N-Dimensional non-abelian dilatonic, stable black holes and their Born-Infeld extension
aa r X i v : . [ g r- q c ] M a y N-Dimensional non-abelian dilatonic, stable black holes and theirBorn-Infeld extension
S. Habib Mazharimousavi ∗ , M. Halilsoy † , and Z. Amirabi ‡ Department of Physics, Eastern Mediterranean University,G. Magusa, north Cyprus, Mersin-10, Turkey ∗ [email protected] † [email protected] and ‡ [email protected] Abstract
We find large classes of non-asymptotically flat Einstein-Yang-Mills-Dilaton (EYMD) andEinstein-Yang-Mills-Born-Infeld-Dilaton (EYMBID) black holes in N-dimensional spherically sym-metric spacetime expressed in terms of the quasilocal mass. Extension of the dilatonic YM solutionto N-dimensions has been possible by employing the generalized Wu-Yang ansatz. Another metricansatz, which aided in finding exact solutions is the functional dependence of the radius functionon the dilaton field. These classes of black holes are stable against linear radial perturbations. Inthe limit of vanishing dilaton we obtain Bertotti-Robinson (BR) type metrics with the topology of
AdS × S N − . Since connection can be established between dilaton and a scalar field of Brans-Dicke(BD) type we obtain black hole solutions also in the Brans-Dicke-Yang-Mills (BDYM) theory aswell. . INTRODUCTION Recently we have found black hole solutions in the Einstein-Yang-Mills (EYM) theory byextending the Wu-Yang ansatz to higher dimensions [1, 2]. Both the YM charge and thedimensionality of the space time played crucial roles to determine the features of such blackholes. It was found long ago, within the context of Dirac monopole theory that Wu-Yangansatz solves the static, spherically symmetric YM equations for N = 4 dimensional flatspace time[3]. The SO (3) gauge structure was derived from the abelian electromagnetic (em)potential such that the internal and space time indices were mixed together in the potential.By a similar analogy we extend this idea to - nowadays fashionable − N dimensional spacetimes where SO ( N −
1) is obtained through a non-abelian gauge transformation from theem potential within the static, spherically symmetric metric ansatz. The YM gauge potentialis chosen to depend only on the angular variables and therefore they become independentof time ( t ) and the radial coordinate ( r ). Upon this choice the YM potential becomesmagnetic type and by virtue of the metric ansatz the YM equations are easily satisfied.Such a choice renders the duality principle to be automatically absent in the theory. Wenote that by invoking the Birkhoff’s theorem of general relativity t and r can be interchangedappropriately in the metric, while the YM potential preserves its form. The fact that thesolutions obtained by this procedure pertain to genuine non- abelian character is obviousfrom comparison with the other known exact EYM solutions . The EYM solutions obtainedby other ansaetze[4, 5, 6, 7, 8, 9] constructed directly from the non-abelian character andthose obtained by our generalized Wu-Yang ansatz [1, 2] are the same. We admit, however,that although our method yields exact solutions it is restricted to spherical symmetry alone.Their solutions, on the other hand [4, 5, 6, 7], apply to less symmetric cases which at bestcan be expressed in infinite series, and in certain limit, such as vanishing of a function, theycoincide with ours. Among other types, particle-like [8] and magnetic monopole [9] solutionsare discussed in even dimensions. To make a comparison between EM and EYM solutionswe refer to the different r powers in the solutions found so far. Specifically, the logarithmicterm in the metric for N = 5 , EYM theory, for instance, is not encountered in the N = 5 , EM theory [1]. For N = 4 it was verified on physical grounds that although the metricremained unchanged, the geodesics particles felt the non-abelian charges [10]. We note thatRef. [10] constitutes the proper reference to be consulted in obtaining a YM solution