Noncommutative Solitonic Black Hole
aa r X i v : . [ h e p - t h ] A ug Noncommutative Solitonic Black Hole
Ee Chang-Young † , , Kyoungtae Kimm † , , Daeho Lee † , , and Youngone Lee ‡ , † Department of Physics and Institute of Fundamental Physics,Sejong University, Seoul 143-747, Korea ‡ Institute of Basic Sciences, Daejin University, Pocheon, Gyeonggi 487-711, Korea
ABSTRACT
We investigate solitonic black hole solutions in three dimensional noncommutative space-time. We do this in gravity with negative cosmological constant coupled to a scalar field.Noncommutativity is realized with the Moyal product which is expanded up to first order inthe noncommutativity parameter in two spatial directions. With numerical simulation westudy the effect of noncommutativity by increasing the value of the noncommutativity pa-rameter starting from commutative solutions. We find that even a regular soliton solution inthe commutative case becomes a black hole solution when the noncommutativity parameterreaches a certain value. [email protected] [email protected] [email protected] [email protected] any candidate theories for quantum gravity, such as string theory and loop quantumgravity, suggest that spacetime may not be commutative at sufficiently high energy scales [1,2]. Meanwhile, black holes in the early universe have been observed recently [3, 4]. Since theenergy density of the early universe was very high, it would be interesting to know the effectof noncommutativity on the formation of a black hole. Black holes in three dimensionalspacetime have been extensively studied. One of the reasons is that gravity models in threedimensions are relatively easier to treat than models in four spacetime dimensions. Thefinding of the BTZ black hole solution [5] has raised a lot of interest in the subject.In [6], the global vortex solution was studied by considering gravity with negative cosmo-logical constant coupled to a complex scalar field in three dimensional commutative space-time. There, the model Lagrangian with a global U (1) symmetry was given by S = Z d x √− g (cid:20) − πG N ( R + 2Λ) + 12 g µν ∂ µ ¯ φ∂ ν φ − λ φφ − v ) (cid:21) , (1)where φ ( x ) is a complex scalar field. The obtained solution was a cylindrically symmetricglobal U(1) vortex solution which smoothly connects the false vacuum at the origin to thetrue vacuum at spatial infinity. Depending upon the ratio of the cosmological constant tothe Plank scale, the model supports a spacetime with a regular soliton or a charged blackhole.In this paper, we investigate the effect of noncommutativity on the above model andwant to see whether global vortex solutions with black hole configuration are allowed in thenoncommutative case. For this purpose we use the same Lagrangian as in (1) except for theMoyal product between the field variables. For computational purpose, here we use the triadand spin connection instead of the metric as in [7]. Since it is hard to obtain an analyticsolution even in the commutative case [6], our approach to find the solution is basicallynumerical. Our analysis is performed up to first order in the noncommutativity parameter.In this paper we work with the noncommutative polar coordinates (ˆ t, ˆ r, ˆ ϕ ) defined by thefollowing commutation relation [ ˆ ρ, ˆ ϕ ] = 2 iθ, , (2)2here ˆ ρ ≡ ˆ r . One reason for using the above commutation relation is that rotational sym-metry is more apparent in the polar coordinates than in the Cartesian coordinates. Anotherquite important reason is that the above commutation