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

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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 tunneling 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 Larrañaga 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 √ 2 −2φ 2 S0 = d x −g R − 2 (∇φ) − 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 √ σ SM = d x −g (16πAσJ + LF luid) . (4)

The ﬁeld equations obtained from the complete action, S = S0 + SM , are

−2φ αβ β ∇α e F = −4πJ (5) 1 ∇2φ + 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) = 2∇ φ∇ φ − g (∇φ)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) = −√ 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 dynamical, 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 theory [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 corresponds 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 ϕϕ 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 Larrañaga

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 horizon (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 coordinate 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, ϕ) 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˜ = −dω˜, where ω 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) − iε,

Z ω Z rout 2r Im I = −Im drdω.˜ (27) 0 rin r − 2M(v) + iε

The r-integral can be evaluated by deforming the contour around the pole in the r-plane, Z ω Im I = 4π M(v)dω˜ = 4πM(v)ω. (28) 0 This result let us we conclude that the emission rate is

Γ ∼ e−8πM(v)ω, (29) where one can clearly see that there is a Boltzmann distribution exp(−βω) 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). 352 Edward Larra˜naga

Thus, the Bekenstein-Hawking entropy associated with the apparent horizon is

1 h i S = A = πR2 = π r2 − 2r D(v) (31) AH 4 AH AH AH AH h 2 i SAH = π 4M (v) − 4M(v)D(v) . (32)

Using Eq. (15), this equation can be put in terms of the electric charge,

2 −2φ0 2 SAH = 4πM (v) − 2πe Q (v). (33)

From this relation one can see that the electric charge contributes to the entropy although it doesn’t contribute to the temperature. By diﬀerentiation we obtain −2φ0 dSAH = 8πM(v)dM(v) − 4πe Q(v)dQ, (34) or 1 e−2φ0 Q(v) dM(v) = dS + dQ. (35) 8πM(v) AH 2M(v) Hence, we have obtained the ﬁrst law of thermodynamics for the emission process at the apparent horizon of the black hole,

dM(v) = TAH (v)dSAH (v) + ΦAH (v)dQ(v), (36)

Q(v) −2φ0 Q(v) −2φ0 where ΦAH (v) = e = e is the re-scaled Maxwell electrostatic 2M(v) rAH potential.

3.2 Entropy spectrum In Ref. [2], Bekenstein showed how the event horizon area of a black hole is an adiabatic invariant and therefore it may be quantized by Ehrenfest principle. When there is an absorption or an emission of a particle through the horizon, the minimum change in the area of the horizon is proportional toh ¯ and then the area spectrum should be linearly quantized. This result has been shown for Schwarzschild [15], Kerr [24] and Vaydia [3] black holes in general realtivity, as well as for the GMGHS and Sen black holes in string theory [13], among other solutions. Once we established the thermodynamic relations above, we will use the adiabatic invariant action and the Bohr-Sommerfeld quantization rule,

X Z A = pidqi = 2πnh,¯ (37) i where pi is the conjugate momentum of the qi coordinate, to obtain the entropy spectrum of the radiating black hole. Thermodynamical properties ... 353

First, note that the metric (12) can be Euclideanized by the transformation v → iτ. Since we will consider radial null trajectories, the variables in this action will be the Euclidean time, q0 = τ, and the radial coordinate, q1 = r. Introducing the Hamiltonian and the Hamilton’s equations, the action becomes

X ZZ pi X ZZ H dH˜ A = dp˜idqi = dq dqi, (38) o o i i i dτ and expanding the sum,

ZZ H ZZ H ˜ ˜ dH A = dHdτ + dr dr. (39) o o dτ Using the deﬁnition of the Euclidean time, dr = ir˙ (40) dτ the action reduces to Z rout Z H dH˜ A = 2 dr, (41) rin o ir˙ where we have introduced the limits for the r-integration just as before. The integration with the r-coordinate gives

Z H Z H dH˜ A = 2 4πM(v)dH˜ = , (42) o o TAH and the integration with H is easily done by considering the ﬁrst law of thermodynamics in Eq. (36) and the Hamiltonian [23, 22]

H = M − ΦQ. (43)