from2n EM solution, which is stated as a theorem therein. Our study shows that the distinctionbetween the abelian and non-abelian contributions becomes more transparent for N > . Let us note that throughout this paper by the non-Abelian field we imply YM field whosehigher dimensional version is obtained by the generalized Wu-Yang ansatz.It is well-known that in general relativity the field equations admit solutions which,unlike the localized black holes can have different properties. From this token we cite thecosmological solutions of de-Sitter ( dS )/ Anti de-Sitter ( AdS ), the conformally flat andBertoti-Robinson (BR) type solutions [11, 12], beside others in higher dimensions. In theEM theory the conformally flat metric in N = 4 is uniquely the BR metric whose topology is AdS × S . This extends to higher dimensions as
AdS × S N − which is no more conformallyflat. The N = 4 , BR solution can be obtained from the extremal Reissner-Nordstrom (RN)black hole solution through a limiting process. The latter represents a supersymmetricsoliton solution to connect different vacua of supergravity. For this reason the BR geometrycan be interpreted as a ”throat” region between two asymptotically flat space times. Also,since the source is pure homogenous electromagnetic (em) field it is called an ”em universe”,which is free of singularities. Its high degree of symmetry and singularity free propertiesmake BR space time attractive from both the string and supergravity theory points of view. We recall that even for a satisfactory shell model interpretation of an elementary particle,BR space time is proposed as a core candidate [13]. All these aspects ( and more), webelieve, justifies to make further studies on the BR space times, in particular for
N > C ,in a particular limit can be shown to correspond to the quasilocal mass. Thus, keeping both Q = 0 = C and a non-zero dilaton gives us an asymptotically non-flat black hole model.3he case C = 0 (without dilaton), yields a metric which is analogous to the BR metric[15].Next, we extend our action to include the non-Abelian Born-Infeld (BI) interactionwhich we phrase as Einstein-Yang-Mills-Born-Infeld-Dilaton (EYMBID) theory. As it iswell-known string / supergravity motivated non-linear electrodynamics due to Born andInfeld [16] received much attention in recent years. Originally it was devised to eliminatedivergences due to point charges, which recovers the linear Maxwell’s electrodynamics in aparticular limit (i.e. β → ∞ ). Now it is believed that BI action will provide significantcontributions for the deep rooted problems of quantum gravity. The BI action containsinvariants in special combinations under a square root term in analogy with the string the-ory Lagrangian. Since our aim in this paper is to use non-Abelian fields instead of the emfield we shall employ the YM field which by our choice will be magnetic type. Some of thesolutions that we find for the EYMBID theory represent non-asymptotically flat black holes.Unfortunately for an arbitrary dilatonic parameter the solutions become untractable. Oneparticular class of solutions on which we shall elaborate will be again the BR type solutionsfor a vanishing dilaton. We explore the possibility of finding conformally flat space time bychoosing particular BI parameter β. After studying black holes in the dilatonic theory we proceed to establish connectionwith the Brans-Dicke (BD) scalar field through a conformal transformation and exploreblack holes in the latter as well. Coupling of BD scalar field with YM field follows underthe similar line of consideration.The organization of the paper is as follows. In Sec. II we introduce the EYMD gravity, itsfield equations, their solutions and investigate their stability. The Born-Infeld (BI) extensionfollows in Sec. III . Sec. IV confines black holes in the Brans-Dicke-YM theory. The paperis completed with conclusion in Sec. V . II. FIELD EQUATIONS AND THE METRIC ANSATZ FOR EYMD GRAVITY
