Black Hole solutions in Einstein-Maxwell-Yang-Mills-Gauss-Bonnet Theory
aa r X i v : . [ g r- q c ] N ov Black Hole solutions inEinstein-Maxwell-Yang-Mills-Gauss-Bonnet Theory
S. Habib Mazharimousavi ∗ and M. Halilsoy † Department of Physics, Eastern Mediterranean University,G. Magusa, north Cyprus, Mersin-10, Turkey ∗ [email protected] and † [email protected] Abstract
We consider Maxwell and Yang-Mills (YM) fields together, interacting through gravity bothin Einstein and Gauss-Bonnet (GB) theories. For this purpose we choose two different sets ofMaxwell and metric ansaetze. In our first ansatz, asymptotically for r → N >
4) theMaxwell field dominates over the YM field. In the other asymptotic region, r → ∞ , however,the YM field becomes dominates. For N = 3 and N = 4, where the GB term is absent, werecover the well-known Ba˜nados-Teitelboim-Zanelli (BTZ) and Reissner-Nordstr¨om (RN) metrics,respectively. The second ansatz corresponds to the case of constant radius function for S N − partin the metric. This leads to the maximally symmetrical, everywhere regular Bertotti-Robinson(BR) type solutions to represent our cosmos both at large and small. . INTRODUCTION We introduce Maxwell field alongside with Yang-Mills (YM) field in general relativityand present spherically symmetric black hole solutions in any higher dimensions. These twogauge fields, one Abelian the other non-Abelian, are coupled through gravity and to ourknowledge they have not been considered together in a common geometry so far. Beinglinear, a multitude of Maxwellian fields can be superimposed at equal ease, but for the YMfield there is no such freedom. Our treatment of YM field in this paper like Maxwell fieldis completely classical (i.e. non-quantum). From physics standpoint, electromagnetism haslong range effects and dominates outside the nuclei of natural matter. YM field on the otherhand is confined to act inside nuclei, however, the existence of exotic and highly dense matterin our universe encourages us to use YM field in a broader sense. To obtain exact solutionsthe Maxwell field is chosen pure electric while the YM field is pure magnetic. Historically,starting with the Reissner-Nordstr¨om (RN) metric, its higher dimensional extensions knownas the Einstein-Maxwell (EM) black holes are well-known by now. Einstein-Yang-Mills(EYM) black holes in higher dimensions have also attracted interest in more recent works[1, 2, 3]. In this paper we combine these two foregoing black holes (i.e. EM and EYM)under the common title of EMYM black holes. Further, we add to it the Gauss-Bonnet(GB) term overall which we phrase as the EMYMGB black holes[17](and the referencestherein). Such black holes will be characterized by mass ( M ), Maxwell charge ( q ), YMcharge ( Q ), GB parameter α and cosmological constant (Λ) . For a broad class of blackhole solutions we investigate the thermodynamics and other properties as manifestationsof these physical parameters. All these parameters naturally have imprints an planetarymotion, gravitational lensing, ripping apart of stars and ultimately on the accurate pictureof our cosmos. It is remarkable that exact solutions to such a highly non-linear theorycan be found for two different Maxwell and metric ansaetze. Before adding the GB termwe find the EMYM solutions in which the relative contributions from the Maxwell /YMcharges can be compared. It is shown that for r → r → ∞ the YM term becomes dominant. Although this may sound contradictoryto the short rangeness of the YM field we may attribute this to the decisive role played2y the higher dimensionality of spacetime. Such behaviors, we speculate, have relevance inconnection with the mini (super-massive) black holes as well as asymptotically (anti)-deSitter space times. The solutions for N > N = 3 ,
