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Canonical Quantization of the BTZ Black Hole using Noether Symmetries

The well-known BTZ black hole solution of (2+1) Einstein's gravity, in the presence of a cosmological constant, is treated both at the classical and quantum level. Classically, the imposition of the two manifest local Killing fields of the BTZ geometry at the level of the full action results in a mini-superspace constraint action with the radial coordinate playing the role of the independent dynamical variable. The Noether symmetries of this reduced action are then shown to completely determine the classical solution space, without any further need to solve the dynamical equations of motion. At a quantum mechanical level, all the admissible sets of the quantum counterparts of the generators of the above mentioned symmetries are utilized as supplementary conditions acting on the wave-function. These additional restrictions, in conjunction with the Wheeler-DeWitt equation, help to determine (up to constants) the wave-function which is then treated semiclassically, in the sense of Bohm. The ensuing space-times are, either identical to the classical geometry, thus exhibiting a good correlation of the corresponding quantization to the classical theory, or are less symmetric but exhibit no Killing or event horizon and no curvature singularity, thus indicating a softening of the classical conical singularity of the BTZ geometry.


Introduction
Due to its simplicity, three dimensional Einstein gravity is widely considered as an interesting model to explore many aspects of General Relativity. At the classical level, it is known that any vacuum solution of Einstein's equations represents a flat spacetime ( [1], [2]), while the existence of a cosmological constant (again in the absence of matter) leads to other maximally symmetric solutions, i.e. de Sitter/anti-de Sitter manifolds [3]. However, the study of three dimensional geometry proved to be highly non trivial with the discovery of the BTZ black hole [4]. The latter emerges in the case of pure gravity and under the presence of a negative cosmological constant −l 2 , i.e. the action describing the system assumes the form Since then many aspects of the properties of the BTZ spacetime have been explored [5]- [8], but it is its canonical quantum description that motivates the present work.
Other methods have also been presented in the literature, regarding the quantization of 2+1 geometries. These, mainly focus in solving the constraints and using them to derive a Hamiltonian description in a reduced form that results in a system of finite degrees of freedom [9]- [12]. In this paper we study the classical and quantum description of the BTZ geometry in a different perspective: At the classical level we are led to a finite dimensional system by imposing on a general three dimensional line element the two manifest local isometries of the BTZ black hole (∂ t and ∂ φ ), and then inserting the resulting reduced metric in the action (1.1); the result is a mini-superspace model in which a 2+1 decomposition in the direction of the radial component r is considered. This method was firstly exhibited in the case of four dimensional static, spherically symmetric space-times in [13] and [14]. A study of the Noether symmetries of the reduced Lagrangian leads to enough integrals of motion so that the BTZ metric is acquired without any need to solve the dynamical equations of motion and thus identify parts of the anti-de Sitter maximal manifold. At the quantum level, we proceed with the canonical quantization of the model; except of the quantum quadratic constraint we use the quantum analogues of the existing Noether symmetries for the reduced system by imposing them on the wave function as eigen-operators.
Due to the reparametrization invariance under arbitrary changes of the radial coordinate (r = f (r)) there is a non-trivial problem for finding the maximum number of conserved quantities: In the theory of regular systems the infinitesimal criterion for the determination of Noether symmetries reads £ ξ L = dF dt where L is the Lagrangian and F an arbitrary function of the configuration space variables and the dynamical parameter [15]. However, since mini-superspace Lagrangians are singular in nature, one has to modify the aforementioned criterion in order to acquire all the possible Noether symmetries generated by the configuration space vector ξ [16] (for specific examples see [17]- [19]). In short, the required change is that the action of the generator ξ on the Lagrangian does not have to be strictly zero (or equal to a total derivative), but it suffices to be equal to a multiple of the constraint. Thus, for a mini-superspace Lagrangian of the form the criterion reads £ ξ L = ω(q) ∂L ∂N , with ω(q) allowed to be any arbitrary function. In [16] it was proven that the ξ's that satisfy this condition are given by i.e. they are Killing fields of the scaled mini-supermetric G αβ := V G αβ . More over one can also use the homothecy of the scaled mini-superspace (£ h (V G αβ ) = V G αβ ) to construct a rehonomic integral of motion. With the help of Dirac's theory for constrained systems [20]- [22], one can be led to a Hamiltonian description and proceed with the quantization of the particular system. In phase space, the Noether charges defined by the ξ's obtained by (1.3), become quantities linear in the momenta. In the usual scenario of quantum cosmology one proceeds with the quantization of the system by demanding that the action of the constraint operators being zero on the wave function. But, in the case where Noether charges are present, one can use them to define linear Hermitian eigen-operators that can be enforced as supplementary conditions on the wave function. Thus, simplifying the procedure and also uniquely (up to a constant phase) defining the wave function.
The structure of the paper is as follows: In section 2 we construct the corresponding mini-superspace action, calculate the symmetry generators and derive the classical BTZ solution with the help of the integrals of motion. In the next section we proceed with the canonical quantization of the system and we derive a wave function for each case of possible sets of observables. In section 4, we use Bohm's semiclassical analysis [23], applied on the previously found wave functions and derive the corresponding semiclassical space-time manifolds. Finally, we conclude our analysis, summing our results in the discussion.
2 Classical treatment 2.1 Derivation of the general form of the line element We require that the metric of a (2+1)-dimensional spacetime with an assigned coordinate system (t, r, φ) admits the following Killing vector fields: The ensuing stationary and axisymmetric line element is of the generic form where i, j = 0, 1, 2 and {x 0 , x 1 , x 2 } = {t, r, φ}. In order to further simplify the line element (2.1), we exploit the remaining freedom in changing coordinates in a way that does not introduce t, φ in the metric components. In this spirit, inserting the following coordinate transformation t −→t = t − g 01 g 22 − g 02 g 12 g 2 02 − g 00 g 22 dr, dr, (where it is assumed that g 02 (r) 2 − g 00 (r) g 22 (r) = 0), the finally reduced metric inferred from (2.1) assumes the formg withg 11 (r) = g 11 + g 00 g 2 12 + g 22 g 2 01 − 2 g 01 g 02 g 12 g 2 02 − g 00 g 22 . Note that the metric (2.2) guarantees that the spacetime described by it is invariant under simultaneous reflections of the time and angular coordinate, i.e. (t,φ) → (−t, −φ).
In order to bring (2.2) into a form suitable for our purposes, we choose g 00 (r) = −a(r) 2 , g 02 (r) = c(r), g 22 (r) = b(r) 2 ,g 11 (r) = n(r) 2 where n(r) stands for n(r) 2 = 4Λ 2 (g 11 (g 2 02 − g 00 g 22 ) + g 00 g 2 12 + g 22 g 2 01 − 2 g 01 g 02 g 12 ) and Λ is the cosmological constant. In this parametrization the line element (2.1) in the coordinate system (t, r,φ) assumes the form The above parametrisation has been chosen for several reasons: it simplifies considerable the canonical formulation and makes the potential of the ensuing quadratic constraint constant with respect to a(r), b(r) and c(r). An immediate implication of the latter is that, as rigorously shown in [17] and easily seen from (1.3), the conformal Killing fields that generate the Noether symmetries reduce to Killing fields.