The N (= n + 1) − dimensional action in the EYMD theory is given by ( G = 1) I = − π Z M d n +1 x √− g (cid:18) R − n − ∇ Φ) + L (Φ) (cid:19) − π Z ∂ M d n x √− hK, (1) L (Φ) = − e − α Φ / ( n − Tr ( F ( a ) λσ F ( a ) λσ ) , Tr ( . ) = ( n )( n − / P a =1 ( . ) , (2)Φ refers to the dilaton scalar potential (we should comment that in this work we are inter-ested in a spherical symmetric dilatonic potential, i.e. Φ = Φ ( r )) and α denotes the dilatonparameter while the second term is the surface integral with its induced metric h ij and trace K of its extrinsic curvature. Herein R is the usual Ricci scalar and F ( a ) = F ( a ) µν dx µ ∧ dx ν arethe YM field 2 − forms (with ∧ indicating the wedge product) which are given by [1, 2] F ( a ) = dA ( a ) + 12 σ C ( a )( b )( c ) A ( b ) ∧ A ( c ) (3)with structure constants C ( a )( b )( c ) (see Appendix A) while σ is a coupling constant and A ( a ) = A ( a ) µ dx µ are the potential 1 − forms. Our choice of YM potential A ( a ) follows from thehigher dimensional Wu-Yang ansatz [1, 2] where σ is expressed in terms of the YM charge.Variations of the action with respect to the gravitational field g µν and the scalar field Φ lead,respectively to the EYMD field equations R µν = 4 n − ∂ µ Φ ∂ ν Φ + 2 e − α Φ / ( n − (cid:20) Tr (cid:16) F ( a ) µλ F ( a ) λν (cid:17) −
12 ( n − Tr (cid:16) F ( a ) λσ F ( a ) λσ (cid:17) g µν (cid:21) , (4) ∇ Φ = − αe − α Φ / ( n − Tr ( F ( a ) λσ F ( a ) λσ ) , (5)where R µν is the Ricci tensor. Variation with respect to the gauge potentials A ( a ) yields theYM equations d (cid:0) e − α Φ / ( n − ⋆ F ( a ) (cid:1) + 1 σ C ( a )( b )( c ) e − α Φ / ( n − A ( b ) ∧ ⋆ F ( c ) = 0 (6)in which the hodge star ⋆ means duality. In the next section we shall present solutions tothe foregoing equations in N-dimension. Wherever it is necessary we shall supplement ourdiscussion by resorting to the particular case N = 5. Let us remark that for N = 4 casesince the YM field becomes gauge equivalent to the em field the metrics are still of RN/BR,therefore we shall ignore the case N = 4. A. N-dimensional solution In N (= n + 1) − dimensions, we choose a spherically symmetric metric ansatz ds = − f ( r ) dt + dr f ( r ) + h ( r ) d Ω n − , (7)5here d Ω n − = dθ + n − P i =2 i − Q j =1 sin θ j dθ i , ≤ θ n − ≤ π, ≤ θ k = n − ≤ π. (8)while f ( r ) and h ( r ) are two functions to be determined. Our gauge potential ansatz is [1, 2] A ( a ) = Qr C ( a )( i )( j ) x i dx j , Q = YM magnetic charge, r = n X i =1 x i , (9)2 ≤ j + 1 ≤ i ≤ n, and 1 ≤ a ≤ n ( n − / ,x = r cos θ n − sin θ n − ... sin θ , x = r sin θ n − sin θ n − ... sin θ ,x = r cos θ n − sin θ n − ... sin θ , x = r sin θ n − sin θ n − ... sin θ ,...x n = r cos θ . We note that the structure constant C aij are found similar to the case N = 5 as described inAppendix A. The YM equations (6) are satisfied and the field equations become ∇ Φ = − αe − α Φ / ( n − Tr ( F ( a ) λσ F ( a ) λσ ) (10) R tt = e − α Φ / ( n − f ( n − Tr ( F ( a ) λσ F ( a ) λσ ) (11) R rr = 4 (Φ ′ ) ( n − − e − α Φ / ( n − ( n − f Tr ( F ( a ) λσ F ( a ) λσ ) (12) R θ i θ i = 2 ( n − Q e − α Φ / ( n − h − h e − α Φ / ( n − ( n − Tr ( F ( a ) λσ F ( a ) λσ ) , (13)in which we note that the remaining angular Ricci parts add no new conditions. A properansatz for h ( r ) now is h ( r ) = Ae − α Φ / ( n − (14)( A = constant)which, after knowing Tr ( F ( a ) λσ F ( a ) λσ ) = ( n −
1) ( n − Q h (15)and eliminating f ( r ) from Eq.s (11) and (12) one getsΦ = − ( n − α ln rα + 1 . (16)6pon substitution of Φ and h ( r ) into the Eq.s (10)-(13) we get three new equations( n − (cid:2) r (cid:0) α + 1 (cid:1) f ′ + (cid:0) ( n − α − (cid:1) f (cid:3) − (cid:18) ( n −
1) ( n − Q A (cid:19) (cid:0) α + 1 (cid:1) r “ α ” = 0(17)( n − (cid:2) r (cid:0) α + 1 (cid:1) f ′′ + ( n − α f ′ (cid:3) − (cid:18) ( n −
