• Valerio Faraoni
Part of the Lecture Notes in Physics book series (LNP, volume 907)


This chapter reviews standard tools used in the analysis of horizon and black hole physics, most notably the congruences of null geodesics crossing a horizon. Rindler, event, Killing, apparent, trapping, dynamical, and other horizons are defined, as well as the Kodama vector (in spherical symmetry) and various notions of surface gravity.


Black Hole Event Horizon Spherical Symmetry Apparent Horizon Killing Vector 
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  1. 1.
    Abreu, G., Visser, M.: Kodama time: geometrically preferred foliations of spherically symmetric spacetimes. Phys. Rev. D 82, 044027 (2010)ADSCrossRefGoogle Scholar
  2. 2.
    Andersson, L., Mars, M., Simon, W.: Local existence of dynamical and trapping horizons. Phys. Rev. Lett. 95, 11102 (2005)Google Scholar
  3. 3.
    Ashtekar, A., Corichi, A.: Laws governing isolated horizons: inclusion of dilaton couplings. Class. Quantum Grav.17, 1317 (2000)zbMATHMathSciNetADSCrossRefGoogle Scholar
  4. 4.
    Ashtekar, A., Galloway, G.J.: Some uniqueness results for dynamical horizons. Adv. Theor. Math. Phys. 9, 1 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Ashtekar, A., Krishnan, B.: Dynamical horizons and their properties. Phys. Rev. D 68, 104030 (2003)MathSciNetADSCrossRefGoogle Scholar
  6. 6.
    Ashtekar, A., Krishnan, B.: Isolated and dynamical horizons and their applications. Living Rev. Relat. 7, 10 (2004)ADSGoogle Scholar
  7. 7.
    Ashtekar, A., Beetle, C., Fairhurst, S.: Isolated horizons: a generalization of black hole mechanics. Class. Quantum Grav.16, L1 (1999)zbMATHMathSciNetADSCrossRefGoogle Scholar
  8. 8.
    Ashtekar, A., Beetle, C., Fairhurst, S.: Mechanics of isolated horizons. Class. Quantum Grav. 17, 253 (2000)zbMATHMathSciNetADSCrossRefGoogle Scholar
  9. 9.
    Ashtekar, A., Beetle, C., Lewandowski, J.: Geometry of generic isolated horizons. Class. Quantum Gravity 19, 1195 (2002)zbMATHMathSciNetADSCrossRefGoogle Scholar
  10. 10.
    Ashtekar, A., Beetle, C., Lewandowski, J.: Mechanics of rotating isolated horizons. Phys. Rev. D 64, 044016 (2002)MathSciNetADSCrossRefGoogle Scholar
  11. 11.
    Ashtekar, A., Corichi, A., Krasnov, K.: Isolated horizons: the classical phase space. Adv. Theor. Math. Phys. 3, 419 (2000)MathSciNetGoogle Scholar
  12. 12.
    Ashtekar, A., Fairhurst, S., Krishnan, B.: Isolated horizons: Hamiltonian evolution and the first law. Phys. Rev. D 62, 104025 (2000)MathSciNetADSCrossRefGoogle Scholar
  13. 13.
    Ashtekar, A., Beetle, C., Dreyer, O., Fairhurst, S., Krishnan, B., Lewandowski, J., Wiśnieski, J.: Isolated horizons and their applications. Phys. Rev. Lett. 85, 3564 (2000)MathSciNetADSCrossRefGoogle Scholar
  14. 14.
    Bak, D., Rey, S.-J.: Cosmic holography. Class. Quantum Grav. 17, L83 (2000)zbMATHMathSciNetADSCrossRefGoogle Scholar
  15. 15.
    Barceló, C., Liberati, S., Sonego, S., Visser, M.: Hawking-like radiation does not require a trapped region. Phys. Rev. Lett. 97, 171301 (2006)MathSciNetADSCrossRefGoogle Scholar
  16. 16.
    Baumgarte, T.W., Shapiro, S.L.: Numerical relativity and compact binaries. Phys. Rept. 376, 41 (2003)zbMATHMathSciNetADSCrossRefGoogle Scholar
  17. 17.
    Bellucci, S., Faraoni, V.: Energy conditions and classical scalar fields. Nucl. Phys. B 640, 453 (2002)zbMATHMathSciNetADSCrossRefGoogle Scholar
  18. 18.
