General formalism for the stability of thin-shell wormholes in 2+1 dimensions
aa r X i v : . [ g r- q c ] S e p General formalism for the stability of thin-shell wormholesin 2+1 dimensions
Cecilia Bejarano , ∗ , Ernesto F. Eiroa , , † , Claudio Simeone , , ‡ Instituto de Astronom´ıa y F´ısica del Espacio, Casilla de correo 67, Sucursal 28, 1428,Buenos Aires, Argentina Departamento de F´ısica, Facultad de Ciencias Exactas y Naturales,Universidad de Buenos Aires, Ciudad Universitaria Pabell´on I, 1428, Buenos Aires, Argentina IFIBA–CONICET, Ciudad Universitaria Pabell´on I, 1428, Buenos Aires, Argentina
July 8, 2018
Abstract
In this article we theoretically construct circular thin-shell wormholes in a 2 + 1 dimensionalspacetime. The construction is symmetric with respect to the throat. We present a generalformalism for the study of the mechanical stability under perturbations preserving the circularsymmetry of the configurations, adopting a linearized equation of state for the exotic matter atthe throat. We apply the formalism to several examples.PACS numbers: 04.20.Gz, 04.60.Kz, 04.40.NrKeywords: Lorentzian wormholes; spacetimes with charge; dilaton gravity
Traversable wormhole geometries [1, 2] have been widely studied in the last three decades. Sev-eral articles considering wormholes in low dimensional (2 + 1) spacetimes have appeared in theliterature [3]. In the framework of General Relativity, wormholes must be threaded by matter notsatisfying the energy conditions. A particular class of wormholes consists of those constructed bycutting and pasting two manifolds to obtain a new one with a joining shell at the throat [2]. In re-cent years shells around vacuum (bubbles), around black holes and stars, and supporting traversablewormholes, have received considerable attention [4–8]; spherically symmetric shells have been stud-ied in detail in four and also in more spacetime dimensions [9, 10]. We have recently analyzed fourdimensional spherical shells for Einstein gravity coupled to Born–Infeld electrodynamics in relationwith thin-shell wormholes [11] and shells around vacuum or around black holes [12]. The dynami-cal evolution of collapsing shells in three spacetime dimensions has been presented and applied toseveral examples in Refs. [13]. Shells in a three dimensional background within Einstein–Maxwelltheory have been associated to thin-shell wormholes [14], and the analysis of 2 + 1 − dimensional ∗ e-mail: [email protected] † e-mail: [email protected] ‡ e-mail: [email protected] ds = − f ( r ) dt + f − ( r ) dr + r dθ , (1)where f ( r ) = − M − Λ r − Q ln (cid:18) rr (cid:19) . (2)The dimensionless constant M is identified as the Arnowitt–Deser–Misner (ADM) mass, Q is theelectric charge, and Λ = − l − is the cosmological constant with dimensions [length] − . The metrichas the right signature only if Λ <
0. So in what follows we will assume that Λ <
0. This choicemakes possible a standard horizon structure in which the metric function is always positive beyonda certain radius, and the geometry is asymptotically Anti-de Sitter. This behavior has made thismetric of great interest within the framework of string theory.Another interesting 2 + 1 dimensional geometry, obtained by Chan and Mann [18], is associatedto the existence of a dilaton field coupled to the electromagnetic field. In the presence of a cos-mological constant background, the spherically symmetric solution includes a dilaton with a radiallogarithmic behavior, and a one-parameter family of metrics of the form ds = − f ( r ) dt + 4 r N − dr N γ N f ( r ) + r dθ , (3)with f ( r ) = − MN r N − − r (3 N − N + 8 Q (2 − N ) N . (4)Here M is the mass, Q is the electric charge, Λ is the cosmological constant, γ and N are integrationconstants; the parameter N , in particular, determines the character of the solution (existence of anevent horizon, etc.). If M >
