Charged dilatonic black rings and black saturns and their thermodynamics
aa r X i v : . [ g r- q c ] S e p Charged dilatonic black rings and black saturnsand their thermodynamics
Saskia GrunauInstitut f¨ur Physik, Universit¨at Oldenburg, D–26111 Oldenburg, GermanyOctober 13, 2018
Abstract
In this paper charged black rings and black saturns are constructed in Einstein-Maxwell-dilatontheory in five dimensions. The neutral black ring and black saturn solutions are embedded in sixdimensions and boosted with respect to the time coordinate and the added sixth dimension. Then thecharged solutions are obtained by a Kaluza-Klein reduction. The influence of the charge is studiedby analysing the physical properties and the phase diagram. The different dilatonic solutions arecompared and their thermodynamic stability is considered.
A promising development in the search for a quantum theory of gravity is string theory, which needs morethan four dimensions for its internal consistency. This resulted in wide interest in higher-dimensionalsolutions including higher-dimensional black holes (see e.g. [1]). Myers and Perry found the higher-dimensional version of the Kerr black hole [2] and also supposed the existence of black holes in higherdimensions with a nonspherical horizon topology. A solution with horizon topology S × S , a black ring,was discovered by Emparan and Reall [3]. This showed that the uniqueness theorem does not hold inhigher dimensions.Since then a variety of black ring solutions has been found, e.g. a black ring with two angular momenta[4], a charged black ring [5], and a supersymmetric black ring [6, 7]. Also composite black objects arepossible: a black saturn, a spherical black hole surrounded by a black ring [8], systems of multiple blackrings [9, 10, 11], or black saturns with multiple rings.In the static case, magnetic [12] and electric charge [13] have been added to the black saturn solution.The magnetized black saturn is balanced, but the electrically charged black saturn exhibits either a conicalor a naked singularity.When gravity is coupled to a Maxwell field and a dilaton field, charged dilatonic black hole solutionsemerge. In the case of Kaluza-Klein coupling the solutions can be obtained by adding a spatial dimensionand performing a boost. A dimensional reduction then leads to the charged dilatonic black hole solutions(compare [14] – [24]).This method was applied to the Schwarzschild spacetime [16] to find the static electrically chargedEinstein-Mawell-dilaton black hole and to the Kerr spacetime [20, 22] to find the corresponding rotatingcharged dilatonic black hole. In [23] a rotating dyonic black hole in Kaluza-Klein theory was presented.There two boosts and a rotation were applied to the solution to obtain a rotating electrically and mag-netically charged black hole in Einstein-Mawell-dilaton theory.Also in higher dimensions rotating Einstein-Mawell-dilaton black holes were found [24] by boostingthe higher-dimensional Myers-Perry solution.In five dimensions Einstein-Mawell-dilaton black rings were constructed. The static charged dilatonicblack ring was obtained for arbitrary values of the dilaton coupling constant [25] and additionally a rotatingcharged dilatonic black ring was found in the case of Kaluza-Klein coupling [25, 26]. In the Kaluza-Kleincase also general black ring solutions were constructed: A three-charge black ring [27] and a black ringwith two angular momenta, electric charge and magnetic charge [28, 29]. Furthermore dilatonic solutionswith dipole charges have been obtained, namely a dipole black ring [30], a black saturn with a dipole1ing [31], and a solution describing two concentric rotating dipole black rings [32]. Numerically also aseven-dimensional dilatonic black ring was found [33].In this paper first a brief review of the rotating charged dilatonic black ring will be given and thenthe charged dilatonic black saturn solution will be constructed, using the method of boosting the seedsolution and then performing a dimensional reduction. The influence of the dilaton on the black ringand black saturn spacetime is studied by analysing the physical properties and the phase diagram. Withincreasing charge, the dilatonic solutions need less angular