Gravitational decoupling in 2+1 dimensional space--times with cosmological term
GGravitational decoupling in dimensional space–times with cosmological term
Ernesto Contreras ∗ †
Yachay Tech University,School of Physical Sciences & Nanotechnology,100119-Urcuqu´ı, Ecuador
In this work we implement the Minimal Geometric Deformation method to obtain the isotropicsector and the decoupler matter content of any anisotropic solution of the Einstein field equationswith cosmological constant in 2 + 1 dimensional space–times. We obtain that the solutions of bothsectors can be expressed analytically in terms of the metric functions of the original anisotropicsolutions instead of formal integral as in its 3 + 1 counterpart. As a particular example we study aregular black hole solution and we show that, depending on the sign of the cosmological constant,the solutions correspond to regular black holes violating the null energy condition or to a non–regular black hole without exotic hair. The exotic/non–exotic and the regular/non–regular blackhole dualities are discussed.
I. INTRODUCTION
The interest in the Minimal Geometric Deformation(MGD) [1–32, 36] as a powerful method to decouple theEinstein field equations to obtain new solutions [18, 22–27, 30, 32–35, 37–39] has considerably increased. Amongthe main applications of the method we find studiesof local anisotropies in spherically symmetric systems[25, 26, 37–39], hairy black holes [24] and new anisotropicsolutions in 2+1 dimensional space–times [30, 33].The method has been extended to solve the inverseproblem [32], namely, given any anisotropic solution ofthe Einstein field equations it is possible to recover theisotropic sector and the decoupler matter content which,after gravitational interaction, led to the anisotropic con-figuration. In that work, it was found that, for ananisotropic solution with exotic matter sector (negativeenergy density), the free parameters involved in the MGDcan be setted such that both the isotropic and the decou-pler sectors satisfy all the energy conditions. It was thefirst time that a kind of exotic/non–exotic matter wasfound using the method.As another extension of MGD and the inverse problem,in Ref. [35] the method have been studied consideringEinstein’s equations with cosmological constant and im-plemented in a polytropic black hole which is a solutionwith a matter content satisfying all the energy conditions.The main finding there was that the isotropic sector isdeeply linked with the appearance of exotic matter, al-though it can be located inside the horizon. In this sense,this work shows again how the apparition of exotic mat-ter seems unavoidable but one could, in principle, controlthe energy conditions by tuning the isotropy/anisotropyof a black hole solution.The MGD-decoupling have been implemented in 2+1 cir-cularly symmetric and static spacetimes obtaining that ∗ On leave from Universidad Central de Venezuela † [email protected] both the isotropic and the anisotropic sector fulfil Ein-stein field equations in contrast to the cases studied in3 + 1 dimensions, where the anisotropic sector satisfiesquasi-Einstein field equations. In this sense, the isotropicand the decoupler sector leads to a pair of new solutionof Einsteins equations, one for each source.In this work we study MGD in 2 + 1 circularly symmetricand static spacetimes with different purposes and inter-ests. First, as a difference to the previous work (see Ref.