relation is physically equivalent tothat of the canonical noncommutivity, [ˆ x, ˆ y ] = iθ , up to first order in the nonocmmutativityparameter θ [7]. A would-be usual commutation relation for the noncommutative polar co-ordinates, [ˆ r, ˆ ϕ ] = iθ , is not equivalent to the canonical noncommutivity, [ˆ x, ˆ y ] = iθ , even inthe first order of θ [7].By the Weyl-Moyal correspondence [8], the physics in the noncommutative spacetimecan be described by the physics in the commutative spacetime with the Moyal product. TheMoyal product corresponding to the commutation relation (2) is given by( f ⋆ g )( ρ, ϕ ) = e iθ (cid:16) ∂∂ρ ∂∂ϕ ′ − ∂∂ϕ ∂∂ρ ′ (cid:17) f ( ρ, ϕ ) g ( ρ ′ , ϕ ′ ) (cid:12)(cid:12)(cid:12)(cid:12) ( ρ,ϕ )=( ρ ′ ,ϕ ′ ) . (3)Since the pure gravity action with negative cosmological constant can be written as theChern-Simons action [9], the first two terms of the action (1) can be rewritten in terms ofthe triad e a and the spin connection ω a ( a = 0 , , S = 18 πG N Z (cid:18) ˆ e a ⋆ ∧ ˆ R a + Λ6 ǫ abc ˆ e a ⋆ ∧ ˆ e b ⋆ ∧ ˆ e c (cid:19) + Z d x ˆ e ⋆ ˆ L [ ˆ φ ] , (4)where ˆ e is the determinant of ˆ e aµ and ˆ R a is the curvature 2-form, ˆ R a = d ˆ ω a + ǫ abc ˆ ω b ⋆ ∧ ˆ ω c ,and the Lagrangian ˆ L [ ˆ φ ] for the scalar field is given byˆ L [ ˆ φ ] = −
14 ( ∂ µ ˆ¯ φ ⋆ g µν ⋆ ∂ ν ˆ φ + ∂ µ ˆ φ ⋆ g µν ⋆ ∂ ν ˆ¯ φ ) − λ φ ⋆ ˆ φ − v ) ⋆ ( ˆ¯ φ ⋆ ˆ φ − v ) . (5)We define the noncommutative metric asˆ g µν = 12 η ab (ˆ e aµ ⋆ ˆ e bν + ˆ e bν ⋆ ˆ e aµ ) (6)3uch that it is real and symmetric. The equations of motion are as follows: ǫ µνρ πG N (cid:20) ˆ R a + Λ2 ǫ abc ˆ e b ⋆ ∧ ˆ e c (cid:21) νρ + 16 ǫ µνρ ǫ abc (cid:16) ˆ e bν ⋆ ˆ e cρ ⋆ ˆ L [ ˆ φ ] + ˆ e cρ ⋆ ˆ L [ ˆ φ ] ⋆ ˆ e bν + ˆ L [ ˆ φ ] ⋆ ˆ e bν ⋆ ˆ e bµ (cid:17) + 18 h ˆ e µc ⋆ ˆ e νb ⋆ ∂ ν ˆ φ ⋆ ˆ e ⋆ ∂ ρ ˆ¯ φ ⋆ ˆ e ρa + ˆ e µc ⋆ ∂ ν ˆ φ ⋆ ˆ e ⋆ ∂ ρ ˆ¯ φ ⋆ ˆ e νb ⋆ ˆ e ρa + ˆ e µb ⋆ ∂ ν ˆ φ ⋆ ˆ e ⋆ ∂ ρ ˆ¯ φ ⋆ ˆ e ρc ⋆ ˆ e νa + ˆ e µb ⋆ ˆ e ρc ⋆ ∂ ν ˆ φ ⋆ ˆ e ⋆ ∂ ρ ˆ¯ φ ⋆ ˆ e νa + ( ˆ φ ↔ ˆ¯ φ ) i = 0 , (7)ˆ T a ≡ d ˆ e a + 12 ǫ abc (ˆ ω b ⋆ ∧ ˆ e c + ˆ e b ⋆ ∧ ˆ ω c ) = 0 , (8) ∂ µ (ˆ g µν ⋆ ∂ ν ˆ φ ⋆ ˆ e + ˆ e ⋆ ∂ ν ˆ φ ⋆ ˆ g µν ) = λ ˆ φ ⋆ ( ˆ¯ φ ⋆ ˆ φ ⋆ ˆ e + ˆ e ⋆ ˆ¯ φ ⋆ ˆ φ − v ˆ e ) . (9)In the commutative limit, θ →
0, the equations (7)-(9) reduce to the commutative ones [10, 6]as expected.In the commutative case, a static and rotationally symmetric metric can be put into thefollowing form [6]: ds = − e A ( r ) B ( r ) dt + dr B ( r ) + r dϕ . (10)The triad and spin connection corresponding to this line element are given by [10] e = e A √ B dt, e = 1 √ B dr, e = rdϕω = −√ Bdϕ, ω = 0 , ω = −√ B ddr (cid:16) e A √ B (cid:17) dt. In terms of A , B , and φ , where φ ≡ | φ | ( r ) e inϕ , the commutative equations of motion reducedfrom (7)-(9) are given by A ′ = 8 πG N r ( | φ | ′ ) ,B ′ = 2 | Λ | r − πG N r (cid:20) B ( | φ | ′ ) + n r | φ | + λ | φ | − v ) (cid:21) , | φ | ′′ + (cid:18) A ′ + B ′ B + 1 r (cid:19) | φ | ′ = 1 B (cid:18) n r + λ ( | φ | − v ) (cid:19) | φ | , (11)and these are exactly the same equations that appeared in [6].In line with the Weyl-Moyal correspondence and taking a hint from the commutative met-ric (10), we now make an ansatz for the noncommutative metric in the ( t, ρ, ϕ ) coordinates4 .00 2.05 2.10 2.15 2.200.8200.8250.8300.8350.840 Λ vr È Φ` È(cid:144) v Figure 1: The noncommutative vortex solutions with Λ v = 0 . G v = 1 .
33 are plottedfor the scalar field | ˆ φ | /v . The dashed line is for the case of θ = 0. The thin and thick solidlines are for the cases of θ = 0 . , .
