Thermodynamical properties of topological Born-Infeld-dilaton black holes ∗ Ahmad Sheykhi Department of Physics, Shahid Bahonar University, P.O. Box 76175-132, Kerman, Iran Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), Maragha, Iran Weexaminethe(n+1)-dimensional(n 3)actioninwhichgravityiscoupledtotheBorn- ≥ Infeld nonlinearelectrodynamicandadilatonfield. We constructanew(n+1)-dimensional 8 analytic solution of this theory in the presence of Liouville-type dilaton potentials. These 0 0 solutions which describe charged topological dilaton black holes with nonlinear electrody- 2 namics,haveunusualasymptotics. They areneither asymptoticallyflatnor(anti)-de Sitter. n a The event horizons of these black holes can be an (n 1)-dimensional positive, zero or neg- J − 7 ative constant curvature hypersurface. We also analyze thermodynamics and stability of 2 these solutions and disclose the effect of the dilaton and Born-Infeld fields on the thermal ] h stability in the canonical ensemble. t - p e h I. INTRODUCTION [ 1 The pioneering theory of the non-linear electromagnetic field was proposed by Born and Infeld v 2 in 1934 for the purpose of solving various problems of divergence appearing in the Maxwell theory 1 1 [1]. Although it became less popular with the introduction of QED, in recent years, the Born- 4 . 1 Infeld action has been occurring repeatedly with the development of superstring theory, where 0 8 the dynamics of D-branes is governed by the Born-Infeld action [2, 3]. It has been shown that 0 : charged black hole solutions in Einstein-Born-Infeld gravity are less singular in comparison with v i X the Reissner-Nordstr¨om solution. In other words, there is no Reissner-Nordstr¨om-type divergence r a term q2/r2 in the metric near the singularity while it exist only a Schwarzschild-type term m/r [4, 5]. Intheabsence of adilaton field, exact solutions of Einstein-Born-Infeld theory with/without cosmological constant have been constructed by many authors [6, 7, 8, 9, 10, 11, 12, 13]. In the scalar-tensor theories of gravity, black hole solutions coupled to a Born-Infeld nonlinear electro- dynamics have been studied in [14]. The Born-Infeld action coupled to a dilaton field, appears in the low energy limit of open superstring theory [2]. Although one can consistently truncate such models, thepresenceofthedilaton fieldcannot beignored ifoneconsider couplingof thegravity to other gauge fields, and therefore one remains with Einstein-Born-Infeld gravity in the presence of a dilaton field. Many attempts have been doneto construct solutions of Einstein-Born-Infeld-dilaton ∗ [email protected] 2 (EBId) gravity [15, 16, 17, 18, 19, 20]. The appearance of the dilaton field changes the asymptotic behavior of the solutions to be neither asymptotically flat nor (anti)-de Sitter [(A)dS]. The mo- tivation for studying non-asymptotically flat nor (A)dS solutions of Einstein gravity comes from the fact that, these kind of solutions can shed some light on the possible extensions of AdS/CFT correspondence. Indeed, it has been speculated that the linear dilaton spacetimes, which arise as near-horizon limits of dilatonic black holes, might exhibit holography [21]. Black hole spacetimes which are neither asymptotically flat nor (A)dS have been explored widely in the literature (see e.g. [22, 23, 24, 25, 26, 27, 28, 29, 30]). On the other hand, it is a general belief that in four dimensions the topology of the event horizon of an asymptotically flat stationary black hole is uniquely determined to be the two- sphere S2 [31, 32]. Hawking’s theorem requires the integrated Ricci scalar curvature with respect to the induced metric on the event horizon to be positive [31]. This condition applied to two- dimensional manifolds determines uniquely the topology. The “topological censorship theorem” of Friedmann, Schleich and Witt is another indication of the impossibility of non spherical horizons [33, 34]. However, when the asymptotic flatness and the four dimensional spacetime are given up, there are no fundamental reasons to forbid the existence of static or stationary black holes with nontrivial topologies. It