Hence, integration of Eq. (42) gives the result

A = SAH = 2πn, (44) where we recovered the Bohr-Sommerfeld quantization rule in units withh ¯ = 1. This relation shows that the entropy spectrum is quantized and equally spaced for the apparent horizon, ∆SAH = 2π. (45) This result appears to conﬁrm that the existence of a temperature associated with the black hole horizon implies the existence of underlying degrees of freedom, related here with a quantum of area in Planck’s scale, as the proposal of gravity as an emergent phenomenon predicts [18]. 354 Edward Larra˜naga

4 The Event Horizon

Now we turn our attention to the event horizon of the radiating black hole (12), which has a radius 2M(v) rEH = . (46) 1 − 2r ˙EH Since the event horizon doesn’t coincide with the time-like limit surface, before we proceed to study the tunneling process we have to verify that Lan- dau’s coordinate clock synchronization condition is satisﬁed. Using the event horizon radius, we make the coordinate transformation

η = v (47)

ξ = r − rEH (48) to obtain the new line element

2 2 2 2 ds = γ00dη + 2γ01dηdξ + γ22dθ + γ33dϕ (49) 2M(v) ! 2D(v)! = − 1 − +r ˙ dη2 + 2dηdξ + r2 1 − dΩ2. (50) r EH r

Note that using these coordinates, the event horizon is defined by the sim- ple condition γ00 = 0. According to Landau [12], the difference between the coordinate times of two physical events taking place simultaneously, but at different spatial places, is Z γ ∆T = − 0i dxi. (51) γ00 Hence, if the simultaneity could be transmitted from one place to another independently of the integration path, the metric should satisfy ! ! γ0j γ0i ∂i = ∂j . (52) γ00 γ00

Using the metric in Eq. (50), this relation is identically satisfied, show- ing that the coordinates (η, ξ, θ, ϕ) define correctly the clock synchronization according to Landau. Hence, it is possible to apply the tunneling method to study the thermodynamics at the event horizon. In order to describe the tunneling effect across the event horizon, we will consider an interesting approximated description [16] using outgoing radial null vectors, which are defined in terms of a parameter λ by the relation

dr 1 2M(v)! dv = 1 − . (53) dλ 2 r dλ Thermodynamical properties ... 355

By diﬀerentiation, this equation gives the second order diﬀerential equation

d2r 1 2M(v)! d2v M(v) dr dv M˙ (v) dv !2 = 1 − + − . (54) dλ2 2 r dλ2 r2 dλ dλ r dλ

d2v Considering the parameter λ as linearly proportional to v, i.e. dλ2 = 0, we obtain d2r M(v) dr dv M˙ (v) dv !2 = − . (55) dλ2 r2 dλ dλ r dλ

d2r and restricting to outgoing null radial geodesics, i.e. dλ2 = 0, it gives

M(v) dr dv M˙ (v) dv !2 = . (56) r2 dλ dλ r dλ

Using Eq. (53) one obtains

1 M(v) 2M(v)! M˙ (v) 1 − = . (57) 2 r2 r r which give the possible trajectories

M(v) q r = 1 ± 1 − 16M˙ (v) . (58) 4M˙ (v)

The solution corresponding to the location of the event horizon (+ sign) can be approximated as

h ˙ ˙ 2 ˙ 3 i rEH = 2M(v) 1 + 4M(v) + 32M (v) + O(M ) (59) for small values of M˙ (v). Since this result implies that the event horizon corresponds to a surface near the apparent horizon, we will investigate the Hawking radiation from these kind of surfaces in order to obtain an estimate the thermodynamic characteristics such as temperature and entropy.

4.1 Thermodynamics on surfaces near the apparent horizon

Consider the radiusr ¯ = 2M(v)(1 + δ) = rAH (1 + δ) where δ is an inﬁnitesimal constant (or almost constant) quantity. Introducing the coordinate transformation 1 x = r − δv (60) 2 356 Edward Larra˜naga the line element (12) becomes

2M(v) ! 2D(v)! ds2 = − 1 − − δ dv2 + 2dvdx + r2 1 − dΩ2. (61) r r From here, it is clear that the radiusr ¯ deﬁnes a new apparent horizon which coincides with the new time-like limit surface. Therefore, we consider the radial null geodesics, dx 1 2M(v) ! =x ˙ = 1 − − δ , (62) dv 2 r and put this trajectory into the imaginary part of the action

Z xout Im I = Im pxdx, (63) xin where initial radius xi is slightly inside the new apparent horizon and the ﬁnal radius, xf is slightly outside. This gives