Lagrangian and Noether (conditional) symmetries
According to the discussion in sec. 2.1, the stationary, axisymmetric line element taken as the starting point in the present work is of the form where the˜were dropped for the sake of simplicity. Since we are working in a (2 + 1) decomposition along the r coordinate [13], where space-time is foliated by r (instead of t) hypersurfaces, the field n(r) plays the role of the r-lapse function and the fields a(r), b(r), c(r) are the r-dynamical variables. Notice that the r-lapse function is already reparametrised, i.e.g 11 (r) = n(r)/2Λ(a 2 b 2 + c 2 ) 1/2 , in a way that makes the potential of the quadratic constraint constant with respect to the dependent dynamical variables. The action for (2 + 1)-dimensional geometries with a general cosmological constant is (1.1). For simplicity we consider Λ = l 2 (which means that in our analysis the BTZ black hole emerges for Λ > 0), thus the aforementioned action takes the form where the ′ denotes differentiation with respect to the radial coordinate r. The Lagrangian (2.5) belongs to a particular form of singular Lagrangians: L = 1 2 n G µν q ′ µ q ′ ν + n V (q) with q µ = (a, b, c), a mini-supermetric of the form and a constant (as desired) potential V (q) = 1. It can be easily shown that the Euler-Lagrange equations of (2.5) are identical to Einstein's field equations R µν − 1 2 g µν R = Λg µν , resulting from the extremization of the action (2.4) and evaluated on the metric (2.3).
Let us now turn attention to the conditional symmetries of (2.5). It has been shown in [17] that, in the specific lapse reparametrisation we employ in order to force the potential to depend only on the lapse, the generators of the conditional symmetries are the Killing vector fields of the mini-supermetric (2.6).This metric describes a flat, Lorentzian manifold, thus admitting a six-dimensional isometry group of motions. By a straightforward calculation it can be confirmed that the infinitesimal condition L ξ G µν = 0 is satisfied by the following six generators: (2.7) The above fields satisfy a Lie algebra [ξ K , ξ L ] = C M KL ξ M of which we present only the non-zero structure constants: In addition, (2.6) admits a homothetic vector field that satisfies the condition L h G µν = G µν :

