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9 Solutions with a Cosmological Constant

9.1 Motivation

The motivation for considering higher-dimensional black holes with a cosmological constant arises from the AdS/CFT correspondence [1Jump To The Next Citation Point]. This is an equivalence between string theory on spacetimes asymptotic to AdSd × X, where X is a compact manifold, and a conformal field theory (CFT) defined on the Einstein universe d−2 R × S, which is the conformal boundary of AdSd. The best understood example is the case of type IIB string theory on spacetimes asymptotic to AdS5 × S5, which is dual to 𝒩 = 4 SU (N ) super-Yang-Mills theory on R × S3. Type IIB string theory can be replaced by IIB supergravity in the limit of large N and strong ’t Hooft coupling in the Yang-Mills theory.

Most studies of black holes in the AdS/CFT correspondence involve dimensional reduction on X to obtain a d-dimensional gauged supergravity theory with a negative cosmological constant. For example, one can reduce type IIB supergravity on S5 to obtain d = 5, 𝒩 = 4 SO (6) gauged supergravity. One then seeks asymptotically anti-de Sitter black-hole solutions of the gauged supergravity theory. This is certainly easier than trying to find solutions in ten or eleven dimensions. However, one should bear in mind that there may exist asymptotically AdSd × X black-hole solutions that cannot be dimensionally reduced to d dimensions. Such solutions would not be discovered using gauged supergravity.

In this section we shall discuss asymptotically AdS d black-hole solutions of the d = 4,5,6,7 gauged supergravity theories arising from the reduction of d = 10 or d = 11 supergravity on spheres. The emphasis will be on classical properties of the solutions rather than their implications for CFT. In AdS, linearized supergravity perturbations can be classified as normalizable or non-normalizable according to how they behave near infinity [1]. By “asymptotically AdS” we mean that we are restricting ourselves to considering solutions that approach a normalizable deformation of global AdS near infinity. A non-normalizable perturbation would correspond to a deformation of the CFT, for instance, making it nonconformal. Black-hole solutions with such asymptotics have been constructed, but space prevents us from considering them here.

9.2 Schwarzschild-AdS

The simplest example of an asymptotically AdS black hole is the Schwarzschild-AdS solution [172250Jump To The Next Citation Point]:

2 ds2 = − U(r)dt2 + U (r )− 1dr2 + r2d Ω2 , U (r) = 1 − -μ---+ r-, (117 ) d−2 rd−3 ℓ2
where μ is proportional to the mass, and ℓ is the radius of curvature of the AdS ground state11. The solution has a regular horizon for any μ > 0. Definitions of mass and angular momentum for asymptotically AdS spacetimes have been given in d = 4 [4Jump To The Next Citation Point] and d ≥ 4 [3]. The mass of Schwarzschild-AdS relative to the AdS ground state is [250Jump To The Next Citation Point]
(d − 2)Ωd− 2μ M = -------------. (118 ) 16πG
For d = 4, the stability of Schwarzschild-AdS against linearized gravitational perturbations has been proven in [163]. For d > 4, spherical symmetry enables one to decompose linearized gravitational perturbations into scalar/vector/tensor types. The equations governing each type can be reduced to ODEs of Schrödinger form, and the stability of vector and tensor perturbations can be established [151]. Stability with respect to scalar gravitational perturbations has not yet been established.

It is expected that the Schwarzschild-AdS black hole is the unique, static, asymptotically AdS, black-hole solution of vacuum gravity with a negative cosmological constant, but this has not been proven.

The thermodynamics of Schwarzschild-AdS were discussed by Hawking and Page for d = 4 [134Jump To The Next Citation Point] and Witten for d > 4 [250Jump To The Next Citation Point]. Let r + denote the horizon radius of the solution. For a small black hole, r+ ≪ ℓ, the thermodynamic properties are qualitatively similar to those of an asymptotically-flat Schwarzschild black hole, i.e., the temperature decreases with increasing r+ so the heat capacity of the hole is negative (as r+ is a monotonic function of μ). However, there is an intermediate value of r+ ∼ ℓ at which the temperature reaches a global minimum Tmin and then becomes an increasing function of r+. Hence the heat capacity of large black holes is positive. This implies that the black hole can reach a stable equilibrium with its own radiation (which is confined near the hole by the gravitational potential ∼ r2∕ℓ2 at large r). Note that for T > Tmin there are two black-hole solutions with the same temperature: a large one with positive specific heat and a small one with negative specific heat.