momentum and less horizon area to be balanced.Furthermore, the thermodynamic stability is discussed. The isothermal moment of inertia and specificheat at constant angular momentum of the dilatonic black ring and black saturn cannot be positive atthe same time. Therefore the charged dilatonic black ring and black saturn are not thermodynamicallystable. Let us first briefly recall the method and the extraction of the physical properties.To generate a Kaluza-Klein black hole, the metric of a neutral black object is used as a seed metric.Since we want to construct a five-dimensional object, we take the five-dimensional seed metric and embedit in a six-dimensional spacetime with an extra coordinate U ,d s = d s + d U . (1)The six-dimensional metric is boosted in the t - U -plane with the matrix L = (cid:18) cosh( β ) sinh( β )sinh( β ) cosh( β ) (cid:19) . (2)This boosted metric is still a solution of the (six-dimensional) vacuum Einstein equations. A five-dimensional solution in the Einstein-Maxwell-dilaton theory can be obtained by comparing the boostedmetric to the Kaluza-Klein parametrization of a six-dimensional metric, which readsd s , KK = e α Φ g µν d x µ d x ν + e − α Φ (d U + A µ d x µ ) , (3)where α = √ . From the comparison of the transformed d s and d s , KK one can read off the metriccomponents g µν , the Maxwell potential A µ and the dilaton function Φ corresponding to the new five-dimensional charged dilatonic black hole.The action in five-dimensional Einstein-Maxwell-dilaton theory is S = 116 πG Z d x √− g (cid:18) R −
12 Φ ,µ Φ ,µ −
14 e − h Φ F µν F µν (cid:19) , (4)where the dilaton coupling constant has the Kaluza-Klein value h = √ . The obtained charged dilatonicmetric fulfils the Einstein equations, G µν = 8 πGT µν (5)with T µν = ∂ µ Φ ∂ ν Φ − g µν ∂ λ Φ ∂ λ Φ + e − h Φ (cid:18) F µλ F λν − g µν F λσ F λσ (cid:19) , (6)the Maxwell equations, ∇ µ (cid:0) e − h Φ F µν (cid:1) = 0 , (7)and the dilaton equation, ∇ Φ = − h − h Φ F µν F µν . (8)Regarding the physical properties, one can easily find relations between the physical quantities of thecharged dilatonic black hole and its uncharged seed metric. In the following the physical quantities of thedilatonic black hole will be denoted with an index β and the quantities of the seed metric with an index 0.2he (ADM) mass M , the angular momentum J , the electric charge Q , the magnetic moment M , andthe dilaton charge Σ can be read off from the metric functions at infinity. In spheroidal coordinates ( r, θ )the asymptotic expansion ( r → ∞ ) of the metric functions yields ([24], [34]) g tt ≈ − GM π r , (9) g tψ ≈ − GJπ sin( θ ) r , (10) A t ≈ GQπr , (11) A ψ ≈ − G M π sin( θ ) r (12)Φ ≈ G Σ πr , (13)assuming the black object is five-dimensional and rotates in ψ -direction. The physical quantities can nowbe found by comparing (9) – (13) to the solution in the limit r → ∞ .Then the following relations between the quantities of the charged and the uncharged black object canbe found: M β = (cid:18)
23 sinh( β ) + 1 (cid:19) M , (14) J β = cosh( β ) J , (15) Q = 23 sinh( β ) cosh( β ) M , (16) M = sinh( β ) J (17)Σ = − (cid:18) (cid:19) / sinh( β ) M . (18)Then the charges fulfill the relation [23, 24] Q M − √ Σ = − √ . (19)The area of the horizon is A H = Z q det(d s H ) d ψ d φ d z (20)using the canonical coordinates of [34]. d s H describes the metric at the horizon. Here one gets the relation A βH = cosh( β ) A H . (21)The Hawking temperature can be calculated via the surface gravity T H = κ π , (22)where κ = −
12 ( ∇ µ ξ ν )( ▽ µ ξ ν ) (cid:12)(cid:12)(cid:12)(cid:12) horizon . (23)The Killing vector field is ξ = ∂ t + Ω ∂ ψ . (24)The relation between the horizon angular velocity of the dilatonic black hole and the horizon angularvelocity of the seed metric is Ω β = 1cosh( β ) Ω . (25)Considering the Hawking temperature a similar relation exists: T βH = 1cosh( β ) T H . (26)3he horizon electrostatic potential Ψ el = ξ µ A µ | horizon = sinh( β )cosh( β ) (27)depends only on the boost parameter β , for the metrics considered in this paper. The rotating charged dilatonic black ring was constructed in [27, 25] and analysed in [35] within thequasilocal formalism, but so far its thermodynamical stability was not considered.
