A gravitational collapse singularity theorem consistent with black hole evaporation

Abstract

The global hyperbolicity assumption present in gravitational collapse singularity theorems is in tension with the quantum mechanical phenomenon of black hole evaporation. In this work, I show that the causality conditions in Penrose’s theorem can be almost completely removed. As a result, it is possible to infer the formation of spacetime singularities even in the absence of predictability and hence compatibly with quantum field theory and black hole evaporation.

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

Fig. 1
Fig. 2
Fig. 3

Notes

  1. 1.

    Introduced a complete Riemannian metric h, this result follows from the application of the limit curve theorem to a sequence of h-arc length parametrized causal curves of the form \(\sigma _k(t)=\sigma (t-a_k)\) where \(a_k\) is a diverging sequence. Since the h-arc length of the curves on both sides of the new origins \(\sigma (-a_k)\) diverges, the limit curve that passes through an accumulation point of \(\{\sigma (-a_k)\}\) is inextendible rather than just past inextendible; see the references for details.

References

  1. 1.

    Penrose, R.: Gravitational collapse and space–time singularities. Phys. Rev. Lett. 14, 57 (1965)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Senovilla, J.M.M., Garfinkle, D.: The 1965 Penrose singularity theorem. Class. Quantum Gravity 32, 124008 (2015)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Hawking, S.W.: Black hole explosions? Nature 248, 30 (1974)

    Article  Google Scholar 

  4. 4.

    Hawking, S.W.: Particle creation by black holes. Commun. Math. Phys. 43, 199 (1975)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Wald, R.M.: Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics. The University of Chicago Press, Chicago (1994)

    Google Scholar 

  6. 6.

    Kodama, H.: Inevitability of a naked singularity associated with the black hole evaporation. Prog. Theor. Phys. 62, 1434 (1979)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Wald, R.M.: Black holes, singularities and predictability. In: Quantum Theory of Gravity. Essays in Honor of the 60th Birthday of Bryce S. DeWitt, S. M. Christensen (ed.), pp. 160–168. CRC Press, Boca Raton (1984)

  8. 8.

    Lesourd, M.: Causal structure of evaporating black holes. Class. Quantum Gravity 36, 025007 (2019)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Unruh, W.G., Wald, R.M.: Information loss. Rep. Prog. Phys. 80, 092002 (2017)

    Article  Google Scholar 

  10. 10.

    Tipler, F.J.: General relativity and conjugate ordinary differential equations. J. Differ. Equ. 30, 165 (1978)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Roman, T.A.: On the “averaged weak energy condition” and Penrose’s singularity theorem. Phys. Rev. D 37, 546 (1988)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Graham, N., Olum, K.D.: Achronal averaged null energy condition. Phys. Rev. D 76, 064001 (2007)

    Article  Google Scholar 

  13. 13.

    Wall, A.C.: Proving the achronal averaged null energy condition from the generalized second law. Phys. Rev. D 81, 024038 (2010)

    Article  Google Scholar 

  14. 14.

    Galloway, G.J., Senovilla, J.M.M.: Singularity theorems based on trapped submanifolds of arbitrary co-dimension. Class. Quantum Gravity 27, 152002 (2010)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Fewster, C.J., Galloway, G.J.: Singularity theorems from weakened energy conditions. Class. Quantum Gravity 28, 125009 (2011)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Case, J.S.: Singularity theorems and the Lorentzian splitting theorem for the Bakry–Emery–Ricci tensor. J. Geom. Phys. 60, 477 (2010)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Woolgar, E.: Scalar-tensor gravitation and the Bakry–Èmery–Ricci tensor. Class. Quantum Gravity 30, 085007 (2013)

    Article  Google Scholar 

  18. 18.

    Hawking, S.W.: The occurrence of singularities in cosmology. III. Causality and singularities. Proc. R. Soc. Lond. A 300(1461), 187 (1967)

    Article  Google Scholar 

  19. 19.

    Hawking, S.W., Ellis, G.F.R.: The Large Scale Structure of Space–Time. Cambridge University Press, Cambridge (1973)

    Google Scholar 

  20. 20.

    Borde, A.: Singularities in closed spacetimes. Class. Quantum Gravity 2, 589 (1985)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Galloway, G.J.: Curvature, causality and completeness in space–times with causally complete spacelike slices. Math. Proc. Camb. Phil. Soc. 99, 367 (1986)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Hawking, S.W., Penrose, R.: The singularities of gravitational collapse and cosmology. Proc. R. Soc. Lond. A 314, 529 (1970)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Minguzzi, E.: Lorentzian causality theory. Living Rev. Relativ. 22, 3 (2019). https://doi.org/10.1007/s41114-019-0019-x

    Article  Google Scholar 

  24. 24.