Hamiltonian formulation and the solution space
We now develop the Hamiltonian formulation of the geometries (2.3) and derive the classical solution space by algebraic means. Following [21,22] we first define the rconjugate momenta where π n is a first class primary constraint. Invoking the Legendre transformation for (2.5), we arrive at the Hamiltonian where the functions u n are Lagrangian multipliers and which, in view of the need for preservation of the primary constraint π n during the evolution, i.e. π ′ n = {π n , H} ≈ 0, leads to the first class secondary constraint H ≈ 0. According to [17], each one of the Killing vector fields contracted with the nonvanishing conjugate momenta (2.10), i.e. Q I := ξ µ I π µ , corresponds, on the phase space, to a linear integral of motion: (2.12) The phase space quantities (2.12) form a Poisson algebra {Q K , Q L } = C M KL Q M with structure constants given by (2.8). The Poisson brackets of (2.12) with the Hamiltonian H are strongly vanishing, i.e. {Q I , H} = 0, guaranteeing that the six quantities (2.12) are constants of motion: In addition, from the quadratic constraint (2.11) and (2.12), one can easily read off the first Casimir invariant of the algebra formed by (2.12), namely (2.14) One observes that in this parametrization of the constant potential, Q C is the kinetic part of the Hamiltonian. An appropriate combination of the phase space quantities Q I leads to the second Casimir invariant which vanishes identically when the quantities Q I are expressed via (2.12) in terms of the conjugate momenta. Let us define now the phase space quantity corresponding to the homothetic vector field (2.9): whose Poisson bracket with the Hamiltonian H reads in view of the vanishing of the quadratic constraint (2.11) on the constraint surface. The latter result implies that dQ h dr = n, which as it has been shown in [16] by integration over r becomes a rheonomic integral of the form In the rest of this section, we will show that (2.12) together with (2.17) and the Casimir invariants (2.14)-(2.15) are enough to determine the entire classical solution space of the geometries (2.3). The integrals of motion (2.12), (2.17), and (2.14)-(2.15) become constants on the solution space; thus, the following relations readily follow where κ I , c h are constants; in the derivation of the last relations it was taken into account that H = 0 on the solution space and that Q C = 0. Let us first solve algebraically the five first (I = 1, . . . , 5) equations of (2.18a) together with (2.18b) for a, a ′ , c, c ′ , n, and n dr: , where b remains an arbitrary function of r. It can be easily checked that the consistency conditions for the fields a and c (a ′ = da dr and c ′ = dc dr ) are identically satisfied in view of (2.19a)-(2.19b) and (2.19c)-(2.19d), respectively. The consistency condition for n (n = d dr ndr), on the other hand, requires which, in view of (2.18c) and (2.18d), is identically satisfied. To complete our consistency check, we substitute (2.19) into the equation Q 6 = κ 6 that has not been used in the above derivation; the result reads which is identically satisfied by virtue of (2.18d). Now, it is an easy task to verify that It is obvious from the discussion above that on the solution space there are six constants available and two equations, i.e. (2.18c)-(2.18d), constraining them; thus, we can specify freely four of them. The following choice of the constants κ I (that respects both conditions (2.18c)-(2.18d)) accompanied by a re-definition of the angular coordinate φ →φ = φ − d 2 t brings the line element (2.3) into the form which, when b is chosen as b(r) = r, exactly reproduces the BTZ metric originally introduced in [4]: From (2.22) it is obvious that d is a non-essential constant and can, since it is additively absorbed, be set to zero in order to simplify the expressions of the constants κ I . Thus, by setting d = 0 the constants (2.20) considerably simplify to Notice that by using the above choice of constants one can bring (2.3) into the form (2.22) directly without re-defining the φ coordinate.

