Annales Henri Poincaré

, Volume 16, Issue 9, pp 2131–2162 | Cite as

Characteristic Initial Data and Smoothness of Scri. I. Framework and Results

  • Piotr T. Chruściel
  • Tim-Torben PaetzEmail author


We analyze the Cauchy problem for the vacuum Einstein equations with data on a complete light-cone in an asymptotically Minkowskian space-time. We provide conditions on the free initial data which guarantee existence of global solutions of the characteristic constraint equations. We present necessary-and-sufficient conditions on characteristic initial data in 3 + 1 dimensions to have no logarithmic terms in an asymptotic expansion at null infinity.


Initial Data Cauchy Problem Asymptotic Expansion Constraint Equation Weyl Tensor 
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.


  1. 1.
    Andersson L., Chruściel P.T.: Hyperboloidal Cauchy data for vacuum Einstein equations and obstructions to smoothness of null infinity. Phys. Rev. Lett. 70(19), 2829–2832 (1993)zbMATHMathSciNetCrossRefADSGoogle Scholar
  2. 2.
    Andersson L., Chruściel P.T., Friedrich H.: On the regularity of solutions to the Yamabe equation and the existence of smooth hyperboloidal initial data for Einsteins field equations. Commun. Math. Phys. 149, 587–612 (1992)CrossRefADSGoogle Scholar
  3. 3.
    Bieri, L., Zipser, N.: Extensions of the stability theorem of the Minkowski space in general relativity. AMS/IP Studies in Advanced Mathematics, vol. 45, American Mathematical Society, Providence, RI, pp. xxiv+491 (2009)Google Scholar
  4. 4.
    Blanchet L.: Radiative gravitational fields in general relativity. II. Asymptotic behaviour at future null infinity. Proc. R. Soc. Lond. Ser. A 409, 383–399 (1987)zbMATHMathSciNetCrossRefADSGoogle Scholar
  5. 5.
    Bondi H., van der Burg M.G.J., Metzner A.W.K.: Gravitational waves in general relativity VII: Waves from axi–symmetric isolated systems. Proc. R. Soc. Lond. A 269, 21–52 (1962)zbMATHCrossRefADSGoogle Scholar
  6. 6.
    Cabet, A., Chruściel, P.T., Tagné Wafo, R.: On the characteristic initial value problem for nonlinear symmetric hyperbolic systems, including Einstein equations (2014). arXiv:1406.3009 [gr-qc]
  7. 7.
    Choquet-Bruhat Y.: Un théorème d’instabilité pour certaines équations hyperboliques non linéares. C. R. Acad. Sci. Paris 276, 281–284 (1973)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Choquet-Bruhat, Y., Chruściel, P.T., Martín-García, J.M.: The light-cone theorem. Class. Quantum Grav. 26, 135011 (2009). arXiv:0905.2133 [gr-qc]
  9. 9.
    Choquet-Bruhat, Y., Chruściel, P.T., Martín-García, J.M.: The Cauchy problem on a characteristic cone for the Einstein equations in arbitrary dimensions. Ann. H. Poincaré 12, 419–482 (2011). arXiv:1006.4467 [gr-qc]
  10. 10.
    Christodoulou D., Klainermann S.: The global nonlinear stability of Minkowski space. Princeton University Press, Princeton (1993)zbMATHGoogle Scholar
  11. 11.
    Chruściel, P.T.: The existence theorem for the Cauchy problem on the light-cone for the vacuum Einstein equations. Forum. Math. Sigma 2, e10 (2014). arXiv:1209.1971 [gr-qc]
  12. 12.
    Chruściel, P.T., MacCallum, M.A.H., Singleton, D.: Gravitational waves in general relativity. XIV: Bondi expansions and the “polyhomogeneity” of Scri. Philos. Trans. Roy. Soc. Lond. Ser. A 350, 113–141 (1995). arXiv:gr-qc/9305021
  13. 13.
