Hawking radiation of spin-1 particles from a three-dimensional rotating hairy black hole
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UCLEI, PARTICLES, FIELDS, GRAVITATION, AND ASTROPHYSICS
Hawking Radiation of Spin-1 Particles from a Three-Dimensional Rotating Hairy Black Hole1 I. Sakalli and A. Ovgun* Department of Physics, Eastern Mediterranean University Famagusta, North Cyprus, Mersin-10, Turkey *e-mail: [email protected] Received March 27, 2015
Abstract—We study the Hawking radiation of spin-1 particles (so-called vector particles) from a three-dimensional rotating black hole with scalar hair using a Hamilton–Jacobi ansatz. Using the Proca equation in the WKB approximation, we obtain the tunneling spectrum of vector particles. We recover the standard Hawking temperature corresponding to the emission of these particles from a rotating black hole with scalar hair. DOI: 10.1134/S1063776115090113
1. INTRODUCTION One of the most radical predictions of general relativity is the existence of black holes. According to the seminal works of Hawking [1–3], black holes are not entirely black. That was the surprising claim made by Hawking over forty years ago. Examining the behavior of quantum fluctuations around the event horizon of a black hole, Hawking substantiated the theory that black holes emit thermal radiation, with a constant temperature (so-called Hawking temperature) directly proportional to the surface gravity κ, which is the gravitational acceleration experienced at the black hole horizon: (1) TH = κ , 2π where the system of units with c = G = kB = 1 is used. The works of Hawking and Bekenstein [4] and of others [5–15], rederiving TH in various ways, bring together the normally disparate areas: general relativity, quantum mechanics, and thermodynamics. The enthusiasm for understanding the underlying coordinations between these subjects of physics creates ample motivation for the study of Hawking radiation (see, e.g., [16–27] and references therein). Quantum fluctuations create a virtual particle pair near the black hole horizon. While the particle with negative energy tunnels into the horizon (absorption), the other, having positive energy, flies away to the spatial infinity (emission) and produces Hawking radiation. In the WKB approximation for the emission and absorption probabilities of the tunneling particles, the tunneling rate Γ is [12, 28, 29] 1 The article is published in the original.
Γ=
Pemission E = exp(− 2Im S )exp ⎛⎜ − net ⎞⎟ , Pabsorption ⎝ T ⎠
(2)
where S is the action of the classically forbidden trajectory of a tunneling particle, which has the net energy Enet and temperature T. One of the methods for finding S is the Hamilton–Jacobi method. This method is generally implemented by substituting a suitable ansatz, consistent with the symmetries of the space-time, in the relativistic Hamilton–Jacobi equation. The resulting radial integral always has a pole located at the event horizon. However, using the residue theory, the associated pole can be analytically avoided [30]. Recently, in the framework of the Hamilton– Jacobi method, the Hawking radiation of spin-1 particles described by the Proca equation in 3D nonrot
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