Single-charge rotating black holes in four-dimensional gauged supergravity
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Single-charge rotating black holes in four-dimensionalgauged supergravity
David D. K. Chow
George P. & Cynthia W. Mitchell Institute for Fundamental Physics & Astronomy,Texas A&M University, College Station, TX 77843-4242, USA [email protected]
Abstract
We consider four-dimensional U(1) gauged supergravity, and obtain asymptotically AdS ,non-extremal, charged, rotating black holes with one non-zero U(1) charge. The thermody-namic quantities are computed. We obtain a generalization that includes a NUT parameter.The general solution has a discrete symmetry involving inversion of the rotation parameter,and has a string frame metric that admits a rank-2 Killing–St¨ackel tensor. Introduction
Over the past 20 years, there has been a substantial effort to construct black hole solutionsof supergravity theories, with increasing complexity. Before the advent of the AdS/CFT cor-respondence, such efforts were largely directed at asymptotically flat black holes of ungaugedsupergravities, but have since turned to asymptotically AdS black holes of gauged supergrav-ities. A large variety of solutions are now known, in various dimensions, carrying variousnumbers of independent angular momenta and U(1) charges.In ungauged supergravities, one can start from an uncharged solution, and then introducecharges in an algorithmic fashion by solution generating techniques. The starting point forthese is typically the Myers–Perry solution [1], which generalizes the Kerr solution from D = 4spacetime dimensions to higher dimensions. This results in the 4-charge Cvetiˇc–Youm solutionin D = 4 [2, 3], the 3-charge Cvetiˇc–Youm solution in D = 5 [4], and the 2-charge Cvetiˇc–Youm solution in D ≥ D = 5[6], which has a 3-charge generalization [7]. These examples possess the maximum number ofindependent angular momenta and U(1) charges.In gauged supergravities, there is a potential in the Lagrangian that inhibits such solutiongenerating techniques. One must therefore resort to guesswork, aided by the other knownsolutions. For supersymmetric solutions, rotation is required, which makes the guessworkchallenging. In the simplest case of Einstein gravity with a cosmological constant, the Kerr–AdS solution in D = 4 has long been known [8, 9], and since generalized to higher dimensions[10, 11, 12]. We now have a large catalogue of asymptotically AdS charged and rotating blackhole solutions of gauged supergravities in D = 4 , , ,
7; see [13, 14] for a review of these.However, we still do not know black hole solutions of gauged supergravity in D = 4 , , D = 6 is such a programme complete [15]. Theknown solutions can appear rather complicated, but a hint that more general solutions shouldbe amenable to guesswork is that all known solutions have string frame metrics that possessrank-2 Killing–St¨ackel tensors [14]. This suggests some underlying structure to the geometriesthat is yet to be fully understood.In the hope of taming the contents of this bestiary of solutions, in this paper we turn toarguably the most basic, but not the simplest, example of a charged and rotating asymptoti-cally AdS black hole solution of gauged supergravity. We work in 4 dimensions, where thereis an N = 8 gauged supergravity theory with gauge group SO(8), whose Cartan subgroup isU(1) . The previously discovered solutions of the theory are: static AdS black holes with 4independent U(1) charges [16]; and rotating AdS black holes with the 4 U(1) charges pair-wise equal [3], which includes the Kerr–Newman–AdS solution [8, 9] of the Einstein–Maxwellsystem (the dyonic Kerr–Newman–AdS solution was