was confirmed that black holes in higher dimensions bring rich physics in comparison with the four dimensions. For instance, for five-dimensional asymptotically flat stationary black holes, in addition to the known S3 topology of event horizons, stationary black hole solutions with event horizons of S2 S1 topology (black rings) have been constructed [36]. It × has been shown that for asymptotically AdS spacetime, in the four-dimensional Einstein-Maxwell theory, there exist black hole solutions whose event horizons may have zero or negative constant curvature and their topologies are no longer the two-sphere S2. The properties of these black holes are quite different from those of black holes with usual spherical topology horizon, due to the different topological structures of the event horizons. Besides, the black hole thermodynamics is drastically affected by the topology of the event horizon. It was argued that the Hawking-Page phase transition [37] for the Schwarzschild-AdS black holes does not occur for locally AdS black holes whose horizons have vanishing or negative constant curvature, and they are thermally stable [38]. The studies on the topological black holes have been carried out extensively in many aspects (see e.g. [39, 40, 41, 42, 43, 44, 45, 46, 47]). In this paper, we would like to explore thermodynamical properties of the topological Born- Infeld-dilaton black holes in higher dimensional spacetimes in the presence of Liouville-type po- tentials for the dilaton field. The motivation for studying higher dimensional solutions of Einstein 3 gravity originates from superstringtheory, which is a promising candidate for the unified theory of everything. As the superstring theory can be consistently formulated only in 10-dimensions, the existence of extra dimensions should be regarded as the prediction of the theory. Although for a while it was thought that the extra spatial dimensions would be of the order of the Planck scale, making a geometric description unreliable, but it has recently been realized that there is a way to make the extra dimensions relatively large and still be unobservable. This is if we live on a three dimensional surface (brane) in a higher dimensional spacetime (bulk). In such a scenario, all gravitational objects such as black holes are higher dimensional. Indeed, the large extra dimension scenarios open up new exciting possibilities to relate the properties of higher dimensional black holes to the observable world by direct probing of TeV-size mini-black holes at future high energy colliders [48]. Besides, it was argued that through Hawking radiation of higher dimensional black holes, it is possible to detect these extra dimensions [49]. In the light of all mentioned above, it becomes obvious that further study of black hole solutions in higher dimensional gravity is of great importance. This paperis organized as follows: In section II, we construct a new class of (n+1)-dimensional topological black hole solutions in EBId theory with two liouville type potentials and general dilaton coupling constant, and investigate their properties. In section III, we obtain the conserved and thermodynamic quantities of the (n+1)-dimensional topological black holes and verify that these quantities satisfy the first law of black hole thermodynamics. In section IV, we perform a stability analysis and show that the dilaton creates an unstable phase for the solutions. The last section is devoted to summary and conclusions. II. TOPOLOGICAL DILATON BLACK HOLES IN BORN-INFELD THEORY We examine the (n+1)-dimensional (n 3) action in which gravity is coupled to dilaton and ≥ Born-Infeld fields 1 4 S = dn+1x√ g ( Φ)2 V(Φ)+L(F,Φ) , (1) 16π − R − n 1 ∇ − Z (cid:18) − (cid:19) where is the Ricci scalar curvature, Φ is the dilaton field and V(Φ) is a potential for Φ. The R Born-Infeld L(F,Φ) part of the action is given by e−8αΦ/(n−1)F2 L(F,Φ)= 4β2e4αΦ/(n−1) 1 1+ . (2)  −s 2β2    4 Here, α is a constant determining the strength of coupling of the scalar and electromagnetic fields, F2 = F Fµν, where F = ∂ A ∂ A is the electromagnetic field tensor, and A is the µν µν µ ν ν µ µ − electromagnetic vector potential. β is the Born-Infeld parameter with the dimension of mass. In the limit β , L(F,Φ) reduces to the standard Maxwell field coupled to a dilaton field → ∞ L(F,Φ) = e−4αΦ/(n−1)F2. (3) − On the other hand, L(F,Φ) 0 as β 0. It is convenient to set → → L(F,Φ) = 4β2e4αΦ/(n−1) (Y), (4) L where (Y) = 1 √1+Y, (5) L − e−8αΦ/(n−1)F2 Y = . (6) 2β2 Theequationsof motioncan beobtainedbyvaryingtheaction (1)withrespecttothegravitational field g , the dilaton field Φ and the gauge field A which yields the following field equations µν µ 4 1 = ∂ Φ∂ Φ+ g V(Φ) 4e−4αΦ/(n−1)∂ (Y)F F η Rµν n 1 µ ν 4 µν − YL µη ν − (cid:18) (cid:19) 4β2 + e4αΦ/(n−1)[2Y∂ (Y) (Y)]g , (7) Y µν n 1 L −L − n 1∂V 2Φ = − +2αβ2e4αΦ/(n−1)[2 Y∂ (Y) (Y)], (8) Y ∇ 8 ∂Φ L −L ∂ √ ge−4αΦ/(n−1)∂ (Y)Fµν = 0. (9) µ Y − L (cid:16) (cid:17) In particular, in the case of the linear electrodynamics with (Y) = 1Y, the system of equations L −2 (7)-(9) reduce to the well-known equations of Einstein-Maxwell-dilaton gravity [22]. We would like to find topological solutions of the above field equations. The most general such metric can be written in the form dr2 ds2 = f(r)dt2+ +r2R2(r)h dxidxj, (10) ij − f(r) where f(r) and R(r) are functions of r which should be determined, and h is a function of ij coordinates x which spanned an (n 1)-dimensional hypersurface with constant scalar curvature i − (n 1)(n 2)k. Here k is aconstant andcharacterizes thehypersurface. Without loss of generality, − − one can take k = 0,1, 1, such that the black hole horizon or cosmological horizon in (10) can − 5 be a zero (flat), positive (elliptic) or negative (hyperbolic) constant curvature hypersurface. The electromagnetic field equation (9) can be integrated immediately to give βqe4αΦ/(n−1) F = , (11) tr β2(rR)2n−2+q2 q where q is an integration constant related to the electric charge of the black hole. Defining the electric charge via Q = 1 exp[ 4αΦ/(n 1)] ∗FdΩ, we get 4π − − R qωn−1 Q = , (12) 4π where ωn−1 represents the volume of constant curvature hypersurface described by hijdxidxj . It is worthwhile to note that the electric field is finite at r = 0. This is expected in Born-Infeld theories. Meanwhile it is interesting to consider three limits of (11). First, for large β (where the Born-Infeld action reduces to the Maxwell case) we have F = qe4αΦ/(n−1) as presented in [22]. On tr (rR)n−1 the other hand, if β 0 we get F = 0. Finally, in the absence of the dilaton field (α = 0), it tr → reduces to the case of (n+1)-dimensional Einstein-Born-Infeld theory [11] βq F = . (13) tr β2r2n−2+q2 Our aim here is to construct exact, (n+1)-pdimensional topological solutions of the EBId gravity with an arbitrary dilaton coupling parameter α. The case in which we find topological solutions of physical interest is to take the dilaton potential of the form V(Φ) = 2Λ e2ζ0Φ+2Λe2ζΦ, (14) 0 whereΛ ,Λ,ζ andζ areconstants. Thiskindofpotentialwaspreviouslyinvestigated byanumber 0 0 of authors both in the context of Friedmann-Robertson-Walker (FRW) scalar field cosmologies [50] and dilaton black holes (see e.g. [18, 19, 22, 30, 47]). In order to solve the system of equations (7) and (8) for three unknown functions f(r), R(r) and Φ(r), we make the ansatz R(r)= e2αΦ/(n−1). (15) Using (15), the electromagnetic field (11) and the metric (10), one can show that equations (7) and (8) have solutions of the form k(n 2) α2+1 2b−2γ m 2 Λ 2β2 (α2 +1)2b2γ f(r) = − r2γ + − r2−2γ − (α2 1)(n+α2 2) − r(n−1)(1−γ)−1 (n 1)(α2 n) − (cid:0) (cid:1)− (cid:0) − (cid:1) − 4β2(α2 +1)b2γ r(n−1)(γ−1)+1 r(n+1)(1−γ)−2 1+ηdr, (16) − n 1 − Z p 6 (n 1)α b Φ(r)= − ln( ), (17) 2(1+α2) r where b is an arbitrary constant, γ = α2/(α2 +1), and q2b2γ(1−n) η = . (18) β2r2(n−1)(1−γ) Intheaboveexpression,mappearsasanintegrationconstantandisrelatedtotheADM(Arnowitt- Deser-Misner) mass of theblack hole. According to thedefinition of mass duetoAbbottand Deser [51], the mass of the solution (16) is [47] b(n−1)γ(n 1)ωn−1 M = − m. (19) 16π(α2 +1) In order to fully satisfy the system of equations, we must have 2 2α k(n 1)(n 2)α2 ζ = , ζ = , Λ = − − . (20) 0 α(n 1) n 1 0 2b2(α2 1) − − − Notice that here Λ is a free parameter which plays the role of the cosmological constant. For later convenience, we redefine it as Λ = n(n 1)/2l2, where l is a constant with dimension of length. − − The integral can be done in terms of hypergeometric function and can be written in a compact form as k(n 2) α2+1 2b−2γ m 2Λ α2+1 2b2γ f(r) = − r2γ + r2(1−γ) − (α2 1)(n+α2 2) − r(n−1)(1−γ)−1 (n 1)(α2 n) − (cid:0) (cid:1)− −(cid:0) (cid:1)− 4β2(α2+1)2b2γr2(1−γ) 1 α2 n α2+n 2 1 F , − , − , η . (21) − (n 1)(α2 n) × − 2 1 −2 2n 2 2n 2 − − − (cid:18) (cid:18)(cid:20) − (cid:21) (cid:20) − (cid:21) (cid:19)(cid:19) Onemay notethatasβ thesesolutionsreducetothe(n+1)-dimensionaltopological dilaton −→ ∞ black hole solutions given in [47] k(n 2) α2+1 2b−2γ m 2Λ(α2 +1)2b2γ f(r) = − r2γ + r2(1−γ) − (α2 1)(α2+n 2) − r(n−1)(1−γ)−1 (n 1)(α2 n) − (cid:0) (cid:1)− − − 2q2(α2 +1)2b−2(n−2)γ + r2(n−2)(γ−1). (22) (n 1)(α2 +n 2) − − In the absence of a nontrivial dilaton (α = γ = 0), the solution (21) reduces to m r2 4β2r2 1 n n 2 q2 f(r) = k + + 1 F , , − , (23) − rn−2 l2 n(n 1) × − 2 1 −2 2 2n 2n 2 −β2r2n−2 − (cid:18) (cid:18)(cid:20) − (cid:21) (cid:20) − (cid:21) (cid:19)(cid:19) whichdescribesan(n+1)-dimensionalasymptotically(A)dStopologicalBorn-Infeldblackholewith a positive, zero or negative constant curvature hypersurface [12]. Using the fact that F (a,b,c,z) 2 1 has a convergent series expansion for z < 1, we can find the behavior of the metric for large r. | | 7 6 4 f(r) 2 0 0.5 1 1.5 2 2.5 3 r –2 –4 FIG. 1: The function f(r) versus r for α =0.7, m =2, β = 1, n =4 and q =1. k = 1 (bold line), k = 0 − (continuous line), and k =1 (dashed line). This is given by k(n 2) α2+1 2b−2γ m 2Λ α2+1 2b2γ f(r) = − r2γ + r2(1−γ) − (α2 1)(n+α2 2) − r(n−1)(1−γ)−1 (n 1)(α2 n) − (cid:0) (cid:1)− −(cid:0) (cid:1)− 2q2(α2 +1)2b−2(n−2)γ q4(α2+1)2b−2(2n−3)γ + . (24) (n 1)(α2 +n 2)r2(n−2)(1−γ) − 2β2(n 1)(α2 +3n 4)r2(2n−3)(1−γ) − − − − The last term in the right hand side of the above expression is the leading Born-Infeld correction to the topological black hole with dilaton field [47]. Note that for α = γ = 0, the above expression reduces to m r2 2q2 q4 f(r) = k + + , (25) − rn−2 l2 (n 1)(n 2)r2(n−2) − 2β2(n 1)(3n 4)r2(2n−3) − − − − which has the form of the (n+1)-dimensional topological black hole in (A)dS spacetime in the limit β (see e.g. [41, 42]). → ∞ Physical Properties of the Solutions In order to study the physical properties of the solutions, we first look for the curvature singu- larities. In the presence of a dilaton field, the Kretschmann scalar R Rµνλκ diverges at r = 0, µνλκ it is finite for r = 0 and goes to zero as r . Thus, there is an essential singularity located 6 → ∞ at r = 0. The spacetime is neither asymptotically flat nor (A)dS. It is notable to mention that in the k = 1 cases these solutions do not exist for the string case where α = 1. As one can ± see from Eq. (21), the solution is ill-defined for α = √n. The cases with α < √n and α > √n should be considered separately. In the first case where α < √n, there exist a cosmological horizon for Λ > 0, while there is no cosmological horizons if Λ < 0. Indeed, where α < √n and Λ < 0, 8 4 3 f(r)2 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 r –1 –2 FIG. 2: The function f(r) versus r for m=2, β =1, q =1, n=4 and k =0. α=0.6 (bold line), α=0.75 (continuous line), and α=0.85 (dashed line). 4 3 f(r)2 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 r –1 –2 FIG. 3: The function f(r) versus r for m = 2, α = 0.6 , q = 1, n = 4 and k = 0. β = 1 (bold line), β = 2 (continuous line), and β =15 (dashed line). 10 8 6 m 4 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 rh FIG. 4: The function m(r ) versus r for β = 1, q = 1, n = 4 and k = 0. α = 0 (bold line), α = 0.8 h h (continuous line), and α=1.2 (dashed line). 