0, Λ < / < N <
2, the solution corresponds to a black hole;in this case, the asymptotic behavior is that of Anti-de Sitter spacetime. The particular case with N = 1 corresponds to the metric of Mandal, Sengupta and Wadia (MSW) [19].A recently introduced spherically symmetric solution in 2 + 1 spacetime dimensions is theSchmidt–Singleton metric of the form [20] ds = − Kr dt + dr + r dθ , (5)with K a constant. This solution results from a matter source in the form of a real self-interactingscalar field, which has a logarithmic behavior with the radial coordinate. The most interestingaspect of this geometry is that while its spatial part is flat, the temporal part has a behavior whichcorresponds to the asymptotics of Anti-de Sitter. This unusual difference between the spatial andtemporal parts is associated to the fact that the spacetime curvature is determined only by theradial pressure of the scalar field.In this work we mathematically introduce the shells which support traversable Lorentzian worm-holes by applying the well known cut and paste procedure; then we present a general formalism forthe study of the mechanical stability under perturbations preserving the symmetry of the circularshells. We apply the formalism to the geometries described above. Finally, we discuss the resultsobtained. We set the units so that G = c = 1. 2 General formalism
The mathematical construction of a symmetric wormhole geometry starts from two equal copiesof a 2 + 1 dimensional manifold, which in coordinates x α = ( t, r, θ ), has a metric with the genericform ds = − f ( r ) dt + g ( r ) dr + h ( r ) dθ , (6)where the metric functions f , g and h are non negative from a given radius to infinity. These copiesare cut and pasted at a radius a ; when in the original manifold there is an event horizon with radius r h , we take a > r h . Provided that the flare-out condition is fulfilled, so that the geodesics open upat the throat, the resulting construction is a traversable wormhole. At the radius a where the twocopies of the geometry are joined, the components of the extrinsic curvature tensor read K ± ij = − n ± γ (cid:18) ∂ x γ ∂ξ i ∂ξ j + Γ γαβ ∂x α ∂ξ i ∂x β ∂ξ j (cid:19) , (7)where ξ i = ( τ, θ ) represent the coordinates on the shell, and n ± γ are the normal unit ( n γ n γ = 1)vectors n ± γ = ± (cid:12)(cid:12)(cid:12)(cid:12) g αβ ∂F∂x α ∂F∂x β (cid:12)(cid:12)(cid:12)(cid:12) − / ∂F∂x γ , (8)with F ( r ) = r − a ( τ ). Using the metric functions, the non zero components of n ± γ take the form n t = ∓ ˙ a p g ( a ) f ( a ) , (9) n r = ± p g ( a )[1 + ˙ a g ( a )] . (10)Then, the extrinsic curvature is given by K ± ˆ τ ˆ τ = ∓ p g ( a )2 p a g ( a ) (cid:26) a + ˙ a (cid:20) f ′ ( a ) f ( a ) + g ′ ( a ) g ( a ) (cid:21) + f ′ ( a ) f ( a ) g ( a ) (cid:27) , (11) K ± ˆ θ ˆ θ = ± h ′ ( a )2 h ( a ) s a g ( a ) g ( a ) , (12)where the hats are used to denote that we are working in an orthonormal basis; the dot means aderivative with respect to the proper time τ on the shell, and a prime stands for a derivative withrespect to r . With these definitions we can write down the Lanczos equations [21, 22], which relatethe extrinsic curvature at both sides of the one dimensional surface with the energy-momentumtensor S ˆ i ˆ j = diag( λ, p) on it − [ K ˆ i ˆ j ] + [ K ] g ˆ i ˆ j = 8 πS ˆ i ˆ j . (13)Here the brackets denote the jump of a given quantity across the surface, K is the trace of K ˆ i ˆ j , and λ and p are the energy density and the pressure on the shell. The cut and paste procedure removesthe interior regions r < a and joins the exterior parts of the two identical geometries described by ametric like (6). The jump of the extrinsic curvature components at the surface r = a is associatedwith the linear