    Bardeen, J.M.: Proceedings of International Conference GR5, p. 174. USSR, Tbilisi, Georgia (1968)

  25. 25.

    Borde, A.: Open and closed universes, initial singularities, and inflation. Phys. Rev. D 50, 3692 (1994)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Hiscock, W.A.: Models of evaporating black holes. I. Phys. Rev. D 23, 2813 (1981)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Brown, B.A., Lindesay, J.: Construction of a Penrose diagram for a spatially coherent evaporating black hole. Class. Quantum Gravity 25, 105026 (2008)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Evans, L.C.: Partial Differential Equations. American Mathematical Society, Providence (1998)

    Google Scholar 

  29. 29.

    Minguzzi, E.: \(K\)-causality coincides with stable causality. Commun. Math. Phys. 290, 239 (2009). arXiv:0809.1214

    MathSciNet  Article  Google Scholar 

  30. 30.

    Aké, L., Flores, J., Sánchez, M.: Structure of globally hyperbolic spacetimes with timelike boundary. Rev. Mat. Iberoamericana. To appear (2020). arXiv:1808.04412

  31. 31.

    Clarke, C.J.S., Joshi, P.S.: On reflecting spacetimes. Class. Quantum Gravity 5, 19 (1988)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Beem, J.K., Ehrlich, P.E., Easley, K.L.: Global Lorentzian Geometry. Marcel Dekker Inc., New York (1996)

    Google Scholar 

  33. 33.

    Galloway, G.J.: Maximum principles for null hypersurfaces and null splitting theorems. Ann. Henri Poincaré 1, 543 (2000)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Minguzzi, E.: Limit curve theorems in Lorentzian geometry. J. Math. Phys. 49, 092501 (2008). arXiv:0712.3942

    MathSciNet  Article  Google Scholar 

  35. 35.

    Minguzzi, E.: Non-imprisonment conditions on spacetime. J. Math. Phys. 49, 062503 (2008). arXiv:0712.3949

    MathSciNet  Article  Google Scholar 

  36. 36.

    Caponio, E., Javaloyes, M., Sánchez, M.: On the interplay between Lorentzian causality and Finsler metrics of Randers type. Rev. Mat. Iberoam. 27, 919 (2011)

    MathSciNet  Article  Google Scholar 

  37. 37.

    Caponio, E., avaloyes, M., Sánchez, M.: Wind Finslerian Structures: From Zermelo’s Navigation to the Causality of Spacetimes. ArXiv:1407.5494

  38. 38.

    Mars, M., Senovilla, J.M.M.: Trapped surfaces and symmetries. Class. Quantum Gravity 20, L293 (2003)

    MathSciNet  Article  Google Scholar 

  39. 39.

    Bernal, A.N., Sánchez, M.: On smooth Cauchy hypersurfaces and Geroch’s splitting theorem. Commun. Math. Phys. 243, 461 (2003)

    MathSciNet  Article  Google Scholar 

  40. 40.

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

    Google Scholar 

  41. 41.

    Carter, B.: Global structure of the Kerr family of gravitational fields. Phys. Rev. D 174, 1559 (1968)

    Article  Google Scholar 

  42. 42.

    Senovilla, J.M.M.: Trapped surfaces. Int. J. Mod. Phys. D 20, 2139 (2011)

    MathSciNet  Article  Google Scholar 

  43. 43.

    Newman, R.P.A.C.: Black holes without singularities. Gen. Relativ. Gravit. 21, 981 (1989)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

Useful conversations with Ivan P. Costa e Silva, José Luis Flores, Miguel Sánchez and José Senovilla, and useful comments by the referees are acknowledged. Work partially supported by GNFM-INDAM.

Author information

Affiliations

Authors

Corresponding author

Correspondence to E. Minguzzi.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Minguzzi, E. A gravitational collapse singularity theorem consistent with black hole evaporation. Lett Math Phys (2020). https://doi.org/10.1007/s11005-020-01295-9

Download citation

Keywords

  • Singularity theorems
  • Black holes
  • Causality

Mathematics Subject Classification

  • 83C57
  • 83C75
  • 53C50