General considerations
Here, we will quantize the classical system described in sec. 2 according to Dirac's canonical quantization procedure [21] supplemented by the condition that the wave function must be an eigenfunction of the quantum analogs of the generators of the Noether symmetries (2.12). The method has been extensively described in [17] and applied to the quantization of the Schwarzshild [17] and Reissner-Nordström [18] black holes. Below, we give a brief overview of the method described in [17]. As usual, in the Schrödinger representation the classical dynamical variables become operators where q α = {a, b, c, n}. (In the following we assume that = 1.) The above operators obey the canonical commutation relations [ q α , π β ] = δ α β . Following Dirac's proposal [21] we demand the quantum analogs of the first class constaints (2.10a) and (2.11) to annihilate the wavefuntion, namely π n Ψ(a, b, c, n) = − ∂ ∂n Ψ(a, b, c, n) = 0 ⇒ Ψ = Ψ(a, b, c), where the former condition guarantees that the wave function is lapse-independent and the latter is the unit eigenvalue problem for the Casimir invariant (2.14). The quantum analog of the Casimir invariant Q C in the above expression is given as the most general scalar quadratic Hermitian (under an arbitrary measure µ) operator [17] a choice by which the latter of the above conditions becomes In a similar fashion, we define the quantum analogs of the integrals of motion (2.12) as the most general linear Hermitian (under the same measure µ) operators In [17], it has been proven that linear operators of the form (3.3) satisfy the same algebra as their classical counterparts Q I , i.e.
where the structure constants C M KL are given by (2.8). Now, we can form an eigenvalue problem for each one of the linear operators (3.3) where κ I are the eigenvalues of the operators Q I . For several reasons, see [17], a natural geometric choice of the measure reads It was shown in [24] and [17] that (3.5) together with the quantum algebra of the operators Q I impose certain restrictions on which of the linear operators Q I can be simultaneously applied on the wave function. This selection rule is given by the integrability condition In addition, the number of essential constants of the underlying geometry provide a lower bound on the number of linear operators that must be used to define eigenvalue equations simultaneously on the wave function; in our case the relevant constants involved are two: the mass M and the angular momentum J of the black hole. At this point a clarification concerning the nature of M, J is pertinent: their origin, as explained in [5], can be traced to identifications of parts of the maximal AdS (3) manifold. In that global (topological) sense they are essential for characterizing the BTZ space-time. However, locally they can be absorbed by appropriate coordinate transformations-see [25] for the infinitesimal criterion and Appendix A for the actual construction of the transformation. We make this distinction since, as we will see later, in the semiclassical approximation M and J become also locally essential.
In the following, we will say that the operators Q I satisfying simultaneously the above selection rule form an admissible subalgebra of the full quantum algebra (3.4), otherwise we will say that they form a non-admissible subalgebra. Let us now exemplify the use of the selection rule (3.7). First, consider the three dimensional subalgebra { Q 1 , Q 2 , Q 3 }. Admissible subalgebras: • Three dimensional subalgebras: • Two dimensional subalgebras: Summarising, the system of differential equations (3.2), (3.5) will be solved (for the choice of measure (3.6)) for each one of the admissible subalgebras (3.8)-(3.9c).

3.2
The three dimensional subalgebra { Q 1 , Q 4 , Q 6 } Let us start with the three dimensional subalgebra (3.8). There are three conditional symmetries, for each one of which an eigenvalue problem of the form (3.5) must be defined. For the choice of measure (3.6) and the Killing vector fields (2.7) these eigenvalue problems read In addition, we have to solve the quadratic constaint (3.2) Integrating successively from (3.10a) to (3.10c), one obtains the general solution where c 0 is an arbitrary constant. Inserting the latter expression into (3.11) we get κ 2 1 + 2 κ 4 κ 6 − 4 Λ = 0, which is identically satisfied in view of (2.18c).

3.3
The two dimensional subalgebra { Q 1 , Q 5 } Next, let us consider the two dimensional subalgebra (3.9a). The two conditional symmetries dictate the following two eigenvalue problems which restrict the form of the wave function to where ψ is an arbitrary function of a b. The Hamiltonian constraint (3.11) further restricts the solution to its final form where c 1 , c 2 are arbitrary constants and J ν (z) and Y ν (z) are the Bessel functions of the first and second kind, respectively.

The two dimensional subalgebra { Q 2 , Q 4 }
The eigenvalue problems of the form (3.5) for the two dimensional subalgebra (3.9b) read where c 0 is an arbitrary constant.