[30]), we consider 2 + 1 Einstein’s equation with cosmo-logical constant. This is because, as the BTZ is a vac-uum solution of this configuration, the set of equationscoming from MGD method could serve as the startingpoint to extend interior 2 + 1 solutions to anisotropicdomains taking into account suitable matching condi-tions between the compact objects and a BTZ vacuum.Second, we study the inverse MGD problem to explore,among other aspects, the exotic/non–exotic matter con-tent duality previously reported in the 3 + 1 dimensionalcase [32, 35]. As we shall see later, the inverse methodleads to more tractable expressions to deal with becausethey correspond to exact analytic instead to formal equa-tions as previously reported [32, 35].This work is organized as follows. In the next sectionwe briefly review the MGD-decoupling method. Then,in section III, we obtain the isotropic sector and the de-coupler matter content considering a regular black holeas anisotropic solution. In section IV we study the en-ergy conditions to explore the apparition of exotic materin the solutions and some final comments and conclusionare in the last section. II. EINSTEIN EQUATIONS IN
SPACE–TIME DIMENSIONS
In a previous work we considered the MGD-method withcosmological constant [35]. In this work study the 2 + 1dimensional case. a r X i v : . [ g r- q c ] J a n Let us consider the Einstein field equations R µν − Rg µν + Λ g µν = κ T totµν , (1)and assume that the total energy-momentum tensor isgiven by T ( tot ) µν = T ( m ) µν + θ µν , (2)As usual, the energy–momentum tensor for a perfect fluidis given by T µ ( m ) ν = diag ( − ρ, p, p ) and the decoupler mat-ter content reads θ µν = diag ( − ρ θ , p θr , p θ ⊥ ). In what fol-lows, we shall assume circularly symmetric space–timeswith a line element parametrized as ds = − e ν dt + e λ dr + r dφ , (3)where ν and λ are functions of the radial coordinate r only. Considering Eq. (3) as a solution of the EinsteinField Equations, we obtain κ ˜ ρ = − Λ + e − λ λ (cid:48) r (4) κ ˜ p r = Λ + e − λ ν (cid:48) r (5) κ ˜ p ⊥ = Λ + 14 e − λ (cid:0) − λ (cid:48) ν (cid:48) + 2 ν (cid:48)(cid:48) + ν (cid:48) (cid:1) (6)where the prime denotes derivation with respect to theradial coordinate and we have defined˜ ρ = ρ + ρ θ (7)˜ p r = p + p θr (8)˜ p ⊥ = p + p θ ⊥ . (9)The next step consists in decoupling the Einstein FieldEquations (4), (5) and (6) by implementing the minimaldeformation e − λ = µ + αf. (10)As usual, Eq. 10 leads to two sets of differential equa-tions: one describing an isotropic system sourced bythe conserved energy–momentum tensor of a perfectfluid T µ ( m ) ν an the other set corresponding to Einsteinfield equations sourced by θ µν . Now, as in a previouswork [35], we interpret the cosmological constant as anisotropic fluid, so we include the Λ–term in the isotropicsector and we obtain κ ρ = − r − µ (cid:48) r (11) κ p = 2Λ r + µν (cid:48) r (12) κ p = 4Λ + µ (cid:48) ν (cid:48) + µ (cid:0) ν (cid:48)(cid:48) + ν (cid:48) (cid:1) , (13) for the perfect fluid and κ ρ θ = − f (cid:48) r (14) κ p θr = f ν (cid:48) r (15) κ p θ ⊥ = f (cid:48) ν (cid:48) + f (cid:0) ν (cid:48)(cid:48) + ν (cid:48) (cid:1) , (16)for the anisotropic system [41]. Note that that the addi-tion of the cosmological constant only affects the isotropicsector because Eqs. (14), (15) and (16) remain un-changed.For the inverse problem we shall apply the same strat-egy followed in reference [32], namely, we implement theconstraint ˜ p ⊥ − ˜ p r = α ( p θ ⊥ − p θr ) (17)It is remarkable that, unlike the 3 + 1 case, the solutionfor f ( r ) obtained from the constraint (17) is an exactexpression in terms of ν, λ , instead of a combination offormal integrals as found in [32]. In fact, f = c r e − ν ν (cid:48) + e − λ α (18)where c is a constant of integration. Now, from Eq.(10), we obtain µ = − αc r e − ν ν (cid:48) (19)Note that somehow, the constant c controls the geomet-ric deformation: if c → ρ = − Λ − αc e − ν (2 rν (cid:48)(cid:48) + ν (cid:48) ( rν (cid:48) − ν (cid:48) (20) p = Λ − αc re − ν ν (cid:48) , (21)and the decoupler matter content satisfies ρ θ = 12 (cid:18) c e − ν (2 rν (cid:48)(cid:48) + ν (cid:48) ( rν (cid:48) − ν (cid:48) + e − λ λ (cid:48) αr (cid:19) (22) p θr = c re − ν ν (cid:48) + e − λ ν (cid:48) αr (23) p θ ⊥ = 14 (cid:32) c re − ν ν (cid:48) + e − λ (cid:0) − λ (cid:48) ν (cid:48) + 2 ν (cid:48)(cid:48) + ν (cid:48) (cid:1) α (cid:33) (24)At this point, some comments are in order. First, Eqs.