Advertisement

Afrika Matematika

, Volume 29, Issue 3–4, pp 615–623 | Cite as

On the characteristic initial value problem for the spherically symmetric Einstein–Euler equations

  • Calvin Tadmon
Article
  • 33 Downloads

Abstract

We consider the Einstein equations for a barotropic irrotational perfect fluid with equation of state \(p=\rho \), where p is the pressure of the fluid, and \(\rho \) its mass-energy density. We express \(\rho \) in terms of the potential introduced thanks to the irrotationality of the fluid, obtaining therefore the relativistic analogue of Bernoulli’s law well known in classical Fluids Mechanics. Under the spherical symmetry assumption we reduce the problem to a highly nonlinear evolution partial integrodifferential equation which we solve globally by a fixed point method, under smallness condition on the initial datum and regularity assumption at the center of symmetry. All the investigation is performed in Bondi coordinates so that the problem dealt with is a characteristic initial value problem with initial datum prescribed on a light cone with vertex at the center of symmetry.

Keywords

Characteristic initial value problem Global solution Einstein equations Irrotational perfect fluid Spherical symmetry Bondi coordinates 

Mathematics Subject Classification

35A01 35A02 35A09 35B06 83C10 83C20 83C55 

Notes

Acknowledgements

I acknowledge support from the Abdus Salam International Centre for Theoretical Physics (ICTP, Trieste-Italy) where this work was finalized at the beginning of one of my research visits as Senior Guest Scientist in the Mathematics Group of ICTP. I also express my thankfulness to the anonymous reviewers for their suggestions that helped improve the paper.

References

  1. 1.
    Bestler, H.L., et al.: What’s inside the cone? Numerically reconstructing the metric from observations. JCAP 02, 009 (2014)CrossRefGoogle Scholar
  2. 2.
    Bestler, H.L., et al.: Towards the geometry of the Universe from data. Not. R. Astron. Soc. 453, 2364–2377 (2015)Google Scholar
  3. 3.
    Brauer, U., Karp, L.: Local existence of classical solutions for the Einstein-Euler system using weighted Sobolev spaces of fractional order. C. R. Acad. Sci. Paris Ser. I(345), 49–54 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chae, D.: Global existence of solutions to the coupled Einstein and Maxwell–Higgs system in the spherical symmetry. Ann. Henri Poincaré 4, 35–62 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Choquet-Bruhat, Y.: General Relativity and the Einstein Equations, Oxford Mathematical Monographs. Oxford University Press, Oxford (2009)zbMATHGoogle Scholar
  6. 6.
    Christodoulou, D.: The Formation of Black Holes in General Relativity, EMS Monographs in Mathematics. European Mathematical Society (EMS), Zürich (2009)CrossRefzbMATHGoogle Scholar
  7. 7.
    Christodoulou, D.: Self-gravitating relativistic fluids: a two-phase model. Arch. Ration. Mech. Anal. 130(130), 343–400 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Christodoulou, D.: The problem of a self-gravitating scalar field. Commun. Math. Phys. 105, 337–361 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dossa, M., Tadmon, C.: The Goursat problem for the Einstein-Yang-Mills-Higgs system in weighted Sobolev spaces. C. R. Acad. Sci. Paris Ser. I(348), 35–39 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dossa, M., Tadmon, C.: The characteristic initial value problem for the Einstein-Yang-Mills-Higgs system in weighted Sobolev spaces. Appl. Math. Res. Express 2010(2), 154–231 (2010)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Lefloch, P.G., Stewart, J.M.: The characteristic initial value problem for plane symmetric spacetimes with weak regularity. Class. Quantum Grav. 28(14), 145019 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lim, W.C., et al.: Spherically symmetric cosmological spacetimes with dust and radiation numerical implementation. JCAP 10, 010 (2013)CrossRefGoogle Scholar
  13. 13.
    Rendall, A.D.: Reduction of the characteristic initial value problem to the Cauchy problem and its applications to the Einstein equations. Proc. R. Soc. A 427, 221–239 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Rendall, A.D.: The initial value problem for a class of general relativistic fluid bodies. J. Math. Phys. 33(3), 1047–1053 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Sango, M., Tadmon, C.: On global well-posedness for the Einstein–Maxwell–Euler system in Bondi coordinates. Rend. Sem. Mat. Univ. Padova 131, 179–192 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Tabensky, R., Taub, A.H.: Plane symmetric self-gravitating fluids with pressure equal to energy density. Commun. Math. Phys. 29, 61–77 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Tadmon, C.: Global solutions, and their decay properties, of the spherically symmetric \(su(2)-\)Einstein–Yang–Mills–Higgs equations. C. R. Acad. Sci. Paris Ser. I(349), 1067–1072 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Tadmon, C., Tchapnda, S.B.: On the spherically symmetric Einstein–Yang–Higgs equations in Bondi coordinates. Proc. R. Soc. A 468, 3191–3214 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    van der Walt, P.J., Bishop, N.T.: Observational cosmology using characteristic numerical relativity. Phys. Rev. D 82, 084001 (2010)CrossRefGoogle Scholar
  20. 20.
    van der Walt, P.J., Bishop, N.T.: Observational cosmology using characteristic numerical relativity: characteristic formalism on null geodesics. Phys. Rev. D 85, 044016 (2012)CrossRefGoogle Scholar

Copyright information

© African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of DschangDschangCameroon
  2. 2.The Abdus Salam International Centre for Theoretical PhysicsTriesteItaly

Personalised recommendations