4) dimensional cases. We show that in the latter cases of N = 3 and N = 4 therole of a YM charge is similar to the Maxwell charge as far as the space time metric isconcerned. Namely, one is still the BTZ, while the other is the RN metric. It is found thatfor N = 3 and N = 5 the metric function contains a logaritmic term while in the other caseswe have inverse power low dependence on r . The unprecedented logarithmic potential leadto entirely different Keplerian orbits. Since black holes are scale invariant objects, i. e., theyoccur both at micro/ macro scales as mini/ supermassive black holes, the orbits revealed inone scale is valid in all scales. To show this we investigate the Newtonian approximation (for N = 5) as projected into the polar plane which reveals the role of the YM charge in formingbound states with smaller orbits. Our second ansatz leads, beside the black hole solution,to a Bertotti-Robinson (BR) type metric within the EMYMGB theory. It is known thatextremal RN, which is supersymmetric soliton solution to extended supergravity, interpolatesbetween two maximally symmetric spacetime, namely the flat space at infinity and BR at thehorizon ( r = 0). In other words the near horizon geometry of an extremal RN black hole isidentified as the BR spacetime. This idea can be extended to higher dimensions as p-branes.The topology of N-dimensional BR type YM field is, as in the EM theory, adS × S N − (i.e., anti de Sitter × ( N −
2) sphere) with a marked difference in their radii. The maximalsymmetry of the BR spacetime, without singularity makes it a favorable model to representour homogenous and isotropic universe in the absence of rotation. For the representative ofthe rotating BR universe and its cosmological implications in N = 4 we refer elsewhere [19].The interesting feature here is that the BR parameter consist of all the parameters of thetheory, namely, M, q, Q,
Λ and α. In other words, even the topological GB parameter servesto construct a finely- tuned cosmological BR model with maximal symmetry. It was shownby Gibbons and Townsend [1] that extreme self-gravitating Yang-type monopole also yieldsan adS × S N topological vacuum in the (2 N + 2) dimensional EYM theory. It is our beliefthat our metrics will be useful both in string/ supergravity theories as well as in cosmology.One immediate conclusion we can draw is that we can generate an ”effective cosmologicalconstant Λ eff ” to play the same role, in the absence of a real Λ . To feel confident, we verify3lso that our physical sources in higher dimensions satisfy all the weak, strong, dominantand causality conditions.The paper is organized as follows: In Sec. ( II ) we introduce our action, metric, Maxwell,YM ansatz and the field equations. Sec. ( III ) follows with the solution of the field equationsfor different dimensionalities. The geometry outside a 5-dimensional black hole with itsNewtonian approximation is investigated in Sec. ( IV ). In Sec. ( V ) we add the Gauss-Bonnet (GB) term and present the most general solution. Sec. ( V I ) deals with a new set ofansatz for the Maxwell field and metric function in which we study the black hole propertiesand BR classes inherent in them. Energy and causality conditions are discussed in Sec.(
V II ). The paper ends with concluding remarks in Sec. (
V III ). II. ACTION, FIELD EQUATIONS AND OUR ANSAETZE (RN TYPE)
The action which describes Einstein-Maxwell-Yang-Mills gravity with a cosmological con-stant in N dimensions reads I G = 12 Z M dx N √− g (cid:18) R − ( N −
1) ( N − − F µν F µν − Tr (cid:0) F ( a ) µν F ( a ) µν (cid:1)(cid:19) (1)where in this context we use the following abbreviation, Tr ( . ) = ( N − N − / X a =1 ( . ) . (2)Here, R is the Ricci Scalar while the YM and Maxwell fields are defined respectively by F ( a ) µν = ∂ µ A ( a ) ν − ∂ ν A ( a ) µ + 12 σ C ( a )( b )( c ) A ( b ) µ A ( c ) ν , (3) F µν = ∂ µ A ν − ∂ ν A µ , in which C ( a )( b )( c ) stands for the structure constants of ( N − N − parameter Lie group G and σ is a coupling constant. A ( a ) µ are the SO ( N − A µ represents the usual Maxwell potential. We note that the internal indices { a, b, c, ... } do notdiffer whether in covariant or contravariant form. Variation of the action with respect tothe space-time metric g µν yields the field equations G µν + ( N −
