Are ultra-spinning Kerr-Sen-AdS 4 black holes always super-entropic ?
aa r X i v : . [ g r- q c ] A ug Are ultraspinning Kerr-Sen-AdS black holes always superentropic? Di Wu , , ∗ Puxun Wu , † Hongwei Yu , ‡ and Shuang-Qing Wu Department of Physics and Synergetic Innovation Center for Quantum Effect and Applications,Hunan Normal University, Changsha, Hunan 410081, People’s Republic of China College of Physics and Space Science, China West Normal University,Nanchong, Sichuan 637002, People’s Republic of China (Dated: August 4, 2020)We study thermodynamics of the four-dimensional Kerr-Sen-AdS black hole and its ultraspinning counter-part and verify that both black holes fulfil the first law and Bekenstein-Smarr mass formulas of black holethermodynamics. Furthermore, we derive new Christodoulou-Ruffini-like squared-mass formulas for the usualand ultraspinning Kerr-Sen-AdS solutions. We show that this ultraspinning Kerr-Sen-AdS black hole doesnot always violate the reverse isoperimetric inequality (RII) since the value of the isoperimetric ratio can belarger/smaller than, or equal to unity, depending upon where the solution parameters lie in the parameters space.This property is obviously different from that of the Kerr-Newman-AdS superentropic black hole, which al-ways strictly violates the RII, although both of them have some similar properties in other aspects, such ashorizon geometry and conformal boundary. In addition, it is found that while there exists the same lower boundon mass ( m e > l / √
27 with l being the cosmological scale) both for the extremal ultraspinning Kerr-Sen-AdS black hole and for the extremal superentropic Kerr-Newman-AdS case, the former has a maximal horizon ra-dius r HP = l / √
3, which is the minimum of the latter. Therefore, these two different kinds of four-dimensionalultraspinning charged AdS black holes exhibit some significant physical differences.
I. INTRODUCTION
Black hole is one of the most remarkable and fascinat-ing objects in nature. It has an event horizon beyond whichany event inside has no effect. As for the horizon topol-ogy of black holes, Hawking proved that for four-dimensionalasymptotically flat, stationary black holes satisfying the dom-inant energy condition, their two-dimensional event horizoncross sections have a topology of S sphere [1]. To obtaindifferent horizon topologies, one needs to relax some assump-tions made in Hawking’s uniqueness theorem. Among variousdifferent possibilities, one is to consider higher dimensionalspacetimes. For example, in the five-dimensional asymptot-ically flat spacetimes, apart from the well-known black hole[2] which has the horizon topology of a round S sphere,the black ring [3] owns the S × S horizon topology, whilea rotating black lens solution [4] has the horizon topologyof a lens-space L ( n , ) . On the other hand, if the spacetimeis considered to be asymptotically nonflat, it has been foundthat especially for the four-dimensional anti-de Sitter (AdS)spacetime, the Einstein equation also admits topological so-lutions with their event horizons being Riemann surfaces ofany genus, namely, planar, toroidal and hyperbolic horizons[5–18], and rotating black string solutions in all dimensions[19]. Higher dimensional spacetimes can have even more richhorizon topologies; for instance, the event horizons of the d-dimensional asymptotically AdS black holes are also possibleto be the Einstein manifolds with positive, zero, or negativecurvature [20].Recently, a new class of AdS black holes [21–23], which is ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] considered as ultraspinning since one of their rotation angularvelocities is boosted to the speed of light, has received con-siderable interest and enthusiasm. This kind of black hole,occasionally called the “black spindle” spacetime [24] be-cause of its bottle-shaped horizon [15], has a noncompacthorizon topology since its spherical horizon has two punc-tures at the north and south poles, although it has a finite hori-zon area. The ultraspinning black hole violates the reverseisoperimetric inequality (“RII”) [25, 26], which implies thatthe Schwarzschild-AdS black hole has a maximum upper en-tropy. Due to the fact that the ultraspinning black hole canexceed the maximum entropy