These properties lead to an interesting phase structure for gravity in AdS [134]. At low temperature, T < Tmin, there is no black-hole solution and the preferred phase is thermal radiation in AdS. At T ∼ Tmin, black holes exists but have greater free energy than thermal radiation. However, there is a critical temperature THP > Tmin beyond which the large black hole has lower free energy than thermal radiation and the small black hole. The interpretation is that the canonical ensemble for gravity in AdS exhibits a (first-order) phase transition at T = Tmin.

In the AdS/CFT context, this Hawking–Page phase transition is interpreted as the gravitational description of a thermal phase transition of the (strongly coupled) CFT on the Einstein universe [249250].

When oxidized, to ten or eleven dimensions, the radius r+ of a small Schwarzschild-AdS black hole will be much less than the radius of curvature (∼ ℓ) of the internal space X. This suggests that the black hole will suffer from a classical Gregory–Laflamme-type instability. The probable endpoint of the instability would be a small black hole localized on X, and therefore would not admit a description in gauged supergravity. Since the radius of curvature of X is typically ℓ and the black hole is much smaller than ℓ, the geometry near the hole should be well approximated by the ten or eleven-dimensional Schwarzschild solution (see e.g., [141]). However, an exact solution of this form is not known.

9.3 Stationary vacuum solutions

If we consider pure gravity with a negative cosmological constant then the most general known family of asymptotically-AdS black-hole solutions is the generalization of the Kerr–Myers–Perry solutions to include a cosmological constant. It seems likely that black rings would exist in asymptotically-AdS spacetimes, but no exact solutions are known.12

The d = 4 Kerr-AdS solution was constructed long ago [27]. It can be parameterized by its mass M and angular momentum J, which have been calculated (using the definitions of [4]) in [113Jump To The Next Citation Point]. The region of the (M, J) plane covered by these black holes is shown in Figure 13View Image. Note that, in AdS, angular momentum is a central charge [108]. Hence regular vacuum solutions exhibit a nontrivial lower bound on their mass: M ≥ |J|∕ℓ. The Kerr-AdS solution never saturates this bound.

View Image

Figure 13: GM ∕ℓ against G |J |∕ ℓ2 for d = 4 Kerr-AdS black holes. The thick curve corresponds to extremal black holes. Black-hole solutions lie on, or above, this curve (which was determined using results in [20Jump To The Next Citation Point]). The thin line is the BPS bound M = |J|∕ℓ.

The Myers–Perry-AdS solution was obtained in [133Jump To The Next Citation Point] for d = 5 and for d > 5 with rotation in a single plane. The general d > 5 solution was obtained in [111112]. They have horizons of spherical topology. There is some confusion in the literature concerning the conserved charges carried by these solutions. A careful discussion can be found in [113]. The solutions are uniquely specified by their mass and angular momenta. For d = 5, the region of (M, J1,J2)-space covered by the Myers–Perry-AdS solution is shown in Figure 14View Image.

View Image

Figure 14: GM ∕ ℓ2 (vertical) against G|Ji|∕ℓ3 for d = 5 Myers–Perry-AdS black holes. Nonextremal black holes fill the region above the surface. The surface corresponds to extremal black holes, except when one of the angular momenta vanishes (in which case there is not a regular horizon, just as in the asymptotically-flat case). This “extremal surface” lies inside the square-based pyramid (with vertex at the origin) defined by the BPS relation M = |J1|∕ℓ + |J2|∕ℓ, so none of the black holes are BPS.

Kerr–Myers–Perry-AdS solutions have the same symmetries as their asymptotically-flat cousins, and exhibit similar enhancement of symmetry in special cases. The integrability of the geodesic equation and separability of the Klein–Gordon equation also extends to this case [20717390].