    Chruściel, P.T., Paetz, T.-T.: The many ways of the characteristic Cauchy problem. Class. Quantum Grav. 29, 145006 (2012). arXiv:1203.4534 [gr-qc]
  14. 14.
    Friedrich H.: On the regular and the asymptotic characteristic initial value problem for Einstein’s vacuum field equations. Proc. R. Soc. Lond. Ser. A 375, 169–184 (1981)zbMATHMathSciNetCrossRefADSGoogle Scholar
  15. 15.
    Friedrich H.: On the hyperbolicity of Einstein’s and other gauge field equations. Commun. Math. Phys. 100, 525–543 (1985)zbMATHMathSciNetCrossRefADSGoogle Scholar
  16. 16.
    Friedrich H.: Hyperbolic reductions for Einstein’s equations. Class. Quantum Grav. 13, 1451–1469 (1996)zbMATHMathSciNetCrossRefADSGoogle Scholar
  17. 17.
    Friedrich H.: Conformal Einstein evolution. The conformal structure of space-time, Lecture Notes in Phys., vol. 604, Springer, Berlin (2002). arXiv:gr-qc/0209018, pp. 1–50
  18. 18.
    Friedrich H.: Conformal structures of static vacuum data. Commun. Math. Phys. 321, 419–482 (2013)zbMATHMathSciNetCrossRefADSGoogle Scholar
  19. 19.
    Klainerman S., Nicolò F.: On local and global aspects of the Cauchy problem in general relativity. Class. Quantum Grav. 16, R73–R157 (1999)zbMATHCrossRefADSGoogle Scholar
  20. 20.
    Klainerman, S., Nicolò, F.: The evolution problem in general relativity. Progress in Mathematical Physics, vol. 25, Birkhäuser, Boston, MA, (2003)Google Scholar
  21. 21.
    Klainerman S., Nicolò F.: Peeling properties of asymptotically flat solutions to the Einstein vacuum equations. Class. Quantum Grav. 20, 3215–3257 (2003)zbMATHCrossRefADSGoogle Scholar
  22. 22.
    Valiente Kroon, J.A.: On the existence and convergence of polyhomogeneous expansions of zero-rest-mass fields. Class. Quantum Grav. 17, 4365–4375 (2000). arXiv:gr-qc/0005087
  23. 23.
    Valiente Kroon, J.A.: A new class of obstructions to the smoothness of null infinity. Commun. Math. Phys. 244, 133–156 (2004). arXiv:gr-qc/0211024
  24. 24.
    Lindblad, H., and Rodnianski, I.: Global existence for the Einstein vacuum equations in wave coordinates. Commun. Math. Phys. 256, 43–110 (2005). arXiv:math.ap/0312479.
  25. 25.
    Newman E.T., Penrose R.: An approach to gravitational radiation by a method of spin coefficients. J. Math. Phys. 3, 566–578 (1962)MathSciNetCrossRefADSGoogle Scholar
  26. 26.
    Paetz, T.-T.: Characteristic initial data and smoothness of Scri. II. Asymptotic expansions and construction of conformally smooth data sets. J. Math. Phys. (2014). arXiv:1403.3560 [gr-qc] (in press)
  27. 27.
    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, 221–239 (1990)zbMATHMathSciNetCrossRefADSGoogle Scholar
  28. 28.
    Sachs R.K.: Gravitational waves in general relativity VIII. Waves in asymptotically flat space-time. Proc. R. Soc. Lond. A 270, 103–126 (1962)zbMATHMathSciNetCrossRefADSGoogle Scholar
  29. 29.
    Tamburino L.A., Winicour J.H.: Gravitational fields in finite and conformal Bondi frames. Phys. Rev. 150, 1039–1053 (1966)CrossRefADSGoogle Scholar
  30. 30.
    Torrence R.J., Couch W.E.: Generating fields from data on \({{\fancyscript{H}}^-\cup{\fancyscript{H}}^+}\) and \({{\fancyscript{H}}^-\cup{\fancyscript{I}}^-}\). Gen. Rel. Grav. 16, 847–866 (1984)zbMATHMathSciNetCrossRefADSGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Gravitational PhysicsUniversity of ViennaViennaAustria

Personalised recommendations