bound, it is also dubbed “super-entropic”. Remarkably, it has been shown [21] that the su-perentropic black hole solution can be alternatively obtainedby taking a simple ultraspinning limit from the usual rotat-ing AdS one. This solution generating procedure is very sim-ple: first recast the rotating AdS black hole in the frame ro-tating at infinity, then boost one rotation angular velocity tothe velocity of light, and finally compactify the correspondingazimuthal direction. Up to date, a lot of new superentropicblack hole solutions [27–31] from the known rotating AdSblack holes have been obtained so far. Very recently, it hasbeen found that the superentropic black hole can also be ob-tained by running a conical deficit from the usual rotating AdSblack hole [32]. In addition, other aspects of the superentropicblack holes, including thermodynamic properties [21, 27, 29–31, 33–35], horizon geometry [23, 27, 29], geodesic motion[24], Kerr/CFT correspondence [28–30], and so on, have beeninvestigated consequently.Although there has been much progress in the last fewyears in constructing superentropic black hole solutions andstudying their physical properties, ultraspinning black holesin gauged supergravities remain to be the virgin territory andthus need to be explored deeply, which motivates us to con-duct the present work. Since the most famous rotating chargedblack hole in the four-dimensional low-energy heterotic stringtheory is the Kerr-Sen solution [36], we first consider its gen-eralization by including a nonzero negative cosmological con-stant, namely the Kerr-Sen-AdS black hole, and then get itsultraspinning counterpart. Along the way, we also addresstheir thermodynamical properties and show that the obtainedthermodynamical quantities perfectly obey both the extendedfirst law and the Bekenstein-Smarr mass formulas for bothblack holes.The remaining part of this paper is organized as follows.In Sec. II, we make a recapitulation about the Kerr-Senblack hole, and then turn to the Kerr-Sen-AdS black holesolution in the four-dimensional gauged Einstein-Maxwell-dilaton-axion (EMDA) theory and investigate its thermody-namics. In Sec. III, after the ultraspinning Kerr-Sen-AdS black hole solution is constructed, its horizon topology andconformal boundary, thermodynamical properties, bounds onthe mass and horizon radius of extremal ultraspinning chargedAdS solutions, and the RII are subsequently discussed. In do-ing so, we establish novel Christodoulou-Ruffini-like squared-mass formulas for the Kerr-Sen-AdS black hole and its ul-traspinning counterpart. Differentiating this formula with re-spect to its individual thermodynamical variable gives the ex-pected thermodynamical quantities which satisfy both the firstlaw and the Bekenstein-Smarr mass formula without applyingthe chirality condition ( J = Ml ). Then, we discuss the re-duced form of the mass formulas after imposing the chiralitycondition. We demonstrate that the RII does not always holdtrue for this ultraspinning Kerr-Sen-AdS black hole, sincethe value of the isoperimetric ratio can be larger/smaller than,or equal to unity, depending upon the range of the solutionparameters. This marks a striking difference from the Kerr-Newman-AdS superentropic black hole. Finally, we end upwith our summaries in Sec. IV. II. KERR-SEN BLACK HOLE AND ITS ADS VERSIONA. A brief review of Kerr-Sen black hole
By using a solution generating technique with the Kerrblack hole as the seed solution, Sen [36] obtained a new so-lution of the four-dimensional rotating charged black hole,which is named as the Kerr-Sen black hole. It is an exactsolution to the four-dimensional low-energy heterotic stringtheory, also known as the EMDA theory, whose Lagrangianhas two different but completely equivalent forms, L = √− g h R − ( ∂φ ) − e − φ F − e − φ H i (1) = √− g h R − ( ∂φ ) − e φ ( ∂χ ) − e − φ F i + χ ε µνρλ F µν F ρλ , (2)where R is the Ricci scalar, φ is the dilaton scalar field, F µν is the Faraday-Maxwell electromagnetic tensor and F = F µν F µν , χ is the axion pseudoscalar field dual to thethree-form antisymmetric tensor: H = − e φ ⋆ d χ , and H = H µνρ H µνρ , and ε µνρλ is the four-dimensional Levi-Civita an-tisymmetric tensor density.The Kerr-Sen black hole solution can be