Taking the neutral rotating black ring metric of [3] (or more precisely, in the form presented in [27]) asseeds, a charged dilatonic rotating black ring can be constructed with the method described above. Themetric in ring coordinates isd s = − V β ( x, y ) − / F ( x ) F ( y ) h d t + ( R √ λν cosh( β ))(1 + y )d ψ i + V β ( x, y ) / R ( x − y ) (cid:20) − F ( x ) (cid:18) G ( y )d ψ + F ( y ) G ( y ) d y (cid:19) + F ( y ) (cid:18) G ( x ) d x + G ( x ) F ( x ) d φ (cid:19)(cid:21) , (28)where R is a scaling parameter, F ( ζ ) = 1 − λζ , (29) G ( ζ ) = (1 − ζ )(1 − νζ ) (30)and V β ( x, y ) = cosh( β ) − sinh( β ) F ( x ) F ( y ) . (31)The coordinate ranges are − ≤ x ≤ −∞ < y ≤ − λ − < y < ∞ . The parameters ν and λ vary in 0 ≤ ν < λ ≤
1. The static charged dilatonic black ring can be obtained by setting ν = 0.The nonvanishing parts of the vector potential are A t = sinh( β ) cosh( β )( F ( y ) − F ( x ))cosh( β ) F ( y ) − sinh( β ) F ( x ) , (32) A ψ = sinh( β ) F ( x ) R √ λν (1 + y )sinh( β ) F ( x ) − cosh( β ) F ( y ) , (33)and the dilaton function is Φ = − √
63 ln (cid:18) cosh( β ) − sinh( β ) F ( x ) F ( y ) (cid:19) . (34) The physical quantities are Ω = 1 R cosh( β ) r νλ (1 + λ ) (35) M = 3 πR G λ (1 + λ )(1 + ν ) (cid:18) β ) (cid:19) , (36) J = cosh( β ) πR G √ λν (1 + λ ) / (1 + ν ) , (37)4 = πR G sinh( β ) cosh( β ) λ (1 + λ )1 + ν , (38) M = − πR G sinh( β ) √ λν (1 + λ ) / (1 + ν ) (39)Σ = πR G r
23 sinh( β ) λ (1 + λ )1 + ν , (40) A H = 8 π R cosh( β ) p λ ( λ − ν )( λ − ν )(1 + λ )(1 + ν )(1 − ν ) , (41) T H = 1cosh( β ) 1 − ν πR p λ ( λ − ν ) , (42)Ψ el = sinh( β )cosh( β ) . (43)From (42) it can be seen that the only extremal ( T H = 0 ) black ring solution is ν = 1. This choice of ν corresponds to a naked singularity [27].In general the charged dilatonic black ring exhibits a conical singularity, which can be placed either at x = − x = +1. The excess or deficit angle can be computed in ring coordinates as δ = 2 π − (cid:12)(cid:12)(cid:12)(cid:12) ∆ φ∂ x √ g φφ √ g xx (cid:12)(cid:12)(cid:12)(cid:12) x = ± . (44)In the following the conical singularity is chosen to be at x = +1, so that ∆ φ = 2 π √ λ ν and δ = 2 π (cid:18) − (1 − ν ) √ λ (1 + ν ) √ − λ (cid:19) . (45)The charge parameter β does not influence the conical singularity. In order for the black ring to bebalanced ( δ = 0), the rotation parameters ν and λ have to fulfil the condition λ = 2 ν ν . (46)The sign of the mechanical moment of inertia I divides the charged dilatonic black ring solution intoa fat branch ( I >
0) and a thin branch (
I < M and δ constant the mechanical moment ofinertia is defined by 1 I = ∂ Ω ∂J (cid:12)(cid:12)(cid:12)(cid:12) M,δ = 2 G (1 − ν )(1 + ν ) R π cosh( β )(1 + λ ) (3 λν + λ − ν ) . (47)In a fixed Q ensemble equation (47) gives the same result as in a fixed Ψ el ensemble. The sign of I is notinfluenced by the charge parameter β and the scaling parameter R .Figure 1 shows the sign of the conical deficit/excess angle (a) and the sign of the mechanical momentof inertia (b) in the λ - ν -parameter space.For convenience the following scaled quantities are introduced: a H = 316 r π A H ( GM ) / , (48) j = 27 π J GM , (49) q = QM , (50) τ = r π √ GM T H . (51)The scaled charge depends only on β and there is an upper limit q = 1 (and a lower limit q = − a) Sign of δ . The solution contains a conical deficit(blue area) or conical excess (green area). Thebalanced solutions ( δ = 0) are on the blackcurve in between the blue and green area. (b) Sign of the mechanical moment of inertia. I >
I < δ =0).Figure 1: Parameter space of the rotating charged dilatonic black ring picture (a) shows how the a H - j -diagram changes for different values of β and the right picture (b) showsa three-dimensional a H - j - q -diagram. As β and accordingly q grows, the curve in the phase diagram getsshifted to lower a H and j . The charge comes with an additional force which helps to stabilize the blackring. So when the charge is increased less angular momentum is needed to keep the black ring balanced.Figure 3(a) shows j - q -plots for different values of the parameter ν , in the extremal case ν = 1 thesolution is a naked singularity. As for spherical dilatonic black holes [24], the j - q -plots form a cusp at j = 0.In figure 3(b) a H - q -diagrams are depicted . The left picture shows diagrams for different values of theparameter ν . a H reaches its highest value for a given q if ν = 0 .
