A variational approach to relativistic fluid balls

Abstract

This paper deals with spherically symmetric static solutions of the relativistic field equation which represents a perfect fluid ball with given equation of state and given baryon number. We prove existence of solutions for baryon numbers that do not exceed a limiting number determined by the equation of state. The proof is based on energy minimization under the given baryon number constraint.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Auchmuty, J.F.G., Beals, R.: Variational solutions of some nonlinear free boundary problems. Arch. Rat. Mech. Anal. 43, 255–271 (1971)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Baym, G., Pethick, C., Sutherland, P.: The ground state of matter at high densities: equation of state and stellar models. Astrophys. Journ. 170, 299–317 (1971)

    Article  Google Scholar 

  3. 3.

    Baym, G., Bethe, H.A., Pethick, C.: Neutron star matter. Nucl. Physics A 175, 225–271 (1971)

    Article  Google Scholar 

  4. 4.

    Below, J. von, Kaul, H.: On the rotating star problem in general relativity. Report No 218 Sonderforschungsbereich DFG 382, 2006

  5. 5.

    Benci, V., Rabinowitz, P.H.: Critical point theorems for indefinite functionals. Invent. math. 52, 241–273 (1979)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Chandrasekhar, S.: An introduction to the study of stellar structure. University. of Chicago Press, chicago (1939)

    Google Scholar 

  7. 7.

    Fjällborg, M., Heinzle, J.M., Uggla, C.: Self-gravitating stationary spherically symmetric systems in relativistic galactic dynamics. Math. Proc. Cambridge Philos. Soc. 143(3), 731–752 (2007)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Harrison, B.K., Thorne, K.S., Wakano, M., Wheeler, J.A.: Gravitation theory and gravitational collapse. Univ. of Chicago Press, chicago (1965)

    Google Scholar 

  9. 9.

    Hartle, J.B., Sharp, D.H.: Variational principle for the equilibrium of a relativistic rotating star. Astrophys. Journ. 147, 317–339 (1967)

    Article  Google Scholar 

  10. 10.

    Hawking, S.H., Ellis, G.F.R.: The large scale structure of space-time. Cambridge Univ. Press, (1973)

  11. 11.

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

    MathSciNet  Article  Google Scholar 

  12. 12.

    Hsu, C.-H., Lin, S.-S., Makino, T.: On spherically symmetric solutions of the relativistic Euler equation. J. Differential Equations 201, 1–24 (2004)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Li, Y.Y.: On uniformly rotating stars. Arch. Rat. Mech. Anal. 115, 357–393 (1991)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Lieb, E.H., Yau, H.-T.: The chandrasekhar theory of stellar collapse as the limit of quantum mechanics. Comm. Math. Phys. 112, 147–174 (1987)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Misner, C.W., Thorne, K.S., Wheeler, J.A.: Gravitation. Freeman, New York (1973)

    Google Scholar 

  16. 16.

    Pandharipande, V.: Hyperionic matter. Nucl. Physics A 178, 123–144 (1971)

    Article  Google Scholar 

  17. 17.

    Pfister, H.: A new and quite general existence proof for static and spherically symmetric perfect fluid stars in general relativity. Class. Quantum Grav. 28, 075006 (2011)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Ramming, T., Rein, G.: Spherically symmetric equilibria for self-gravitating kinetic or fluid models in the nonrelativistic and relativistic case-a simple proof for finite extension. SIAM J. Math. Anal. 45(2), 900–914 (2013)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Rendall, A.D., Schmidt, B.G.: Existence and properties of spherically symmetric static bodies with a given equation of state. Class. Quant. Grav. 8, 985–1000 (1991)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Shapiro, S.L., Teukolsky, S.A.: Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects. John Wiley and Sons, New York (1983)

    Google Scholar 

  21. 21.

    Straumann, N.: General relativity and relativistic astrophysics. Springer, New York (2004)

    Google Scholar 

  22. 22.

    Wu, X.: The structure of rotating neutron stars. Ph.D. Thesis, Department of Physics, Universität Tübingen (1992)

Download references

Acknowledgements

The authors are grateful to Simon Brendle, Gerhard Huisken, Frank Loose, Herbert Pfister, and Urs Schaudt for helpful conversations.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Joachim von Below.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by M. Struwe.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

von Below, J., Kaul, H. A variational approach to relativistic fluid balls. Calc. Var. 60, 31 (2021). https://doi.org/10.1007/s00526-020-01898-z

Download citation

Mathematics Subject Classification

  • 83C05
  • 35R35
  • 74G65