expressed in theBoyer-Lindquist coordinates as [37, 38] ds = − ∆Σ (cid:0) dt − a sin θ d ϕ (cid:1) + Σ∆ dr + Σ d θ + sin θΣ (cid:2) adt − ( r + br + a ) d ϕ (cid:3) , A = qr Σ (cid:0) dt − a sin θ d ϕ (cid:1) , φ = ln (cid:16) r + a cos θΣ (cid:17) , χ = ba cos θ r + a cos θ , (3)where ∆ = r + ( b − m ) r + a , Σ = r + br + a cos θ , in which b = q / m is the dilatonic scalar charge, the parame-ters m and q are the mass and electric charge of the black hole,respectively, and its angular momentum is J = ma .Although the metric and gauge field of the Kerr-Sen blackhole have almost the same forms as those of the Kerr-Newmanblack hole (therefore they own many very similar physicalproperties, such as geometric feature [37], quantum thermalproperty and thermodynamical four laws [38], instability ofthe bound state of the charged mass scalar field and CFT holographic duality of the scattering process, etc.), there aresome significant differences between them. For example, theKerr-Sen black hole is a nonvacuum, nonalgebraically specialsolution in the four-dimensional low-energy heterotic stringtheory. In addition to the metric and an Abelian vector field,the Kerr-Sen solution contains another two nongravitationalfields: an antisymmetric third-order tensor field (or a dual ax-ion pseudoscalar field), and a dilaton scalar field, which areabsent from the Kerr-Newman solution. As for the Petrov-Pirani classification, the Kerr-Newman black hole, which isan exact electric vacuum solution to the Einstein-Maxwell the-ory, belongs to the family of type-D, while the Kerr-Sen so-lution is of type-I [39]. While the electrostatic potentials ofthe stringy left- and right-movers of the Kerr-Sen black holeare identical, they are unequal in the Kerr-Newman case [40].Of course, there are still many other salient different aspectsbetween them, for instance the capture region of scatteredphotons, the emission probability of black hole evaporationprocess, the magnetic induction ratio, and so on (see Refs.[41, 42] and references therein). B. Kerr-Sen-AdS black hole solution We now turn to include a nonzero negative cosmologicalconstant into the Kerr-Sen solution and present a simple formof the Kerr-Sen-AdS black hole, which is an exact solutionto the gauged EMDA theory, whose Lagrangian has the fol-lowing form¯ L = √− g n R − ( ∂ ¯ φ ) − e φ ( ∂ ¯ χ ) − e − ¯ φ F + l (cid:2) + e − ¯ φ + e ¯ φ ( + ¯ χ ) (cid:3)o + ¯ χ ε µνρλ F µν F ρλ , (4)with l being the cosmological scale. Distinct from the un-gauged case, now the above Lagrangian receives a potentialterm contributed from the dilation and axion fields, so it is im-possible to reexpress it in the dualized version in terms of thethree-form field that appeared in the ungauged Lagrangian.Written in terms of the Boyer-Lindquist coordinates andadapted to the frame rotating at infinity, the Kerr-Sen-AdS black hole solution can be given by the following exquisiteforms: d ¯ s = − ∆ r Σ (cid:16) dt − a sin θΞ d ϕ (cid:17) + Σ∆ r dr + Σ∆ θ d θ + ∆ θ sin θΣ (cid:16) adt − r + br + a Ξ d ϕ (cid:17) , ¯ A = qr Σ (cid:16) dt − a sin θΞ d ϕ (cid:17) , ¯ φ = ln (cid:16) r + a cos θΣ (cid:17) , ¯ χ = ba cos θ r + a cos θ , (5)where Σ = r + br + a cos θ as before, and now we have ∆ r = (cid:16) + r + brl (cid:17) ( r + br + a ) − mr , ∆ θ = − a l cos θ , Ξ = − a l . Obviously, the above solution (5) consistently reduces to theKerr-Sen black hole solution (3) when the AdS radius l tendsto infinity.It should be pointed out that more general solutions havebeen already constructed [43, 44] in the special case of thepairwise equal charge parameters of the four-dimensionalgauged STU supergravity theory. As the gauged EMDA the-ory is a more special case of that theory, therefore the abovesolution can be included as a special case obtained in [43, 44];however, here we present it in a slightly different and moresuitable form. C. Thermodynamics