Z xout Z H dH˜ Z xout Z H 2 Im I = Im dx = Im dHdx.˜ (64) 2M(v) xin 0 x˙ xin 0 1 − r − δ Doing the x-integration just as before, Z ω M(v) M(v) Im I = 4π dω˜ = 4π ω, (65) 0 (1 − δ)2 (1 − δ)2 and therefore the emission rate is this time

− 8πM(v) ω Γ ∼ e−2Im S = e (1−δ)2 . (66)

As always, this equation let us identify the temperature associated with the new apparent horizon, (1 − δ)2 T¯ = . (67) AH 8πM(v) On the other hand, the entropy will be proportional to the area of the new apparent horizon,

¯ ¯2 h 2 i SAH = πRAH = π r¯AH − 2¯rAH D(v) (68) ¯ h 2 2 i SAH = π 4M (v)(1 + δ) − 4M(v)D(v)(1 + δ) , (69) or, using Eq. (15), in terms of the electric charge as

¯ 2 2 −2φ0 2 SAH = 4πM (v)(1 + δ) − 2πe Q (v)(1 + δ). (70) Thermodynamical properties ... 357

By diﬀerentiation, we obtain the expression

¯ 2 −2φ0 dSAH = 8πM(v)dM(1 + δ) − 4πe Q(v)dQ(1 + δ) (71) which corresponds to the first law of thermodynamics on the new apparent horizon, ¯ ¯ ¯ dM(v) = TAH dSAH + ΦAH (v)dQ, (72) where we have identified the temperature in Eq. (67) up to the first order in δ and the last term includes the electric potential which is defined now as

−2φ0 −2φ0 ¯ e Q(v) e Q(v) ΦAH (v) = = . (73) 2M(v)(1 + δ) r¯AH

4.2 Thermodynamics on the event horizon The results of the above section let us obtain immediately the thermodynamic quantities at the event horizon by taking, according to Eqs. (46) and (59), the parameter δ = 4M˙ . Hence, Eqs. (67) and (70) give the temperature and entropy associated with the event horizon of the radiating black hole as

(1 − 4M˙ )2 T = (74) EH 8πM(v) 2 ˙ 2 −2φ0 2 ˙ SEH = 4πM (v)(1 + 4M) − 2πe Q (v)(1 + 4M) (75) and the corresponding ﬁrst law is

dM(v) = TEH dSEH + ΦEH (v)dQ. (76)

The obtained results show that it is possible to formulate a consistent thermodynamics on the apparent horizon of the radiating black hole from string theory and that the same formulation works on nearing surfaces, including its event horizon. Since the thermodynamics at rEH is treated as a perturbation near the surface r = rAH , one may conclude that the Hawking radiation comes from the apparent horizon, just as stated for the general relativity theory in Refs. [25] and [17]. Now, lets move our attention to the fact that the construction presented so ˙ far assumes that the term M, or equivalentlyr ˙EH , is a constant. When it is considered the possibility of a non-constant M˙ , the ﬁrst law of thermodynamics must be revised. Taking the diﬀerential of Eq. (75) gives

˙ 2 −2φ0 ˙ dSEH = 8πM(v)(1 + 4M) dM − 4πe Q(v)(1 + 4M)dQ h i +8π 4M 2(v)(1 + 4M˙ ) − e−2φ0 Q2(v) dM.˙ (77) 358 Edward Larra˜naga

Thus, we have

dSEH Q(v) dM = + e−2φ0 dQ 8πM(v)(1 + 4M˙ )2 2M(v)(1 + 4M˙ ) " M(v) Q2(v) # + −4 + e−2φ0 dM,˙ (78) (1 + 4M˙ ) M(v)(1 + 4M˙ )2 or identifying the temperature, the electric potential and the radius of the event horizon,

" 8M 2(v) 4M(v)Q2(v) # −2φ0 dM = TEH dSEH + ΦEH dQ + 2 − + 2 e dr˙EH . (79) rEH rEH Finally, using Eq. (15) this equation is written in terms of the dilaton charge as 16M 2(v) " 2D(v)# dM = TEH dSEH + ΦEH dQ − 1 − dr˙EH . (80) rEH rEH

This equation shows that taking into account a variable M˙ gives as a result the last term of Eq. (80), which represents a growth of entropy due to the non-equilibrium of the black hole with the external Vaidya-type spacetime.

Acknowledgements. This work was supported by the Universidad Na- cional de Colombia, Hermes Grant Code 41673.

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Received: October 1, 2018; Published: October 25, 2018

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