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Thermodynamic Properties on the Apparent and Event Horizons of a Dynamical Charged Black Hole from Heterotic String Theory

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Thermodynamic Properties on the Apparent and Event Horizons of a Dynamical Charged Black Hole from Heterotic String Theory Advanced Studies in Theoretical Physics Vol. 12, 2018, no. 8, 347 - 360 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/astp.2018.8936 Thermodynamic Properties on the Apparent and Event Horizons of a Dynamical Charged Black Hole from Heterotic String Theory Edward Larra~naga National Astronomical Observatory, National University of Colombia Ciudad Universitaria, Bogot´a,111156, Colombia Copyright c 2018 Edward Larra~naga. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduc- tion in any medium, provided the original work is properly cited. Abstract Thermodynamic characteristics such as temperature and entropy on the apparent and event horizons of a radiating charged black hole from low energy effective heterotic string theory are obtained using the tun- neling of massless shells approach. The corresponding entropy spectrum is deduced and the first law of thermodynamics is studied to show that it holds on both of these surfaces and that, in a general case, there may be a grow in entropy. Keywords: Hawking radiation, Thermodynamics, Dynamical black hole, String Theory, Apparent horizon, Event horizon 1 Introduction Hawking radiation may be thought as a tunneling process in which particles move through the event horizon of a static black hole [20, 21, 19]. Some authors have shown that this treatment can also be extended to describe the Hawking radiation emitted by the horizons of some non-stationary black holes which satisfy Landau's coordinate clock synchronization condition [10, 11, 25, 17]. In this work we use the tunneling approach to study the thermodynamic properties of a radiating black hole from the low energy heterotic string theory, 348 Edward Larra~naga obtaining the temperature, entropy and first law of thermodynamics associated with its apparent horizon, as well as with its event horizon. We also calcu- late the associated entropy spectrum with the help of the Bohr-Sommerfeld quantization rule and the adiabatic invariance, to confirm that the apparent horizon area spectrum of this black hole is independent of its parameters, just as was proposed, in general terms, by Bekenstein [2]. 2 A Radiating Black Hole from String Theory The black hole that we will consider is a solution of the field equations com- ing from the low energy effective action of the heterotic string theory in four dimensions. This is given by Z q 1 1 S = d4x −ge~ −φ −R + H Hαβρ − g~αβ@ φ∂ φ + F F αβ ; (1) 0 12 αβρ α β 8 αβ where R is the Ricci scalar,g ~αβ is the metric that arises naturally in the σ model and F represents the Maxwell field, Fαβ = @αAβ − @βAα, associated with a U (1) subgroup of E8 × E8. There is also a dilaton field, φ, and an antisymmetric tensor gauge field, 1 H = @ B + @ B + @ B − (A F + A F + A F ) ; (2) αβρ α βρ β ρα ρ αβ 4 α βρ β ρα ρ αβ which will not be considered here, i.e. Hαβρ = 0. When working in the −φ conformal Einstein frame, the metric becomes gαβ = e g~αβ and the action takes the simpler form Z 4 p 2 −2φ 2 S0 = d x −g R − 2 (rφ) − e F : (3) To the action above we will also add a minimal coupling of the Maxwell field to a current J α and an extra matter Lagrangian which will account for a null fluid surrounding the black hole, Z 4 p σ SM = d x −g (16πAσJ + LF luid) : (4) The field equations obtained from the complete action, S = S0 + SM , are −2φ αβ β rα e F = −4πJ (5) 1 r2φ + e−2φF F αβ = 0 (6) 2 αβ Gαβ = 8πTαβ (7) where the energy-momentum tensor is decomposed as (Dilaton) (EM) (F luid) Tαβ = Tαβ + Tαβ + Tαβ (8) Thermodynamical properties ... 349 with 1 h i T (Dilaton) = 2r φr φ − g (rφ)2 (9) αβ 8π α β αβ e−2φ 1 T (EM) = 2F F σ − g F 2 (10) αβ 8π ασ β 2 αβ 2 @L T (F luid) = −p F luid + g A J σ − 2A J : (11) αβ −g @gαβ αβ σ (α β) A particular exact solution of these field equations representing a dynami- cal, electrically charged, spherically symmetric black hole emitting or absorbing a null fluid was reported in Ref. [1]. This new solution generalize the Vaidya and Bonnor-Vaidya solutions as well as the GMGHS black hole of string the- ory [7, 8, 5, 6]. In advanced Eddington-Finkelstein coordinates, (v; r; θ; φ), the corresponding line element is 2M(v)! 