Now we are in a position to investigate thermodynamics ofthe Kerr-Sen-AdS black hole. In the framework of the ex-tended phase space [45, 46] (see Ref. [47] for a fairly com-prehensive review), thermodynamic quantities associated withthe above solution (5) can be computed through the standardmethod and have the following expressions:¯ M = m Ξ , ¯ J = ma Ξ , ¯ Q = q Ξ , ¯ T = ( r + + b )( r + + br + + l + a ) − ml π ( r + + br + + a ) l , ¯ S = π ( r + + br + + a ) Ξ , ¯ Ω = a Ξ r + + br + + a , ¯ Φ = qr + r + + br + + a . (6) It is easy to very that these thermodynamic quantities (6)satisfy the Bekenstein-Smarr mass formulas¯ M = T ¯ S + Ω ¯ J + ¯ Φ ¯ Q − V ¯ P . (7)Here ¯ V is the thermodynamic volume¯ V = π Ξ ( r + + b )( r + + br + + a ) , (8)which is conjugate to the pressure ¯ P = / ( π l ) . Unfortu-nately, the first law, however, boils down to a differential iden-tity only d ¯ M = ¯ T d ¯ S + ¯ Ω d ¯ J + ¯ Φ d ¯ Q + ¯ V d ¯ P + ¯ Jd Ξ / ( a ) . (9)The reason for this is simply because we have just adopted therotating frame at infinity, not the rest frame at infinity.The transformation of the above Kerr-Sen-AdS solutioninto the frame rest at infinity can be easily done by taking asimple coordinate transformation ϕ → ϕ − at / l . After a te-dious calculation of the thermodynamic quantities in this restframe, it is not difficult to find that only the mass, the angu-lar velocity and the thermodynamic volume are different fromthose given in Eq. (6) and can be written as follows: e M = ¯ M + al ¯ J , e Ω = ¯ Ω + al , e V = ¯ V + π a ¯ J . (10)In this situation, now thermodynamic quantities can indeedsatisfy both the standard forms of the first law and theBekenstein-Smarr mass formula simultaneously, d e M = ¯ T d ¯ S + e Ω d ¯ J + ¯ Φ d ¯ Q + e V d ¯ P , e M = T ¯ S + e Ω ¯ J + ¯ Φ ¯ Q − e V ¯ P . (11)It is not difficult to check that the above differential andintegral mass formulas can be deduced from the followingChristodoulou-Ruffini-like squared-mass formulas: e M = (cid:16) + P ¯ S (cid:17)(cid:20)(cid:16) + P ¯ S (cid:17) ¯ S π + π ¯ J S + ¯ Q (cid:21) . (12) III. ULTRASPINNING KERR-SEN-ADS BLACK HOLEA. The ultraspinning solution
To construct the ultraspinning version from the above Kerr-Sen-AdS black hole solution (5), we just need to performthree steps: (i) redefine the angle coordinate ϕ by multiply-ing it with a factor Ξ ; (ii) take the a → l limit; and (iii) thencompactify the ϕ direction with a period of the dimensionlessparameter µ . Having finished these steps, we then obtain theultraspinning Kerr-Sen-AdS black hole solution, d ˆ s = − ˆ ∆ r ˆ Σ (cid:0) dt − l sin θ d ϕ (cid:1) + ˆ Σ ˆ ∆ r dr + ˆ Σ sin θ d θ + sin θ ˆ Σ (cid:2) ldt − ( r + br + l ) d ϕ (cid:3) , ˆ A = qr ˆ Σ (cid:0) dt − l sin θ d ϕ (cid:1) , ˆ φ = ln (cid:16) r + l cos θ ˆ Σ (cid:17) , ˆ χ = bl cos θ r + l cos θ , (13)whereˆ ∆ r = ( r + br + l ) l − − mr , ˆ Σ = r + br + l cos θ . B. Horizon geometry and conformal boundary
In this subsection, we first focus on other basic proper-ties, such as the horizon geometry and conformal boundary ofthe ultraspinning Kerr-Sen-AdS black hole. To ensure thatthe geometry is free of any closed timelike curve (CTC), oneneeds to check whether the inequality g ϕϕ > g ϕϕ = mrl ˆ Σ sin θ , we find that g ϕϕ is al-ways positive (due to m > r > Σ >
0) in the entirespacetime and thus the spacetime is free of CTC.To investigate the geometry of the event horizon, we con-sider the constant ( t , r ) surface on which the induced metricreads d ˆ s h = ˆ Σ + sin θ d θ + mr + l ˆ Σ + sin θ d ϕ , (14)where ˆ Σ + = ˆ Σ | r + . This metric appears to be singular at θ = θ = π . To examine whether the metric is ill-defined at θ = θ = π , one can analyze it in two limits: θ → θ → π . As an example, let us consider the small anglecase θ ∼
0, (similarly for the θ ∼ π case). By introducing anew variable k = l ( − cos θ ) for a small angle θ , and notingthat sin θ ≃ k / l , the two-dimensional cross section (14) forsmall k becomes d ˆ s h = (cid:0) r + + br + + l (cid:1)(cid:16) dk k + k l d ϕ (cid:17) , (15)which is clearly a metric of constant, negative curvature on aquotient of the hyperbolic space H . This result is very simi-lar to that of the Kerr-Newman-AdS superentropic black hole[27]. Due to the symmetry, the θ = π limit gives rises to thesame result. Apparently, the space is free from pathologiesnear the north and south poles. Topologically, the event hori-zon is a sphere with two punctures, and occasionally is calledas the black spindle. This implies that the above ultraspinningKerr-Sen-AdS black hole enjoys a finite area but noncompacthorizon.Next, we want to investigate the