These solutions reduce to the Schwarzschild-AdS solution in the limit of zero angular momentum. It has been shown that the only regular stationary perturbations of the Schwarzschild-AdS solution are those that correspond to taking infinitesimal angular momenta in these rotating solutions [162]. Hence, if other stationary vacuum black-hole solutions exist (e.g., black rings) then they are not continuously connected to the Schwarzschild-AdS solution.

These solutions exhibit an important qualitative difference from their asymptotically-flat cousins. Consider the Killing field tangent to the null-geodesic generators of the horizon:

∂ ∂ V = ---+ Ωi----. (119 ) ∂t ∂ φi
In asymptotically-flat spacetime, this Killing field is spacelike far from the black hole, which implies that it is impossible for matter to co-rotate rigidly with the hole (i.e., to move on orbits of V). However, in AdS, if Ωi ℓ ≤ 1 then V is timelike everywhere outside the horizon. This implies that rigid co-rotation is possible; the Killing field V defines a co-rotating reference frame. Consequently, there exists a Hartle–Hawking state describing thermal equilibrium of the black hole with co-rotating thermal radiation [133Jump To The Next Citation Point].

The dual CFT interpretation is of CFT matter in thermal equilibrium rotating around the Einstein universe [133Jump To The Next Citation Point]. There is an interesting phase structure, generalizing that found for Schwarzschild-AdS [133Jump To The Next Citation Point1320Jump To The Next Citation Point135Jump To The Next Citation Point]. For sufficiently large black holes, one can study the dual CFT using a fluid mechanics approximation, which gives quantitative agreement with black-hole thermodynamics [14].

What happens if Ωiℓ > 1? Such black holes are believed to be classically unstable. It was observed in [135Jump To The Next Citation Point] that rotating black holes in AdS may suffer from a super-radiant instability, in which energy and angular momentum are extracted from the black hole by super-radiant modes. However, it was proven that this cannot occur if Ωiℓ ≤ 1. But if Ωiℓ > 1 then an instability may be present. This makes sense from a dual CFT perspective; configurations with Ωi ℓ > 1 would correspond to CFT matter rotating faster than light in the Einstein universe [133]. The existence of an instability was first demonstrated for small d = 4 Kerr-AdS black holes in [23]. A general analysis of odd-dimensional black holes with equal angular momenta reveals that the threshold of instability is at Ω â„“ = 1 i [177Jump To The Next Citation Point], i.e., precisely where the stability argument of [135] fails. The endpoint of this classical bulk instability is not known.

In d = 4, Figure 13View Image reveals (using Ω = dM ∕dJ) that all extremal Kerr-AdS black holes have Ω â„“ > 1 and are, therefore, expected to be unstable. We have checked that d = 5 extremal Myers–Perry-AdS black holes also have Ωi ℓ > 1 and so they too should be classically unstable. However, the instability should be very slow when the black-hole size is much smaller than the AdS radius ℓ, and one expects it to disappear as ℓ → ∞: it takes an increasingly long time for the super-radiant modes to bounce back off the AdS boundary.

Finally, we should mention a subtlety concerning the use of the term “stationary” in asymptotically AdS spacetimes [177]. Consider the AdS5 metric

( 2) ( 2)− 1 ds2 = − 1 + r- dt2 + 1 + r- dr2 + r2(dθ2 + sin2θ dφ2 + cos2θ dψ2) . (120 ) ℓ2 ℓ2
This admits several types of globally-timelike Killing fields. For example, there is the “usual” generator of time translations k = ∂∕∂t, which has unbounded norm, but there is also the “rotating” Killing field V = ∂∕∂t + ℓ− 1∂ ∕∂φ + ℓ−1∂ ∕∂ψ, which has constant norm. On the conformal boundary, k is timelike and V is null. Hence, from a boundary perspective, particles following orbits of V are rotating at the speed of light. These two different types of timelike Killing vector field allow one to define two distinct notions of stationarity for asymptotically-AdS spacetimes. So far, all known black hole solutions are stationary with respect to both definitions because they admit global Killing fields analogous to ∂ ∕∂φ, ∂∕∂ ψ. However, it is conceivable that there exist AdS black holes (with less symmetry than known solutions) that are stationary only with respect to the second definition, i.e., they admit a Killing field that behaves asymptotically like V but not one behaving asymptotically like k. From a boundary CFT perspective, such black holes would rotate at the speed of light.