given explicitly in [17]).We shall write down a rotating solution for which only 1 of the 4 U(1) charges is non-zero. We compute the thermodynamical quantities and find that there are no supersymmetricsolutions. Then, we generalize the solution to include a NUT parameter, and can performelectric/magnetic duality. Like the uncharged solution, there is a discrete symmetry thatincludes the rotation parameter being inverted through the AdS radius. We write downa rank-2 Killing St¨ackel tensor for the string frame metric, which implies separability of thestring frame Hamilton–Jacobi equation for geodesic motion and of the Klein–Gordon equation.2 AdS black hole solution gauged supergravity is a consistent truncation of the maximal N = 8,SO(8) gauged supergravity. It is N = 2 supergravity coupled to 3 abelian vector multiplets.The full bosonic Lagrangian for U(1) gauged supergravity was given in [18]. Without axions,which suffices for our purposes here, the truncated bosonic Lagrangian was given in [16].We shall further truncate and consider black hole solutions with a single U(1) charge. Thebosonic fields of this truncation are a graviton, a vector and a scalar, and the Lagrangian is L = R ⋆ − ⋆ d ϕ ∧ d ϕ − e ϕ ⋆ F (2) ∧ F (2) + 3 g (e ϕ + e − ϕ ) ⋆ , (2.1)where F (2) = d A (1) . We have used a non-canonical normalization for the scalar kinetic termto avoid awkward √ G ab = ( ∇ a ϕ ∇ b ϕ − ∇ c ϕ ∇ c ϕ g ab ) + e ϕ ( F ac F bc − F cd F cd g ab ) + g (e ϕ + e − ϕ ) g ab , ∇ a (e ϕ F ab ) = 0 , (cid:3) ϕ − e ϕ F ab F ab + g (e ϕ − e − ϕ ) = 0 . (2.2)A charged and rotating black hole solution isd s = 1 √ Hρ Ξ (cid:18) − ∆ θ (∆ θ ∆ r − V r a sin θ ) d t − mrc p a g s ∆ θ a sin θ t d φ +(∆ θ e V r a − ∆ r sin θ ) a sin θ d φ (cid:19) + √ H (cid:18) ρ ∆ r d r + ρ ∆ θ d θ (cid:19) ,A (1) = 2 mrsHρ (cid:18) c ∆ θ d t Ξ − a p a g s sin θ d φ Ξ (cid:19) ,ϕ = log H, (2.3)where∆ r = r + a − mr + g r ( r + 2 ms r + a ) , ∆ θ = 1 − a g cos θ, ρ = r + a cos θ,V r = (1 + g r )(1 + g r + 2 ms rg ) , e V r = (1 + r /a )(1 + r /a + 2 ms r/a ) ,H = 1 + 2 mrs ρ , s = sinh δ, c = cosh δ, Ξ = 1 − a g . (2.4)The Boyer–Lindquist-type coordinates used are asymptotically static. If m = 0, then thecoordinate change Ξˆ r sin ˆ θ = ( r + a ) sin θ, ˆ r cos ˆ θ = r cos θ, (2.5)gives d s = − (1 + g ˆ r ) d t + dˆ r g ˆ r + ˆ r dˆ θ + sin ˆ θ d φ . (2.6)This is simply anti-de Sitter spacetime, and we see that t and φ are canonically normalized.The solution has 4 parameters: a mass parameter m ; a rotation parameter a ; a chargeparameter δ ; and a gauge-coupling constant g . In the absence of rotation, with a = 0, thesolution reduces to a particular case of the 4-charge static solution [16], but with only 1 ofthe 4 charges non-zero. In the absence of charge, with δ = 0, the solution reduces to the4-dimensional Kerr–AdS metric [8, 9]. In the absence of gauging, with δ = 0, the solutionreduces to the 4-charge Cvetiˇc–Youm solution [2, 3], but with only 1 of the 4 charges non-zero.To find this solution, we have been helped by the structure of these limits. We have alsobeen helped by the structure of the black hole solution in 5-dimensional gauged supergravitycarrying a single non-zero rotation parameter and a single non-zero U(1) charge [19].3 Thermodynamics