conformal boundary ofthe ultraspinning Kerr-Sen-AdS black hole. Multiplying the metric (13) with the conformal factor l / r and taking the r → ∞ limit, we find that the boundary metric has the form ds bdry = − dt + l sin θ dtd ϕ + l d θ / sin θ , (16)which is the same one as that of the superentropic Kerr-Newman-AdS black hole [27]. It is easy to see that the co-ordinate ϕ is null on the conformal boundary. In the small k = l ( − cos θ ) limit, we can reexpress the conformal bound-ary metric (16) as ds bdry = − dt + kdtd ϕ + dk / ( k ) , (17)which can be interpreted as an AdS written as a Hopf-likefibration over H . It means again that the metric has nothingpathological near two poles θ = θ = π . C. Mass formulas
Now we shall investigate thermodynamics of the ultraspin-ning Kerr-Sen-AdS black hole. Its fundamental thermody-namic quantities can be obtained through the standard methodand are given below M = µ m π , J = µ ml π , Q = µ q π , T = r + + b π l − m π ( r + + br + + l ) = r + + br + − l π r + l , S = µ (cid:0) r + + br + + l (cid:1) , Ω = lr + + br + + l , Φ = qr + r + + br + + l . (18)Note that the angular momentum and the mass satisfy a chiral-ity condition: J = Ml . Usually, the mass and angular momen-tum are computed by the conformal completion method [48].However, the angular momentum can also be evaluated cor-rectly by the Komar integral, while the mass can be obtainedvia the Komar integral after the subtraction of a divergencearising from the zero-mass background. Also, it is worthyto point out that the angular velocity Ω is that of the eventhorizon because the ultraspinning black hole is rotating at thevelocity of light at infinity.Within the framework of the extended phase space, it can beverified that the above thermodynamical quantities completelysatisfy both the first law and the Bekenstein-Smarr mass for-mula: dM = T dS + Ω dJ + Φ dQ + VdP + Kd µ , (19) M = T S + Ω J + Φ Q − VP , (20)with the thermodynamic volume and a new chemical potentialas follows V = ( r + + b ) S = µ ( r + + b )( r + + br + + l ) , (21) K = − m ( r + + br + − l ) π ( r + + br + + l ) = l − r + ( r + + b ) π r + l , (22)which are conjugate to the pressure P = / ( π l ) and the di-mensionless parameter µ , respectively.As was done in a previous work [34], here we propose toestablish the following simple relations M = µΞ ¯ M π , J = µΞ ¯ J π , Q = µΞ ¯ Q π , Ω = ¯ ΩΞ , S = µΞ ¯ S π , V = µΞ ¯ V π , T = ¯ T , Φ = ¯ Φ , P = ¯ P , (23)and take the ultraspinning limit a → l . Then we can see thatthe above thermodynamic quantities given in Eq. (18) for theultraspinning Kerr-Sen-AdS black hole can also be obtaineddirectly from those of its corresponding usual black hole. Thisfurther confirms that our previous method advised in Ref. [34]is a very effective and convenient routine to simply obtain theexpected thermodynamic quantities of all ultraspinning blackholes from those of their usual counterparts by taking the ul-traspinning limit properly.In Ref. [34], we have derived a new Christodoulou-Ruffini-like squared-mass formula for the Kerr-Newman-AdS super-entropic black hole. Here we hope to seek a similar one forthe ultraspinning Kerr-Sen-AdS black hole. Since the eventhorizon equation ( ˆ ∆ r + =
0) can be rewritten as S / ( π l ) = µ Mr + , (24)then after using 3 / l = π P , we get r + = PS / ( µ M ) . Now,we can substitute it into the entropy S = µ ( r + + br + + l ) / J = Ml ) as well as b = π Q / ( µ M ) to arrive at an identity M = PS µ (cid:16) P S + π Q (cid:17) + µ J S , (25)which is the expected Christodoulou-Ruffini-like squared-mass formula for the ultraspinning Kerr-Sen-AdS black hole.It is interesting to note that using this squared-mass formula,it is very convenient to study black hole chemistry and pos-sible thermodynamical phase transition of this ultraspinningKerr-Sen-AdS black hole.Leaving aside the chirality condition ( J = Ml ), it is obvi-ous that the thermodynamical quantities S , J , Q , P and µ in Eq.