5, this is also the point where j reachesits lowest value. If ν = 1, the horizon area vanishes.The temperature of the charged dilatonic black ring versus the parameter ν and the temperatureversus the horizon area and the charge can be seen in figure 4. The charge has only small influence on thetemperature. (a) a H versus j for different values of β
0 1 2 3 4 5 6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 q j a H q (b) a H versus j versus q Figure 2: Phase diagram of the charged dilatonic black ring a) j - q -diagram for different values of the parameter ν . (b) a H - q -diagram for different values of the parameter ν .Figure 3: j - q -diagrams and a H - q -diagrams of the charged dilatonic black ring(a) Temperature versus ν for different values of β .
0 0.5 1 1.5 2 2.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 q τ a H q (b) Temperature versus a H versus q .Figure 4: Temperature diagrams of the charged dilatonic black ring In a (stationary and asymptotically flat) spacetime that contains a conical singularity, the first law ofthermodynamics gets an extra term A d P (see [36], [37]). P is the pressure caused by the conical singularityand A /T H is the area spanned by the conical singularity.For the charged dilatonic spacetimes considered in this paper, the first law isd M = T H d S + Ωd J + Ψ el d Q − A d P . (52)The parameter A for the charged dilatonic black ring is A = πR √ λ (1 − λ ) / − ν . (53)To analyse the thermodynamical stability two quantities are considered (compare [37]), the specific heatat constant angular momentum C J = T H ∂S∂T H (cid:12)(cid:12)(cid:12)(cid:12) J, A (54)and the isothermal moment of inertia ǫ = ∂J∂ Ω (cid:12)(cid:12)(cid:12)(cid:12) T H , A . (55)7o ensure thermodynamical stability both quantities need to be positive.In a fixed Q ensemble the specific heat and the isothermal moment of inertia read C J = 2 π R cosh( β ) √ λ (1 + λ )( λ − ν ) / [6 cosh( β ) λ ( λν − λ + ν −
1) + λ (1 − ν ) + 2( λν + λ − ν )] G ( ν − β ) λ ( λν + 14 λν − ν + λ − ν − − λ ( ν − ν − − λν (1 + ν ) + 8 ν ](56) ǫ = πR cosh( β ) λ (1 + λ ) G (1 + ν ) [2 λ cosh( β ) ( λ − ν −
1) + λ (3 − ν ) − ν ] × [2 λ cosh( β ) ( − λν − λν + 13 ν + 2 ν + λ + 1) + λ ( ν + 12 ν + 3) − λν (1 + ν ) + 8 ν ] (57)and in a fixed Ψ el ensemble the two quantities are C J = 2 π R cosh( β ) √ λ (1 + λ )( λ − ν ) / [ λ (3 ν −
5) + 2 λ (4 ν − − ν ] G ( ν − λ ( ν + 16 ν −
1) + 2 λ ( − ν + 4 ν − − ν ] (58) ǫ = πR cosh( β ) λ (1 + λ ) [ λ (1 − ν − ν ) + 2 λ (7 ν − ν + 1) + 8 ν ]2 G (1 + ν ) [ λ (1 + ν ) + 2 λ (1 − ν ) − ν ] . (59)In the fixed Ψ el ensemble the sign of C J and ǫ is the same for β = 0 and β = 0, whereas in the fixed Q ensemble, the charge parameter β influences the sign of C J and ǫ . Nevertheless for all β the chargeddilatonic black ring is not thermodynamically stable, since C J and ǫ cannot be positive at the same time.Figures 5 and 6 show C J and ǫ for different values of β in the fixed Q ensemble (the plots for β = 0also correspond to the fixed Ψ el ensemble, since both ensembles are equal at β = 0). (a) β = 0 (b) β = 1 (c) β = 5Figure 5: Sign of the specific heat at constant angular momentum of the rotating charged dilatonic black ring in the fixed Q ensemble. C J is positive in the green area and negative in the blue area. The balanced solutions can be found along the dashedcurve.(a) β = 0 (b) β = 0 . β = 1 . Q ensemble. ǫ is positivein the green area and negative in the blue area. The balanced solutions can be found along the dashed curve. The charged dilatonic black saturn
