Holographic Description of 2-Dimensional Quantum Black Holes

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1 Author: Facultat de Física, Universitat de Barcelona, Diagonal 645, 0808 Barcelona, Spain. Advisor: Crisitano Gerani Abstract: In this work we obtain and analyse a black hole solution on the 1-brane of the Randall- Sundru II braneworld in the 3-diensional BTZ black hole bulk. Interpreting the results in the light of the conjectured duality between d-diensional quantu-corrected black hole solutions and black holes on the brane in the AdS d+1 braneworld, we have found that there exists a lower bound for the size of a -diensional quantu black hole. At the end, we suggest that this lower bound could be an inforon, i.e. a renant of the quantu black hole the existence of which has been proposed in the literature as a possible solution of the black hole inforation paradox. I. INTRODUCTION In the fraework of quantu field theory in curved spacetie QFTCS), there is a natural way of taking into account, at least to a certain order, the effect of quantu fields on the spacetie. More specifically, what one expects is that the back-reaction effects of quantu fields on the gravitational field are governed by the seiclassical Einstein s equations [1] R µν 1 Rg µν + Λg µν = κ d ψ T µν ψ, 1) where ψ is a state corresponding to a certain configuration of the present quantu fields. The constant κ d is the fundaental ass scale in d diensions and it has units of [κ d ] = E d [1]. In the two diensional case the Einstein tensor G µν = R µν 1 Rg µν vanishes identically and κ d becoes diensionless so we can set it equal to one. In this case, 1) becoes Λg µν = ψ T µν ψ ) and the dynaics of the etric is of purely quantu origin. This fact will be crucial for our work. A black hole solution of ) with a conforal field theory CFT) was found in []. In the Schwarzschild gauge it reads ds = λ x + µ x 1)dt + dx λ x + µ x 1, 3) where λ, µ > 0 depend on paraeters of the CFT, such as the nuber of fields. However, in this paper we do not study 3) directly in the fraework of QFTCS. Instead, we will take an holographic point of view. Specifically, our contribution consists of studying the properties of a black hole solution on the brane of the AdS 3 braneworld and ap the results obtained into the quantu black hole 3), assuing the validity of the holographic conjecture stated in [3]. Explicitly, the conjecture states that The black hole solutions localised in the brane in the AdS d+1 braneworld which are found by solving the classical bulk equations in AdS d+1 with the brane boundary conditions, correspond to quantu-corrected black holes in d diensions, rather than classical ones. A braneworld in AdS d+1 is a odel in which the observable universe, i.e. the universe where we live in, is a d-diensional slice the brane) ebedded in an asyptotically AdS d+1 spacetie the bulk). While quantu fields are trapped on the brane, gravity can access the bulk. In our particular case, we work on the Randall-Sundru type II RSII) [4] braneworld in the 3-diensional BTZ black hole bulk [5]. II. THE BACK HOE ON A 1-BRANE IN BTZ In this section we obtain a black hole solution on the 1-brane of the RSII braneworld. To do so, we follow a siilar procedure to that perfored in [6]. A. The BTZ Black Hole The only asyptotically AdS black hole solution known in +1 diensions is the BTZ black hole, and it plays the role of the bulk in our braneworld. It is a +1)- diensional solution of the Einstein-Maxwell equations, but in this work we consider the non-rotating, neutral version given by ds = F r)dt + dr F r) + r dθ 4) where F r) = r r > 0), is the AdS 3 length, M := /κ 3 is the ass of the black hole being 1/κ 3 the 3-diensional Planck ass and θ θ + π. In order to understand soe parts of the following subsection, it is iportant to notice that the singularity at r = 0 of 4), hidden by the horizon at r =, is not a singularity in the curvature of the spacetie, but a singularity in its causal structure. Indeed, the BTZ black hole is a quotient of AdS 3 by the identification of points by eans of a discrete subgroup of its isoetry group,