(25) can be regarded formally as independent thermodynami-cal variables and constitute a whole set of extensive variablesfor the fundamental functional relation M = M ( S , J , Q , P , µ ) . As did in Refs. [34, 40, 49, 50], differentiating the abovesquared-mass formula (25) with respect to S , J , Q , P , and µ , It should be emphasized that not all of them are truly independent by virtueof the existence of the chirality condition J = Ml after the rotation param-eter a has been set to the AdS radius l . However, let us ignore this relationtemporarily and take a viewpoint that they look like completely indepen-dent of each other at this moment so that there are enough parameters tohold fixed when performing the following partial derivative manipulations.A detailed discussion about the impact of the chirality condition on themass formulas is presented in the next subsection. respectively, yields their corresponding conjugate quantitiesas expected. In doing so, one can arrive at the differential firstlaw (19) and the integral Bekenstein-Smarr relation (20), withthe conjugate thermodynamic potentials correctly given by thecommon Maxwell relations as follows.Differentiation of the squared-mass formula (25) with re-spect to the entropy S leads to the conjugate Hawking temper-ature: T = ∂ M ∂ S (cid:12)(cid:12)(cid:12)(cid:12) ( J , Q , P , µ ) = − M S + P µ M (cid:16) P S + π Q (cid:17) = r + + br + − l π r + l , (26)and the corrected angular velocity and the electrostatic poten-tial, which are conjugate to J and Q , respectively, are givenby Ω = ∂ M ∂ J (cid:12)(cid:12)(cid:12)(cid:12) ( S , Q , P , µ ) = µ J SM = lr + + br + + l , (27) Φ = ∂ M ∂ Q (cid:12)(cid:12)(cid:12)(cid:12) ( S , J , P , µ ) = π PQ µ M S = qr + r + + br + + l . (28)Similarly for the pressure P and the dimensionless quantity µ ,one can get the thermodynamical volume and a new chemicalpotential V = ∂ M ∂ P (cid:12)(cid:12)(cid:12)(cid:12) ( S , J , Q , µ ) = S µ M (cid:16) P S + π Q (cid:17) = µ ( r + + b )( r + + br + + l ) , (29) K = ∂ M ∂µ (cid:12)(cid:12)(cid:12)(cid:12) ( S , J , Q , P ) = M µ − PS µ M (cid:16) P S + π Q (cid:17) = l − r + ( r + + b ) π r + l . (30)All the above conjugate quantities reproduce those expres-sions previously given in Eqs. (18), (21), and (22). Anyway,with all these conjugate variables derived from the squared-mass formula (25), the differential first law (19) is triviallysatisfied while the integral mass formula (20) is easily checkedto be completely obeyed too. D. Chirality condition and reduced mass formulas
Now, let us make a careful discuss about the impact of thechirality condition ( J = Ml ) on the thermodynamical relationsof the ultraspinning Kerr-Sen-AdS black hole. Due to theexistence of the chirality condition, three thermodynamicalquantities ( M , J , P ) are not completely independent, there ex-ists a constraint relation among them J = M / ( π P ) , (31)which means that the ultraspinning Kerr-Sen-AdS black holeis actually a degenerate thermodynamical system. After tak-ing into account the chirality condition physically, the first law(19) and the Bekenstein-Smarr relation (20) should be con-strained by the condition (31), and actually depict a degener-ate thermodynamical system.Considering J as a redundant variable (although it is a realmeasurable quantity) and eliminating J from the differentialand integral mass formulas in favor of l = / ( π P ) , the firstlaw (19) and the Bekenstein-Smarr relation (20) now reduceto the following nonstandard forms (so named for their ther-modynamic quantities cannot constitute the ordinary canoni-cal conjugate pairs due to the existence of a factor ( − Ω l ) infront of dM and M ): ( − Ω l ) dM = T dS + V ′ dP + Φ dQ + Kd µ , ( − Ω l ) M = ( T S − V ′ P ) + Φ Q , (32)where V ′ = V − J Ω P = V − π Ω M l . In the same way, the squared-mass formula (25) degeneratesto M (cid:16) − µ π PS (cid:17) = PS µ (cid:16) P S + π Q (cid:17) . (33)In doing so, one actually views the enthalpy M as the fun-damental functional relation M = M ( S , Q , P , µ ) . Similar tothe strategy adopted before, the above nonstandard differen-tial and integral mass formulas can be derived from Eq. (33)by exploiting the standard Maxwell rule. Alternately, one per-haps prefers to eliminating P instead of J via Eq. (31). Asthe resulted expressions are rather complicated, we will notpresent them here. E. Bounds on the mass and horizon radius of extremalultraspinning black holes