(11), (12) and (13) correspond to Einstein field equationswith cosmological constant for a perfect fluid. Second,Eqs. (14), (15) and (16) corresponds to Einstein fieldequations without cosmological constant and anisotropicdecoupler fluid. Note that this set of equations to-gether with those of the isotropic sector allows us todecouple Einstein’s equation with cosmological constantin 2 + 1 dimensional space–times for any anisotropicfluid. What is more, the above expressions can be usedto extend isotropic solutions embedded in a BTZ vac-uum to anisotropic domains after the implementationof suitable matching conditions. Finally, that the in-verse problem leads to exact analytical expressions entailsthe “isotropization” of a broader set of systems than its3 + 1 dimensional counterpart which is given in termsof formal integrals. In this sense, the inverse problemin 2 + 1 dimensions can be implemented starting fromany anisotropic solutions at difference to the 3 + 1 casewhere depending on the particular form of the anisotropicsolution, the inverse problem would yield to formal in-stead exact analytical solutions. In the next section weshall implement the inverse problem using a well knownregular and circularly symmetric black hole solution asanisotropic system. III. ISOTROPIC SECTOR OF A REGULARBLACK HOLE IN 2+1 DIMENSIONS
In this section we shall illustrate the inverse MGD prob-lem using as anisotropic solution a regular black hole met-ric. The reason to consider a regular black hole solutionis twofold: to explore the conditions for the apparition ofexotic matter and to study if the MGD inverse problemcan affect the regularity of the solution.Let us consider the regular black hole solution [40] withmetric functions e ν = − M − Λ r − q log (cid:0) a + r (cid:1) (25) e − λ = − M − Λ r − q log (cid:0) a + r (cid:1) (26)where M , Λ, a and q are free parameters. This geometryis sustained by a matter content given by˜ ρ = q π ( a + r ) (27)˜ p r = − q π ( a + r ) (28)˜ p ⊥ = q (cid:0) r − a (cid:1) π ( a + r ) (29)This solution corresponds to a black hole whenever − M − Λ r − q log (cid:0) a + r (cid:1) = 0 leads to two real roots (orone real root in the extreme case) for some values of theparameters { M, Λ , q, a } [40]. What is more, the solutionis regular everywhere, which can be deduced from theinvariants R = 2 q (cid:0) a + r (cid:1) ( a + r ) + 6Λ (30) Ricc = K = 4 q (cid:0) a + 2 a r + r (cid:1) ( a + r ) + 8Λ q (cid:0) a + r (cid:1) ( a + r ) + 12Λ , (31) where R , Ricc and K correspond to the Ricci, Riccisquared and Kretschmann scalar respectively.From now on we shall apply the inverse MGD problem toobtain the isotropic generator and the decoupler mattercontent associated with this regular black hole solution.From Eq. (18), the decoupler function f reads f = − c (cid:0) a + r (cid:1) (cid:0) q log (cid:0) a + r (cid:1) + M + Λ r (cid:1) a + r ) + q ) − q log (cid:0) a + r (cid:1) + M + Λ r α (32)Now, from Eq. (19), the radial metric function of theisotropic sector is given by µ = αc (cid:0) a + r (cid:1) (cid:0) q log (cid:0) a + r (cid:1) + M + Λ r (cid:1) a + r ) + q ) (33)Replacing the above result in the set of isotropic Einsteinequations, Eqs. (11), (12) and (13), the perfect fluidreads ρ = − αc a r (cid:0) q (cid:0) a r + r (cid:1) + Λ a r + 2 M q + q (cid:1) πλ r − αc q a r log a r + 4Λ λ r πλ r (34) p = αc (cid:0) a + r (cid:1) π (Λ ( a + r ) + q ) + Λ8 π (35)where a r := a + r (36)Λ r := Λ a r + q (37)At this point some comments are in order. First, notethat, as in the 3 + 1 case, the inverse problem