2 {e πn ξ } n Z, generated in turn by an specific Killing vector, ξ [7]. In order to preserve causality after perforing the identification, one has to restrict the physical spacetie to be the region of AdS 3 /P e πn ξ P ) in which ξ is spacelike, i.e. g ξ, ξ) > 0. In conclusion, BT Z = {AdS 3 /P e πn ξ P ) g ξ, ξ) > 0}, and there exists a gauge in which the induced etric on BTZ is 4), ξ = θ, and r > 0 correspond to the region g ξ, ξ) > 0. Analytical prolongation beyond r = 0 would lead to closed tielike curves because of the identification, and for this reason r = 0, that lies at an affine-finite distance, is a physical singularity. It is iportant to have it clear for the following subsection. The RSII set up in the BTZ bulk is constructed by splitting up the bulk into two separated parts, gluing two copies of one of the parts along a 1-brane and perforing a Z -syetry on the etric w.r.t. the brane a irror), that is, if x Z y then gx) = gy). In the following subsection we write the action of the theory, solve the equations of otion and interpret the solution for the brane. B. Solution for the Bulk and the Brane et us take the convention that greek indices run fro zero to two, while latin indices run fro zero to one. We shall assue the existence of two charts {x ±α }, each at one side of the brane Σ, a chart {y a } on Σ and the first junction condition [h ab ] := h + ab Σ) h ab Σ) = 0 in order to have a distributionally well defined curvature, where h ab is the induced etric on Σ. We define the noral vectors n ±α to point away fro Σ into the adjacent space. Notice that under these assuptions the extrinsic curvatures satisfy K + αβ Σ) = K αβ Σ) := K αβσ) [8]. Our bulk is present at both sides of the boundary brane Σ, so two Gibbons-Hawking-York boundary ters [9] together with the bulk Einstein-Hilbert action are required for the gravitational part of the action. The 1-brane action is proportional to its world-volue which is an area, since a 1-brane is a string actually), being the proportionality constant its tension σ. Hence, the action of the theory reads S[g αβ, h αβ ] = 1 κ d 3 x gr + 3 )+ + 1 κ d y hk + + K ) σ 3 Σ κ d y h 5) 3 Σ where 1/κ 3 is the 3-diensional Planck ass. The action variation w.r.t. g αβ and h αβ give note we are using the projector h αβ rather than the induced etric h ab ) δs = κ 3 κ 3 Σ d 3 x gr αβ 1 Rg αβ 1 g αβ)δg αβ + d y h K αβ Kh αβ + σh αβ )δh αβ 6) where A := 1/)A + Σ)+A Σ)). The bulk equations of otion are nothing but Einstein s equations and are solved by 4). et us now rewrite the boundary equation in the for K αβ = σh αβ. 7) To solve 7) for the 1-brane we begin with the static ansatz Σ : 0 = Θr, θ) := θ Ψr), Ψ Ψ + π 8) which leads to the noral vector n α = ± µθ µθ µ Θ ±A0, Ψ r), 1), where Ψ r) := d dr Ψr) and A := r. Now we prolongate n F r)rψ r)) α to the rest of the +1 spacetie and copute the extrinsic curvature. Plugging the result into 7), the solution for Ψr) is iediately found, giving two branes that are syetric w.r.t. the x-axis and that wrap around an infinite nuber of ties see FIG.1), σ log 4 +σ ) r 1 σ )+σ 4 r Ψ ± r) = ±. 9) Therefore, in order to have a brane defined at r one has to ipose σ < 1. Choosing y 0 = t, y 1 = r) FIG. 1: Ψ ± solutions, where x := r cos θ), and y := r sin θ). on Ψ + and Ψ, the pull-back of the etric reads ) r ds = dt + = φr) r dr 10) being φr) = αr βr +α ), α = σ 3 and β = 4σ 1 σ ). Nevertheless, 10) is not a black hole yet. To see why, let us recall that the BTZ solution has a causal singularity at r = 0 instead of a curvature one. In 10) the curvature is also well behaved at r = 0, so if analytic prolongation beyond r = 0 does not lead to closed tielike curves, then there are no physical reasons to prohibit such an extension and r = 0 would not be a singularity. Indeed, oving to the Schwarzschild gauge Treball de Fi de Grau Barcelona, June 017