energy density λ = − π h ′ ( a ) h ( a ) s a g ( a ) g ( a ) (14)3nd to the pressure p = 18 π s g ( a )1 + ˙ a g ( a ) (cid:26) a + ˙ a (cid:20) f ′ ( a ) f ( a ) + g ′ ( a ) g ( a ) (cid:21) + f ′ ( a ) f ( a ) g ( a ) (cid:27) . (15)It is useful to introduce the conservation equation [22], which is obtained using the “ADM” constrain(also called Codazzi-Mainardi equation) and taking into account the Lanczos equations − ∇ i S ij = (cid:20) T αβ ∂x α ∂ξ j n β (cid:21) , (16)where the operator ∇ stands for the covariant derivative and T αβ denotes the bulk energy-momentum tensor. Defining the one dimensional area A = 2 π p h ( a ), we can write the conservationequation in a form that relates the energy on the shell with the work done by the pressure and theenergy flux ddτ ( λ A ) + p d A dτ = − ˙ aλ A (cid:20) f ′ ( a ) f ( a ) + g ′ ( a ) g ( a ) + h ′ ( a ) h ( a ) − h ′′ ( a ) h ′ ( a ) (cid:21) . (17)In the case where g ( r ) = [ f ( r )] − and h ( r ) = r in a neighborhood of r = a , it is easy to see thatthe factor between the brackets vanishes and the flux term is zero. By using that λ ′ = ˙ λ/ ˙ a , we canwrite the condition (17) in the form λ ′ h ( a ) h ′ ( a ) + λ + p − λh ( a )2 h ′ ( a ) (cid:20) f ′ ( a ) f ( a ) + g ′ ( a ) g ( a ) + h ′ ( a ) h ( a ) − h ′′ ( a ) h ′ ( a ) (cid:21) . (18)As in the case of 3 + 1 dimensions (see Ref. [7]), if there exists an equation of state in the form p = p ( λ ) or p = p ( a, λ ), the expression (18) is a first order differential equation that can be recastin the form λ ′ = F ( a, λ ), for which always exists a unique solution with a given initial condition,provided that F has continuous partial derivatives, so it can be (formally) integrated to obtain λ ( a ). Then, from Eq. (14) we obtain the equation of motion for the shell˙ a + V ( a ) = 0 , (19)where we have defined the potential V ( a ) by V ( a ) = 1 g ( a ) − (cid:20) πλ h ( a ) h ′ ( a ) (cid:21) . (20)The first and the second derivatives of the potential, using Eq. (18) successively, have the form V ′ ( a ) = (cid:20) g ( a ) (cid:21) ′ + 64 π [ W ( − R + W T )] , (21)and V ′′ ( a ) = (cid:20) g ( a ) (cid:21) ′′ − π (cid:26) ( R − W T ) ( R − W T ) − W T ′ + λ ( η − (cid:20) S − W h ′′ ( a ) h ′ ( a ) + W T (cid:21)(cid:27) (22)where we have introduced η = p ′ /λ ′ , and the functions R = λ − p, S = λ + p, T = f ′ ( a ) f ( a ) + g ′ ( a ) g ( a ) + h ′ ( a ) h ( a ) , W = λ h ( a ) h ′ ( a ) .
4n the static configurations, all the equations are evaluated at a fixed radius a ; in this case wehave the energy density and pressure for the static configuration λ = − π h ′ ( a ) h ( a ) p g ( a ) , (23) p = 18 π f ′ ( a ) f ( a ) p g ( a ) . (24)Because the geometry must fulfill the flare-out condition that the geodesics open up at the wormholethroat, then h ′ ( a ) > η can be interpreted as the squared velocity of sound (then 0 ≤ η ≤ a can be written in terms of the metric functionsin the form V ′′ ( a ) = − f ( a ) f ′ ( a ) g ′ ( a ) + 2 g ( a ) (cid:8) [ f ′ ( a )] − f ( a ) f ′′ ( a ) (cid:9) f ( a )] [ g ( a )] + η h ( a ) [2 g ( a ) h ′′ ( a ) − g ′ ( a ) h ′ ( a )] − g ( a )[ h ′ ( a )] g ( a )] [ h ( a )] . (25)From Eq. (19) it is clear that a stable static solution with throat radius a satisfies V ( a ) = V ′ ( a ) =0, and V ′′ ( a ) >
0. Thus the stability analysis of the wormhole configurations is essentially theanalysis of the sign of the second derivative of the potential.
In this Section we will apply the formalism to some relevant examples. The first and the secondwormhole geometries are locally indistinguishable from the exterior regions of the associated blackhole metrics, while the spatial part of the third one is flat. The three cases considered share theasymptotic Anti–de Sitter behavior of the temporal part of the metric.