9.4 Gauged supergravity theories

In order to discuss charged anti-de Sitter black holes we will need to specify which gauged supergravity theories we are interested in. The best-understood examples arise from the dimensional reduction of d = 10 or d = 11 dimensional supergravity theories on spheres to give theories with maximal supersymmetry and non-Abelian gauge groups. However, most work on constructing explicit black hole solutions has dealt with consistent truncations of these theories, with reduced supersymmetry, in which the non-Abelian gauge group is replaced by its maximal Abelian subgroup. Indeed, there is no known black-hole solution with a nontrivial non-Abelian gauge field obeying normalizable boundary conditions.

There is a consistent dimensional reduction of d = 11 supergravity on 7 S to give d = 4, 𝒩 = 8, SO (8 ) gauged supergravity [67]. This non-Abelian theory can be consistently truncated to give d = 4, 𝒩 = 2, U (1)4 gauged supergravity, whose bosonic sector is Einstein gravity coupled to four Maxwell fields and three complex scalars [55Jump To The Next Citation Point]. The scalar potential is negative at its global maximum. The AdS4 ground state of the theory has the scalars taking constant values at this maximum. One can truncate this theory further by taking the scalars to sit at the top of their potential, and setting the Maxwell fields equal to each other. This gives minimal d = 4, 𝒩 = 2 gauged supergravity, whose bosonic sector is Einstein–Maxwell theory with a cosmological constant. The embedding of minimal d = 4, 𝒩 = 2 gauged supergravity theories into d = 11 supergravity can be given explicitly [31Jump To The Next Citation Point], and is much simpler than the embedding of the non-Abelian 𝒩 = 8 theory.

The d = 11 supergravity theory can also be dimensionally reduced on S4 to give d = 7, 𝒩 = 2, SO (5 ) gauged supergravity [201202].

The d = 10 massive IIA supergravity can be dimensionally reduced on S4 to give d = 6 𝒩 = 2 SU (2 ) gauged supergravity [59Jump To The Next Citation Point]. This theory has half-maximal supersymmetry.

It is believed that the d = 10 type IIB supergravity theory can be consistently reduced on 5 S to give d = 5, 𝒩 = 4, SO (6) gauged supergravity, although this has been established only for a subsector of the full theory [62]. This theory can be truncated further to give d = 5, 𝒩 = 1, U (1)3 gauged supergravity with three vectors and two scalars. Again, setting the scalars to constants and making the vectors equal gives the minimal d = 5 gauged supergravity, whose bosonic sector is Einstein–Maxwell theory with a negative cosmological constant and a Chern–Simons coupling. The explicit embeddings of these Abelian theories into d = 10 type IIB supergravity are known [31Jump To The Next Citation Point55Jump To The Next Citation Point].

It is sometimes possible to obtain a given lower-dimensional supergravity theory from several different compactifications of a higher-dimensional theory. For example, minimal d = 5 gauged supergravity can be obtained by compactifying type IIB supergravity on any Sasaki-Einstein space Yp,q [18]. More generally, if there is a supersymmetric solution of type IIB supergravity that is a warped product of AdS5 with some compact manifold X5, then type IIB supergravity can be dimensionally reduced on X5 to give minimal d = 5 gauged supergravity [104Jump To The Next Citation Point]. An analogous statement holds for compactifications of d = 11 supergravity to give minimal 𝒩 = 2, d = 4 gauged supergravity or minimal d = 5 gauged supergravity [103104].

9.5 Static charged solutions

The d = 4 Reissner–Nordström-AdS black hole is a solution of minimal 𝒩 = 2 gauged supergravity. It is parameterized by its mass M and electric and magnetic charges Q,P. This solution is stable against linearized perturbations within this (Einstein–Maxwell) theory [164Jump To The Next Citation Point]. Compared with its asymptotically-flat counterpart, perhaps the most surprising feature of this solution is that it never saturates a BPS bound. If the mass of the black hole is lowered, it will eventually become extremal, but the extremal solution is not BPS. If one imposes the BPS condition on the solution, then one obtains a naked singularity rather than a black hole [221185].

