Abstract
We revisit the tunneling picture for the Hawking effect in light of the charged Nariai manifold, because this general relativistic solution, which displays two horizons, provides the bonus to allow the knowledge of exact solutions of the field equations. We first perform a revisitation of the tunneling ansatz in the framework of particle creation in external fields à la Nikishov, which corroborates the interpretation of the semiclassical emission rate \({\varGamma }_{emission}\) as the conditional probability rate for the creation of a couple of particles from the vacuum. Then, particle creation associated with the Hawking effect on the Nariai manifold is calculated in two ways. On the one hand, we apply the Hamilton–Jacobi formalism for tunneling, in the case of a charged scalar field on the given background. On the other hand, the knowledge of the exact solutions for the Klein–Gordon equations on Nariai manifold, and their analytic properties on the extended manifold, allow us a direct computation of the flux of particles leaving the horizon, and, as a consequence, we obtain a further corroboration of the semiclassical tunneling picture from the side of S-matrix formalism.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
For the sake of completeness, one should write \(\mathrm {Im} W = \frac{1}{2} \sum _{\omega ,l,m} \log (1+<n_{\omega ,l,m}>)\), which takes into account the full dependence on quantum numbers, and one realizes that the label \(\omega \) introduced in (10) is split, with some abuse of language, into \(\omega ,l,m\), where \(\omega \) is the energy, and l, m are the usual quantum numbers for angular momentum.
To be precise this is true only at finite, since they have different singularities at infinity.
One may wonder if the flux of \(J_r\), which is meaningful in the external region \(U<0\), is still meaningful also in the black hole region \(U>0\). We observe that the current involves substantially Wronskian relations also in the inner region, and so is conserved also there, even if its physical interpretation is not perspicuous.
References
Hawking, S.W.: Commun. Math. Phys. 43, 199 (1975) [Erratum-ibid 46, 206 (1976)]
Hartle, J.B., Hawking, S.W.: Phys. Rev. D 13, 2188 (1976)
Damour, T., Ruffini, R.: Phys. Rev. D 14, 332 (1976)
Srinivasan, K., Padmanabhan, T.: Phys. Rev. D 60, 024007 (1999). arXiv:gr-qc/9812028
Visser, M.: Int. J. Mod. Phys. D 12, 649 (2003). arXiv:hep-th/0106111
Parikh, M.K., Wilczek, F.: Phys. Rev. Lett. 85, 5042 (2000). arXiv:hep-th/9907001
Akhmedov, E.T., Akhmedova, V., Singleton, D.: Phys. Lett. B 642, 124 (2006). arXiv:hep-th/0608098
Vanzo, L., Acquaviva, G., Di Criscienzo, R.: Class. Quant. Grav. 28, 183001 (2011). arXiv:1106.4153 [gr-qc]
Parikh, M.K.: Int. J. Mod. Phys. D 13, 2351 (2004) [Gen. Relativ. Gravit. 36, 2419 (2004)]. arXiv:hep-th/0405160
Moretti, V., Pinamonti, N.: Commun. Math. Phys. 309, 295 (2012). arXiv:1011.2994 [gr-qc]
Chowdhury, B.D.: Pramana 70, 593 (2008) [Pramana 70, 3 (2008)]. doi:10.1007/s12043-008-0001-8. arXiv:hep-th/0605197
Akhmedov, E.T., Akhmedova, V., Pilling, T., Singleton, D.: Int. J. Mod. Phys. A 22, 1705 (2007). arXiv:hep-th/0605137
Belgiorno, F., Cacciatori, S.L., Dalla Piazza, F.: JHEP 0908, 028 (2009). arXiv:0906.1520 [gr-qc]
Belgiorno, F., Cacciatori, S.L., Dalla Piazza, F.: Class. Quant. Grav. 27, 055011 (2010). arXiv:0909.1454 [gr-qc]
Kofman, L.A., Sakhni, V., Starobinski, A.A.: Sov. Phys. JETP 58, 1090 (1983)
Massar, S., Parentani, R.: Nucl. Phys. B 575, 333 (2000). arXiv:gr-qc/9903027
Akhmedov, E.T., Pilling, T., Singleton, D.: Int. J. Mod. Phys. D 17, 2453 (2008). arXiv:0805.2653 [gr-qc]
Akhmedova, V., Pilling, T., de Gill, A., Singleton, D.: Phys. Lett. B 666, 269 (2008). arXiv:0804.2289 [hep-th]
Akhmedova, V., Pilling, T., de Gill, A., Singleton, D.: Phys. Lett. B 673, 227 (2009). arXiv:0808.3413 [hep-th]
Vanzo, L.: Europhys. Lett. 95, 20001 (2011). arXiv:1104.1569 [gr-qc]
Shankaranarayanan, S., Padmanabhan, T., Srinivasan, K.: Class. Quant. Grav. 19, 2671 (2002). arXiv:gr-qc/0010042
Shankaranarayanan, S., Srinivasan, K., Padmanabhan, T.: Mod. Phys. Lett. A 16, 571 (2001). arXiv:gr-qc/0007022
Kerner, R., Mann, R.B.: Phys. Rev. D 73, 104010 (2006). arXiv:gr-qc/0603019
Sannan, S.: Gen. Relativ. Gravit. 20, 239 (1988)
Nikishov, A.I., Eksp, Zh: Teor. Fiz. 57, 1210 (1969)
Damour, T.: Klein paradox and vacuum polarization. In: Ruffini, R. (ed.) Proceedings of first Marcel Grossmann Meeting on General Relativity (Trieste, 1975), p. 459. North-Holland, Amsterdam (1977)
Kim, S.P., Hwang, W.Y.P.: arXiv:1103.5264 [hep-th]
Stephens, C.R.: Ann. Phys. 193, 255 (1989)
Romans, L.J.: Nucl. Phys. B 383, 395 (1992)
Mann, R.B., Ross, S.F.: Phys. Rev. D 52, 2254 (1995)
Bousso, R.: Phys. Rev. D 60, 063503 (1999)
Medved, A.J.M.: Phys. Rev. D 66, 124009 (2002). arXiv:hep-th/0207247
Parikh, M.K.: Phys. Lett. B 546, 189 (2002). arXiv:hep-th/0204107
Zhang, J.Y., Zhao, Z.: Nucl. Phys. B 725, 173 (2005)
Kim, S.P.: JHEP 0711, 048 (2007). arXiv:0710.0915 [hep-th]
Farmany, A., Dehghani, M., Setare, M.R., Mortazavi, S.S.: Phys. Lett. B 682, 114 (2009)
Rahman, M.A., Hossain, M.I.: Phys. Lett. B 712, 1 (2012). arXiv:1205.1216 [gr-qc]
de Gill, A., Singleton, D., Akhmedova, V., Pilling, T.: Am. J. Phys. 78, 685 (2010). doi:10.1119/1.3308568. arXiv:1001.4833 [gr-qc]
Gamelin, T.W.: Complex Analysis, Undergraduate Texts in Mathematics. Springer, Berlin (2001)
Bezerra, V.B., Vieira, H.S., Costa, A.A.: Class. Quant. Grav. 31, 045003 (2014). arXiv:1312.4823 [gr-qc]
Vieira, H.S., Bezerra, V.B., Muniz, C.R.: Ann. Phys. 350, 14 (2014). arXiv:1401.5397 [gr-qc]
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Belgiorno, F., Cacciatori, S.L. & Dalla Piazza, F. Tunneling method for Hawking radiation in the Nariai case. Gen Relativ Gravit 49, 109 (2017). https://doi.org/10.1007/s10714-017-2275-y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10714-017-2275-y