We first consider in our wormhole construction the BTZ metric with f ( r ) given by Eq. (2), g ( r ) = f − ( r ) and h ( r ) = r . We assume a negative cosmological constant Λ, so that the asymp-totic behavior of the geometry is that of the 2 + 1 dimensional Anti–de Sitter universe ( AdS ).Wormholes connecting two copies of the exterior region (i.e. radii beyond the horizon) of thisgeometry have already been studied in Ref. [14]. We will revisit the construction and the linearizedstability analysis of this geometry using our general formalism in order to make straightforwardthe comparison with the new results presented below for wormholes associated to other sphericallysymmetric 2 + 1 geometries.In the wormhole geometry resulting from the cut and paste procedure, the features of theoriginal metric, i.e. its horizon structure, determine the form of the stability regions in parameterspace as the charge increases. Then that structure must be detailed. The position r h of theevent horizon is given by the largest real positive solution of the equation f ( r ) = 0, which gives5 - a (cid:144) r Η È Q È = - a (cid:144) r Η È Q È = - a (cid:144) r Η È Q È = Q ci - a (cid:144) r Η È Q È = Q ci - a (cid:144) r Η È Q È = - a (cid:144) r Η È Q È = Q c ii - a (cid:144) r Η È Q È = Q c ii - a (cid:144) r Η È Q È = Figure 1: BTZ wormhole geometry: stability regions (in gray) when two critical values of the chargeexist. The values of the parameters are Λ r = − M = 0 .
5, so that Q ic = 0 .
432 and Q iic = 1 . r h = p − M/ Λ when Q = 0, and it can be numerically solved for Q = 0. For low values of themass, i.e. M < − Λ r , two critical values of the charge Q exist: Q ic and Q iic are such that for acharge smaller than Q ic and for a charge larger than Q iic an event horizon exists in the originalmetric, while for 0 < Q ic < | Q | < Q iic there is a naked singularity. For M = − Λ r , there is only onecritical value of the charge, Q ic = Q iic = √− Λ r , and beyond that value, i.e. M > − Λ r , an eventhorizon always exists in the original metric for any value of the charge. The critical values of thecharge are obtained as the positive real solutions of the equation Q − Q ln (cid:2) − Q / (Λ r ) (cid:3) − M = 0.In the dynamic case, the energy density and the pressure are given respectively by Eqs. (14) and(15), with the corresponding metric functions replaced. Because in this case g ( r ) = f ( r ) − and h ( r ) = r , the conservation equation (17) simplifies to ddτ ( λ A ) + p d A dτ = 0 , (26)which gives the condition aλ ′ + λ + p = 0 . (27)The equation of motion of the shell (19) is obtained in terms of the potential by replacing themetric functions in Eq. (20).The energy density and the pressure in the static case, with throat radius a , take the form λ = − p − M − Λ a − Q ln( a /r )4 πa (28)and p = − Λ a + Q πa p − M − Λ a − Q ln( a /r ) . (29)6 - a (cid:144) r Η È Q È = - a (cid:144) r Η È Q È = - a (cid:144) r Η È Q È = Q c - a (cid:144) r Η È Q È = Q c - a (cid:144) r Η È Q È = - a (cid:144) r Η È Q È = Figure 2: BTZ wormhole geometry: stability regions (in gray) for the case in which there is onlyone value of the critical charge. The values of the parameters are Λ r = − M = 1, so that Q ic = Q iic = 1. - a (cid:144) r Η È Q È = - a (cid:144) r Η È Q È = - a (cid:144) r Η È Q È = - a (cid:144) r Η È Q È = Figure 3: BTZ wormhole geometry: stability regions (in gray) when there is no critical charge, soan event horizon always exists in the original manifold. The values of the parameters are Λ r = − M = 1 . a , reads V ′′ ( a ) = 2 a " − Λ a + Q + (cid:0) Λ a + Q (cid:1) M + Λ a + 2 Q ln( a /r ) + (cid:2) M − Q + 2 Q ln( a /r ) (cid:3) η . (30)Figures 1 to 3 illustrate the behavior of the stability regions ( V ′′ ( a ) >
0) with an increase ofthe charge, for a fixed value of the adimensionalized cosmological constant Λ r and three differentvalues of the mass M . In particular, only in the case in which there is no horizon in the originalmetric, stability could be possible for vanishing or small negative η . In all cases a null chargemakes stability incompatible with | η | <
1. The evolution of the regions is not monotonous: as alarger charge is considered, so that the original geometry has an event horizon, the stability regions7ecover a form similar to those corresponding to low values of the charge. The analysis here extendsthe previous works [14] to a larger range of the relevant parameters.