    Bengtsson, I., Senovilla, J.M.M.: Region with trapped surfaces in spherical symmetry, its core, and their boundaries. Phys. Rev. D 83, 044012 (2011)ADSCrossRefGoogle Scholar
  19. 19.
    Ben-Dov, I.: Outer trapped surfaces in Vaidya spacetimes. Phys. Rev. D 75, 064007 (2007)MathSciNetADSCrossRefGoogle Scholar
  20. 20.
    Bergmann, P.G.: Comments on the scalar tensor theory. Int. J. Theor. Phys. 1, 25 (1968)CrossRefGoogle Scholar
  21. 21.
    Booth, I.: Black hole boundaries. Can. J. Phys. 83, 1073 (2005)ADSCrossRefGoogle Scholar
  22. 22.
    Booth, I., Fairhurst, S.: The first law for slowly evolving horizons. Phys. Rev. Lett. 92, 011102 (2004)ADSCrossRefGoogle Scholar
  23. 23.
    Booth, I., Fairhurst, S.: Isolated, slowly evolving, and dynamical trapping horizons: Geometry and mechanics from surface deformations. Phys. Rev. D 75, 084019 (2007)MathSciNetADSCrossRefGoogle Scholar
  24. 24.
    Booth, I., Brits, L., Gonzalez, J.A., Van den Broeck, V.: Marginally trapped tubes and dynamical horizons. Class. Quantum Grav. 23, 413 (2006)zbMATHADSCrossRefGoogle Scholar
  25. 25.
    Born, M.: Die theorie des starren elektrons in der kinematik des relativitätsprinzips. Ann. Physik (Leipzig) 335, 1 (1909)Google Scholar
  26. 26.
    Brans, C., Dicke, R.H.: Mach’s principle and a relativistic theory of gravitation. Phys. Rev. 124, 925 (1961)zbMATHMathSciNetADSCrossRefGoogle Scholar
  27. 27.
    Carroll, S.M.: Spacetime and Geometry—An Introduction to General Relativity (Addison-Wesley, San Francisco, 2004)zbMATHGoogle Scholar
  28. 28.
    Chriusciel, P.T.: Uniqueness of stationary, electro-vacuum black holes revisited. Helv. Physica Acta 69, 529 (1996)ADSGoogle Scholar
  29. 29.
    Chu, T., Pfeiffer, H.P., Cohen, M.I.: Horizon dynamics of distorted rotating black holes. Phys. Rev. D 83, 104018 (2011)ADSCrossRefGoogle Scholar
  30. 30.
    Clifton, T.: Properties of black hole radiation from tunnelling. Class. Quantum Grav. 25, 175022 (2008)MathSciNetADSCrossRefGoogle Scholar
  31. 31.
    Collins, W.: Mechanics of apparent horizons. Phys. Rev. D 45, 495 (1992)MathSciNetADSCrossRefGoogle Scholar
  32. 32.
    Corichi, A., Sudarsky, D.: When is \(S = A/4\)? Mod. Phys. Lett. A 17, 1431 (2002)zbMATHMathSciNetADSCrossRefGoogle Scholar
  33. 33.
    Csizmadia, P., Rácz, I.: Gravitational collapse and topology change in spherically symmetric dynamical systems. Class. Quantum Grav. 27, 015001 (2010)ADSCrossRefGoogle Scholar
  34. 34.
    Davies, P.C.W.: Scalar production in Schwarzschild and Rindler metrics. J. Phys. A 8, 609 (1975)ADSCrossRefGoogle Scholar
  35. 35.
    Di Criscienzo, R., Nadalini, M., Vanzo, L., Zerbini, S., Zoccatelli, G.: On the Hawking radiation as tunneling for a class of dynamical black holes. Phys. Lett. B 657, 107 (2007)zbMATHMathSciNetADSCrossRefGoogle Scholar
  36. 36.
    Di Criscienzo, R., Hayward, S.A., Nadalini, M., Vanzo, L., Zerbini, S.: Hamilton-Jacobi method for dynamical horizons in different coordinate gauges. Class. Quantum Grav. 27, 015006 (2010)ADSCrossRefGoogle Scholar
  37. 37.
    Dyer, C.C., Honig, E.: Conformal Killing horizons. J. Math. Phys. 20, 409 (1979)zbMATHMathSciNetADSCrossRefGoogle Scholar
  38. 38.
    d’Inverno, R.: Introducing Einstein’s Relativity. Oxford University Press, Oxford (2002)Google Scholar
  39. 39.