1) ( N − g µν = T µν , (4)4here the stress-energy tensor is the superposition of the Maxwell and YM parts, namely T µν = (cid:18) F λµ F νλ − F λσ F λσ g µν (cid:19) + Tr (cid:20) F ( a ) λµ F ( a ) νλ − F ( a ) λσ F ( a ) λσ g µν (cid:21) . (5)Variation with respect to the gauge potentials A ( a ) µ and A µ yield the respective YM andMaxwell equations F ( a ) µν ; µ + 1 σ C ( a )( b )( c ) A ( b ) µ F ( c ) µν = 0 , F µν ; µ = 0 , (6)whose integrability equations in order are ∗ F ( a ) µν ; µ + 1 σ C ( a )( b )( c ) A ( b ) ∗ µ F ( c ) µν = 0 , ∗ F µν ; µ = 0 , (7)in which ∗ means duality [4]. The N-dimensional spherically symmetric line element is chosenas ds = − f ( r ) dt + dr f ( r ) + r d Ω N − , (8)in which the S N − line element will be expressed in the standard spherical form d Ω N − = dθ + N − P i =2 i − Q j =1 sin θ j dθ i , (9)where 0 ≤ θ N − ≤ π, ≤ θ i ≤ π, ≤ i ≤ N − . For the YM field we employ the magnetic Wu-Yang ansatz [3, 5, 6] where the potential1-forms are expressed by A ( a ) = Qr ( x i dx j − x j dx i ) , Q = charge, r = N − X i =1 x i , (10)2 ≤ j + 1 ≤ i ≤ N − , and 1 ≤ a ≤ ( N −
1) ( N − / , The Maxwell potential 1-form is chosen as A = (cid:26) qr N − dt, N ≥ q ln ( r ) dt, N = 3 (11)for the electric charge q .The following sections will be devoted to the EMYM equations (i.e.Eq. (4)) and their solutions for all dimensionalities N ≥
3. In the last two sections we addGB theory (for N ≥
5) into our formalism and find new combined solutions for differentMaxwell and metric ansaetze. The energy-momentum tensor for the Maxwell and YM fieldsfor N ≥ T a Max b = − ( N − q r N − diag [1 , , − , − , .., − , (12)5 a YM b = − ( N −
3) ( N − Q r diag [1 , , κ, κ, .., κ ] , (13) κ = N − N − . III. EMYM SOLUTION FOR N ≥ The basic field equation that incorporates all relevant expressions is given by r N − (cid:18) f ′ + Λ3 ( N − r (cid:19) + ( f − r + Q + 2 q ( N − r − N ) ( N −
2) = 0 , (14)where f ′ ≡ dfdr ,which admits the integral f ( r ) = 1 − Mr N − − Λ3 r + ( N −
3) 2 q ( N − r N − − ( N − Q ( N − r , (15)for the constant of integration M as the mass parameter.It is observed that this solution is not valid for N = 5 and N = 3 , for these particularcases therefore different solutions will be found. It is valid, however, for N ≥ , whichimplies that the signs of Maxwell and YM terms are opposite. This may have interestingconsequences pertaining to the confinement of a system that possesses both type of charges.We express (for Λ = 0) f ( r ) = 1 + 2Φ ( r ) (16)where Φ ( r ) stands for the Newtonian-like potential, which we identify asΦ ( r ) = − Mr N − + ( N − q ( N − r N − − ( N − Q N − r . (17)We can define the active force through F = −∇ Φ , which yields F ( r ) = − M ( N − r ( N − + 2 q ( N − ( N − r N − − ( N − Q ( N − r . (18)The signs of the Maxwell and YM terms reveal that while the former is repulsive the latterbecomes attractive. One can easily show that for r → ∞ the YM term dominates (letΛ = 0), namely lim r →∞ f ( r ) → − ( N − Q ( N − r . (19)6or r → + we have the opposite case,lim r → + f ( r ) → N −
3) 2 q ( N − r N − , (20)which may be interpreted as an ”asymptotic independence ” from one type of charge (orthe other )in different limits. For the mini black holes this has the striking effect that theHawking temperature depends only on the electric charge.To find the radius of possible horizon(s), we equate the metric function f ( r ) to zero,which leads to the following equation6 ( N −
3) ( N − q − M ( N −
2) ( N − r N − − Λ ( N −
2) ( N − r N − + (21)3 ( N −
2) ( N − r n − − N −
2) ( N − r N − = 0 . This equation in six dimensional spacetime, without the cosmological constant and zeromass has an exact real solution r h = vuut √ ∆ + 2 ˜ Q √ ∆ + ˜ Q ! (22)where ∆ = 8 ˜ Q − q + 12 r q (cid:16) q − Q (cid:17) , (23)˜ Q = 6 Q , ˜ q = 12 q , ˜ q ≥ Q . A. The case N=5