does notmodify the position of the killing horizon. In fact, thehorizon appears whenever − M − Λ r − q log (cid:0) a + r (cid:1) =0 which, as discussed above, leads to one or two real rootsfor the black hole solution. Second, the regularity of thesolution depends on the positivity of the parameter Λ.In fact, the invariants R = − αc a r (cid:32) (cid:0) M q + 3 q (cid:1) r − q log a r r − q (cid:0) a + 4 r (cid:1) r − a r r (cid:33) (38) Ricc = K = α c a r (cid:0) F + 2Λ r (cid:1) r (39)where F = (cid:0) q log a r + 2Λ q a r + Λ a r + 2 M q + q (cid:1) , (40)reveal that the solution is regular everywhere wheneverΛ r = Λ (cid:0) a + r (cid:1) + q (cid:54) = 0, which can be satisfied if Λ > > − M + q log( a + r H ) r H (41)from where it must be imposed that r H + a < q | log( a + r H ) | > M , with r H the horizon radius. Inthis case, the solution corresponds to a regular isotropicblack hole solution.Note that, in the case Λ < r c as frequently found in the application of MGD.This result would lead to a naked singularity for r c >r H or to a non regular black hole solution for r c < r H .In the last case, we say that the isotropic sector of theregular black hole corresponds to a non-regular black holesolution.Now we focus our attention into the decoupler sector. Inthis case, the metric functions are { ν, f } and the decou-pler matter content reads ρ θ = c a r (cid:0) q (cid:0) a r + r (cid:1) + Λ a r + 2 M q + q (cid:1) r + q αa r + c q a r log a r r + Λ α (42) p θr = − c a r r − q αa r + Λ α (43) p θ ⊥ = q (cid:0) r − a (cid:1) αa r − c a r r − Λ α (44)The above solution corresponds to an anisotropic regularblack hole solution for Λ >
0. In fact, the solution hasa killing horizon when − M − Λ r − q log (cid:0) a + r (cid:1) = 0.The invariants are given by R = H ( a + r ) (Λ ( a + r ) + q ) (45) Ricc = H ( a + r ) (Λ ( a + r ) + q ) (46) K = H ( a + r ) (Λ ( a + r ) + q ) (47)where H , H and H are (too long) regular functions interms of polynomials of r and log( a + r ). Note that, asdiscussed above, the regularity of the solution dependson the sign of Λ. More precisely, for Λ < IV. ENERGY CONDITIONS
In this section we study the energy conditions of theobtained solution for the distinct cases outlined be-fore.
A. Case I. Λ > As previously discussed, this case corresponds to regularblack hole solutions for the isotropic and decoupler sectorwith the horizon radius located at − M − Λ r H − q log( a + r H ) = 0. Given the nature of the solution, a numericalanalysis is mandatory. However, setting suitable valuesof M, Λ , q, a to obtain real horizons and Λ >
0, we inferthat the behaviour of the energy density can be writtenas lim r → ρ = Aαc − B (48)lim r →∞ ρ = − Cαc − B (49)with A , B , C real and positive numbers. In particular,for Λ = 2, M = 1, q = 1 and a = 0 . r → ρ = 0 . αc − . r →∞ ρ = − . αc − . αc is a positive (negative) quantitysuch that lim r → ρ > r → ρ < r →∞ ρ > r →∞ ρ <
0) necessarily. In figure 1 we showthe profile of the energy density for some values of αc . - - r ρ ( r ) FIG. 1: Energy density for for αc = −
200 (black solid line), αc = −
140 (dashed blue line) αc = 400 (short dashed redline) and αc = 500 (dotted green line). For the decoupler sector we obtain that the negative val-ues for the energy density can be avoided with a suitablechoice of the parameter. Without loss of generality, letus set Λ = 2, M = 1, q = 1 and a = 0 . r → ρ θ = 102 .α − . c (52)lim r →∞ ρ θ = 2 α + c α and c positive values, the exotic mattercontent can be avoided whenever c ≤ . α . In figure2 we show the profile of ρ θ for α = 1 and some values for c . r ρ ( r ) FIG. 2: Energy density for for c = 1 (black solid line), c = 10 (dashed blue line) c = 50 (short dashed red line)and c = 100 (dotted green line). We would like to conclude this section by emphasizingthat for Λ > M, q, Λ , a , theapparition of exotic matter in the decoupler sector canbe circumvented for certain values of the parameters in-volved. In the next section we shall study the energyconditions for Λ < B. Case II. Λ < In this case, Λ < r H + a < q | log( a + r H ) | > M or r + a >