In the case of the symmetric wormhole connecting two charged Chan-Mann solutions, having acosmological constant and a dilaton field, the metric functions correspond to f ( r ) given by Eq. (4), g ( r ) = 4 r N − N − γ − N f − ( r ) and h ( r ) = r . The associated scalar field is φ = 2 kN ln (cid:18) rβ (cid:19) , (31)with β = γ / (2 − N ) and k = ± p N (2 − N )(2 B ) − ( B is a constant). When 2 / < N < ≤ | Q | ≤ Q c or a naked singularity if | Q | > Q c , where thecritical value of the charge is given by Q c = √− Λ (cid:20) ( N − N − M N Λ (cid:21) N/ (3 N − . (32)When 0 ≤ | Q | ≤ Q c , for the wormhole construction we take two copies of the region r ≥ a > r h ,where r h is the horizon radius determined by the greatest positive root of the function f ( r ). When | Q | > Q c there is no horizon and then no restriction exists on the possible radius of the throat. Theenergy density and the pressure for the dynamic case are obtained by replacing the correspondingmetric functions in Eqs. (14) and (15). In this case, Eqs. (17) and (18) have, respectively, the form ddτ ( λ A ) + p d A dτ = − π ˙ aλ (cid:18) N − (cid:19) (33)and aλ ′ + λ + p = − λ (cid:18) N − (cid:19) . (34)One obtains the equation of motion of the throat by introducing the metric functions in Eq. (20)and replacing this potential in Eq. (19).The static configurations with shell radius a , have the energy density and the pressure givenby λ = − γ /N √ N πa /N s − M a − /N − a N − Q − N (35)and p = γ /N √ − N πa /N p N (3 N − M (2 − N )(2 − N ) a − /N − N a q M (2 − N )(2 − N ) a − /N − − N ) a + 4(3 N − Q . (36)The second derivative of the potential evaluated at a has the form V ′′ ( a ) = ( ψ + χ η ) a − (4+ N ) /N γ /N , (37)8 .9 1.2 1.5 - a (cid:144) ΒΗ È Q È = - a (cid:144) ΒΗ È Q È = - a (cid:144) ΒΗ È Q È = Q c - a (cid:144) ΒΗ È Q È = Q c - a (cid:144) ΒΗ È Q È = - a (cid:144) ΒΗ È Q È = Figure 4: Chan-Mann wormhole geometry: stability regions (in gray) for the case N = 9 / β = − M β (2 − N ) /N = 0 .
5, for which Q c = 0 . - a (cid:144) ΒΗ È Q È = - a (cid:144) ΒΗ È Q È = - a (cid:144) ΒΗ È Q È = Q c - a (cid:144) ΒΗ È Q È = Q c - a (cid:144) ΒΗ È Q È = - a (cid:144) ΒΗ È Q È = Figure 5: Chan-Mann wormhole geometry: stability regions (in gray) for the case N = 9 / β = − M β (2 − N ) /N = 1 .
5, for which Q c = 0 . ψ = (2 − N ) h M ( N + 2) a /N + 4 a Q i N − a Λ (cid:0) N − N + 4 (cid:1) N (3 N − a ( N − N − (cid:0) a Λ + Q (cid:1) N h M ( N − N − a /N + 4 a Λ( N −
2) + 4 a (3 N − Q i (38)9 .2 0.3 0.40246 a (cid:144) ΒΗ È Q È = a (cid:144) ΒΗ È Q È = a (cid:144) ΒΗ È Q È = Q c a (cid:144) ΒΗ È Q È = Q c a (cid:144) ΒΗ È Q È = a (cid:144) ΒΗ È Q È = Figure 6: Chan-Mann wormhole geometry: stability regions (in gray) for the case N = 1 (MSWmetric), Λ β = − M β (2 − N ) /N = 0 .
5, for which Q c = 0 . a (cid:144) ΒΗ È Q È = a (cid:144) ΒΗ È Q È = a (cid:144) ΒΗ È Q È = Q c a (cid:144) ΒΗ È Q È = Q c a (cid:144) ΒΗ È Q È = a (cid:144) ΒΗ È Q È = Figure 7: Chan-Mann wormhole geometry: stability regions (in gray) for the case N = 1 (MSWmetric), Λ β = − M β (2 − N ) /N = 1 .
