Evolution of the Einstein Equations to Future Null Infinity

  • Oliver RinneEmail author
  • Vincent Moncrief
Conference paper
Part of the Springer Proceedings in Physics book series (SPPHY, volume 157)


We describe recent progress with a formulation of the Einstein equations on constant mean curvature surfaces extending to future null infinity. Long-time stable numerical evolutions of an axisymmetric gravitationally perturbed Schwarzschild black hole have been obtained. Here we show how matter can be included in our formulation. We study late-time tails for the spherically symmetric Einstein–Yang–Mills equations both for initial data that disperse and that collapse to a black hole.



O.R. gratefully acknowledges support from the German Research Foundation through a Heisenberg Fellowship and research grant RI 2246/2. V.M. was supported by NSF grant PHY-0963869 to Yale University.


  1. 1.
    Friedrich, H.: Cauchy problems for the conformal vacuum field equations in general relativity. Commun. Math. Phys. 91, 445 (1983). doi: 10.1007/BF01206015 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Moncrief, V., Rinne, O.: Regularity of the Einstein equations at future null infinity. Class. Quantum Grav. 26, 125010 (2009). doi: 10.1088/0264-9381/26/12/125010 ADSCrossRefMathSciNetGoogle Scholar
  3. 3.
    Andersson, L., Chruściel, P., Friedrich, H.: On the regularity of solutions to the Yamabe equation and the existence of smooth hyperboloidal initial data for Einstein’s field equations. Commun. Math. Phys. 149, 587 (1992). doi: 10.1007/BF02096944 ADSCrossRefzbMATHGoogle Scholar
  4. 4.
    Rinne, O., Moncrief, V.: Hyperboloidal Einstein-matter evolution and tails for scalar and Yang–Mills fields, Class. Quantum Grav. 30, 095009 (2013). doi: 10.1088/0264-9381/30/9/095009
  5. 5.
    Rinne, O.: An axisymmetric evolution code for the Einstein equations on hyperboloidal slices. Class. Quantum Grav. 27, 035014 (2010). doi: 10.1088/0264-9381/27/3/035014 ADSCrossRefMathSciNetGoogle Scholar
  6. 6.
    Brill, D., Cavallo, J., Isenberg, J.: K-surfaces in the Schwarzschild space–time and the construction of lattice cosmologies. J. Math. Phys. 21, 2789 (1980). doi: 10.1063/1.524400
  7. 7.
    Bondi, H., van der Burg, M., Metzner, A.: Gravitational waves in general relativity. VII. Waves from axi-symmetric isolated systems. Proc. R. Soc. London, Ser. A 269, 21 (1962). doi: 10.1098/rspa.1962.0161 ADSCrossRefzbMATHGoogle Scholar
  8. 8.
    Pürrer, M., Aichelburg, P.: Tails for the Einstein–Yang–Mills system. Class. Quantum Grav. 26, 035004 (2009). doi: 10.1088/0264-9381/26/3/035004 ADSCrossRefGoogle Scholar
  9. 9.
    Zenginoğlu, A.: A hyperboloidal study of tail decay rates for scalar and Yang–Mills fields. Class. Quantum Grav. 25, 175013 (2008). doi: 10.1088/0264-9381/25/17/175013 ADSCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Max Planck Institute for Gravitational Physics (Albert Einstein Institute)PotsdamGermany
  2. 2.Department of Mathematics and Department of PhysicsYale UniversityNew HavenUSA

Personalised recommendations