Skip to main content
Log in

Rotating, Stationary, Axially Symmetric Spacetimes with Collisionless Matter

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

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 \({\mathbb{R}^5}\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Andersson L., Beig R., Schmidt B.G.: Static self-gravitating elastic bodies in Einstein gravity. Commun. Pure Appl. Math. 61, 988–1023 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  2. Andersson L., Beig R., Schmidt B.G.: Rotating elastic bodies in Einstein gravity. Commun. Pure Appl. Math. 63, 559–589 (2009)

    MathSciNet  Google Scholar 

  3. Andréasson H.: Global foliations of matter spacetimes with Gowdy symmetry. Commun. Math. Phys. 206, 337–366 (1999)

    Article  ADS  MATH  Google Scholar 

  4. Andréasson, H.: The Einstein–Vlasov System/Kinetic Theory. Living Rev. Relativ. 14 (2011)

  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)

    Article  ADS  MATH  Google Scholar 

  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)

  7. Batt J., Faltenbacher W., Horst E.: Stationary spherically symmetric models in stellar dynamics. Arch. Rational Mech. Anal. 93, 159–183 (1986)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  8. Deimling, K.: Nonlinear Functional Analysis. Berlin, New York: Springer, 1985

  9. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Berlin, New York: Springer, 1977

  10. Heilig, Heilig : On Lichtenstein’s analysis of rotating Newtonian stars. Ann. Inst. Henri Poincaré, Physique théorique 60, 457–487 (1994)

    MATH  Google Scholar 

  11. Heilig U.: On the existence of rotating stars in general relativity. Commun. Math. Phys. 166, 457–493 (1995)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  13. Lichtenstein, L.: Gleichgewichtsfiguren rotierender Flüssigkeiten. Berlin: Springer, 1933

  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)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  15. Rein G.: Stationary and static stellar dynamic models with axial symmetry. Nonlinear Anal. Theory Methods Appl. 41, 313–344 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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)

    Article  MATH  MathSciNet  Google Scholar 

  18. Schaudt U.: On the Dirichlet problem for the stationary and axisymmetric Einstein equations. Commun. Math. Phys. 190, 509–540 (1998)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  19. Wald, R.: General Relativity. Chicago: Chicago University Press, 1984

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Gerhard Rein.

Additional information

Communicated by P. T. Chruściel

Rights and permissions

Reprints and permissions

About this article

Cite this article

Andréasson, H., Kunze, M. & Rein, G. Rotating, Stationary, Axially Symmetric Spacetimes with Collisionless Matter. Commun. Math. Phys. 329, 787–808 (2014). https://doi.org/10.1007/s00220-014-1904-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-014-1904-5

Keywords

Navigation