The outer black hole horizon is located at the largest root of ∆ r ( r ), say at r = r + . Its angularvelocity Ω is constant over the horizon and is obtained from the Killing vector l = ∂∂t + Ω ∂∂φ (3.1)that becomes null on the horizon. The angular momentum J is given by the Komar integral J = 116 π Z S ∞ ⋆ d K, (3.2)where K is the 1-form obtained from the Killing vector ∂/∂φ . The electrostatic potential,which is also constant over the horizon, is Φ = l · A (1) | r = r + . The conserved electric charge is Q = 116 π Z S ∞ e ϕ ⋆ F (2) . (3.3)The horizon area A is obtained by integrating the square root of the determinant of theinduced metric on a time slice of the horizon. The surface gravity κ , again constant over thehorizon, is given by l b ∇ b l a = κl a evaluated on the horizon. As usual, we take the temperatureto be T = κ/ π and the entropy to be S = A/ T d S + Ω d J + Φ d Q is an exact differential, and so we may integrate thefirst law of black hole mechanics,d E = T d S + Ω d J + Φ d Q, (3.4)to obtain an expression for the thermodynamic mass E .There are several other definitions of mass for asymptotically AdS spacetimes in the liter-ature. The AMD (Ashtekar–Magnon–Das) mass is one such definition, for 4 dimensions [20]and higher [21]. One introduces a conformally rescaled metric g ab = Ω g ab , with Ω = 0 anddΩ = 0 on the conformal boundary. Its Weyl tensor is C abcd , and we define n a = ∂ a Ω. For anasymptotic Killing vector field K , which here is K = ∂/∂t , there is an associated conservedquantity. In 4 dimensions, the AMD mass is E = 18 πg Z Σ dΣ a Ω n c n d C acbd K b , (3.5)where dΣ a is the area element of the S section of the conformal boundary. The AMD masshas previously been used to compute the mass of various asymptotically AdS rotating blackholes [22, 23, 24]. For definiteness, we take Ω = 1 /gr for our solution. As r → ∞ , one findsthat the Weyl tensor component C trtr behaves as C trtr = m [2 + (1 + a g ) s ]2Ξ g r (2Ξ + a g sin θ ) + O (cid:18) r (cid:19) . (3.6)The conformal boundary has metricd s = − ∆ θ Ξ d t + 1 g ∆ θ d θ + sin θ Ξ g d φ . (3.7)Substituting these into (3.5), we can compute the AMD mass, and find that it agrees withthe thermodynamic mass. 4n summary, we find the thermodynamic quantities E = m [2 + (1 + a g ) s ]2Ξ ,S = π p ( r + a )( r + a + 2 ms r + )Ξ , T = r − a + g r (3 r + a + 4 ms r + )4 πr + p ( r + a )( r + a + 2 ms r + ) ,J = p a g s cma Ξ , Ω = p a g s a (1 + g r ) c ( r + a ) ,Q = msc , Φ = 2 mscr + r + a + 2 ms r + . (3.8)Supersymmetric AdS black holes are known [25, 26]. In this single-charge case, the BPScondition, up to choices of signs, is [26] E − gJ − Q = 0 . (3.9)This leads to the relation (2e δ + 1)(e δ − a g + (3e δ + 1) = 0 , (3.10)which has no solutions. Hence, there are no supersymmetric solutions with a single U(1)charge. A more general solution that includes a NUT parameter ℓ isd s = 1 √ H ( r + y )Ξ (cid:18) − ( V y R − V r Y ) d t − c p a g s ( mrY + ℓyR ) a t d φ +( e V r Y − e V y R ) a d φ (cid:19) + √ H (cid:18) r + y R d r + r + y Y d y (cid:19) ,A (1) = 2 mrsH ( r + y ) c (1 − g y ) d t Ξ − p a g s ( a − y ) a d φ Ξ ! + 2 ℓysH ( r + y ) c (1 + g r ) d t Ξ − p a g s ( r + a ) a d φ Ξ ! ,ϕ = log H, (4.1)where R = r + a − mr + g r ( r + 2 ms r + a ) , Y = a − y + 2 ℓy + g y ( y + 2 ℓs y − a ) ,V r = (1 + g r )(1 + g r + 2 ms rg ) , e V r = (1 + r /a )(1 + r /a + 2 ms r/a ) ,V y = (1 − g y )(1 − g y − ℓs yg ) , e V y = (1 − y /a )(1 − y /a − ℓs y/a ) ,H = 1 + 2( mr + ℓy ) s r + y , s = sinh δ, c = cosh δ, Ξ = 1 − a g . (4.2)Without any NUT parameter, making the coordinate change y = a cos θ and renaming R ( r )as ∆ r ( r ) recovers the AdS black hole (2.3). 