5, for which Q c = 0 . χ = M ( N + 2) a /N − a (cid:2) a Λ( N − + (4 − N ) Q (cid:3) ( N − N − . (39)The stability results (which correspond to ( V ′′ ( a ) >
0) are shown for some representative10alues of the parameters in Figs. 4 to 7. For fixed values of the parameters N , the adimensionalizedcosmological constant Λ β , and the adimensionalized mass M β (2 − N ) /N , the largest ranges of theparameter η compatible with stability take place for values of the charge near the critical one; inparticular, large ranges of η including values within the interval [0 ,
1) correspond to stable solutionsif the charge is slightly above the critical value. In general, stable configurations with 0 ≤ η < | Q | > Q c . The behavior with M β (2 − N ) /N for a fixed parameter N and Λ β only showsa change in the range of a /β for which configurations are stable, without changing the rangeof the parameter η . For given Λ β and M β (2 − N ) /N , decreasing values of N (within the range2 / < N <
2) move the stability regions to smaller a /β , and for a fixed | Q | < Q c to higherpositive values of the parameter η . Now we consider the particular case of a wormhole connecting two exterior geometries with theSchmidt–Singleton metric given by f ( r ) = Kr , g ( r ) = 1 , h ( r ) = r . The associated scalar field isgiven by φ = 1 √ κ ln (cid:18) rr (cid:19) , (40)where κ is the coupling constant in the Liouville potential of the field and r is a constant. Themetric presents no event horizon, so there is no restriction to the possible radius for the wormholethroat. After applying the formalism, we have the energy density and pressure given by the simpleexpressions λ = − πa p a , (41) p = 14 π √ a (cid:20) ¨ a + 1 + ˙ a a (cid:21) . (42)The conservation equation for this case is ddτ ( λ A ) + p d A dτ = − π ˙ aλ (43)which gives the condition λ ′ a + 2 λ + p = 0 . (44)The resulting equation of motion is ˙ a + V ( a ) = 0 where the potential has the form V ( a ) = 1 − (4 πaλ ) . (45)The simplicity of V ( a ) makes possible a fully analytical treatment; no plots are needed in order toobtain the conditions required to render the wormhole construction stable under radial perturba-tions. Using Eq. (44), the first and second derivatives of V ( a ) read V ′ ( a ) = 32 π aλ ( λ + p ) , (46) V ′′ ( a ) = − π (cid:8) λ + (2 λ + p ) [( η + 1) λ + p ] (cid:9) . (47)In the static case the energy density and pressure are λ = − πa , p = 14 πa . (48)11ence, the second derivative of the potential evaluated at the radius of the static configuration isgiven by V ′′ ( a ) = − η + 1) a . (49)This result implies that the mechanical stability under radial perturbations is possible only for theinterval η < −
1. Had the ring been constituted by normal matter, the requirement of negativevalues of the parameter η would rule out stability. But the matter of the ring does not fulfillthe energy conditions, i.e. we are dealing with an exotic fluid, then the result η < − We have developed a general formalism for the construction and the analysis of stability of the staticconfigurations corresponding to circular thin-shell wormholes in 2 + 1 spacetime dimensions. Wehave used the usual cut and paste procedure in the construction, and we have adopted a linearizedequation of state for the exotic matter at the throat. The stability analysis then has been reducedto the study of the sign of the second derivative of an effective potential evaluated at the throatradius a . We have applied this formalism to three examples: the charged BTZ, Chan–Mann, andSchmidt–Singleton geometries. Though the steps followed are the same, the difficulties are differentin each case. In particular, in two examples the energy flux in the right hand side of the energyconservation equation does not vanish, while in the other one (BTZ) it is identically zero.In all cases we have found the stability regions in terms of the adimensionalized throat radiusand the parameter η associated with the equation of state. In the BTZ wormhole, for low valuesof the mass and when the charge is very close or between the two critical values (correspondingto the extremal black hole in the original metric) small values of η , in particular with 0 ≤ η < N (with 2 / < N < N shiftthe regions of stability to smaller adimensionalized throat radius, and to higher positive values ofthe parameter η when the charge is lower than the critical one. The Schmidt–Singleton wormholespacetime, which has no charge and admits the simplest treatment, is stable only when η < − Acknowledgments
This work has been supported by Universidad de Buenos Aires and CONICET.
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