For N = 5, the master equation (14) admits the solution f ( r ) = 1 − Mr − Λ3 r + 4 q r − Q ln ( r ) r , (24)which involves an unusual logarithmic term. This is an asymptotically flat black hole solutionin which finding the exact radius of the horizon of the possible black holes, leads us to thefollowing non-algebric equationΛ3 r − r + 2 M r + 2 Q r ln r − q = 0 . (25)7his equation, even without cosmological constant does not give an analytical solution. Byusing numerical method, we plot the root ( r h ) of the above equation in Fig. (1), to showthe contribution of the Maxwell and YM charges to the construction of such possible blackholes. The Hawking temperature T H can be written as T H = κ π = 14 π | f ′ ( r h ) | = 14 π (cid:12)(cid:12)(cid:12)(cid:12) r h −
23 Λ r h − q r h − Q r h (cid:12)(cid:12)(cid:12)(cid:12) (26)where r h is the radius of the event horizon and κ stands for the surface gravity. Thecorresponding Newtonian-like YM force term in this case has the form f orce ∼ Q r (1 − r ) , (27)which implies that it is attractive (repulsive) for r > √ e ( r < √ e ).The Maxwell term remainsalways repulsive. We notice also that for r → ∞ (for Λ = 0) the YM term dominates overthe Maxwell lim r →∞ f ( r ) → − Q ln ( r ) r . (28)In the limit r → + , on the other hand we obtainlim r → + f ( r ) → q r , (29)which is in confirm with the behavior (20) for N ≥ . B. The case N=4
In 4-dimensional case the solution for the metric function f ( r ) is given by f ( r ) = 1 − Mr − Λ3 r + ( q + Q ) r , (30)in which both the Maxwell and YM charges have similar feature and the metric is thewell-known RN de-Sitter. Extremal YM black hole for instance, follows in analogy with the4-dimensional RN black hole. Both charges act repulsively against the attractive property ofmass. The black hole solution for f ( r ) = 0 given in Eq. (30) without cosmological constant,has two roots r ± = M ± p M − ( q + Q ) (31)8n which the mass and charges must satisfy the constraint M ≥ ( q + Q ) to have the event( r + ) and Cauchy ( r − ) horizons. Here the thermodynamic properties of the solution (30) isexactly same as the four dimensional RN black holes and therefore we just comment that theroles of the YM and Maxwell charges in the metric are not distinguished from each other. C. The case N=3
In 3-dimensional space time we adopt the Maxwell potential 1-form [7] A = q ln ( r ) dt, ( q = electric charge), (32)and introduce the YM gauge potential 1-forms accordingly as A (1) = Q cos ( φ ) ln ( r ) dt, (33) A (2) = Q sin ( φ ) ln ( r ) dt,A (3) = − Qdφ, ( Q = YM charge),which satisfy the Maxwell and YM equations, respectively. The corresponding EMYMequation for f ( r ) , independent from Eq. (14) becomes3 rf ′ + 2Λ r + 6 (cid:0) Q + q (cid:1) = 0 , (34)which is readily integrated as f ( r ) = − M −
13 Λ r − (cid:0) q + Q (cid:1) ln ( r ) , (35)and the line element is ds = − f ( r ) dt + dr f ( r ) + r dθ . (36)A negative cosmological constant leads to a black hole solution [7] f ( r ) = − M + 13 | Λ | r − (cid:0) q + Q (cid:1) ln ( r ) (37)in which the possible radii of the horizons are given by r + = exp " − LamberW − , − | Λ | e − MQ q Q + q ) ! − M Q + q ) (38) r − = exp " − LamberW , − | Λ | e − MQ q Q + q ) ! − M Q + q ) (39)9n which LambertW ( k, x ) stands for the Lambert function [8]. The energy density, givenby ǫ = T tt = Q + q r (40)may be used to calculate the total energy of the black hole. This shows that the energydiverges logarithmically. The surface gravity, κ defined by κ = (cid:18) − g tt g ij g tt,i g tt,j (cid:19) r = r h = (cid:18) f ′ ( r ) (cid:19) r = r + (41)gives κ = (cid:12)(cid:12)(cid:12)(cid:12) | Λ | r + − Q + q r + (cid:12)(cid:12)(cid:12)(cid:12) . (42)Finally we find the Hawking temperature at event horizon as T H = κ π = 12 π (cid:12)(cid:12)(cid:12)(cid:12) | Λ | r + − Q + q r + (cid:12)(cid:12)(cid:12)(cid:12) (43)and a plot of T H is given in Fig. (2) in terms of the mass, Λ and ( Q + q ) . In analogy with the N = 4 case the squared charges are simply superposed, and themetric is still the BTZ metric. It is observed that for q = 0 = Q the same (BTZ) metricdescribes an EYM black hole in 3-dimensions. Addition of rotation to the metric, which isbeyond our scope here, may add new features to differ for the Maxwell and YM fields. IV. THE GEOMETRY OUTSIDE THE 5-DIMENSIONAL EMYM BLACK HOLE
