Radiating spherical collapse for an inhomogeneous interior solution

  • Eduardo BittencourtEmail author
  • Vanessa P. Freitas
  • José M. Salim
  • Grasiele B. Santos
Research Article


We analyse the problem of gravitational collapse considering the matching of an exterior region described by the Vaidya’s metric and an interior region described by a spherically symmetric shear-free inhomogeneous geometry sourced by a viscous fluid. We establish initial and final conditions for the process in order that the outcome be a nonsingular object, when this is possible, and check how it depends on the fulfillment of the energy conditions. We then apply explicitly the matching procedure to the cases of linear and nonlinear Lagrangians describing electromagnetic fields inside the star, and analyse how the different behaviours for the scale factor of the interior geometry produce singular or nonsingular final stages of the collapse depending on the range where the initial conditions lie.


Gravitational collapse Inhomogeneous models Viscous fluids 



The authors are in debt with R. Klippert for his valuable comments on this manuscript. GBS would like to thank the PCI program at the Brazilian Center for Research in Physics–CBPF, where part of this work was developed, for financial support. VPF thanks FAPERJ for the Grant E-26/200.279/2015.


  1. 1.
    Vaidya, P.V.: The gravitational field of a radiating star. Proc. Ind. Acad. Sci. A 33, 264 (1951)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Fayos, F., Jaen, X., Llanta, E., Senovilla, J.M.M.: Interiors of Vaidya’s radiating metric: gravitational collapse. Phys. Rev. D 45, 2732 (1992)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Santos, N.O.: Non-adiabatic radiating collapse. Mon. Not. R. Astron. Soc. 216, 403 (1985)ADSCrossRefGoogle Scholar
  4. 4.
    Bonnor, W.B., De Oliveira, A.K.G., Santos, N.O.: Radiating spherical collapse. Phys. Rep. 181, 269 (1989)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Fayos, F., Jaen, X., Llanta, E., Senovilla, J.M.M.: Matching of the Vaidya and Robertson-Walker metric. Class. Quantum Gravity 8, 2057 (1991)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Lasky, P., Lun, A.: Spherically symmetric gravitational collapse of general fluids. Phys. Rev. D 75, 024031 (2007)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Darmois, G.: Les équations de la Gravitation Einsteinienne Mémorial des Sciences Mathematiques, vol. 25. Gauthier-Villars, Paris (1927)zbMATHGoogle Scholar
  8. 8.
    Bonnor, W.B., Vickers, P.A.: Junctions conditions in general relativity. Gen. Relativ. Gravit. 13, 29 (1981)ADSCrossRefGoogle Scholar
  9. 9.
    Hawking, S.W., Ellis, G.F.R.: The Large Scale Structure of Space-Time. Cambridge University Press, Cambridge (1973)CrossRefGoogle Scholar
  10. 10.
    Joshi, P.S., Malafarina, D.: Recent developments in gravitational collapse and spacetime singularities. Int. J. Mod. Phys. D 20, 2641 (2011)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Mbonye, M.R., Kazanas, D.: Nonsingular black hole model as a possible end product of gravitational collapse. Phys. Rev. D 72, 024016 (2005)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Dymnikova, I.: Spherically symmetric space-time with the regular de Sitter center. Int. J. Mod. Phys. D 12, 1015 (2003)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Tavakoli, Y., Escamilla-Rivera, C., Fabris, J.C.: The final state of gravitational collapse in Eddington-inspired Born-Infeld theory. Ann. Phys. (Berlin) 529, 1600415 (2017)ADSCrossRefGoogle Scholar
  14. 14.
    Malafarina, D.: Classical collapse to black holes and quantum bounces: a review. Universe 3(2), 48 (2017)ADSCrossRefGoogle Scholar
  15. 15.
    Fayos, F., Senovilla, J.M.M., Torres, R.: General matching of two spherically symmetric spacetimes. Phys. Rev. D 54, 4862 (1996)ADSMathSciNetCrossRefGoogle Scholar
  16. 16.
    Fayos, F., Torres, R.: A class of interiors for Vaidya’s radiating metric: singularity-free gravitational collapse. Class. Quantum Gravity 25, 175009 (2008)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Bittencourt, E., Salim, J.M., Santos, G.B.: Magnetic fields and the Weyl tensor in the early universe. Gen. Relativ. Gravit. 46, 1790 (2014)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Bittencourt, E., Klippert, R., Santos, G.B.: Dynamical wormhole definitions confronted. Class. Quantum Gravity 35, 155009 (2018)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Herrera, L., Santos, N.O.: Local anisotropy in self-gravitating systems. Phys. Rep. 286, 53 (1997)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Hernandez Jr., W.C., Misner, C.W.: Observer time as a coordinate in relativistic spherical hydrodynamics. Astrophys. J. 143, 452 (1966)ADSCrossRefGoogle Scholar
  21. 21.
    Cahill, M.E., McVittie, G.C.: Spherical symmetry and mass-energy in general relativity. II. Particular cases. J. Math. Phys. 11, 1382 (1970)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Bondi, H., van der Burg, M.G.J., Metzner, A.W.K.: Gravitational waves in general relativity. VII. Waves from axi-symmetric isolated systems. Proc. R. Soc. Lond. A 269, 21 (1962)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Tupper, B.O.J.: The equivalence of electromagnetic fields and viscous fluids in general relativity. J. Math. Phys. 22, 2666 (1981)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Tupper, B.O.J.: The equivalence of perfect fluid space-times and magnetohydrodynamic space-times in general relativity. Gen. Relativ. Gravit 15, 47 (1983)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    Tupper, B.O.J.: The equivalence of perfect fluid space-times and viscous magnetohydrodynamic space-times in general relativity. Gen. Relativ. Gravit 15, 849 (1983)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Adler, R.J., Bjorken, J.D., Chen, P., Liu, J.S.: Simple analytic models of gravitational collapse. Am. J. Phys. 73, 1148 (2005)ADSCrossRefGoogle Scholar
  27. 27.
    Mena, F.C., Oliveira, J.M.: Radiative gravitational collapse to spherical, toroidal and higher genus black holes. Ann. Phys. 387, 135 (2017)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    Novello, M., Bergliaffa, S.E.P.: Bouncing cosmologies. Phys. Rep. 463, 127 (2008)ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    De Lorenci, V.A., Klippert, R., Novello, M., Salim, J.M.: Nonlinear electrodynamics and FRW cosmology. Phys. Rev. D 65, 063501 (2002)ADSCrossRefGoogle Scholar
  30. 30.
    Berezin, V.A., Dokuchaev, V.I., Eroshenko, YuN: On maximal analytical extension of the Vaidya metric. Class. Quantum Gravity 33, 145003 (2016)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    Bittencourt, E., Freitas, V.P., Salim, J.M., Santos, G.B.: Nonsingular gravitational collapse: two-fluid approach (in preparation)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Eduardo Bittencourt
    • 1
    Email author
  • Vanessa P. Freitas
    • 2
  • José M. Salim
    • 2
  • Grasiele B. Santos
    • 1
    • 2
  1. 1.Universidade Federal de ItajubáItajubáBrazil
  2. 2.Centro Brasileiro de Pesquisas FísicasRio de JaneiroBrazil

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