1. For example, wecan choose the values M = 2, q = 1, a = 1 from whereΛ = − . r H = 2 and the critical radius is at r c = 0 . αc >
0, the asymptotic behaviour of theenergy density is given bylim r → r c ρ = ∞ (54)lim r →∞ ρ = 0 . αc + 0 . . (55)In figure in figure 3 we show the profile of ρ as a functionof the radial coordinate for different values of αc > M = 2, q = 1, a = 1 and Λ = − . r →∞ ρ θ = − . α − . c (56) r ρ ( r ) FIG. 3: Energy density for for αc = 0 . αc = 0 . αc = 0 . αc = 1 (dotted green line). r ρ ( r ) FIG. 4: Energy density for for c = − c = − c = − c = − so that for α < c < α = − r → r c ρ θ → ∞ (57)and we obtain that the exotic matter can be avoided. Infigure 4 we show the profile of the energy density ρ θ for α = − c .At his point a couple of comments are in order. First,note that in both cases (isotropic and decoupler sector)the exotic content can be avoided. Second, for suitablechoices of the parameters, the solution corresponds to anon–regular black hole containing a non–vanishing criti-cal radius. In this sense, we conclude that although theexotic/non–exotic duality can be circumvented for Λ < V. CONCLUSIONS
In this work we have extended the Minimal GeometricDeformation method in 2 + 1 dimensional space–timesto decouple the Einstein field equations including cosmo-logical constant. We obtained that the isotropic sectorobeys Einstein’s equation with cosmological constant butthe decoupler part consists in a system without cosmo-logical term. In this sense, we can combine any 2 + 1isotropic, static and circularly symmetric interior solu-tion of the Einstein field equations with cosmologicalconstant embedded in a BTZ vacuum with certain de-coupler matter solution and suitable matching conditionsto obtain new anisotropic interior solutions in the threedimensional realm.We showed that the inverse problem leads to exact ana-lytical solutions for the decoupling and the isotropic met-ric in terms of the original anisotropic solution instead toformal integrals obtained in the 3 + 1 counterpart. Thescope of this result to obtain analytical solutions is broad.Indeed, it can be implemented taking into account anyanisotropic solution as the starting point because it doesnot involve formal integrals as the 3 + 1 case. As a par-ticular example we implemented the inverse problem toa regular 2 + 1 black hole solution. We obtain that for apositive cosmological constant the isotropic sector corre- sponds to a regular isotropic black hole in presence of a“exotic” hair (negative energy density), and the decou-pler sector is a regular anisotropic black hole which, un-der certain circumstances, can be surrounded by a mattercontent with positive energy density so that the appari-tion of exotic matter can be avoided. For negative cosmo-logical constant both the isotropic and the decoupler sec-tor corresponds to non–regular black hole solution wherethe existence of exotic hair can be avoided with a suit-able choice of the free parameters. It is worth mentioningthat on one hand the non–regular black hole solution issingular in a non–vanishing radius as often occur in theimplementation of the Minimal Geometric Deformationprotocol. On the other hand, the exotic matter can beavoided but the price that it has to be paid is that the thesolutions are not regular anymore. In this sense, the kindof exotic/non-exotic matter duality appearing in previousworks transmute to a regular/non-regular duality in thecases where the exotic content can be avoided.
VI. ACKNOWLEDGEMENT
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