Trapping horizon and negative energy

  • Pei-Ming HoEmail author
  • Yoshinori Matsuo
Open Access
Regular Article - Theoretical Physics


Assuming spherical symmetry and the semi-classical Einstein equation, we prove that, for the observers on top of the trapping horizon, the vacuum energy-momentum tensor is always that of an ingoing negative energy flux at the speed of light with a universal energy density ℰ ≃ − 1/(2κa2), (where a is the areal radius of the trapping horizon), which is responsible for the decrease in the black hole’s mass over time. This result is independent of the composition of the collapsing matter and the details of the vacuum energy-momentum tensor. The physics behind the universality of this quantity ℰ and its surprisingly large magnitude will be discussed.


Black Holes Effective Field Theories 


Open Access

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  1. [1]
    P.C.W. Davies, S.A. Fulling and W.G. Unruh, Energy momentum tensor near an evaporating black hole, Phys. Rev. D 13 (1976) 2720 [INSPIRE].Google Scholar
  2. [2]
    R. Parentani and T. Piran, The internal geometry of an evaporating black hole, Phys. Rev. Lett. 73 (1994) 2805 [hep-th/9405007] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    P.-M. Ho and Y. Matsuo, On the near-horizon geometry of an evaporating black hole, JHEP 07 (2018) 047 [arXiv:1804.04821] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    P.-M. Ho, Y. Matsuo and S.-J. Yang, Vacuum energy at apparent horizon in conventional model of black holes, arXiv:1904.01322 [INSPIRE].
  5. [5]
    V.P. Frolov and G.A. Vilkovisky, Spherically symmetric collapse in quantum gravity, Phys. Lett. B 106 (1981) 307 [INSPIRE].MathSciNetCrossRefGoogle Scholar
  6. [6]
    T.A. Roman and P.G. Bergmann, Stellar collapse without singularities?, Phys. Rev. D 28 (1983) 1265 [INSPIRE].MathSciNetGoogle Scholar
  7. [7]
    S.A. Hayward, Formation and evaporation of regular black holes, Phys. Rev. Lett. 96 (2006) 031103 [gr-qc/0506126] [INSPIRE].
  8. [8]
    S.D. Mathur, The information paradox: a pedagogical introduction, Class. Quant. Grav. 26 (2009) 224001 [arXiv:0909.1038] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    D. Marolf, The black hole information problem: past, present and future, Rept. Prog. Phys. 80 (2017) 092001 [arXiv:1703.02143] [INSPIRE].CrossRefGoogle Scholar
  10. [10]
    H. Kawai, Y. Matsuo and Y. Yokokura, A self-consistent model of the black hole evaporation, Int. J. Mod. Phys. A 28 (2013) 1350050 [arXiv:1302.4733] [INSPIRE].CrossRefzbMATHGoogle Scholar
  11. [11]
    H. Kawai and Y. Yokokura, Phenomenological description of the interior of the Schwarzschild black hole, Int. J. Mod. Phys. A 30 (2015) 1550091 [arXiv:1409.5784] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    P.-M. Ho, Comment on self-consistent model of black hole formation and evaporation, JHEP 08 (2015) 096 [arXiv:1505.02468] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  13. [13]
    H. Kawai and Y. Yokokura, Interior of black holes and information recovery, Phys. Rev. D 93 (2016) 044011 [arXiv:1509.08472] [INSPIRE].MathSciNetGoogle Scholar
  14. [14]
    P.-M. Ho, The absence of horizon in black-hole formation, Nucl. Phys. B 909 (2016) 394 [arXiv:1510.07157] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    P.-M. Ho, Asymptotic black holes, Class. Quant. Grav. 34 (2017) 085006 [arXiv:1609.05775] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    H. Kawai and Y. Yokokura, A model of black hole evaporation and 4D Weyl anomaly, Universe 3 (2017) 51 [arXiv:1701.03455] [INSPIRE].CrossRefGoogle Scholar
  17. [17]
    C.R. Stephens, G. ’t Hooft and B.F. Whiting, Black hole evaporation without information loss, Class. Quant. Grav. 11 (1994) 621 [gr-qc/9310006] [INSPIRE].
  18. [18]
    O. Lunin and S.D. Mathur, AdS/CFT duality and the black hole information paradox, Nucl. Phys. B 623 (2002) 342 [hep-th/0109154] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    O. Lunin and S.D. Mathur, Statistical interpretation of Bekenstein entropy for systems with a stretched horizon, Phys. Rev. Lett. 88 (2002) 211303 [hep-th/0202072] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    A. Almheiri, D. Marolf, J. Polchinski and J. Sully, Black holes: complementarity or firewalls?, JHEP 02 (2013) 062 [arXiv:1207.3123] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    S.L. Braunstein, Black hole entropy as entropy of entanglement, or its curtains for the equivalence principle, [arXiv:0907.1190v1].
  22. [22]
    S.L. Braunstein, S. Pirandola and K. Życzkowski, Better late than never: information retrieval from black holes, Phys. Rev. Lett. 110 (2013) 101301 [arXiv:0907.1190] [INSPIRE].CrossRefGoogle Scholar
  23. [23]
    G. ’t Hooft, What happens in a black hole when a particle meets its antipode, arXiv:1804.05744 [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Department of Physics and Center for Theoretical PhysicsNational Taiwan UniversityTaipeiR.O.C.
  2. 2.Department of PhysicsOsaka UniversityToyonakaJapan

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