1) ( n − Q A (cid:19) (cid:0) α + 1 (cid:1) r “ − α − α ” = 0(18) (cid:0) α + 1 (cid:1) ( n − (cid:0) Q − A (cid:1) r + A α (cid:0) α + 1 (cid:1) f ′ r “ α α ” + α (cid:0) ( n − α − (cid:1) A f r “ α α ” = 0 . (19)Eq. (17) yields the integral for f ( r ) f ( r ) = Ξ − (cid:16) r + r (cid:17) ( n − α α ! r α , (20)Ξ = ( n − n − α + 1) Q (21)and the equations (18) and (19) imply that A must satisfy the following constraint A = Q (cid:0) α + 1 (cid:1) . (22)One may notice that, with the solution (20), (7) becomes a non-asymptotically flat metricand therefore the ADM mass can not be defined. Following the quasilocal mass formalismintroduced by Brown and York [17] it is known that, a spherically symmetric N-dimensionalmetric solution as ds = − F ( R ) dt + dR G ( R ) + R d Ω N − , (23)admits a quasilocal mass M QL defined by [18, 19] M QL = N − R N − B F ( R B ) ( G ref ( R B ) − G ( R B )) . (24)Here G ref ( R ) is an arbitrary reference function, which guarantees having zero quasilocalmass once the matter source is turned off and R B is the radius of the spacelike hypersurfaceboundary. Applying this formalism to the solution (20), one obtains the horizon r + in termsof M QL as r + = (cid:18) α + 1) M QL ( n −
1) Ξ α A n − (cid:19) . (25)7aving the radius of horizon, one may use the usual definition of the Hawking temperatureto calculate T H = 14 π | f ′ ( r + ) | = Ξ4 π [( n − α + 1]( α + 1) ( r + ) γ (26)where Ξ and r + are given above and γ = − α α .In order to see the singularity of the spacetime we calculate the scalar invariants, whichare tedious for general N, for this reason we restrict ourselves to the case N = 5 alone. Thescalar invariants for N = 5 are as follows R = ω r α α + σ r α α , (27) R µν R µν = ω r α α + ω r α α + σ r α α , (28) R µναβ R µναβ = ω r α α + ω r α α + σ r α α (29)where ω i and σ i are some constants andlim α → ω i = 0 , lim α → σ = 2 Q , (30)lim α → σ = 20 Q , lim α → σ = 33 Q . These results show that, for non-zero dilaton field (i.e. α = 0), the origin is singular whereasfor α = 0 (as a limit), we have a regular spacetime. Although these results have been foundfor N = 5, it is our belief that for a general N >
1. Linear dilaton
Setting α = 1 , gives the linear dilaton solution (20) as f ( r ) = ( n − n − Q (cid:18) − (cid:16) r + r (cid:17) n − (cid:19) r, h ( r ) = 2 Q r (31) r + = − n M QL ( n −
2) ( | Q | ) n − ! . (32)One can use the standard way to find the high frequency limit of Hawking temperature atthe horizon, which means that T H = 14 π | f ′ ( r + ) | = ( n − πQ . (33)8urthermore, M QL is an integration constant which is identified as quasilocal mass, so onemay set this constant to be zero to get the line element ds = − Ξ rdt + dr Ξ r + 2 Q rd Ω n − , (34)Ξ = ( n − n − Q . (35)By a simple transformation r = e Ξ ρ this line element transforms into ds = Ξ e Ξ ρ (cid:18) − dt + dρ + 2 ( n − Q ( n − d Ω n − (cid:19) (36)which represents a conformal M × S n − space time with the radius of S n − equal to q n − n − Q .
2. BR limit of the solution
In the zero dilaton limit α = 0 , we express our metric function in the form of f ( r ) = Ξ ◦ ( r − r + ) r, Ξ ◦ = ( n − Q , (37) h = A ◦ = Q . (38)In N (= n + 1) − dimensions we also set r + = 0 , r = ρ and τ = Ξ ◦ t, to transform the metric(7) into ds = Q ( n − (cid:18) − dτ + dρ ρ + ( n − d Ω n − (cid:19) . (39)This is in the BR form with the topological structure AdS × S n − , where the radius of the S n − sphere is √ n − . AdS × S N − topology for < α < In this section we shall show that, the general solution given in Eq. (20), for some specificvalues for 0 < α < , may also represent a conformally flat space time. To this end, we set r + = 0 , and apply the following transformation r = (cid:18) Ξ 1 − α α ρ (cid:19) − α − α , (40)Ξ = ( n − n − α + 1) Q , (41)9o get ds = (Ξ) − α − α (cid:18) − α α (cid:19) − − α ρ − α − α − dτ + dρ ρ + Ξ A (cid:18) − α α (cid:19) d Ω n − ! . (42)To have a conformally flat space time, we impose Ξ A (cid:16) − α α (cid:17) to be one, i.e.( n −