In the following, we would like to establish some new in-equalities on the mass and horizon radius of the extremalultraspinning black holes. We begin with the extremal su-perentropic Kerr-Newman-AdS black hole for which ˇ ∆ r =( r + l ) / l − mr + q . Without loss of generality, here andhereafter, we shall assume that both the mass parameter andthe AdS scale are positive.The location of the event horizon of the extremal superen-tropic Kerr-Newman-AdS black hole is determined by ˇ ∆ r e = ˇ ∆ ′ r e =
0, which gives m e = r e ( r e + l ) l , q e = ( r e + l )( r e − l ) l . (34)By virtue of positiveness of q e , it is evident that the followinginequalities hold r e > l √ , m e > l √ , (35)which means that the scale of Hawking-Page phase transi-tion: r HP = l / √ black hole, whose mass isbounded from the lower limit: 8 l / √
27. Now we turn to consider the extremal case of an ultraspin-ning Kerr-Sen-AdS black hole. Its horizon is determined byˆ ∆ r e = ˆ ∆ ′ r e =
0, which yields m e = ( r e + b e )( r e + b e r e + l ) l , b e = l − r e r e . (36)By the requirement: b e = q e / ( m e ) > r e >
0, itis clear that we must have a distinct inequality:0 r e l √ , (37)which means that the scale of Hawking-Page phase transi-tion r HP = l / √ black hole. Substitut-ing the inequality (37) into Eq. (36), one can find that theextremal mass still has the same lower bound: m e = ( l − r e ) l r e > l √ . (38)Therefore, although the extremal Kerr-Newman-AdS su-perentropic black hole and the extremal ultraspinning Kerr-Sen-AdS black hole share the same lower mass bound, theHawking-Page phase transition scale r HP = l / √ black holes. F. RII
Almost a decade ago, it is conjectured [25] that the AdSblack hole satisfies the following RII: R = h ( D − ) V A D − i / ( D − ) (cid:16) A D − A (cid:17) / ( D − ) > , (39)where A D − = π [( D − ) / ] / Γ [( D − ) / ] is the area of the unit ( D − ) sphere and A = S is the horizon area. Equality is at-tained for the Schwarzschild-AdS black hole, which impliesthat the Schwarzschild-AdS black hole has the maximum en-tropy. In other words, it indicates that for a given entropy, theSchwarzschild-AdS black hole occupies the least volume, andhence is most efficient in storing information.It is straightforward to check whether the ultraspinningKerr-Sen-AdS black hole satisfies this RII or not. It is read-ily known that the area of the unit two-dimensional sphere is A = µ , the thermodynamic volume is V = ( r + + b ) S / A = S = µ ( r + + br + + l ) . Conse-quently, the isoperimetric ratio now reads R = (cid:16) r + + b µ A (cid:17) (cid:16) µ A (cid:17) = (cid:16) r + + br + + b r + + br + + l (cid:17) . (40)Obviously, the value range of R is uncertain. If b < l (namely, q < ml ), then R <
1. In this case, the ultraspin-ning Kerr-Sen-AdS black hole violates the RII and is super-entropic. Otherwise, if b > l , one then has R >
1. In thissituation, the ultraspinning Kerr-Sen-AdS black hole obeysthe RII and is subentropic. Since the ratio of R depends uponthe values of the solution parameters ( q , m and l ), thus onecan see that the ultraspinning Kerr-Sen-AdS black hole isnot always superentropic. Only when the parameters satisfy q < ml does it violate the RII, while the Kerr-Newman-AdS superentropic black hole always violates the RII [21].As far as this point is concerned, the ultraspinning Kerr-Sen-AdS black hole and Kerr-Newman-AdS superentropic blackhole exhibit yet another markedly different property. This isone of the main results that we have obtained in this paper. IV. CONCLUSIONS
In this paper, we have studied some interesting proper-ties of the Kerr-Sen-AdS black hole and in particular, itsultraspinning cousin in the four-dimensional gauged EMDAtheory. After a brief review of the famous Kerr-Sen blackhole solution, we presented its exquisite generalization to in-clude a nonzero negative cosmological constant, namely theKerr-Sen-AdS black hole. Then its ultraspinning cousin wasconstructed via employing a simple a → l limit procedure.The expressions of these solutions, namely their metric, theAbelian gauge potential, the dilaton scalar, and the axionpseudoscalar fields are very convenient for investigating theirthermodynamical properties. With these solutions at hand,all thermodynamic quantities that can be computed throughthe standard method are verified to fulfil both the differentialand integral