2D(v)! ds2 = − 1 − dv2 + 2dvdr + r2 1 − dΩ2; (12) r r together with the Maxwell and dilaton fields, Q(v) F = dv ^ dr (13) r2 2D(v)! e2φ = e2φ0 1 − ; (14) r where Q(v) is the electric charge, D(v) is the dilatonic charge and φ0 corre- sponds to the asymptotic value of the dilaton field φ. The functions M(v), Q(v) and D(v) are not independent because they are subject to the constraint Q2(v) = 2D(v)M(v)e2φ0 : (15) Therefore this solution is parameterized by only two quantities, say M(v) and Q(v). The Maxwell field equations restricts the radial current to be [M(v)D(v)]0 J α = − δα (16) 4πr[r − 2D(v)]Q(v) r and the fluid energy-momentum tensor is characterized by its energy density, θθ '' µ = Tvv, ρ = −Tvr, and its pressure, P = T = T (for a detailed description of the characteristics of a null fluid see Ref. [9].), 2 [r2D00(v) − M(v)D0(v) + [r − D(v)]M 0(v)] µ = (17) r2[r − 2D(v)] 2D0(v) ρ = (18) r[r − 2D(v)] P = 0: (19) 350 Edward Larra~naga Note that we may, arbitrarily, take the asymptotic value of the dilaton as φ0 = 0, and it can be recovered easily by performing the simultaneous shift of the dilaton, φ ! φ + φ0 together with a re-scaling of the Maxwell field, φ0 Aα ! e Aα. 3 Thermodynamics on the Apparent Horizon by Tunneling of Massless Shells In this section we will study the Hawking radiation and temperature associated with the apparent horizon of the dynamical black hole from string theory. From the line element in Eq. (12), the condition 2M(v)! gvv = − 1 − = 0; (20) rT defines both the time-like limit surface (at the radius rT ) and the apparent hori- zon (at the radius rAH ). Thus, the radiating black hole has rT = rAH = 2M(v). Since these two surfaces coincide, the apparent horizon satisfies Landau's co- ordinate clock synchronization condition [12] and it is possible to follow the proposal of Parikh [20] to analyze the tunneling of massless shells to obtain the thermal spectrum emitted by this black hole [25, 17] . In the Wentzel- Kramers-Brillouin (WKB) approximation, the emission rate, Γ, is the square of the tunneling amplitude, Γ ∼ e−2Im I ; (21) so we will consider the imaginary part of the action describing the radial motion of a mass-less shell, Z rout Im I = Im prdr; (22) rin where pr is the radial momentum and the integration is performed from an initial radius rin, corresponding to the place of pair creation, slightly inside the apparent horizon, and a final radius, rout, slightly outside the apparent horizon. Because of these integration limits, we need to work in coordinates which are not singular at the apparent horizon, precisely as the coordinates (v; r; θ; ') used in the line element (12). Introducing the Hamiltonian, H, and the Hamilton equation, dH =r; _ (23) dpr we can write the imaginary part of the action as Z rout Z pr Z rout Z H dH~ Im I = Im dp~rdr = Im dr: (24) rin 0 rin 0 r_ Thermodynamical properties ... 351 Since we are considering the tunneling of massless shells, we need to cal- culate radial null geodesics. These are defined by the conditions ds2 = 0 and dΩ = 0 in the line element (12), giving the equation for outgoing trajectories, dr 1 2M(v)! =r _ = 1 − ; (25) dv 2 r and therefore Eq. (24) takes the form Z rout Z H 2r Im I = Im dHdr:~ (26) rin 0 r − 2M(v) In order to integrate this relation, we use energy conservation, which let us write dH~ = −d!~, where ! represents the energy of the emitted shell, together with Feynman's description [20], which let us displace the energy from M(v) to M(v) − i", Z ! Z rout 2r Im I = −Im drd!:~ (27) 0 rin r − 2M(v) + i" The r-integral can be evaluated by deforming the contour around the pole in the r-plane, Z ! Im I = 4π M(v)d!~ = 4πM(v)!: (28) 0 This result let us we conclude that the emission rate is Γ ∼ e−8πM(v)!; (29) where one can clearly see that there is a Boltzmann distribution exp(−β!) associated with the apparent horizon and that the corresponding temperature is 1 1 T = = : (30) AH β 8πM(v) Note that this temperature depends only on the mass M(v) (not on the electric charge Q(v)) and it has the same functional form as those reported in Refs. [25, 17, 14] and [4] for the Vaidya solution in general relativity. 3.1 Entropy and the first law of thermodynamics In the thermodynamic description of black holes, entropy is proportional to the area of the apparent horizon. For the radiating black hole from string theory in q Eq. (12), the area is calculated using the areal radius R(v; r) = r2 − 2rD(v).
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