2) (1 − α ) (( n − α + 1) ( α + 1) = 1 (43)and therefore yields, α = n − n − . The line element (42) takes the form of a conformally flatspace time, namely ds = a ( ρ ) (cid:18) − dτ + dρ ρ + d Ω n − (cid:19) , (44) a ( ρ ) = 2 n − n − ( n − (cid:18) Q n − (cid:19) n − n − ( n − n − n − ρ − α − α . (45) B. Linear Stability of the EYMD black holes
In this chapter we follow a similar method used by Yazadjiev [14] to investigate thestability of the possible EYMD black hole solutions, introduced previously, in terms of alinear radial perturbation. Although this method is applicable to any dimensions we confineourselves to the five-dimensional black hole case given by Eq. (7). To do so we assume thatour dilatonic scalar field Φ ( r ) changes into Φ ( r ) + ψ ( t, r ) , in which ψ ( t, r ) is very weakcompared to the original dilaton field and we call it the perturbed term. As a result wechoose our perturbed metric as ds = − f ( r ) e Γ( t,r ) dt + e χ ( t,r ) dr f ( r ) + h ( r ) d Ω . (46)One should notice that, since our gauge potentials are magnetic, the YM equations (6) aresatisfied. The linearized version of the field equations (10-13) plus one extra term of R tr aregiven by R tr : 32 χ t ( t, r ) h ′ ( r ) h ( r ) = 43 ∂ r Φ ( r ) ∂ t ψ ( t, r ) (47) ∇ ◦ ψ − χ ∇ ◦ Φ + 12 (Γ − χ ) r Φ ′ f = 4 α e α Φ Q ( α + 1) ψ (48) R θθ : (2 − R ◦ θθ ) χ − hh ′ f (Γ − χ ) r = 8 α α + 1) ψ (49)10n which a lower index ◦ represents the quantity in the unperturbed metric. First equationin this set implies χ ( t, r ) = − α ψ ( t, r ) (50)which after making substitutions in the two latter equations and eliminating the (Γ − χ ) r one finds ∇ ◦ ψ ( t, r ) − U ( r ) ψ ( t, r ) = 0 (51)where U ( r ) = 4 e α Φ Q (1 + α ) = 4 Q (1 + α ) r α α . (52)To get these results we have implicitly used the constraint (22) on A. Again by imposingthe same constraint , one can show that U ( r ) is positive. It is not difficult to apply theseparation method on (51) to get ψ ( t, r ) = e ± ǫt ζ ( r ) , ∇ ◦ ζ ( r ) − U eff ( r ) ζ ( r ) = 0 , U eff ( r ) = (cid:18) ǫ f + U ( r ) (cid:19) , (53)where ǫ is a constant. Since U eff ( r ) is positive one can easily show that, for any real valuefor ǫ there exists a solution for ζ ( r ) which is not bounded. In other words by the linearperturbation our black hole solution is stable for any value of ǫ. As a limit of this proof, onemay set α = 0 , which recovers the BR case.We remark that with little addition this method can be easily extended to any higherdimensions. This implies that the N-dimensional EYMD black holes are stable under thelinear perturbation. III. FIELD EQUATIONS AND THE METRIC ANSATZ FOR EYMBID GRAVITY
The N (= n + 1) − dimensional action in the EYMBI-D theory is given by ( G = 1) I = − π Z M d n +1 x √− g (cid:18) R − n − ∇ Φ) + L ( F , Φ) (cid:19) − π Z ∂ M d n x √− hK, (54) L ( F , Φ) = 4 β e α Φ / ( n − − s Tr ( F ( a ) λσ F ( a ) λσ ) e − α Φ / ( n − β = (55)4 β e α Φ / ( n − L ( X ) , where L ( X ) = 1 − √ X, X = Tr ( F ( a ) λσ F ( a ) λσ ) e − α Φ / ( n − β , Tr ( . ) = n ( n − / P a =1 ( . ) , (56)11hile the rest of the parameters are defined as before. Variations of the EYMBID actionwith respect to the gravitational field g µν and the scalar field Φ lead respectively to thecorrespondence EYMBID field equations R µν = 4 n − ∂ µ Φ ∂ ν Φ − e − α Φ / ( n − (cid:16) Tr (cid:16) F ( a ) µλ F ( a ) λν (cid:17) ∂ X L ( X ) (cid:17) + (57)4 β n − e α Φ / ( n − K ( X ) g µν , ∇ Φ = 2 αβ e α Φ / ( n − K ( X ) , (58)where we have abbreviated K ( X ) = 2 X∂ X L ( X ) − L ( X ) (59)( ∂ X L ( X ) = − √ X ).Variation with respect to the gauge potentials A ( a ) yields the new relevant YM equations d (cid:0) e − α Φ / ( n − ⋆ F ( a ) ∂ X L ( X ) (cid:1) + 1 σ C ( a )( b )( c ) e − α Φ / ( n − ∂ X L ( X ) A ( b ) ∧ ⋆ F ( c ) = 0 . (60)It is remarkable to observe that the field equations (57-59) in the limit of β → ∞ , reduceto the Eq.s (4-6), which are the field equations for the EYMD theory. Also in the limit of β →
0, Eq.s (57-59) give R µν = 4 n − ∂ µ Φ ∂ ν Φ , (61) ∇ Φ = 0 (62)which refer to the gravity coupled with a massless scalar field.