mass formulas. Moreover, new Christodoulou-Ruffini-like squared-mass formulas are displayed for thesefour-dimensional black holes, from which all expected con-jugate partners are derived via differentiating them with re-spect to their corresponding thermodynamic variables and areshown to consist of the ordinary canonical conjugate pairs thatappear in the standard forms of black hole thermodynamics.Furthermore, we adopted the method proposed in Ref. [34]to demonstrate that all thermodynamical quantities of the ul-traspinning Kerr-Sen-AdS black hole can be attained via ap-plying the same ultraspinning limit to those of their corre-sponding predecessors. After that, we have made a detaileddiscussion about the impact of the chirality condition on theactual thermodynamics of this ultraspinning black hole. Tosome extent, these aspects are very similar to those of theKerr-Newman-AdS superentropic black hole.What attracts us the most in this work is to peer whetherthere are some other properties peculiar to the ultraspinningKerr-Sen-AdS black hole. After investigating its horizon ge-ometry and conformal boundary, we arrived at the conclu-sion that both of them are still similar to those of the Kerr-Newman-AdS superentropic black hole.However, when turning to investigate the extremal ul-traspinning black holes and the RII, we indeed discoveredthat the ultraspinning Kerr-Sen-AdS and superentropic Kerr-Newman-AdS black holes exhibit some significant physicaldifferences.A summary of three novel consequences obtained in thispaper are listed in the following order: 1. New Christodoulou-Ruffini-like squared-mass formu-las are presented both for the usual Kerr-Sen-AdS black hole and for its ultraspinning counterpart. To thebest of our knowledge, they do not appear in the litera-ture before, and are useful to study black hole chemistryand possible thermodynamical phase transition of theseAdS black holes.2. Remarkably, it have been found that the ultraspinningKerr-Sen-AdS black hole is not always superentropic,since the RII is violated only in the space of the solutionparameters that satisfy the condition q < ml . Once q > ml , the ultraspinning Kerr-Sen-AdS black holebecomes sub-entropic. This property is in sharp con-trast with the Kerr-Newman-AdS superentropic blackhole which always violates the RII [21].3. We have established some new inequalities on the massand horizon radius of the extremal ultraspinning Kerr-Sen-AdS and extremal superentropic Kerr-Newman-AdS black holes. It is observed that while bothextremal black holes share the same mass minimumbound m e > l / √
27, the scale of Hawking-Page phasetransition r HP = l / √ black hole never exceeds the Hawking-Page scale r HP , while that of the the extremal superentropic Kerr-Newman-AdS black hole always exceeds the samescale r HP . This might be taken as a direct signatureto distinguish these two extremal ultraspinning chargedAdS black holes, which hints that it is possible to judgethem via the observation of their shadow sizes.There are two promising further topics to be pursued in thefuture. As mentioned above, one intriguing topic is to explorethe black hole shadows of two ultraspinning charged AdS so-lutions, and this might shed light on our knowledge of blackholes in gauged supergravity theories. And the other is to ex-tend the present work to the more general dyonic case. Wehope to report the related progress along these directions soon. ACKNOWLEDGMENTS
This work is supported by the National Natural ScienceFoundation of China under Grants No. 11775077, No.11690034, No. 11435006, No. 11675130, and No. 11275157,and by the Science and Technology Innovation Plan of Hunanprovince under Grant No. 2017XK2019.
Notes added .—After submission of this paper for publica-tion, we have reconsidered the work, in particular about thebound on the horizon radius of extremal ultraspinning Kerr-Sen-AdS black hole by making a shift on the radial coordi-nate r = ρ − b , so that the metric looks like more similar to thesuperentropic Kerr-Newman-AdS solution as now we havethe structure function ˘ ∆ ρ = ( ρ + l − b ) l − − m ρ + q .Repeating the computations as did in the Sec. III E, we obtainagain m e > l / √
27 but now ρ e > l / √
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