It is evident from solution (24) that, ξ α = (1 , , , ,
0) and η α = (0 , , , ,
1) are twoof the Killing vectors associated with the symmetry under displacements in the direction t,and rotation angle ψ. Accordingly the conserved quantities may be written as e = − ξ · u = − g αβ ξ α u β = f ( r ) u t (44) ℓ = η · u = g αβ η α u β = r sin ( θ ) sin ( φ ) u ψ (45)where u = (cid:0) u t , u r , u θ , u φ , u ψ (cid:1) is the five-velocity while e and ℓ are the energy density andangular momentum per unit mass, respectively. We restrict the particle to stay on the plane θ = π , φ = π with u θ = u φ = 0 , such that by applying u · u = − g αβ u α u β = − f ( r ) (cid:0) u t (cid:1) + 1 f ( r ) ( u r ) + r (cid:0) u ψ (cid:1) = − u t = dtdτ , u r = drdτ , u θ = dθdτ , u φ = dφdτ , u ψ = dψdτ and τ is the proper time measured bythe observer moving with the particle. Putting (44) and (45) into (46), one gets12 (cid:18) drdτ (cid:19) + 12 (cid:20)(cid:18) ℓ r + 1 (cid:19) f ( r ) − (cid:21) = e −
12 (47)or equivalently 12 (cid:18) drdτ (cid:19) + V eff ( r ) = E (48)where E is the density of total energy per unit mass, V eff ( r ) is the effective potential forradial motion of the particle and f ( r ) is given in (24). In Fig. (3) we plot V eff ( r ) in termsof r for different values of q, Q and ℓ but zero value for M. Since the particle is restricted toremain in a plane which only r and ψ would change, we rewrite the equation (48) in termsof r and ψ as follows: 12 (cid:18) dψdτ (cid:19) (cid:18) drdψ (cid:19) + V eff ( r ) = E (49)or equivalently from Eq. (49) drdψ = ± r r ℓ ( E − V eff ( r )) , (50) V eff ( r ) = 12 (cid:20)(cid:18) ℓ r + 1 (cid:19) f ( r ) − (cid:21) , in which ± depends to the initial direction of the motion and we set it to be positive. In thefollowing subsection, instead of studying the exact geodesics equation (50), we shall restrictourselves to the relatively simpler Newtonian approximation. A. Newtonian motion
In the weak field limit, (and setting Λ to be zero) one may use f ( r ) = 1 + 2Φ ( r ) whichimplies Φ ( r ) = − Mr + 2 q r − Q ln ( r ) r , (51)and consequently the radial force on a unit mass particle is given by F r = − ddr Φ ( r ) = − (cid:18) Mr − q r − Q r + 2 Q ln ( r ) r (cid:19) . (52)The radial equation of motion, therefore, may be written as d rdt = ℓ r − (cid:18) Mr − q r − Q r + 2 Q ln ( r ) r (cid:19) , (53)11here ℓ is the angular momentum per unit mass. As usual, one may start with the followingsubstitution u = 1 r (54)to reduce the last equation into the form d udψ + (cid:18) Q − Mℓ (cid:19) u + 2 Q ℓ u ln u − q ℓ u = 0 . (55)As a particular case we set 1 + Q − Mℓ = 0 , ˜ Q = √ Qℓ and ˜ q = q qℓ to get d udψ + ˜ Q u ln u − ˜ q u = 0 , (56)such that the inverse of the solution of this equation is plotted in the Fig. (4) (i.e., r = u versus ψ ) . Obviously, from the figure, one observes that, the roles of Maxwell and YMcharges outside the black hole are in contrast with each other. Next, we consider Eq. (55)and set the mass of the black hole to be zero. By adjusting charges in terms of the angularmomentum, we express the equation in the form d udψ + Q ! u + ˜ Q u ln u − ˜ q u = 0 . (57)One notices that, if ˜ Q = 0 , this reduces to the Duffing type equation as, d udψ + u − ˜ q u = 0 (58)which can be considered for small ˜ q , as a perturbed simple harmonic oscillator. In our work,however, we are interested to consider both terms (i.e., Maxwell and YM terms). From Eq.(55) we observe that u=1 forms a circular orbit provided the condition (with M = 0) Q − q = 2 M − ℓ (59)holds. We investigate the stability of this orbit by choosing u = 1 + a cos βψ (60)in which β and a are constants such that a ≪ . By substitution we obtain β = ± √ ℓ r Q − q (61)12o that stability of the circular orbits are attained provided Q > q . Thus, a dominatingYM charge gives rise to a deeper well and stable orbits. The foregoing argument can easilybe extended to cover elliptical orbits as well, which will not be repeated here. Let us notethat the possibilities involved in a complete analysis of the Eq. (55) may reveal differentbehaviors as well.