3 through ρr) = the etric reads 1 σ )r + σ 4 1 σ 11) ds = F ρ)dt + dρ F ρ), 1) with F ρ) = 1 σ ρ 1 σ ). However, this spacetie is nothing but AdS with cosological constant [10] Λ = 1 σ 13) and of course it does not contain nor singularities neither horizons. C. Black Hole on the Brane In order to have a black hole, we shall construct the 1-brane using parts of both branches Ψ + and Ψ. The radius at which Ψ + and Ψ intersect are given by 4σ e nπ r n = e nπ 4σ 1 σ ) 14) with n Z and Ψ ± approach infinity with asyptotic angle θ σ ) = ± 1 log 4σ 1 σ ) ). 15) Indeed, when θ σ )/π Z the branches becoe parallel for large r, the denoinator in 14) vanishes for n = θ σ )/π and therefore the last intersection point lies at infinity. It is easy to check that drn dn < 0 σ,, n), so when θ σ )/π / Z there exists a last intersecting point given by r nax, being [ ] 1 n ax σ ) = π θ σ ) 16) where the operator [ ] takes the first integer coing after the R-nuber it contains. For r > r nax the branches do not intersect and approach infinity with asyptotic angle given by 15). The 1-brane of our braneworld, Σ, can be constructed cutting out the parts of Ψ + and Ψ at which r < r nax, and gluing the reaining branches, that we will refer to as Σ + and Σ respectively, in the last intersection point r = r nax see FIG.1). et us first study the branch Σ +. It is convenient to consider the Schwarzschild gauge 1) here. In Σ, ρ > ρr nax ), so introducing the coordinate x := ρ ρr nax ), 0 < x <, the etric on Σ + reads ds Σ + = λ x +Mx N)dt dx + λ x + Mx N 17) where λ := Λ, M := 1 r nax 1 σ ) + σ and N := r nax ). Perforing the sae procedure on Σ but defining x := ρ ρr nax )), < x < 0 instead, allows us to write down the etric for Σ in the copact for ds Σ = fx)dt + dx fx), 18) where fx) := λ x + M x N. The horizon of 18) lies at x h = M + M + λ N λ, 19) and it will only exist if N > 0, i.e. if the last intersecting point is inside the BTZ horizon, r nax <. Provided that N > 0, we can rescale the coordinates as x := x N, t := Nt, and the etric becoes ds Σ = f x)d t + d x f x) 0) where f x) := λ x +µ x 1, µ := M N fro now on we reove the tildes). Using standard conforal copactification techniques to study the causal structure of 0), we obtained the Kruskal and Penrose diagras shown in FIG.. Inspection of these diagras leads one to con- FIG. : Kruskal diagra left) and Penrose diagra right). clude that 0) contains a black and white) hole region. This is precisely the black hole studied in the following sections of this paper. The curves x = 0 in the Penrose diagra are not straight lines because we have cut the brane precisely at x = 0 and have not allowed it to reach r = 0. Actually, x = 0 is a geoetrical geodesic of spacelike character. This eans that a free falling particle can be at rest instantaneous) at x = 0. Moreover, the extrinsic curvature of {t, x = 0)} Σ is precisely K Σ = µ. For these reasons, and other interesting ones discussed in [6] we shall interpret our black hole 0) as a singularity caused by the presence of a particle with ass µ sitting at x = 0. In the following sections, we go beyond the work done in [6] by studying the properties of the black hole on the brane and apping the results into the corresponding quantu black hole, assuing the validity of the holographic conjecture. Treball de Fi de Grau 3 Barcelona, June 017