9 8 6 m 4 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 rh FIG. 5: The function m(r ) versus r for β = 1, α = 0.8, n = 4 and k = 0. q = 0 (bold line), q = 1 h h (continuous line), and q =1.5 (dashed line). the spacetimes associated with the solution (21) exhibit a variety of possible casual structures de- pending on the values of the metric parameters (see Figs. 1-3). For simplicity in these figures, we kept fixed l = b = 1. These figures show that our solutions can represent topological black hole, with two event horizons, an extreme topological black hole or a naked singularity provided the parameters of the solutions are chosen suitably. In the second case where α > √n, the spacetime has a cosmological horizon. One can obtain the casual structure by finding the roots of f(r) = 0. Unfortunately, because of the nature of the exponent in (21), it is not possible to find analytically the location of the horizons. To have further understanding on the nature of the horizons, as an example for k = 0, we plot in Figs. 4 and 5 the mass parameter m as a function of the horizon radius for different values of dilaton coupling constant α and charge parameter q. Again, we have fixed l = b = 1, for simplicity. It is easy to show that the mass parameter of the black hole can be expressed in terms of the horizon radius r as h m(r ) = k(n−2) α2 +1 2b−2γr n−2+γ(3−n)+ 2Λ α2+1 2b2γrn(1−γ)−γ h − (α2 1)(n+α2 2) h (n 1)(α2 n) h − (cid:0) (cid:1)− −(cid:0) (cid:1)− 4β2(α2+1)2b2γ n(1−γ)−γ 1 α2 n α2+n 2 r 1 F , − , − , η .(26) −(n 1)(α2 n) h × −2 1 −2 2n 2 2n 2 − − − (cid:18) (cid:18)(cid:20) − (cid:21) (cid:20) − (cid:21) (cid:19)(cid:19) These figures show that for a given value of α, the number of horizons depend on the choice of the value of the mass parameter m. We see that, up to a certain value of the mass parameter m, there are two horizons, and as we decrease the m further, the two horizons meet. In this case we get an extremal black hole with mass m (see the next section). Figure 4 shows that with ext increasing α, the m also increases. It is worth noting that in the limit r 0 we have a nonzero ext h → value for the mass parameter m. This is in contrast to the Schwarzschild black holes in which the mass parameter goes to zero as r 0. As we have shown in figure 5, this is due to the effect of h → 10 the charge parameter q and the nature of the Born-Infeld field, and in the case q = 0, the mass parameter m goes to zero as r 0. In summary, the metric of Eqs. (10) and (21) can represent h → a charged topological dilaton black hole with inner and outer event horizons located at r− and r , provided m > m , an extreme topological black hole in the case of m = m , and a naked + ext ext singularity if m < m . ext III. THERMODYNAMICS OF TOPOLOGICAL DILATON BLACK HOLE Wenowwouldliketostudythethermodynamicalpropertiesofthesolutionswehavejustfound. Thetemperature of theblack hole can beobtained by continuing the metric to its Euclidean sector via t = iτ and requiring the absence of conical singularity at the horizon. It is a matter of − calculation to show that ′ κ f (r ) + T = = , (27) + 2π 4π where κ is the surface gravity. The temperature is then T = (α2+1)b2γr+1−2γ k(n−2)(n−1)b−4γr4γ−2+Λ 2β2(1 1+η ) + − 2π(n 1) 2(α2 1) + − − + − (cid:18) − (cid:19) = k(n−2)(α2 +1)b−2γr2γ−1+ (n−α2)mr (n−1)(γ−1) q2(α2p+1)b2(2−n)γr2(2−n)(1−γ)−1 − 2π(α2 +n 2) + 4π(α2 +1) + − π(α2+n 2) + − − 1 n+α2 2 3n+α2 4 F , − , − , η , (28) 2 1 + × 2 2n 2 2n 2 − (cid:18)(cid:20) − (cid:21) (cid:20) − (cid:21) (cid:19) where η = η(r = r ). There is also an extreme value for the mass parameter in which the + + temperature of the event horizon of black hole is zero. It is a matter of calculation to show that 2k(n 2)(α2 +1)2b−2γ (2−n)(γ−1)+γ 4q2(α2+1)2b2(2−n)γ (3−n)(1−γ)−1 m = − r + r ext (n α2)(α2+n 2) + (n α2)(α2 +n 2) + − − − − 1 n+α2 2 3n+α2 4 F , − , − , η . (29) 2 1 + × 2 2n 2 2n 2 − (cid:18)(cid:20) − (cid:21) (cid:20) − (cid:21) (cid:19) The entropy of the topological black hole typically satisfies the so called area law of the entropy whichstates that theentropy of theblack holeis aquarter oftheevent horizon area[52]. Thisnear universal law applies to almost all kinds of black holes, including dilaton black holes, in Einstein gravity [53]. It is a matter of calculation to show that the entropy of the topological black hole is S = b(n−1)γωn−1r+(n−1)(1−γ). (30) 4 The electric potential U, measured at infinity with respect to the horizon, is defined by U = Aµχµ|r→∞−Aµχµ|r=r+, (31)