V. EMYM BLACK HOLES IN GB GRAVITY
In this section we use our previous ansaetze and find solutions with the GB term. Thenew action is modified now as I G = 12 Z M dx N √− g (cid:18) R + α L GB − ( N −
1) ( N − − F µν F µν − Tr (cid:0) F ( a ) µν F ( a ) µν (cid:1)(cid:19) , (62)where the new term L GB = R µναβ R µναβ − R αβ R αβ + R is the GB Lagrangian with theconstant GB parameter α. The Maxwell and YM ansatz are chosen as in the previous section.The EMYMGB equation that helps us to determine f ( r ) is given by (cid:18) ∆ − Λ3 ( N − r (cid:19) ( N − r (2 N − − N − q r = 0 , (63)∆ = (cid:16) − r + 2 ∼ αr ( f ( r ) − (cid:17) f ′ ( r ) + ∼ α ( N −
5) ( f ( r ) − − r ( N −
3) ( f ( r ) − − Q ( N − . Solution for f ( r ) follows as f ± ( r ) = r α ± Ψ , N = 51 + r ∼ α (1 ± Υ) , N ≥ s M α + (cid:18) Λ3 + 18 α (cid:19) r α + Q ln ( r ) α − q αr , (65)Υ = s ∼ α Λ3 + 4 ∼ αQ ( N − N − r + 8 M ∼ αr N − − N − q ∼ α ( N − r N − , (66)in which ∼ α = ( N −
3) ( N − α and M is a constant of integration to represent mass. Inthe limit α → f − ( r ) reduces to the ones in the previous section , asit should and for positive branch α can not be zero. For Q = 0 our result reduces to the13ne known before for the EMGB theory [9]. Similarly for q = 0 we recover the resultsobtained previously [10, 11, 12]. In Fig. (5) we plot the radius of the event horizon of the 5-dimensional EMYMGB solution in terms of Maxwell and YM charges. The different effectsof these charges we use to emphasize. For N ≥ eff ˜= − α (cid:16) ± q α (cid:17) , N = 5 − ∼ α (cid:18) ± q ∼ α (cid:19) , N ≥ ∼ α →
0, the negative branch, admits Λ eff → Λ and the positivebranch in the limit of Λ = 0, gives Λ eff = − ∼ α . For the case N = 5, on the other hand, the range of r determines the sign of YM term.Although the ± signs determine the role of both terms we prefer the choice ( − ) under whichin the limit α → N ≥
5. We observe on the other hand that in lower dimensions ( N = 3 ,
4) their roles remainindistinguishable. The asymptotic solutions reveal that the physical results are independentfrom one charge or the other. It is our belief that this may be helpful in understanding theproblem of confinement (i.e., accretion, collapse) versus the electric and YM charges.