4 III. ANAYSIS OF THE BACK HOE SOUTION ON Σ Here we prove that the black hole 0) on Σ is not allowed for soe values of the tension of the brane, σ. Since the physical agnitudes in 0) are σ and µ, let us set = 1 and regard note that it is diensionless) as a fixed paraeter. For convenience, we shall work with σ rather than σ. Notice that now σ 0, 1). The key point to see that the black hole 0) is not allowed for soe σ is to realise that by variating the tension new intersecting points IP) between Ψ + and Ψ can appear fro the infinity. Indeed, 16) is a step function of σ and its unit jups at certain values of σ are translated to jups of the last IP r nax σ ) fro a finite value to infinity see discussion below 15)). That is, by variating σ we can ake appear new intersecting points fro infinity. However, as discussed in Section II C the last IP has to be inside the BTZ horizon, r nax <, in order to have a black hole on Σ. If we keep variating σ once a new IP has appeared fro infinity, it could eventually end up crossing the BTZ horizon and hence we would have a black hole on Σ again, but the tensions at which r nax > do not allow such a black hole. et us now find the values of σ at which the black hole on Σ is not allowed. First, we have to obtain the tensions σn at which new IPs appear fro infinity solving n = 1 π θ σn). Before doing it, though, we shall perfor a brief analysis of the function 1 π θ σ ). It has a axiu at σ = 1/ and satisfies 1 π θ σ ) < 1 > 0, σ 0, 1)), which eans that n ax 1 > 0, σ 0, 1)). The function exhibits three different behaviours depending on : for < 1 it is always negative while for > 1 it vanishes twice and widens its positive range as increases. In the critical case = 1 it vanishes only once at its axiu see FIG.3). For reasons that will becoe clear in the next section, we can restrict ourselves to study the case > 1 only. In this situation, there exist two solutions for each n < 1, given by σ n± = 1 1 ± 1 exp nπ ) ). 1) For the case r n=1 σ ), we see that r 1 σ ) < σ 0, 1) and furtherore it is the last IP in σ σ 0, σ 0+). Hence, the first result is that the tensions σ 0, σ 0+) allow a black hole on Σ. Studying r 0 σ ) in [σ 0+, 1], it is easy to see that it is onotonically decreasing and >, so the second result is that in [σ 0+, 1] the black hole on Σ is forbidden. A siilar although a bit ore tedious) analysis perfored in the lower range [0, σ 0 ] leads one to conclude that the forbidden intervals of tension are n=0 [σ nh, σ n ] ) [σ 0+, 1] ) FIG. 3: Behaviour of 1 π θ vs σ for different bulk asses. where σnh = exp nπ ) 1 + exp nπ ) ) ) exp nπ ) 3) is the tension at which the IP r n σ ) passes through the BTZ horizon, r n σ nh ) =. In the following section we conclude our contribution assuing the validity of the holographic conjecture in order to regard 0) as a quantu black hole, and then study and interpret the iplications of the result ). A. Consequences and Results on the Quantu Black Hole One necessary condition to apply the holographic conjecture is that the bulk ust be classical. In our case, this eans 1. Hence, before aking any reference to the conjecture, we shall rewrite the result in ) as an expansion to a certain order in a paraeter ɛ, which in turn has to be a power of 1/. Soon we will see that in order to obtain an analytical expansion in ɛ the appropriate paraeter is ɛ = 1/. Furtherore, choosing this paraeter yields to a better physical interpretation of the expansion, since the BTZ black hole teperature is T BT Z [5]. Then, an expansion in ɛ is actually an expansion in the inverse of the teperature of the black hole living in the bulk. As a first study, we shall restrict our expansion to the first order in ɛ or the leading order in its inverse, and leave to future work the analysis of higher order corrections. Before perforing the expansion of ), let us reark that since n ranges fro 0 to we are not allowed to truncate the expansion of the exponential and we have to retain it. Expanding the other factors to leading order, the forbidden intervals ) read n=0 [ exp nπɛ) exp nπɛ) ɛ, ɛ 4 4 ]) ] [1 ɛ, 1. 4) Treball de Fi de Grau 4 Barcelona, June 017