15, respectively.which satisfy the commutation relation (2) as follows: d ˆ s = − e A ( ρ ) ˆ B ( ρ ) dt + dρ ρ ˆ B ( ρ ) + ρ dϕ . (12)A noncommutative triad for the above metric compatible with the definition of the metric(6) can be chosen as ˆ e = e ˆ A ( ρ ) q ˆ B ( ρ ) dt, ˆ e = dρ q ρ ˆ B ( ρ ) , ˆ e = √ ρdϕ. (13)With the above choice of the triad, the noncommutative spin connection can be determinedfrom the noncommutative torsion free condition (8):ˆ ω = − q ˆ B ( ρ ) dϕ, ˆ ω = 0 , ˆ ω = − q ρ ˆ B ( ρ ) ddρ (cid:18) e ˆ A ( ρ ) q ˆ B ( ρ ) (cid:19) dt. (14)Now we expand the metric functions ˆ A, ˆ B, and the scalar field ˆ φ for static global vorticeswith vorticity n in terms of θ up to first order as follows. e A ( ρ ) ˆ B ( ρ ) = e A ( ρ ) ˜ B ( ρ ) + θ ˜ F ( ρ ) + O ( θ ) , ˆ B ( ρ ) = ˜ B ( ρ ) + θ ˜ G ( ρ ) + O ( θ ) , ˆ φ ( ρ, ϕ ) = ( ˜ φ ( ρ ) + θ ˜Φ( ρ )) e inϕ + O ( θ ) . A ( ρ ) ≡ A ( r ) , ˜ B ( ρ ) ≡ B ( r ) , ˜ φ ( ρ ) ≡ | φ | ( r ) , ˜ F ( ρ ) ≡ F ( r ) , ˜ G ( ρ ) ≡ G ( r ) , ˜Φ( ρ ) ≡ Φ( r ) , and from (7)-(9), we get the original commutative equations (11) in the zeroth order of θ ,and the following three equations for F ( r ) , G ( r ) , Φ( r ) in the first order of θ : a ( r ) F ( r ) + a ( r ) F ′ ( r ) + a ( r ) G ( r ) + a ( r ) G ′ ( r )+ a ( r )Φ( r ) + a ( r )Φ ′ ( r ) + a ( r )Φ ′′ ( r ) + a ( r ) = 0 ,b ( r ) G ( r ) + b ( r ) G ′ ( r ) + b ( r )Φ( r ) + b ( r )Φ ′ ( r ) + b ( r ) = 0 ,c ( r ) F ( r ) + c ( r ) F ′ ( r ) + c ( r ) G ( r ) + c ( r )Φ( r ) + c ( r )Φ ′ ( r ) + c ( r ) = 0 . (15)All the coefficients in these equations are functions of the known commutative solution, | φ | , A and B , and are given in the appendix.The numerical analysis was performed up to first order in the noncommutativity param-eter θ for the vorticity n = 1. The cosmological constant, the Newton constant, and theradial coordinate are scaled to Λ v = | Λ | /λv , G v = 8 πG N v , and √ λvr , respectively. Inorder to solve (15) we impose the Dirichlet condition at the origin for F and G which is - - Λ vr B ` Figure 2: The solutions for noncommutative metric are plotted for ˆ B at values Λ v = 0 . G v = 1 .
33. The dashed line is for the case of θ = 0. The thin and thick solid lines arefor the cases of θ = 0 . , .
2, respectively. We note that there is a fine split at r = 0.6 Λ vr B ` Figure 3: The noncommutative black hole solutions with G v = 1 .
33 are plotted for ˆ B . Thedashed and thin solid lines are for the cases of Λ v = 0 . , .
08, respectively, at θ = 0. Thethick solid line is for the case of Λ v = 0 .
08 at θ = 0 . r )Φ(0) = Φ( ∞ ) = 0 . The effects of turning on θ on the global vortex solutions is shown in Fig. 1. This showsthat the scalar field concentrates inwards as θ grows. The gradient of the scalar field around r = 0 becomes steeper. In Fig. 2, the metric function ˆ B corresponding to the global vorticesin Fig. 1 is drawn. Note that the commutative solution ( θ = 0) has no horizon. On theother hand, when the value of the noncommutativity parameter θ reaches 0 . θ becomes larger than thisvalue, for instance θ = 0 .
2, it becomes a nonextremal black hole solution. When the startingcommutative solution is a nonextremal black hole solution with Λ v = 0 .
08, the correspondingnoncommutative solution for θ = 0 . θ for already-black holes in commutative spacetime is shown. As the noncommutativity parameter θ getsbigger, the area of the outer horizon increases while the separation between the inner andouter horizons grows. This effect is what we would expect when the mass of a nonextremalblack hole increases in the commutative case.In order to see whether the singularities of the metric at zeros are coordinate artifactsor true physical singularities, the Kretschmann scalars ˆ R µνρσ ˆ R µνρσ for the solutions having7 Λ vr K r e t s c h m a nn s ca l a r Figure 4: The plot of the noncommutative Kretschmann scalar with Λ v = 0 . G v = 1 . θ = 0. The thin and thick solid lines are for the cases of θ = 0 . , .
