8102, Observatoire de Paris-MeudonMeudonFrance

Personalised recommendations