VI. EMYMGB SOLUTION FOR A SPECIFIC ANSATZ FOR N ≥ (BR TYPE) In this section we choose our metric ansatz as ds = − f ( r ) dt + dr f ( r ) + h d Ω N − , N ≥ , (68)in which h = constant, is to be expressed in terms of the parameters of the theory. Whilethe YM field will be chosen as before, our Maxwell field will be different. For the presentpurpose let our Maxwell 1-form be given by the choice A = qrdt, (69)where the constant q is related to the electric charge. This choice has the feature that theonly non-vanishing electromagnetic field 2-form F = − qdt ∧ dr, (70)14ields a uniform electric field. The non-vanishing Maxwell energy-momentum tensor com-ponents T a Max b are T a Max b = − q diag [1 , , − , − , .., − . (71)The non-zero YM energy-momentum components T a YM b are T a YM b = − ( N −
3) ( N − Q h diag [1 , , κ, κ, .., κ ] , N ≥ , (72) κ = N − N − . The field equations (4) with the GB and Λ terms, on the premise that h = constant, reducesto (cid:2) h + 2 α ( N −
3) ( N − (cid:3) f ′′ − ( N −
3) ( N −
4) [1 + α ( N −
5) ( N − N −
1) ( N −
2) Λ h + ( N −
3) ( N − h Q − q h = 0 , N ≥ . (73)The solution for f ( r ) can be expressed as f ( r ) = C r + C r + C (74)where C and C are integration constants while C is a constant depending on the param-eters of the theory. Explicitly we have C = (cid:26)
12 ( N −
3) ( N −
4) [1 + α ( N −
5) ( N − −
16 ( N −
2) ( N −
1) Λ h − ( N −
3) ( N − Q h + q h (cid:27) / (cid:2) h + 2 α ( N −
3) ( N − (cid:3) , N ≥ . (75)The constant h is also expressible by23 h = ( N −
3) ( N − ± p ( N −
3) ( N −
2) [ K ( N −
2) + 8 q L ]6 q + ( N −
1) ( N −
2) Λ , (76)in which we have abbreviated K = 43 ( N − (cid:2) ( N −
5) ( N − α − Q (cid:3) Λ + ( N − , (77) L = ( N −
5) ( N − α − Q , N ≥ , and (+) and ( − ) signs are chosen in the Maxwell and YM limits, respectively. From theseexpressions we obtain the EMYM limit by setting α = 0. Similarly the EMGB and EYMGBlimits can be obtained by setting Q = 0 and q = 0 , respectively. For C = 0 = C we can15ave the roots of f ( r ) = 0 , which signals the horizons for black holes. The abundance ofparameters in the EMYMGB theory creates a large class of possibilities admitting variousblack hole solutions which we shall not pursue here. By choosing C = C = 0 and con-straining h = βC , for a suitable constant β ( > , followed by a redefinition of time we castthe line element into ds = h β (cid:18) − dt + dr r + βd Ω N − (cid:19) , (78)which is of the static BR form. For α = 0 = Λ = q (as a limit ) we arrive at [13, 14] ds = Q N − (cid:18) − dt + dr r + ( N − d Ω N − (cid:19) . (79)In general since β = 1 conformal flatness is not satisfied. Only for N = 4 we have thecase of exact BR which is conformally flat. However in general we have shown that in theEMYMGB theory we construct a metric which is of BR type, albeit it fails to satisfy theconformal flatness. In the pure Maxwell limit, ( q = 0) , Q = Λ = α = 0 and adopting C = C = 0 , we obtain (after scaling) ds = 1 C (cid:20) − dt + dr r + ( N − d Ω N − (cid:21) , (80)which is also of similar type with the constant C = 2 q (cid:0) N − N − (cid:1) . This is in agreement with thehigher dimensional BR metric in the EM theory [13]. We recall that following the methodof Ginsparg and Perry [15] the N-dimensional YM-BR solution can be expressed in the form ds = Q ( N − (cid:2) − sinh χdT + dχ + ( N − d Ω N − (cid:3) . (81)The transformation that takes us to this result in the limit ǫ → f ( r ) = N − Q ( r + t ) ( r − t ) , t = Q ( N − ǫ T, r = ǫ cosh χ, (82)while the magnetic type YM field remains unchanged. VII. ENERGY AND CAUSALITY CONDITIONSA. N ≥ − dimensions (RN type) The energy conditions (EC) of the matters associated with the energy momentum tensorgiven by (12) and (13) for N ≥ − dimensions, i.e. T ab = T aMax b + T aY M b (83)16an be studied by using the definition of the energy-density of the matter ρ [18] ρ = − T tt = − T rr = ( N − q r N − + ( N −
3) ( N − Q r , (84)the principal pressures p i p i = T ii (no sum convention) (85)and the effective pressure p eff = 1 N − N − X i =1 p i . (86)
1. Weak Energy Condition (WEC)
The WEC may be expressed as ρ ≥ ρ + p i ≥ , ( i = 1 , ...N −
1) (87)which holds true in any dimensions N ≥ , by the energy momentum tensor given by (83).
