Abstract
Tunnelling processes through black hole horizons have recently been investigated in the framework of WKB theory, discovering an interesting interplay with Hawking radiation. In this paper, we instead adopt the point of view proper of QFT in curved spacetime, namely, we use a suitable scaling limit towards a Killing horizon to obtain the leading order of the correlation function relevant for the tunnelling. The computation is done for a certain large class of reference quantum states for scalar fields, including Hadamard states. In the limit of sharp localization either on the external side or on opposite sides of the horizon, the quantum correlation functions appear to have thermal nature. In both cases the characteristic temperature is referred to the surface gravity associated with the Killing field and thus connected with the Hawking one. Our approach is valid for every stationary charged rotating non-extremal black hole. However, since the computation is completely local, it covers the case of a Killing horizon which just temporarily exists in some finite region, too. These results provide strong support to the idea that the Hawking radiation, which is detected at future null infinity and needs some global structures to be defined, is actually related to a local phenomenon taking place even for local geometric structures (local Killing horizons), existing just for a while.
Similar content being viewed by others
References
Akhmedov E.T., Pilling T., Singleton D.: Subtleties in the quasi-classical calculation of Hawking radiation. Int. J. Mod. Phys. D 17, 2453–2458 (2008)
Akhmedov E.T., Pilling T., de Gill A., Singleton D.: Temporal contribution to gravitational WKB-like calculations. Phys. Lett. B 666, 269–271 (2008)
Angheben M., Nadalini M., Vanzo L., Zerbini S.: Hawking radiation as tunneling for extremal and rotating black holes. JHEP 0505, 014 (2005)
Boyer R.H.: Geodesic Killing orbits and bifurcate Killing horizons. Proc. Soc. London A311, 245–252 (1969)
Buchholz D., Verch R.: Scaling algebras and renormalization group in algebraic quantum field theory. Rev. Math. Phys. 7, 1195 (1995)
Buchholz D.: Quarks, gluons, color: Facts or fiction?. Nucl. Phys. B469, 333 (1996)
Dappiaggi, C., Moretti, V., Pinamonti, N.: Rigorous construction and Hadamard property of the Unruh state in Schwarzschild spacetime. Adv. Theor. Math. Phys., Issue 2 (2011), to appear, http://arXiv.org/abs/0907.1034v1 [gr-qc], 2009
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. B657, 107–111 (2007)
Fredenhagen K., Haag R.: On the derivation of Hawking radiation associated with the formation of a black hole. Commun. Math. Phys. 127, 273 (1990)
Haag R., Narnhofer H., Stein U.: On quantum field theory in gravitational background. Commun. Math. Phys. 94, 219 (1984)
Hayward S.A., Di Criscienzo R., Vanzo L., Nadalini M., Zerbini S.: Local Hawking temperature for dynamical black holes. Class. Quant. Grav. 26, 062001 (2009)
Hawking S.W.: Particle Creation by Black Holes. Commun. Math. Phys. 43, 199 (1975)
Kay B.S., Wald R.M.: Theorems On The Uniqueness And Thermal Properties Of Stationary, Nonsingular, Quasifree States On Space-Times With A Bifurcate Killing Horizon. Phys. Rept. 207, 49 (1991)
Kerner R., Mann R.B.: Fermions tunnelling from black holes. Class. Quant. Grav. 25, 095014 (2008)
Medved A.J.M., Vagenas E.C.: On Hawking Radiation as Tunneling with Back-Reaction. Mod. Phys. Lett. A20, 2449–2454 (2005)
Parikh M.K., Wilczek F.: Hawking Radiation as Tunneling. Phys. Rev. Lett. 85, 5042 (2000)
Racz I., Wald R.M.: Global extensions of space-times describing asymptotic final states of black holes. Class. Quant. Grav. 13, 539 (1996)
Racz I., Wald R.M.: Extension of space-times with Killing horizon. Class. Quant. Grav. 9, 2643 (1992)
Wald R.M.: General Relativity. Chicago University Press, Chicago, IL (1984)
Wald R.M.: Quantum field theory in curved space-time and black hole thermodynamics. The University of Chicago Press, Chicago, IL (1994)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Y. Kawahigashi
Rights and permissions
About this article
Cite this article
Moretti, V., Pinamonti, N. State Independence for Tunnelling Processes Through Black Hole Horizons and Hawking Radiation. Commun. Math. Phys. 309, 295–311 (2012). https://doi.org/10.1007/s00220-011-1369-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-011-1369-8