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Communications in Mathematical Physics

, Volume 329, Issue 2, pp 787–808 | Cite as

Rotating, Stationary, Axially Symmetric Spacetimes with Collisionless Matter

  • Håkan Andréasson
  • Markus Kunze
  • Gerhard Rein
Article

Abstract

The existence of stationary solutions to the Einstein–Vlasov system which are axially symmetric and have non-zero total angular momentum is shown. This provides mathematical models for rotating, general relativistic and asymptotically flat non-vacuum spacetimes. If angular momentum is allowed to be non-zero, the system of equations to solve contains one semilinear elliptic equation which is singular on the axis of rotation. This can be handled very efficiently by recasting the equation as one for an axisymmetric unknown on \({\mathbb{R}^5}\).

Keywords

Angular Momentum Symmetric Function Implicit Function Theorem Bianchi Identity Total Angular Momentum 
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.

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References

  1. 1.
    Andersson L., Beig R., Schmidt B.G.: Static self-gravitating elastic bodies in Einstein gravity. Commun. Pure Appl. Math. 61, 988–1023 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Andersson L., Beig R., Schmidt B.G.: Rotating elastic bodies in Einstein gravity. Commun. Pure Appl. Math. 63, 559–589 (2009)MathSciNetGoogle Scholar
  3. 3.
    Andréasson H.: Global foliations of matter spacetimes with Gowdy symmetry. Commun. Math. Phys. 206, 337–366 (1999)ADSCrossRefzbMATHGoogle Scholar
  4. 4.
    Andréasson, H.: The Einstein–Vlasov System/Kinetic Theory. Living Rev. Relativ. 14 (2011)Google Scholar
  5. 5.
    Andréasson H., Kunze M., Rein G.: Existence of axially symmetric static solutions of the Einstein–Vlasov system. Commun. Math. Phys. 308, 23–47 (2011)ADSCrossRefzbMATHGoogle Scholar
  6. 6.
    Bardeen, J.: Rapidly rotating stars, disks, and black holes. In: DeWitt, C., DeWitt, B.S. (eds.) Black Holes/Les Astres Occlus. Les Houches (1972)Google Scholar
  7. 7.
    Batt J., Faltenbacher W., Horst E.: Stationary spherically symmetric models in stellar dynamics. Arch. Rational Mech. Anal. 93, 159–183 (1986)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Deimling, K.: Nonlinear Functional Analysis. Berlin, New York: Springer, 1985Google Scholar
  9. 9.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Berlin, New York: Springer, 1977Google Scholar
  10. 10.
    Heilig, Heilig : On Lichtenstein’s analysis of rotating Newtonian stars. Ann. Inst. Henri Poincaré, Physique théorique 60, 457–487 (1994)zbMATHGoogle Scholar
  11. 11.
    Heilig U.: On the existence of rotating stars in general relativity. Commun. Math. Phys. 166, 457–493 (1995)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Lichtenstein L.: Untersuchung über die Gleichgewichtsfiguren rotierender Flüssigkeiten, deren Teilchen einander nach dem Newtonschen Gesetze anziehen. Erste Abhandlung. Homogene Flüssigkeiten. Allgemeine Existenzsätze. Math. Z. 1, 229–284 (1918)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Lichtenstein, L.: Gleichgewichtsfiguren rotierender Flüssigkeiten. Berlin: Springer, 1933Google Scholar
  14. 14.
    Pfister H., Schaudt U.: The boundary value problem for the stationary and axisymmetric Einstein equations is generically solvable. Phys. Rev. Lett. 77, 3284–3287 (1996)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Rein G.: Stationary and static stellar dynamic models with axial symmetry. Nonlinear Anal. Theory Methods Appl. 41, 313–344 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Rein, G., Rendall, A.: Global existence of solutions of the spherically symmetric Vlasov–Einstein system with small initial data. Commun. Math. Phys. 150, 561–583 (1992). [Erratum: Comm. Math. Phys. 176, 475–478 (1996)]Google Scholar
  17. 17.
    Rein G., Rendall A.: Compact support of spherically symmetric equilibria in non-relativistic and relativistic galactic dynamics. Math. Proc. Camb. Phil. Soc. 128, 363–380 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Schaudt U.: On the Dirichlet problem for the stationary and axisymmetric Einstein equations. Commun. Math. Phys. 190, 509–540 (1998)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Wald, R.: General Relativity. Chicago: Chicago University Press, 1984Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Håkan Andréasson
    • 1
  • Markus Kunze
    • 2
  • Gerhard Rein
    • 3
  1. 1.Mathematical Sciences, Chalmers University of TechnologyGöteborg UniversityGöteborgSweden
  2. 2.Mathematisches InstitutUniversität KölnKölnGermany
  3. 3.Fakultät für Mathematik, Physik und InformatikUniversität BayreuthBayreuthGermany

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