3.5
The two dimensional subalgebra { Q 3 , Q 6 } Finally, we will consider the two dimensional subalgebra (3.9c). The two eigenvalue problems read 17b) The two differential equations above allow the solution for an arbitrary function ψ(b). Taking into account the quadratic constraint (3.11) one arrives at the general solution where c 0 is an arbitrary constant.

Semiclassical analysis 4.1 General considerations
To make a connection between the quantum solutions (3.12), (3.14), (3.16), and (3.18) of the various admissible subalgebras and the classical solution space (2.22), we develop here a semiclassical analysis of these quantum results in the spirit of [18] following the original Bohmian approximation [23]. We start by defining the general form of the wave function a,b,c) , where Ω(a, b, c) and S(a, b, c) are the amplitude and the phase of the wave function, respectively. Inserting (4.1) into the Hamiltonian constraint (3.11) and taking the imaginary part, one arrives at the continuity equation The real part reads where we define the quantum potential (its name will be shortly justified) Notice that for a vanishing quantum potential (4.3) resembles the Hamiltonian constraint (2.11); in that case, a direct inspection of their elements leads to the identification {π a , π b , π c } = {∂ a S, ∂ b S, ∂ c S}, which when expressed, through (2.10b)-(2.10d), in terms of the variables of the configuration space provide us with the following semiclassical equations of motion (4.5) From the discussion above one expects that when the quantum potential vanishes, V = 0, the semiclassical equations (4.5) must reproduce the classical solution (2.22). Otherwise, when V = 0, the solution of the semiclassical equations must differ from the classical one because of quantum effects introduced by the quantum potential.

4.2
The three dimensional subalgebra { Q 1 , Q 4 , Q 6 } Let us start with the wave function (3.12), Ψ = c 0 e 1 2 (κ6a 2 +2 κ 4 b 2 +2 κ 1 c) , of the three dimensional subalgebra (3.8). Inserting the numerical values (2.23) of the κ I 's corresponding to the classical solution space, the above wave function reduces to A direct comparison of the latter with (4.1) leads to the identification which satisfy the continuity equation (4.2) and lead to a vanishing quantum potential V = 0. Thus, it is expected that the solution to the following semiclassical equations of motion will reproduce the classical solution (2.22). Notice that in the derivation of (4.7) we chose n(r) = −2 Λ r. (4.8) The above gauge fixing of the lapse function is compatible with its expression (2.19f) on the solution space and, in addition, also satisfies the Hamiltonian constraint. Thus, our choice of the lapse (4.8) does not lead to inconsistencies. Now, we will solve the system (4.7). The last of the equations (4.7c) leads to The other two can also be directly integrated

The two dimensional subalgebra { Q 1 , Q 5 }
Let us now move to the solution (3.14) of the two dimensional subalgebra (3.9a). After substitution of the numerical values (2.23), the corresponding wave function looks like Because of the appearance of the Bessel functions in the above wave function we cannot decisively conclude about the form of its amplitude and phase. To bring (4.11) into the form (4.1) we have to study its behaviour for large and small values of r separately.

Asymptotic behaviour for large values of r
Notice that the asymptotic behaviour of the dynamical fields (2.22) is a(r) ∼ r, b(r) ∼ r, c(r) ∼ J 2 ; thus, the arguments of the Bessel functions in (4.11) in the limit of large r are also large. The behaviour of the Bessel functions for large values of its arguments is the following [26] Now, inserting the above asymptotic expressions of the Bessel functions into (4.11) one gets (4.12) Notice that the simple choice of the constants c 2 = c 1 brings the expression in the square brackets into an exponential form; thus, we arrive at the final form of the wave function in the asymptotic limit where c 3 is a complex constant. From (4.13) one can easily read the amplitude and the phase, namely which satisfy the continuity equation but, unlike the previous case, give rise to a non-zero quantum potential