    Eardley, D.: Black hole boundary conditions and coordinate conditions. Phys. Rev. D 57, 2299 (1998)MathSciNetADSCrossRefGoogle Scholar
  40. 40.
    Eling, C., Guedens, R., Jacobson, T.: Nonequilibrium thermodynamics of spacetime. Phys. Rev. Lett. 96, 121301 (2006)MathSciNetADSCrossRefGoogle Scholar
  41. 41.
    Fairhurst, S., Krishnan, B.: Distorted black holes with charge. Int. J. Mod. Phys. D 10, 691 (2001)zbMATHMathSciNetADSCrossRefGoogle Scholar
  42. 42.
    Figueras, P., Hubeny, V.E., Rangamani, M., Ross, S.F.: Dynamical black holes and expanding plasmas. J. High Energy Phys. 0904, 137 (2009)ADSCrossRefGoogle Scholar
  43. 43.
    Fodor, G., Nakamura, K., Oshiro, Y., Tomimatsu, A.: Surface gravity in dynamical spherically symmetric space-times. Phys. Rev. D 54, 3882 (1996)MathSciNetADSCrossRefGoogle Scholar
  44. 44.
    Fulling, S.A.: Nonuniqueness of canonical field quantization in Riemannian space-time. Phys. Rev. D 7, 2850 (1973)ADSCrossRefGoogle Scholar
  45. 45.
    Haijcek, P.: Origin of Hawking radiation. Phys. Rev. D 36, 1065 (1987)MathSciNetADSCrossRefGoogle Scholar
  46. 46.
    Hawking, S.W.: Gravitational radiation in an expanding universe. J. Math. Phys. 9, 598 (1968)ADSCrossRefGoogle Scholar
  47. 47.
    Hawking, S.W.: Black holes in general relativity. Commun. Math. Phys. 25, 152 (1972)MathSciNetADSCrossRefGoogle Scholar
  48. 48.
    Hawking, S.W., G.Ellis, F.R.: The Large Scale Structure of Space-Time. Cambridge University Press, Cambridge (1973)Google Scholar
  49. 49.
    Hayward, S.A.: Quasilocal gravitational energy. Phys. Rev. D 49, 831 (1994)MathSciNetADSCrossRefGoogle Scholar
  50. 50.
    Hayward, S.A.: General laws of black hole dynamics. Phys. Rev. D 49, 6467 (1994)MathSciNetADSCrossRefGoogle Scholar
  51. 51.
    Hayward, S.A.: Gravitational energy in spherical symmetry. Phys. Rev. D 53, 1938 (1996)MathSciNetADSCrossRefGoogle Scholar
  52. 52.
    Hayward, S.A.: Unified first law of black-hole dynamics and relativistic thermodynamics. Class. Quantum Grav. 15, 3147 (1998)zbMATHMathSciNetADSCrossRefGoogle Scholar
  53. 53.
    Hayward, S.A.: Formation and Evaporation of Nonsingular Black Holes. Phys. Rev. Lett. 96, 031103 (2006)ADSCrossRefGoogle Scholar
  54. 54.
    Hayward, S.A., Di Criscienzo, R., Nadalini, M., Vanzo, L., Zerbini, S.: Local Hawking temperature for dynamical black holes. Class. Quantum Grav. 26, 062001 (2009)ADSCrossRefGoogle Scholar
  55. 55.
    Hernandez, W.C., Misner, C.W.: Observer time as a coordinate in relativistic spherical hydrodynamics. Astrophys. J. 143, 452 (1966)ADSCrossRefGoogle Scholar
  56. 56.
    Hiscock, W.A.: Gravitational entropy of nonstationary black holes and spherical shells. Phys. Rev. D 40, 1336 (1989)zbMATHMathSciNetADSCrossRefGoogle Scholar
  57. 57.
    Jacobson, T.: Thermodynamics of spacetime: the Einstein equation of state. Phys. Rev. Lett. 75, 1260 (1995)zbMATHMathSciNetADSCrossRefGoogle Scholar
  58. 58.
    Jang, K.-X., Feng, T., Peng, D.-T.: Hawking radiation of apparent horizon in a FRW universe as tunneling beyond semiclassical approximation. Int. J. Theor. Phys. 48, 2112 (2009)CrossRefGoogle Scholar
  59. 59.