The black saturn solution was found by Elvang and Figueras in 2007 [8]. The solution describes a sphericalblack hole ( S ) surrounded by a black ring ( S × S ). If one takes the metric of [8] as seeds and appliesthe above methods, one gets the metric of a charged dilatonic black saturn. In canonical coordinates, themetric is given by ds = − V β ( ρ, z ) − / H y H x (cid:20) d t + cosh( β ) (cid:18) ω ψ H y + q (cid:19) d ψ (cid:21) + V β ( ρ, z ) / H x (cid:26) k P (cid:0) d ρ + d z (cid:1) + G y H y d ψ + G x H x d φ (cid:27) , (60)where V β ( ρ, z ) = cosh( β ) − H y H x sinh( β ) . (61)The nonvanishing parts of the vector potential are A t = sinh( β ) cosh( β )( H x − H y )cosh( β ) H x − sinh( β ) H y , (62) A ψ = sinh( β )( ω ψ + qH y )sinh( β ) H y − cosh( β ) H x (63)and the dilaton function is Φ = − √
63 ln (cid:18) cosh( β ) − sinh( β ) H y H x (cid:19) . (64)The metric funtions are G x = ρ µ µ µ ,G y = µ µ µ ,H x = F − (cid:0) M + c M + c M + c c M + c c M (cid:1) ,H y = F − µ µ (cid:18) M µ µ − c M ρ µ µ − c M µ µ ρ + c c M + c c M µ µ (cid:19) ,ω ψ = 2 F √ G x (cid:16) c R p M M − c R p M M + c c R p M M − c c R p M M (cid:17) ,P = ( µ µ + ρ ) ( µ µ + ρ )( µ µ + ρ ) F = µ µ ( µ − µ ) ( µ − µ ) ( ρ + µ µ )( ρ + µ µ )( ρ + µ µ )( ρ + µ µ ) × ( ρ + µ µ )( ρ + µ µ ) Y i =1 ( ρ + µ i )where M = µ µ ( µ − µ ) ( µ − µ ) ( ρ + µ µ ) ( ρ + µ µ ) ( ρ + µ µ ) ,M = µ µ µ µ µ ρ ( µ − µ ) ( µ − µ ) ( µ − µ ) ( ρ + µ µ ) ,M = µ µ µ µ ρ ( µ − µ ) ( µ − µ ) ( ρ + µ µ ) ( ρ + µ µ ) ,M = 2 µ µ µ µ µ ( µ − µ )( µ − µ )( µ − µ )( ρ + µ )( ρ + µ )( ρ + µ µ )( ρ + µ µ )( ρ + µ µ ) ,M = µ µ µ µ ( µ − µ ) ( ρ + µ µ ) ( ρ + µ µ ) and µ i = p ρ + ( z − a i ) − ( z − a i ) ,R i = p ρ + ( z − a i ) . a i with i = 1 , . . . , κ i = a i +2 − a L , i = 1 , . . . , L with L = a − a [8], where the κ i are ordered in the following way0 ≤ κ ≤ κ < κ ≤ . (66)Analogously the coordinate z can be scaled z = z − a L (67)and a new dimensionless parameter c = c c (1 − κ ) can be introduced. The parameters and the coordinate z were scaled in order to simplify the expressions of the physical quantities.The parameter c is fixed at | c | = L r κ κ κ (68)in order to remove singularities at ρ = 0, z = 0 resulting from the construction of the black saturn in [8].To ensure the asymptotic flatness of the solution, the parameters q and k have to be chosen as q = L r κ κ κ c κ c , (69) k = 1 | κ c | . (70) To calculate the mass, charge, magnetic moment, angular momentum and dilaton charge of the chargeddilatonic black saturn the asymptotic coordinates of [8] are used: ρ = 12 r sin 2 θ (71) z = 12 r cos 2 θ . (72)The physical quantities can be obtained by taking the limit r → ∞ . The (ADM) mass, charge, magneticmoment, angular momentum and dilaton charge of the charged dilatonic black saturn are M = (cid:18)
23 sinh( β ) + 1 (cid:19) πL G κ (1 − κ + κ ) − κ κ ( κ − κ ) c + κ [ κ − κ κ (1 + κ − κ )] c κ [1 + κ c ] , (73) J = cosh( β ) πL G κ [1 + κ c ] r κ κ κ (cid:8) κ − c κ [( κ − κ )(1 − κ + κ ) + κ (1 − κ )]+ c κ κ [( κ − κ )( κ − κ ) + κ (1 + κ − κ − κ )] − c κ κ [ κ − κ κ (2 + κ − κ − κ )] (cid:9) , (74) Q = πL G sinh( β ) cosh( β ) κ (1 − κ + κ ) − κ κ ( κ − κ ) c + κ [ κ − κ κ (1 + κ − κ )] c κ [1 + κ c ] , (75) M = πL G sinh( β ) 1 κ [1 + κ c ] r κ κ κ (cid:8) κ − c κ [( κ − κ )(1 − κ + κ ) + κ (1 − κ )]+ c κ κ [( κ − κ )( κ − κ ) + κ (1 + κ − κ − κ )] − c κ κ [ κ − κ κ (2 + κ − κ − κ )] (cid:9) , (76)Σ = − πL G r