Behaviour close to the origin r = 0
Let us now see how is the wave function (4.11) behaving near the origin r = 0. In view of (2.22) the dynamical fields in the limit of small r's behave like a(r) ∼ M, b(r) ∼ r, c(r) ∼ J 2 ; thus, likewise the arguments of the Bessel functions in (4.11) are also small close to the origin. By inserting into (4.11) the expression of the Bessel functions for small arguments [26] where Γ is the gamma function, one obtains which after a re-definition of the constants reduces to where c 3 is a complex constant. Now, using the relation y x = cos(x ln y) + sin(x ln y) and some trigonometric identities, one can bring the above expression into the explicitly exponential form Ψ ∼ 2 c 3 cos (M ln(a b)) e M ln( a b ) . which as expected satisfies the continuity equation and leads to the non-vanishing potential The semiclassical equations of motion (4.5) read where the gauge (4.8) was again used. Integration of (4.24c) yields c(r) = c 1 . This line element represents a homogeneous space-time, admitting the three Killing fields ∂ t , ∂ φ and −M∂ t + 2 r ∂ r + M∂ φ . Notice that, since there is no classical curvature singularity, we would expect the semiclassical geometry to share the same property; indeed, for the above metric, there is no central singularity at r = 0. An interesting observation is that, as earlier mentioned, both M and J appearing in the metric are now also locally essential constants; the Ricci scalar is R = − M 2 2 and the Kretchmann scalar is R ijkl R ijkl = − 1 4 (2J 2 − 3) M 4 . Another property is that the Killing horizon of the classical black hole has disappeared.

4.4
The two dimensional subalgebra { Q 2 , Q 4 } We will continue our semiclassical study with the wave function (3.16) of the two dimensional subalgebra (3.9b). After substituting the numerical values (2.23) into (3.16) the wave function reduces to which is already in exponential form; therefore, one can define The above amplitude and phase satisfy the continuity equation, result to a vanishing quantum potential V = 0, and lead to the semiclassical equations of motion where an integration constant was set to zero. Finally, by substituting the expressions (4.31)-(4.32) into (4.30b) and integrating, one arrives at the solution b(r) = r 2 + c 2 .
where, as before, an integration constant was set to zero. Finally, taking into account the expressions for b(r) and c(r), the last remaining equation (4.35b) of the above system admits the solution a(r) = Λ r 2 + c 2 .

Discussion
In the present work the method of canonical quantization has been implemented for the BTZ geometry. The procedure begins at the classical level with the construction of a "proper" mini-superspace description. Accordingly, a Lagrangian function that describes a system dynamically equivalent to that of the Einstein field equations of 2+1 gravity in the presence of a cosmological constant must be derived. The resulting Lagrangian is of course singular, thus one has to take into account the modified infinitesimal criterion (1.3) in order to acquire all the variational symmetries of the action. To simplify the procedure, we chose to work in the constant potential parametrization. In this parametrization the ξ's of (1.3) become Killing fields of the corresponding mini-supermetric, which is three dimensional, flat and exhibits a homothecy. The six Killing fields and the homothetic field are linked to six autonomous (2.12) and a rehonomic (2.16) integrals of motion, respectively. In phase space we use these seven conserved quantities, not only to derive the BTZ solution algebraically, but also to read off the two Casimir invariants of the six dimensional algebra spanned by the Q I 's.
The adoption of the constant potential parametrization becomes imperative at the quantum level. There the non-identically zero Casimir invariant is manifested as the kinetic part of the Hamiltonian. Hence, the Wheeler-DeWitt operator becomes compatible with any choice of the eigen-operators Q I that satisfy the integrability conditions (3.7). There exist four independent maximal Abelian subalgebras of the Q I 's that lead to an equal number of different wave functions that also satisfy the constraint HΨ = 0, one for each set of "measurable" quantities.
Using Bohm's approximation for the derivation of quantum trajectories, one can assign to each wave function a corresponding semiclassical geometry. Our analysis shows that for the subalgebras (3.8), (3.9b), and (3.9c) the corresponding geometries are identical to the classical one due to the fact that the quantum potential is zero. A possible explanation of this result follows from the particular lapse parametrization we are using here, which identifies the kinetic part of the quantum quadratic constraint with the Casimir invariant of the algebra of the charges (see the relevant section in [18] for a complete discussion). However, the simultaneous measurement of Q 1 and Q 5 leads to quantum corrections at both limits r → +∞ and r → 0: In the asymptotic limit, one obtains a semiclassical geometry that is asymptotically AdS (R = −6Λ − M 2 2Λr 4 ), which is in accordance with the classical geometry, where R = −6Λ. The existence of a curvature singularity at r = 0, in this semiclassical approximation, is not alarming since the region of validity of the solution rests in the range of large values of r. In the vicinity of the origin, r → 0, the semiclassical geometry (4.28) is obtained. A quick inspection of the line element (4.28) entails that the aforementioned "macroscopic singularity" does not exist anymore. Moreover, this homogeneous semiclassical space-time is characterized by two essential constants, i.e. M and J. The latter can be interpreted as an indication that at small scales the contribution of the mass and of the angular momentum exceeds by far the corresponding contribution of the cosmological constant Λ.