5e have been guided towards this more general solution, from the solution without anyNUT parameter, by symmetry. In particular, if we let x = i r and replace m → − i m , thenthe solution is symmetric under the simultaneous interchange of ℓ and m and of x and y . Wealso note that the metric determinant has a very simple expression, with √− g = √ H ( r + y ) a Ξ . (4.3)In the absence of charge, with δ = 0, the solution reduces to the 4-dimensional Kerr–NUT–AdS metric [8, 9]. In the absence of gauging, with δ = 0, the solution reduces to the4-charge Cvetiˇc–Youm solution generalized to include a NUT parameter [3], but with only 1of the 4 charges non-zero; this 1-charge solution has also been rederived by a different method[27]. The field equations (2.2) admit the electric/magnetic duality symmetry F (2) → e ϕ ⋆ F (2) , ϕ → − ϕ. (5.1)We can perform this transformation on the general solution (4.1). Using the orientation ε tryφ = 1, the vector and scalar become A (1) = 2 mysr + y (cid:18)p a g s (1 + g r ) d t Ξ − c ( r + a ) a d φ Ξ (cid:19) − ℓrsr + y (cid:18)p a g s (1 − g y ) d t Ξ − c ( a − y ) a d φ Ξ (cid:19) ,ϕ = − log H. (5.2)As before, we recover solutions in [2, 3] for g = 0, in [16] for a = 0, and in [8, 9, 17] for δ = 0. The Kerr–NUT–AdS solutions [28] of Einstein gravity in arbitrary dimensions possess discreteinversion symmetries, under which a rotation parameter a i is inverted through the AdS radius1 /g . This type of symmetry had previously been noted in 5 dimensions [29]. A metric withover-rotation, | a i g | >
1, is mapped to a metric with under-rotation, | a i g | <
1, under such asymmetry.The inversion symmetry persists for the general solution with a single U(1) charge above,not only for the metric, but for all fields. Under the transformation a → ag , r → rag , y → yag , m → ma g , ℓ → ℓa g , φ → gt, gt → φ, s → ags. (5.3)the solution is invariant. The peculiar-looking factors of p a g s that appear in thesolution are interchanged with c = √ s under the inversion transformation.6 .3 Killing tensor A general feature of charged and rotating supergravity black hole solutions that generalizethe Kerr or Myers–Perry solution seems to be that their string frame metrics admit rank-2Killing–St¨ackel tensors [14]. These are symmetric tensors K ab that satisfy ∇ ( a K bc ) = 0.Consider here the string frame metric d e s , which is related to the original Einstein framemetric d s by d s = √ H d e s . The inverse string frame metric is, including the NUT param-eter, (cid:18) ∂∂ e s (cid:19) = − a r + y e V r R − e V y Y ! ∂ t − c p a g s ( r + y ) (cid:18) mrR + ℓyY (cid:19) a ∂ t ∂ φ + 1( r + y ) (cid:18) V y Y − V r R (cid:19) a ∂ φ + Rr + y ∂ r + Yr + y ∂ y . (5.4)The components ( r + y ) e g ab are additively separable as a function of r plus a function of y .For the string frame metric, a rank-2 Killing–St¨ackel tensor is given by e K = e K ab ∂ a ∂ b = a e V y Y ∂ t − c p a g s ℓyaY ∂ t ∂ φ + V y a Y ∂ φ + Y ∂ y − y (cid:18) ∂∂ e s (cid:19) , (5.5)and there is a separability structure. This induces a rank-2 conformal Killing–St¨ackel tensor,with components e K ab , for the Einstein frame metric. The Hamilton–Jacobi equation forgeodesic motion and the massive Klein–Gordon equation separate for the string frame metric,whereas only the Hamilton–Jacobi equation for null geodesics and the massless Klein–Gordonequation separate for the Einstein frame metric. We have presented an asymptotically AdS rotating black hole solution of 4-dimensional U(1) gauged supergravity that possesses 1 non-zero U(1) charge, and studied some of its proper-ties. It is arguably the most basic of the charged and rotating AdS black hole solutions ofgauged supergravity. Therefore, we hope that it will provide further insights into construct-ing more general solutions with the maximum number of independent angular momenta andU(1) charges in not only four dimensions, but also higher dimensions. Such gravitationalbackgrounds would be useful for studying the AdS/CFT correspondence. References [1] R.C. Myers and M.J. Perry, “Black holes in higher dimensional space-times,”
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