23 sinh( β ) κ (1 − κ + κ ) − κ κ ( κ − κ ) c + κ [ κ − κ κ (1 + κ − κ )] c κ [1 + κ c ] . (77)10he horizon area is computed for each horizon separately: A BH H = 4 L π cosh( β ) s − κ ) (1 − κ )(1 − κ ) κ κ (1 − κ )(1 − κ ) κ (1 − κ ) c (1 + κ c ) , (78) A BR H = 4 L π cosh( β ) s κ ( κ − κ ) κ ( κ − κ )(1 − κ ) − ( κ − κ ) c + κ κ (1 − κ ) κ c (1 + κ c ) ! . (79)The horizon area of the whole black object is A H = A BHH + A BRH . From this one gets the entropy of thecharged dilatonic black saturn S = A H G .The Hawking temperatures of the spherical black hole and the black ring are T BH H = 12 Lπ cosh( β ) s (1 − κ )(1 − κ )2(1 − κ ) (1 + κ c ) κ κ (1 − κ )(1 − κ ) κ (1 − κ ) c , (80) T BR H = 12 Lπ cosh( β ) s κ (1 − κ )( κ − κ )2 κ ( κ − κ ) (1 + κ c ) − ( κ − κ ) c + κ κ (1 − κ ) κ c ! . (81)The horizon angular velocities areΩ BH = 1 L cosh( β ) (1 + κ c ) r κ κ κ (cid:18) κ (1 − κ ) − κ (1 − κ )(1 − κ ) c κ (1 − κ ) + κ κ (1 − κ )(1 − κ ) c (cid:19) , (82)Ω BR = 1 L cosh( β ) (1 + κ c ) r κ κ κ (cid:18) κ − κ (1 − κ ) c κ − κ ( κ − κ ) c + κ κ (1 − κ ) c (cid:19) . (83)It is interesting to see how the total charge is divided between the spherical black hole and the blackring. Each charge can be computed separately using a Gauß integral Q i = 116 πG Z H i e h Φ ∗ F , (84)where H i is the black hole or black ring horizon and F = d A . In terms of the metric components this canbe written as Q i = 116 πG Z H i e h Φ g φφ g zz √− det g ( g tψ ∂ ρ A ψ − g ψψ ∂ ρ A t ) d ψ d φ d z . (85)So the individual charges of the spherical part and the ring part of the dilatonic black saturn are Q BH = πL G κ (1 − κ ) + κ κ (1 − κ )(1 − κ ) c κ (1 + c κ ) , (86) Q BR = πL G κ [1 − (1 − κ ) c ][ κ − κ ( κ − κ ) c + κ κ (1 − κ ) c ] κ (1 + c κ ) . (87)The two charges sum up to the total ADM charge Q . Additionally there is a relation between the charges Q i and the Komar masses of the sphere ( M BH ) and the ring ( M BR ) of the uncharged black saturn (see[8] for the Komar masses), Q BH = 23 sinh( β ) cosh( β ) M BH , (88) Q BR = 23 sinh( β ) cosh( β ) M BR . (89) Next we check the φ - and ψ -rod for conical singularities. In canonical coordinates the deficit or excessangle of a conical singularity is (compare equation (44)) δ = 2 π − ∆ η lim ρ → ∂ ρ √ g ηη √ g ρρ = 2 π − ∆ η lim ρ → r g ηη ρ g ρρ , (90)11here η = φ, ψ and ∆ η is the period of the angular coordinate φ or ψ . For a balanced black saturn δ = 0holds, so that the period of an angular coordinate is (see also [8], [34])∆ η = 2 π lim ρ → s ρ g ρρ g ηη . (91)The charged dilatonic black saturn has a semi-infinite ψ -rod at z ∈ [1 , ∞ ] and a semi-infinite φ -rod at z ∈ [ −∞ , κ ]. These rods fix the periods to be ∆ ψ = ∆ φ = 2 π to ensure asymptotic flatness (for the rodstructure of a black saturn see [8]). Using equation (90) on both rods, each time one gets the condition | c | = L q κ κ κ which is already known from equation (68).If this condition holds, there is still a conical singularity left sitting between the black ring and the S black hole in the equatorial plane. By requiring ∆ φ = 2 π on the finite φ -rod at z ∈ [ κ , κ ] this conicalsingularity disappears. Here equation (90) gives c = 1 κ (cid:20) κ − κ √ κ (1 − κ )(1 − κ )( κ − κ ) − (cid:21) . (92)The conditions for balanced solutions are the same for the charged dilatonic and the neutral blacksaturn.Since the black saturn consists of two black objects, one has to consider under which circumstancesthe solution is in thermodynamical equilibrium. Clearly, in equilibrium the Hawking temperature of bothobjects has to be the same. In addition both objects should have the same angular velocity (see [37], [38])and the same horizon electrostatic potential. The latter is already fulfilled, so one just needs to considerΩ BH = Ω BR , (93) T