A. N-dimensional solution In N (= n + 1) − dimensions, we again, adopt the metric ansatz (7) and our YM potentialsare given by Eq. (9). N-dimensional YM equations (60) are satisfied while the field equations12mply the following set of four equations ∇ Φ = 2 αβ e α Φ / ( n − K ( X ) (63) R tt = − β e α Φ / ( n − f ( n − K ( X ) (64) R rr = 4 (Φ ′ ) ( n −
1) + 4 β e α Φ / ( n − ( n − f K ( X ) (65) R θ i θ i = − n − Q e − α Φ / ( n − h ∂ X L + 4 h β e α Φ / ( n − ( n − K ( X ) . (66)in which X is defined by (56). We use the same ansatz for h ( r ) as Eq. (14)which gives X = ( n −
1) ( n − Q β A (67)and therefore, after eliminating f ( r ) from Eq.s (64) and (65), leads to (16). Upon substitu-tion of Φ and h ( r ) into the Eq.s (63)-(66) we find the following equations( n − (cid:2) r (cid:0) α + 1 (cid:1) f ′ + (cid:0) ( n − α − (cid:1) f (cid:3) + 4 β K ( X ) (cid:0) α + 1 (cid:1) r “ α ” = 0 (68)( n − (cid:2) r (cid:0) α + 1 (cid:1) f ′′ + ( n − α f ′ (cid:3) + 8 β K ( X ) (cid:0) α + 1 (cid:1) r “ − α − α ” = 0 (69) (cid:0) α + 1 (cid:1) (cid:0) β A K ( X ) − (4 Q ∂ X L + A ) ( n −
1) ( n − (cid:1) r + (70)( n − A α (cid:0) α + 1 (cid:1) f ′ r “ α α ” + ( n − α (cid:0) ( n − α − (cid:1) A f r “ α α ” = 0 . Eq. (68) yields the integral for f ( r ) f ( r ) = Ξ − (cid:16) r + r (cid:17) ( n − α α ! r α , (71)Ξ = − β ( α + 1) K ( X )( n −
1) (( n − α + 1) (72)in which r + is an integration constant connected to the quasi local mass i.e., r + = (cid:18) α + 1) M QL ( n −
1) Ξ α A n − (cid:19) (73)and K ( X ) is abbreviated as in (59). This solution satisfies Eq. (69), but from Eq. (70) A must satisfy the constraint4 K ( X ) β A (cid:0) α − (cid:1) + ( n −
1) ( n − (cid:0) Q ∂ X L + A (cid:1) = 0 . (74)13 . Linear dilaton In the linear dilaton case i.e., α = 1 , Eq. (71) yields f ( r ) = Ξ − (cid:16) r + r (cid:17) ( n − ! r, h ( r ) = A √ r , r + = (cid:18) M QL ( n −
1) Ξ A n − (cid:19) (75)in which A = 2 Q s − Q cri Q , Ξ = 2 ( n − n − Q cri − s − Q cri Q (76)where Q cri = ( n −
1) ( n − β (77)and Q ≥ Q cri . In this case one may set Ξ = A = 1 to get ds = − − (cid:16) r + r (cid:17) ( n − ! rdt + 1 (cid:18) − (cid:0) r + r (cid:1) ( n − (cid:19) r dr + rd Ω n − . (78)
2. BR limit of the solution
In the zero dilaton limit α = 0 , we express our metric functions (71) in the form f ( r ) = Ξ ◦ ( r − r + ) r, Ξ ◦ = 8 β ( n − n −
1) ( n −
2) + 8 β Q , (79) h = A ◦ = Q − ( n −
1) ( n − β . (80)In N (= n + 1) − dimensions we also set r + = 0 , r = ρ and τ = Ξ ◦ t, to transform the metric(7) into ds = 1Ξ ◦ (cid:18) − dτ + dρ ρ + Ξ ◦ A ◦ d Ω n − (cid:19) . (81)This is in the BR form with the topological structure AdS × S N − , where the radius of thesphere is √ Ξ ◦ A ◦ . It can be shown thatΞ ◦ A ◦ = ( n − (cid:18) β Q − ( n −
1) ( n − n −
1) ( n −
2) + 8 β Q (cid:19) (82)which, in the limit of β → ∞ , becomeslim β →∞ Ξ ◦ A ◦ = ( n −
2) (83)14uch that, the solution (81) becomes the BR type solution of EYMD theory (see Eq. (39)).We set now Ξ ◦ A ◦ = 1, to obtain a conformally flat metric. This claims that( n − (cid:18) β Q − ( n −
1) ( n − n −
1) ( n −