5 That is, to leading order in ɛ the forbidden intervals squeeze to becoe a discrete spectru of forbidden tensions, which constitutes the first result of this work. However, it is clearly a second order correction and we leave its study to future work. Staying in the first order in ɛ, we see that all the forbidden tensions lie at the unphysical points σ = 0 and σ = 1 and, therefore, do not lead to any physical consequence. This is equivalent to say that the allowed range σ0, σ0+) covers the whole interval σ 0, 1) in a first order expansion in ɛ. et us now study the consequences of this fact on the ass µ, that we shall ake diensionless defining µ := µ λ recall 1/λ is the AdS length). For σ σ0, σ0+), µ is given by µσ r 1 σ ) = )) r 1 σ )) + σ 5) and it is a onotonically increasing function of σ. Evaluating µ ± := µσ σ 0±) ) and perforing the expansion to the first order in ɛ and the leading order in its inverse, we have µ + = ɛ, µ = ɛ. 6) The existence of an upper and a lower bound in the ass µ, and the fact that one is the inverse of the other, is the second result we have obtained and it is rather surprising. Studying a possible invariance or duality µ 1/ µ is left for future work. In order to understand the bounds µ ± we shall evaluate the size of the black hole horizon in both cases, again to first order in ɛ. At µ we obtain of the separation line between the quantu and classical regies. Evaluating the horizon at µ + gives x h µ + ) = 1 ɛ λ 4. 8) This is even ore interesting because fro the holographic point of view we are already in the fully quantu regie and there is still a lower bound x h µ + ) for the size of the quantu black hole. Motivated by the results on the black hole inforation loss proble done in [11, 1], we suggest that the lower bound for the size of the horizon could be interpreted as the existence of an inforon, i.e. a renant of the -diensional quantu black hole proposed in already existent works as a possible solution of the black hole inforation paradox. However, this is out of the scope of this paper and we leave the study of the black hole in these lines to future work. IV. CONCUSIONS We have obtained a black hole solution on the 1-brane of the RSII braneworld and studied its causal structure. Analysing the solution we have found that the black hole on Σ is forbidden for a spectru of tensions. Studying the consequences of this in the large liit, we have obtained the intriguing result of a lower bound for the size of the black hole. Interpreting the results in the light of the holographic conjecture, we suggest that the lower bound could be understood as an inforon. x h µ ) = 1 λ. 7) Now we are in conditions to assue the holographic conjecture and we shall interpret the last result in its light. What 7) is telling us is that the black hole on Σ can not be larger than the AdS length 1 λ. At the sae tie, we know that the horizon of classical black holes in AdS is larger than the AdS length. Therefore, if we take the holographic point of view and regard 0) as a quantu black hole, our result in 7) is a natural recovery Acknowledgents I would specially like to thank y advisor Dr.Cristiano Gerani for a patient and very professional guidance. I would also like to thank y colleagues Biel Cardona and Francesc Cunillera for useful discussion. Finally, I ost sincerely thank y faily for being constant supporters and always taking care of e. [1] N.D. Birrell and P.C.W. Davies, Quantu fields in curved space, Cabridge University Press, New York 198). [] R. B. Mann, A. Shiekh and. Tarasov, Nucl. Phys. B 341, ). [3] R. Eparan, A. Fabbri and N. Kaloper, JHEP 008, ). [4]. Randall and R. Sundru, Phys. Rev. ett. 83, ). [5] M. Bañados, C. Teitelboi and J. Zanelli, Phys. Rev. ett. 69, ). [6] C. Gerani and G.P. Procopio, Phys. Rev. D 74, ). [7] M. Bañados, M. Henneaux, C. Teitelboi and J. Zanelli, Phys. Rev. D 48, ). [8] S. C. Davis, Phys. Rev. D ). [9] G.W. Gibbons and S. W. Hawking, Phys. Rev. D 15, ). [10] A. Achúcarro, M. Ortiz, Phys. Rev. D 48, ). [11] S. B. Giddings, Phys. Rev. D ). [1] S. B. Giddings, Phys. Rev. D ). Treball de Fi de Grau 5 Barcelona, June 017