BH H = T BR H . (94)Equation (93) leads to c = − κ (1 − κ ) κ κ (1 − κ ) (95)and together with (94) one finds κ = κ + κ (1 − κ )1 − κ − κ . (96)If (92) is not imposed the conical singularity in thermodynamical equilibrium is δ = 2 π (cid:18) − κ (1 − κ ) ( κ − κ )( κ + κ − r κ κ [ κ κ ( κ + κ − ] (cid:19) . (97)Here the relations for the charged dilatonic black saturn and the relations for the neutral black saturn arethe same (see [37]). As in [37] figure 7 shows the parameter space of the charged dilatonic black saturn.In the coloured region the charged dilatonic black saturn is in thermodynamical equilibrium. The sign of δ can be seen in figure 7(a). On the black curve, the solution is balanced. On the left of the black curvethe solution has a conical deficit ( δ < δ > I >
I < β .12 a) Sign of δ . The solution contains a conical deficit(blue area) or conical excess (green area). Thebalanced solutions ( δ = 0) are on the blackcurve in between the blue and green area. (b) Sign of the mechanical moment of inertia. I >
I < δ = 0).Figure 7: Parameter space of the charged dilatonic black saturn. In the coloured region the solution is in thermodynamical equilib-rium. Here the conditions for balance and thermodynamical equilibrium are imposed, so that κ and β are theonly free parameters.The phase diagram of a rotating charged dilatonic black saturn is shown in figure 8, the left pictureshows how the a H - j -diagram changes for different values of β and the right picture shows a three-dimensional a H - j - q -diagram. As β and accordingly q grows, the curve in the phase diagram gets shiftedto lower a H and j . The charge comes with an additional force which helps to stabilize the black saturn.So when the charge is increased less angular momentum is needed to keep the black saturn balanced.As before scaled quantities are used (see equations (48) – (51)). The scaled charge depends only on β and has the upper limit q = 1 (and the lower limit q = − j - q -plots for different values of the parameter κ . In the extremal case κ = 1 thesolution is a naked singularity. As before the j - q -plots form a cusp at j = 0. The j - q -diagrams of thecharged dilatonic black saturn are similar to those of the charged dilatonic black ring, in fact they areexactly the same in the extremal cases ν = 1 and κ = 1, respectively. a H - q -diagrams for different values of the parameter κ are depicted in figure 9(b). a H reaches itshighest value for a given q if κ ≈ . j reaches its lowest value. If κ = 1, the horizon area vanishes. 9(c) shows a comparison of the a H - q -diagrams of the charged dilatonicblack ring and the charged dilatonic black saturn, the parameters ν and κ are chosen in such a way that a H is maximal for each q .The temperature of the charged dilatonic black saturn versus the parameter κ and the temperatureversus the horizon area and the charge can be seen in figure 10. The charge has only small influence onthe temperature. 13 a) a H versus j for different values of β
0 1 2 3 4 5 6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 q j a H q (b) a H versus j versus q Figure 8: Phase diagram of the charged dilatonic black saturn(a) j - q -diagram for different values of theparameter κ . (b) a H - q -diagram for different values of theparameter κ . (c) Comparison of the a H - q -diagrams.Case κ ≈ . ν = 0 . q , a H has its maximal value (and j its minimal value).Figure 9: j - q -diagrams and a H - q -diagrams of the charged dilatonic black saturn and a comparison to the charged dilatonic blackring (a) Temperature versus κ for different values of β .