2) + 8 β Q (cid:19) = 1 (84)and consequently we find β = ( n − ( n − Q ( n − , (85) ds = 2 Q ( n − (cid:18) − dτ + dρ ρ + d Ω (cid:19) . (86)This particular choice of β casts the EYMBI metric into a conformally flat form with thetopology of AdS × S AdS × S N − topology for < α < As one may show, for 0 < α < r + = 0 , a similar transformation as (40), here alsoleads to the line element ds = (Ξ) − α − α (cid:18) − α α (cid:19) − − α ρ − α − α − dτ + dρ ρ + Ξ A (cid:18) − α α (cid:19) d Ω n − ! . (87)Again we set Ξ A (cid:16) − α α (cid:17) = 1 which gives the conformally flat line element ds = a ( ρ ) (cid:18) − dτ + dρ ρ + d Ω n − (cid:19) , (88)with a ( ρ ) = (Ξ) − α − α (cid:18) − α α (cid:19) − − α ρ − α − α . (89) B. Linear Stability of the EYMBID black holes
Similar to the proof given in Sec. (
II.B ), here also we study the stability of the possibleblack holes in EYMBID theory which undergoes a linear perturbation. Again we give adetailed study for the 5-dimensional black holes which is extendible to any higher dimensions.Our perturbed metric is same as we adapted in Eq. (46). The linearized field equations plus15he extra term of R tr are given now by R tr : ( n − χ t ( t, r ) h ′ ( r ) h ( r ) = 43 ∂ r Φ ( r ) ∂ t ψ ( t, r ) (90) ∇ ◦ ψ − χ ∇ ◦ Φ + 12 (Γ − χ ) r Φ ′ f = − n − α β e n − α Φ (cid:0) L ( X ◦ ) + 4 X ◦ ∂ X ◦ L ( X ◦ ) (cid:1) ψ (91) R θθ : (2 − R ◦ θθ ) χ − hh ′ f (Γ − χ ) r = 169 αA β (2 X ◦ ∂ X ◦ L ( X ◦ ) − L ( X ◦ )) ψ (92)in which our conventions are as before. The first equation in this set implies that χ ( t, r ) = − α ψ ( t, r ) (93)which, after we make substitutions in the two latter equations and eliminating the (Γ − χ ) r we find ∇ ◦ ψ ( t, r ) − U ( r ) ψ ( t, r ) = 0 (94)where U ( r ) = 83 β e α Φ (cid:2) L ( X ◦ ) − X ◦ ∂ X ◦ L ( X ◦ ) − α (cid:0) L ( X ◦ ) + 4 X ◦ ∂ X ◦ L ( X ◦ ) (cid:1)(cid:3) . (95)To get these results we have implicitly used the constraint (74) on A . Again by imposingthe same constraint , one can show that U ( r ) is positive definite. We follow the separationmethod to get ψ ( t, r ) = e ± ǫt ζ ( r ) , ∇ ◦ ζ ( r ) − U eff ( r ) ζ ( r ) = 0 , U eff ( r ) = (cid:18) ǫ f + U ( r ) (cid:19) , (96)where ǫ is a constant. Here also the fact that U eff ( r ) > β → ∞ this reduces to the case of EYMD black holesolution whose stability was already verified before. IV. BLACK HOLES IN THE BDYM THEORY In N (= n + 1) − dimensions we write the Brans-Dicke-Yang-Mills (BDYM) action as I = − π Z M d n +1 x √− g (cid:18) φR − ωφ ( ∇ φ ) + L m (cid:19) − π Z ∂ M d n x √− hK, (97) L m = − Tr ( F ( a ) λσ F ( a ) λσ ) ,
16n which ω is the coupling constant, and φ stands for the BD scalar field with the dimensions G − ( G is the N − dimensional Newtonian constant [21]). Variation of the BDYM’s actionwith respect to the g µν gives φG µν = ωφ (cid:18) ∇ µ φ ∇ ν φ − g µν ( ∇ φ ) (cid:19) + 2 (cid:18) Tr (cid:16) F ( a ) µλ F ( a ) λν (cid:17) − g µν Tr (cid:16) F ( a ) λσ F ( a ) λσ (cid:17)(cid:19) +(98) ∇ µ ∇ ν φ − g µν ∇ φ, while variation of the action with respect to the scalar field φ and the gauge potentials A ( a ) yields ∇ φ = − n −
32 [( n − ω + n ] Tr (cid:16) F ( a ) λσ F ( a ) λσ (cid:17) , (99)and d (cid:0) ⋆ F ( a ) (cid:1) + 1 σ C ( a )( b )( c ) A ( b ) ∧ ⋆ F ( c ) = 0 , (100)respectively.We follow now the routine process to transform BDYM action into the EYMD action[21].For this purpose, one can use a conformal transformation (variables with a caret ˆ . denotethose in the Einstein frame)ˆ g µν = φ n − g µν and ˆΦ = ( n − α ln φ. (101)This transforms (97) intoˆ I = − π Z M d n +1 x p − ˆ g (cid:18) ˆ R − n − (cid:16) ˆ ∇ ˆΦ (cid:17) − e − α ˆΦ / ( n − Tr (cid:16) ˆ F ( a ) λσ ˆ F ( a ) λσ (cid:17)(cid:19) − π Z ∂ M d n x q − ˆ h ˆ K, (102)where ˆ α = n − p ( n − ω + n . (103)This transformed action is similar to the EYMD action given by (1). Variation of this actionwith respect to the ˆ g µν , ˆΦ and ˆA ( a ) givesˆ R µν = 4 n − ˆ ∂ µ Φ ˆ ∂ ν Φ + 2 e − α ˆΦ / ( n − (cid:20) Tr (cid:16) ˆ F ( a ) µλ ˆ F ( a ) λν (cid:17) −