    Kavanagh, W., Booth, I.: Spacetimes containing slowly evolving horizons. Phys. Rev. D 74, 044027 (2006)MathSciNetADSCrossRefGoogle Scholar
  60. 60.
    Kodama, H.: Conserved energy flux from the spherically symmetric system and the back reaction problem in the black hole evaporation. Progr. Theor. Phys. 63, 1217 (1980)ADSCrossRefGoogle Scholar
  61. 61.
    McClure, M.L., Dyer, C.C.: Asymptotically Einstein-de Sitter cosmological black holes and the problem of energy conditions. Class. Quantum Grav. 23, 1971 (2006)zbMATHMathSciNetADSCrossRefGoogle Scholar
  62. 62.
    McClure, M.L., Anderson, K., Bardahl, K.: Cosmological versions of Vaidya’s radiating stellar exterior, an accelerating reference frame, and Kinnersley’s photon rocket. Preprint arXiv:0709.3288Google Scholar
  63. 63.
    McClure, M.L., Anderson, K., Bardahl, K.: Nonisolated dynamical black holes and white holes. Phys. Rev. D 77, 104008 (2008)MathSciNetADSCrossRefGoogle Scholar
  64. 64.
    Misner, C.W., Sharp, D.H.: Relativistic equations for adiabatic, spherically symmetric gravitational collapse. Phys. Rev. 136, 571 (1964)MathSciNetADSCrossRefGoogle Scholar
  65. 65.
    Morris, M.S., Thorne, K.S.: Wormholes in space-time and their use for interstellar travel: a tool for teaching general relativity. Am. J. Phys. 56, 395 (1988)zbMATHMathSciNetADSCrossRefGoogle Scholar
  66. 66.
    Mukohyama, S., Hayward, S.A.: Quasilocal first law of black hole dynamics. Class. Quantum Grav. 17, 2153 (2000)zbMATHMathSciNetADSCrossRefGoogle Scholar
  67. 67.
    Nielsen, A.B.: Black holes without boundaries. Int. J. Mod. Phys. D 17, 2359 (2009)ADSCrossRefGoogle Scholar
  68. 68.
    Nielsen, A.B.: Black holes and black hole thermodynamics without event horizons. Gen. Rel. Gravit. 41, 1539 (2009)zbMATHADSCrossRefGoogle Scholar
  69. 69.
    Nielsen, A.B.: The spatial relation between the event horizon and trapping horizon. Class. Quantum Gravity 27, 245016 (2010)ADSCrossRefGoogle Scholar
  70. 70.
    Nielsen, A.B., Firouzjaee, J.T.: Conformally rescaled spacetimes and Hawking radiation. Gen. Rel. Gravit. 45, 1815 (2013)zbMATHMathSciNetADSCrossRefGoogle Scholar
  71. 71.
    Nielsen, A.B., Yeom, D.-H.: Spherically symmetric trapping horizons, the Misner-Sharp mass and black hole evaporation. Int. J. Mod. Phys. A 24, 5261 (2009)zbMATHMathSciNetADSCrossRefGoogle Scholar
  72. 72.
    Nielsen, A.B., Yoon, J.H.: Dynamical surface gravity. Class. Quantum Grav. 25, 085010 (2008)MathSciNetADSCrossRefGoogle Scholar
  73. 73.
    Nielsen, A.B., Visser, M.: Production and decay of evolving horizons. Class. Quantum Grav. 23, 4637 (2006)zbMATHMathSciNetADSCrossRefGoogle Scholar
  74. 74.
    Nordtvedt, K.: PostNewtonian metric for a general class of scalar tensor gravitational theories and observational consequences. Astrophys. J. 161, 1059 (1970)MathSciNetADSCrossRefGoogle Scholar
  75. 75.
    Parikh, M.K., Wilczek, F.: Hawking radiation as tunneling. Phys. Rev. Lett. 85, 5042 (2000)MathSciNetADSCrossRefGoogle Scholar
  76. 76.
    Penrose, R.: Gravitational collapse and spacetime singularities. Phys. Rev. Lett. 14, 57 (1965)zbMATHMathSciNetADSCrossRefGoogle Scholar
  77. 77.
