Einstein-scalar field equation in LTB space-time: General scheme and special solutions

LTB gravity interacting with scalar field
  • Antonio ZeccaEmail author
Regular Article


A complex massive scalar field minimally coupled to the Einstein field equation in the Lemaître-Tolman-Bondi space-time is considered. The energy momentum tensor of the scalar field is assumed to be the source of the Einstein equation. The spherical symmetry implies that the scalar field is allowed to depend only on the radial and time coordinate. In turn the validity of the scalar field equation ensures the consistency of the Einstein field one. The spherical symmetry of the field also ensures a further symmetry condition required by the spherical symmetry of the Ricci scalar. The explicit equations of the scheme are studied in the case the field depends only either on the radial or on the time coordinate. To that end a first general integration step of the system of equations is useful. In the case of purely time-dependent massless scalar field, the solution results in a Robertson-Walker-like space-time. This is an already known homogeneization effect. Such solution also admits an initial inflationary phase and a late accelerated expansion. In the massive time-dependent field case, a very simple but non-trivial solution is given. Such solution is possible because the field can take complex values. The purely static field case is also considered. It has no non-trivial factorized solutions of the physical radius both in the massive and in the massless case.


  1. 1.
    D. Christodoulou, Commun. Math. Phys. 106, 587 (1986)ADSCrossRefGoogle Scholar
  2. 2.
    D. Christodoulou, Commun. Math. Phys. 105, 337 (1986)ADSCrossRefGoogle Scholar
  3. 3.
    M.W. Choptuik, Phys. Rev. Lett. 70, 337 (1993)CrossRefGoogle Scholar
  4. 4.
    A. Wang, Braz. J. Phys. 31, 188 (2001)ADSCrossRefGoogle Scholar
  5. 5.
    S.M.C.V. Gonsalves, I.G. Moss, Class. Quantum Grav. 14, 2607 (1997)ADSCrossRefGoogle Scholar
  6. 6.
    L.A. Urena-Lopez, J. Phys.: Conf. Ser. 761, 012076 (2016)Google Scholar
  7. 7.
    E.W. Kolb, M.S. Turner, The Early Universe (Addison-Wesley, 1990)Google Scholar
  8. 8.
    A. Randall, Class. Quantum Grav. 21, 2445 (2004)ADSCrossRefGoogle Scholar
  9. 9.
    R. Aguila et al., Eur. Phys. J. C 74, 3158 (2014)ADSCrossRefGoogle Scholar
  10. 10.
    G. Lemaitre, Gen. Relativ. Gravit. 29, 641 (1997)ADSCrossRefGoogle Scholar
  11. 11.
    R.C. Tolman, Proc. Natl. Acad. Sci. U.S.A. 20, 169 (1934)ADSCrossRefGoogle Scholar
  12. 12.
    H. Bondi, Mon. Not. R. Astron. Soc. 107, 410 (1947)ADSCrossRefGoogle Scholar
  13. 13.
    A.E. Romano, Phys. Rev. D 75, 043509 (2007)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    K. Enqvist, Gen. Relativ. Gravit. 40, 451 (2008)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    J.A. Stein-Schabes, Phys. Rev. 35, 235 (1987)Google Scholar
  16. 16.
    A. Zecca, Adv. Stud. Theor. Phys. 12, 9 (2018)CrossRefGoogle Scholar
  17. 17.
    A. Zecca, Adv. Stud. Theor. Phys. 7, 1101 (2013)CrossRefGoogle Scholar
  18. 18.
    A. Krasinski, Inhomogeneous Cosmological Models (Cambridge University Press, Cambridge, 1997)Google Scholar
  19. 19.
    V. Varela, Rev. Bras. Fis. 18, 617 (1988)Google Scholar
  20. 20.
    M. Reiris, Gen. Relativ. Gravit. 49, 46 (2017)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    N.D. Birrell, P.C.W. Davies, Quantum Fields in Curved Space-Time (Cambridge University Press, Cambridge, 1982)Google Scholar
  22. 22.
    L. Landau, E. Lifchitz, Theorie Quantique Relativiste, Premiere Partie (Edition Mir. Moscou, 1972)Google Scholar
  23. 23.
    Giesel, J. Tambornino, T. Thiemann, Class. Quantum Grav. 27, 105013 (2010)ADSCrossRefGoogle Scholar
  24. 24.
    A. Zecca, Nuovo Cimento B 116, 341 (2001)ADSGoogle Scholar
  25. 25.
    S. Weinberg, Cosmology (Oxford University Press, New York, 2008)Google Scholar
  26. 26.
    R. Maartens, Philos. Trans. R. Soc. A 369, 5115 (2011)ADSCrossRefGoogle Scholar
  27. 27.
    G.F.R. Ellis, Gen. Relativ. Gravit. 11, 281 (1979)ADSCrossRefGoogle Scholar

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© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Dipartimento di Fisica dell’Università degli Studi di MilanoMilanoItaly
  2. 2.GNFMGruppo Nazionale per la Fisica MatematicaMilanoItaly

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