12 ( n − Tr (cid:16) ˆ F ( a ) λσ ˆ F ( a ) λσ (cid:17) ˆ g µν (cid:21) , (104)ˆ ∇ Φ = −
12 ˆ αe − α ˆΦ / ( n − Tr ( ˆ F ( a ) λσ ˆ F ( a ) λσ ) , (105)17 (cid:16) e − α ˆΦ / ( n − ⋆ ˆF ( a ) (cid:17) + 1 σ C ( a )( b )( c ) e − α ˆΦ / ( n − ˆA ( b ) ∧ ⋆ ˆF ( c ) = 0 . (106)It is not difficult to conclude that, if we find a solution to the latter equations, by an inversetransformation, we can find the solutions of the related equations of the BDYM theory. Inother words if (cid:16) ˆ g µν , Φ , ˆF ( a ) (cid:17) is a solution of the latter equations, then (cid:0) g µν , φ, F ( a ) (cid:1) = (cid:18) exp (cid:18) − α ( n −
1) ( n −
3) ˆΦ (cid:19) ˆ g µν , exp (cid:18) α ( n −
3) ˆΦ (cid:19) , ˆF ( a ) (cid:19) (107)is a solution of (98-100) and vice versa.One may call (cid:0) g µν , φ, F ( a ) (cid:1) , the reference solution and (cid:16) ˆ g µν , ˆΦ , ˆF ( a ) (cid:17) the target solution.Hence our solution in EYMD would be the target solution i.e. d ˆ s = − ˆ f ( r ) dt + dr ˆ f ( r ) + ˆ h ( r ) d Ω n − , (108)where ˆ f ( r ) = ˆΞ − (cid:18) ˆ r + r (cid:19) ( n − α α r α , ˆ h ( r ) = ˆ Ae − α ˆΦ / ( n − , (109)ˆΞ = ( n − n −
2) ˆ α + 1) ˆ Q , ˆΦ = − ( n − α ln r ˆ α + 1 , ˆ A = ˆ Q (cid:0) ˆ α + 1 (cid:1) , ˆ r + = α + 1) ˆ M QL ( n −
1) ˆΞ ˆ α ˆ A n − ! . Our reference solution would read now ds = − f ( r ) dt + dr f ( r ) + h ( r ) d Ω n − , (110)in which f ( r ) = ˆΞ − (cid:18) ˆ r + r (cid:19) ( n − α α r n − α n − ( ˆ α ) , h ( r ) = ˆ Ae − α ˆΦ( n +1)( n − n − = ˆ Ar ˆ α n +1) ( ˆ α ) ( n − , (111) φ = r − n − α n − ( ˆ α ) , and F ( a ) = ˆF ( a ) = d ˆA ( a ) + 12 σ C ( a )( b )( c ) ˆA ( b ) ∧ ˆA ( c ) (112)where the YM potential is same as (9) with the new charge ˆ Q. Herein one can find theHawking temperature of the BDYM-black hole at the event horizon as T H = ˆΞ [( n −
2) ˆ α + 1]4 π ( ˆ α + 1) (ˆ r + ) − ( n − ( ˆ α − ) − α ( ˆ α ) ( n − ! (113)where ˆ r + is the radius of the event horizon.18 . CONCLUSION A simple class of spherically symmetric solutions to the EYMD equations is obtainedin any dimensions. Magnetic type Wu-Yang ansatz played a crucial role in extending thesolution to N-dimension. For the non-zero dilaton the space time possesses singularity,representing a non-asymptotically flat black hole solution expressed in terms of the quasilocalmass. Particular case of a linear dilatonic black hole is singled out as a specific case. Hawkingtemperature for all cases has been computed which are distinct from the EMD temperatures[22]. Stability against linear perturbations for these dilatonic metrics is proved. It has beenshown that the extremal limit in the vanishing dilaton, results in the higher dimensionalBR space times for the YM field. With the common topology of
AdS × S N − for boththeories, while the radius of S N − for the Maxwell case is ( N − , it becomes ( N − / inthe YM case. As a final contribution in the paper we apply a conformal transformation toderive black hole solutions in the Brans-Dicke-YM theory. It is our belief that these YMBRmetrics, beside the dilatonic ones, will be useful in the string/supergravity theory as muchas the EMBR metrics are. Acknowledgement 1
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