    Pielahn, M., Kunstatter, G., Nielsen, A.B.: Critical analysis of dynamical surface gravity in spherically symmetric black hole formation. Phys. Rev. D 84, 104008 (2011)ADSCrossRefGoogle Scholar
  78. 78.
    Poisson, E.: A Relativist’s Toolkit: The Mathematics of Black-Hole Mechanics. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
  79. 79.
    Rácz, I.: On the use of the Kodama vector field in spherically symmetric dynamical problems. Class. Quantum Grav. 23, 115 (2006)zbMATHADSCrossRefGoogle Scholar
  80. 80.
    Roman, T.A., Bergmann, P.G.: Stellar collapse without singularities? Phys. Rev. D 28, 1265 (1983)MathSciNetADSCrossRefGoogle Scholar
  81. 81.
    Scheel, M.A., Shapiro, S.L., Teukolsky, S.A.: Collapse to black holes in Brans-Dicke theory. 2. Comparison with general relativity. Phys. Rev. D 51, 4236 (1995)Google Scholar
  82. 82.
    Schnetter, E., Krishnan, B.: Non-symmetric trapped surfaces in the Schwarzschild and Vaidya spacetimes. Phys. Rev. D 73, 021502 (2006)MathSciNetADSCrossRefGoogle Scholar
  83. 83.
    Sorkin, R.D.: In: Wiltshire, D. (ed.) Proceedings of the First Australasian Conference on General Relativity and Gravitation, February 1996, Adelaide, pp. 163–174. University of Adelaide (1996). [Preprint arXiv:gr-qc/9701056]Google Scholar
  84. 84.
    Sultana, J., Dyer, C.C.: Conformal killing horizons. J. Math. Phys. 45, 4764 (2004)zbMATHMathSciNetADSCrossRefGoogle Scholar
  85. 85.
    Sultana, J., Dyer, C.C.: Cosmological black holes: A black hole in the Einstein-de Sitter universe. Gen. Rel. Gravit. 37, 1349 (2005)zbMATHMathSciNetADSCrossRefGoogle Scholar
  86. 86.
    Szabados, L.: Quasi-local energy-momentum and angular momentum in GR: a review article. Living Rev. Relat. 7, 4 (2004)ADSGoogle Scholar
  87. 87.
    Tolman, R.C.: Non-Newtonian Mechanics. Some transformation equations. Philos. Mag. 25 (125), 150 (1912)Google Scholar
  88. 88.
    Tung, R.-S.: Stationary untrapped boundary conditions in general relativity. Class. Quantum Grav. 25, 085005 (2008)MathSciNetADSCrossRefGoogle Scholar
  89. 89.
    Unruh, W.G.: Notes on black-hole evaporation. Phys. Rev. D 14, 870 (1976)ADSCrossRefGoogle Scholar
  90. 90.
    Vanzo, L., Acquaviva, G., Di Criscienzo, R.: Tunnelling methods and Hawking’s radiation: achievements and prospects. Class. Quantum Grav. 28, 183001 (2011)ADSCrossRefGoogle Scholar
  91. 91.
    Visser, M.: Dirty black holes: thermodynamics and horizon structure. Phys. Rev. D 46, 2445 (1992)MathSciNetADSCrossRefGoogle Scholar
  92. 92.
    Visser, M.: Gravitational vacuum polarization. I. Energy conditions in the Hartle-Hawking vacuum. Phys. Rev. D 54, 5103 (1996)Google Scholar
  93. 93.
    Visser, M.: Essential and inessential features of Hawking radiation. Int. J. Mod. Phys. D 12, 649 (2003)zbMATHMathSciNetADSCrossRefGoogle Scholar
  94. 94.
    Wagoner, R.V.: Scalar-tensor theory and gravitational waves. Phys. Rev. D 1, 3209 (1970)ADSCrossRefGoogle Scholar
  95. 95.
    Wald, R.M.: General Relativity. Chicago University Press, Chicago (1984)zbMATHCrossRefGoogle Scholar
  96. 96.
    Wald, R.M.: The thermodynamics of black holes. Living Rev. Relat. 4, 6 (2001)ADSGoogle Scholar
  97. 97.
    Wald, R.M., Iyer, V.: Trapped surfaces in the Schwarzschild geometry and cosmic censorship. Phys. Rev. D 44, R3719 (1991)MathSciNetADSCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Valerio Faraoni
    • 1
  1. 1.Physics DepartmentBishop’s UniversitySherbrookeCanada

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