2, respectively.one or two zeros in Fig. 2 are plotted in Fig. 4. This shows that the Kretschmann invariantfor the regions around these zeros behaves regularly. From this result we can say that thezeros are not genuine singularities, rather they correspond to the horizons of black holes.The increase in gravitational mass for different values of θ by using the Hamiltonianformalism of [11] is plotted in Fig. 5. The increase in gravitational mass for a given θ isdefined by ∆ M ≡ H ( θ = 0) − H ( θ = 0) , where H denotes the Hamlitonian. The result shows that the gravitational mass increasesas the noncomutativity parameter θ increases. This linearity shown in Fig. 5 is due to ouranalysis which was performed up to first order in θ . The gravitational mass defined heredepends only on the asymptotic geometrical quantities at spatial infinity. The metric is fullydetermined by the scalar field and its derivatives which are constant at spatial infinity. Sincethe coefficient of θ is constant, it implies the linear dependence of mass on θ .Thus our result sums up as follows. The ‘inward’ behaviour of the global vortex inFig. 1 results in higher peaks of the Kretschmann scalar near r = 0 as in Fig. 4. Thesepeaks correspond to gravitational energy concentrations. We may interpret this as thenoncommutativity of space makes the scalar soliton have higher concentration near the centerthan in the commutative case. This concentrated scalar soliton behaves as the concentration8 æ æ æ æ æ æ æ æ æ æ ΘD M Figure 5: The plot of the gravitational mass increase vs θ with Λ v = 0 . G v = 1 . θ increases. In other words, as one increases the value of the noncommutativity parameter,the spacetime starts to allow a black hole solution at a certain value of the noncommutativityparameter. This is not what we expected before. A ‘phase change’ in the solution spacehappens as the noncommutativity parameter changes.Comparing with the result obtained in [7] for noncommutative BTZ black hole, the roleof the scalar field in the present work on forming a black hole is similar to that of the mag-netic flux B there. The shift of the locations of the horizons of the noncommutative blackholes in that paper depends upon both θ and the magnetic flux B at the origin. The shift ofhorizons obtained in this work depends on both θ and the scalar field. Thus, one may inferthat the global vortex around the origin takes over the role of the magnetic flux in [7]. Acknowledgments
This work was supported by the National Research Foundation (NRF) of Korea grantsfunded by the Korean government (MEST) [R01-2008-000-21026-0 and NRF-2009-0075129(E. C.-Y., K. K. and D. L.), and The Korea Research Foundation Grant funded by the KoreaGovernment(MOEHRD), KRF-2008-314-C00063(Y.L.)9 ppendix
The coefficients appearing in the equations (15) for F ( r ) , G ( r ) , Φ( r ) are given as follows: a ( r ) = − r h ( n − λv r ) φ + λr φ − rB ((1 − rA ′ ) φ ′ + rφ ′′ ) i ,a ( r ) = r Bφ ′ ,a ( r ) = re A h ( n − λv r ) φ + λr φ + rB ((1 + rA ′ ) φ ′ + rφ ′′ ) i ,a ( r ) = e A r Bφ ′ ,a ( r ) = − e A rB ( n − λv r + 3 λr φ ) ,a ( r ) = 2 e A r B ( B + rBA ′ + rB ′ ) ,a ( r ) = 2 e A r B ,a ( r ) = nBe A h ( n + 2 λv r ) φA ′ − λr φ A ′ − r ( A ′ B ′ φ ′ + B ( A ′ φ ′ + φ ′ A ′′ + A ′ φ ′′ )) i ,b ( r ) = 12 h r ( B ′ − r ) + 4 πG N (2( n − λv r ) φ + λr φ + r ( λv − Bφ ′ )) i ,b ( r ) = − rB,b ( r ) = − πG N Bφ ( n − λv r + λr φ ) ,b ( r ) = − πG N r B φ ′ ,b ( r ) = 8 πG N nλrBφ ( φ − v ) φ ′ ,c ( r ) = 12 h r (2Λ r + 2 BA ′ + B ′ ) − πG N (2( n − λv r ) φ + λr φ + r ( λv − Bφ ′ )) i ,c ( r ) = − rB,c ( r ) = − re A ( B ′ + 2 B ( A ′ − πG N rφ ′ )) ,c ( r ) = − πG N e A Bφ ( n − λv r + λr φ ) ,c ( r ) = 16 πG N e A r B φ ′ ,c ( r ) = 8 πG N e A nλrBφ ( φ − v ) φ ′ . References [1] N. Seiberg and E. Witten,
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