2. Strong Energy Condition (SEC)
The SEC states that ρ + N − X i =1 p i ≥ ρ + p i ≥ , ( i = 1 , , ..., N −
1) (88)which for 4 < N ≤ N ≥ r ≤ r sec in which r sec = (cid:18) N − q ( N − Q (cid:19) N − , N ≥ . (89)
3. Dominant Energy Condition (DEC)
In accordance with the DEC, which are given by ρ ≥ | p i | , ( i = 1 , , ..., N −
1) (90)our energy momentum tensor satisfies these for any dimensions
N > . . Causality Condition (CC) We express the CC after knowing that ρ > ≤ p eff < ρ, (91)which is satisfied by the energy momentum tensor given by (83) for N = 4 , . For N ≥ r < r cc where r cc = N − q ( N −
2) ( N − Q ! N − , N ≥ . (92) B. N = 3 , − Dimensions (RN type)
In 4 − dimensions, the energy momentum tensor simply reads T ab = − q + Q r diag [1 , , − , −
1] (93)in which WEC, SEC, DEC and CC are all verified. In 3 − dimensions also the energy mo-mentum tensor which is given by T ab = − q + Q r diag [1 , , −
1] (94)satisfies all the energy and causality conditions. C. N ≥ − dimensions (BR type) To investigate the energy conditions of second type solutions (BR type) we rewrite theenergy momentum tensor of the system in the form of T ab = T aMax b + T aY M b (95)where T a Max b = − q diag [1 , , − , − , .., − , (96) T a YM b = − ˜ Q diag [1 , , κ, κ, .., κ ] , N ≥ ,κ = N − N − Q = ( N −
3) ( N − Q h . N ≥ . The SEC is also verified for N = 4 , , N ≥ , only under the condition q ≥ κ ˜ Q (97)the SEC is satisfied. It is also easy to show that, DEC is satisfied for any dimensions.Finally, the CC is satisfied for N = 4 , , but for N ≥ q ≥ N − N − Q . (98) VIII. CONCLUSION
Our exact solution in the first part of the paper suggests that for N ≥
5, the Maxwell andYM fields compete for dominance in the asymptotic regions. That is, for r → r → ∞ )the Maxwell charge q (the YM charge Q) dominates. This may shed light on the problem ofgravitational confinement (i.e., accretion, collapse) versus the Maxwell and YM charges. Asa drawback of our model the YM field is treated, in analogy with the Maxwell field, entirelyclassical. The Newtonian approximation in the polar plane consisting of the coordinates( r, ψ ) reveals that the YM charge deepen the potential well to form bound states. In lowerdimensions (i.e., N = 3 , N = 4), however the roles of q and Q remain indistinguishable. Thepresence of a logarithmic term for N = 3 and N = 5 is a distinctive property compared toother dimensions. An effective cosmological constant can be defined from the GB parametersfor r → ∞ . In the last part of the paper where we introduced a different ansatz, we presentexact solution to the EMYMGB theory which can represent a variety of black holes. Anotherpossibility is by the choice of parameters to cast the metric into the static BR form whichlacks conformal flatness but may be important in supergravity/string theory[16], as wellas in cosmology. Finally, validity of the energy/ causality conditions are discussed for allsolutions that are obtained in the paper. [1] N. Okuyama and K. I. Maeda, Phys. Rev. D (2003) 104012.G.W. Gibbons and P.K. Townsend, Class. Quant. Grav. (2006) 4873.[2] Y. Brihaye, E. Radu and D. H. Tchrakian, Phys. Rev. D (2007) 024022.
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119 B (2004) 931. X. FIGURE CAPTIONS:
Fig. (1): Radius of the event horizon of the 5-dimensional, EMYM black hole in terms ofthe Maxwell and YM charges, and specific values for M and Λ. The non-black hole regionis shown as NBH.Fig. (2): A plot of T H for 3-dimensional EMYM black hole solution, in terms of Λ and( Q + q ) . We set the mass of the black hole to be unit.Fig. (3): The effective potential versus r . We aim to compare the roles of Maxwell andYM charges, versus their effects in the V eff ( r ) in 5-dimensions . It is seen that it is the YMcharge Q which provides deep potential well apt for the bound states.Fig. (4): A plot of radius of a particle whose initial values are set as: r ( ψ = 0) =1 , drdψ (cid:12)(cid:12)(cid:12) ψ =0 = − .
7. For ˜ Q = 3 and ˜ q < . r = 1 , but for ˜ q ≥ . Q − mℓ = 0 . Fig. (5): Radius of the event horizon of the 5-dimensional, EMYMGB black hole in termsof Maxwell and YM charges, and specific values for M , α and Λ. The non-black hole regionis shown as NBH. (These plot may be compared with those in Fig. (1)).21and Λ. The non-black hole regionis shown as NBH. (These plot may be compared with those in Fig. (1)).21