Annales Henri Poincaré

, Volume 12, Issue 1, pp 49–65 | Cite as

Linear Perturbations for the Vacuum Axisymmetric Einstein Equations

  • Sergio DainEmail author
  • Martín Reiris


In axial symmetry, there is a gauge for Einstein equations such that the total mass of the spacetime can be written as a conserved, positive definite, integral on the spacelike slices. This property is expected to play an important role in the global evolution. In this gauge the equations reduce to a coupled hyperbolic–elliptic system which is formally singular at the axis. Due to the rather peculiar properties of the system, the local in time existence has proved to resist analysis by standard methods. To analyze the principal part of the equations, which may represent the main source of the difficulties, we study linear perturbation around the flat Minkowski solution in this gauge. In this article we solve this linearized system explicitly in terms of integral transformations in a remarkable simple form. This representation is well suited to obtain useful estimates to apply in the non-linear case.


Bessel Function Compact Support Einstein Equation Momentum Space Elliptic System 
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.
    Dain S.: Axisymmetric evolution of Einstein equations and mass conservation. Class. Quantum Grav. 25, 145021 (2008)CrossRefMathSciNetADSGoogle Scholar
  2. 2.
    Dain S.: The inequality between mass and angular momentum for axially symmetric black holes. Int. J. Mod. Phys. D 17(3–4), 519–523 (2008)zbMATHCrossRefMathSciNetADSGoogle Scholar
  3. 3.
    Dain S.: Proof of the angular momentum-mass inequality for axisymmetric black holes. J. Differ. Geom. 79(1), 33–67 (2008)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Dain S., Ortiz Omar E.: Well-posedness, linear perturbations, and mass conservation for the axisymmetric Einstein equations. Phys. Rev. D 81(4), 044040 (2010)CrossRefADSGoogle Scholar
  5. 5.
    Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1966) (Reprint of the second (1944) edition)Google Scholar
  6. 6.
    Weinstein A.: Generalized axially symmetric potential theory. Bull. Am. Math. Soc. 59, 20–38 (1953)zbMATHCrossRefGoogle Scholar
  7. 7.
    Zemanian A.H.: Generalized Integral Transformations. 2nd edn. Dover Publications Inc., New York (1987)zbMATHGoogle Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  1. 1.Facultad de Matemática, Astronomía y FísicaFaMAF, Universidad Nacional de Córdoba, Instituto de Física Enrique Gaviola, IFEG, CONICETCórdobaArgentina
  2. 2.Max Planck Institute for Gravitational Physics, (Albert Einstein Institute)PotsdamGermany

Personalised recommendations