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 q τ a H q (b) Temperature versus a H versus q Figure 10: Temperature-diagrams of charged the dilatonic black saturn .3 Thermodynamical stability To determine the thermodynamical stability of the charged dilatonic black saturn, the specific heat atconstant angular momentum C J and the isothermal moment of inertia ǫ are calulated as discussed insection 3.3. Due to their enormous size, the exact formulas are not displayed here.The results are similar to those of the charged dilatonic black ring. In the fixed Ψ el ensemble the signof C J and ǫ is the same for β = 0 and β = 0, whereas in the fixed Q ensemble, the charge parameter β influences the sign of C J and ǫ . As in the black ring case for all β the charged dilatonic black saturn isnot thermodynamically stable, since C J and ǫ cannot be positive at the same time.Figure 11 and 12 show C J and ǫ for different values of β in the fixed Q ensemble (the plots for β = 0also correspond to the fixed Ψ el ensemble, since both ensembles are equal at β = 0). (a) β = 0 (b) β = 1 (c) β = 5Figure 11: Sign of the specific heat at constant angular momentum of the charged dilatonic black saturn in the fixed Q ensemble. C J is positive in the green area and negative in the blue area. The balanced solutions can be found along the dashedcurve.(a) β = 0 (b) β = 0 . β = 1Figure 12: Sign of the isothermal moment of inertia of the charged dilatonic black saturn in the fixed Q ensemble. ǫ is positive inthe green area and negative in the blue area. The balanced solutions can be found along the dashed curve. Conclusion
In this paper charged black rings and charged black saturns were constructed in five-dimensional Einstein-Maxwell-dilation theory with the Kaluza-Klein value of the dilaton coupling constant. The charged so-lutions have the same excess/deficit angle δ and the same mechanical moment of inertia as their neutralcounterparts. The physical quantities of the charged solutions were calculated and related to the quanti-ties of the corresponding neutral objects. It was shown, that the curve in the phase diagram of a chargeddilatonic black ring or black saturn gets shifted to lower a H and j as the charge parameter β is increased.Concerning the thermodynamical stability, the specific heat at constant angular momentum and theisothermal moment of inertia were considered. Both quantities have to be positive at the same time toensure thermodynamical stability, but neither the charged dilatonic black ring nor the charged dilatonicblack saturn are thermodynamically stable. Note that the neutral black ring and black saturn are alsonot thermodynamically stable [37].For future work it would be interesting to construct charged dilatonic black rings and black saturnswith arbitrary values of the dilaton coupling contant, including the pure Einstein-Maxwell case, where thecoupling constant vanishes. Subsequently, one would investigate their thermodynamic stability.Since currently no analytical methods are available to obtain such more general charged solutions inclosed form, one might resort to a combination of perturbative methods and numerical methods, to ob-tain a rather complete picture for such solutions. In the thin ring limit, where one has widely separatescales in the problem, the blackfold approach represents an excellent approximate method to study theproperties of such solutions in five and more dimensions [39, 40, 41, 42, 43]. In [42] it was shown that theblackfold construction correctly reproduces all physical properties of the rotating charged dilatonic blackring. Interestingly the blackfold construction of a charged black ring also agrees very well with the chargedblack saturn in the ultraspinning limit. In a way this was expected since as j is increased more mass iscarried by the ring part of the black saturn and less by the central black hole, so that the a H - j -curve ofthe black saturn asymptotes the black ring curve in the phase diagram. Figure 13 shows a comparison ofthe blackfold approximation of the charged black ring with the analytic solution of the charged black ringand black saturn. The different (scaled) terms of the Smarr formula are plotted against the scaled charge q = QM .In constrast, when the scales involved make the blackfold approach inappropriate, such as in the fatblack ring regime, numerical methods can yield important new insights into the phase diagram of suchhigher dimensional black objects [44, 45, 46]. I would like to thank Jutta Kunz, Niels Obers and Eugen Radu for helpful discussions, and also gratefullyacknowledge support by the DFG, in particular, within the DFG Research Training Group 1620 “Modelsof Gravity”. 16 a) Scaled electromagnetic term of the Smarr formula as a func-tion of the scaled charge. The red dots correspond to theblackfold approximation of a charged black ring. The blueline corresponds to the analytic solution of a charged blackring/saturn.(b) Scaled horizon area term of the Smarr formula as a functionof the scaled charge.The red dots correspond to the black-fold approximation of a charged black ring. The blue linescorrespond to the analytic solution of a charged black ring.The ultraspinning limit is reached for ν →
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