Static, spherically-symmetric, charged, black-hole solutions of the 𝒩 = 2, d = 4, U(1)4 gauged supergravity theory were obtained in [69]. The solutions carry only electric charges and are parameterized by their mass M and electric charges Qi. Alternatively they can be dualized to give purely magnetic solutions. Once again, they never saturate a BPS bound. One would expect the existence of dyonic solutions of this theory, but such solutions have not been constructed.

Static, spherically-symmetric, charged, black-hole solutions of d = 5, 3 U (1) gauged supergravity were obtained in [9]. They are parameterized by their mass M and electric charges Qi. If the charges are set equal to each other then one recovers the d = 5 Reissner–Nordström solution of minimal d = 5 gauged supergravity. The solutions never saturate a BPS bound.

A static, spherically-symmetric, charged black-hole solution of d = 6, SU (2) gauged supergravity was given in [59]. Only a single Abelian component of the gauge field is excited, and the solution is parameterized by its charge and mass.

Static, spherically-symmetric, charged, black-hole solutions of d = 7, SO (5 ) gauged supergravity are known [55]. They can be embedded into a truncated version of the full theory in which there are two Abelian vectors and two scalars. They are parameterized by their mass and electric charges.

Electrically-charged, asymptotically-AdS, black-hole solutions exhibit a Hawking–Page like phase transition in the bulk, which entails a corresponding phase transition for the dual CFT at finite temperature in the presence of chemical potentials for the R-charge. This has been studied in [31565732].

These black holes exhibit an interesting instability [121122]. This is best understood for a black hole so large (compared to the AdS radius) that the curvature of its horizon can be neglected, i.e., it can be approximated by a black brane. The dual CFT interpretation is as a finite temperature configuration in flat space with finite charge density. For certain regions of parameter space, it turns out that the entropy increases if the charge density becomes nonuniform (with the total charge and energy held fixed). Hence, the thermal CFT state exhibits an instability. Using the AdS/CFT dictionary, this maps to a classical instability in the bulk in which the horizon becomes translationally nonuniform, i.e., a Gregory–Laflamme instability. The remarkable feature of this argument is that it reveals that a classical Gregory–Laflamme instability should be present precisely when the black brane becomes locally thermodynamically unstable. Here, local thermodynamic stability means having an entropy, which is concave down as a function of the energy and other conserved charges (if the only conserved charge is the energy, then this is equivalent to positivity of the heat capacity). The Gubser–Mitra (or “correlated stability”) conjecture asserts that this correspondence should apply to any black brane, not just asymptotically-AdS solutions. See [128] for more discussion of this correspondence.

For finite-radius black holes, the argument is not so clear cut because the dual CFT lives in the Einstein universe rather than flat spacetime, so finite size effects will affect the CFT argument and the Gubser–Mitra conjecture does not apply. Nevertheless, it should be a good approximation for sufficiently large black holes and hence there will be a certain range of parameters for which large charged black holes are classically unstable.13

9.6 Stationary charged solutions

The most general, known, stationary, black-hole solution of minimal d = 4, 𝒩 = 2 gauged supergravity is the Kerr-Newman-AdS solution, which is uniquely parameterized by its mass M, angular momentum J and electric and magnetic charges (Q, P ). The thermodynamic properties of this solution, and implications for the dual CFT were investigated in [20]. An important property of this solution is that it can preserve some supersymmetry. This occurs for a one-parameter subfamily specified by the electric charge: M = M (Q), J(Q ), P = 0 [17121]. Hence supersymmetric black holes can exist in AdS but they exhibit an important qualitative difference from the asymptotically flat case; they must rotate.

Charged rotating black-hole solutions of more general d = 4 gauged supergravity theories, e.g., 𝒩 = 2, 4 U (1) gauged supergravity, should also exist. Electrically charged, rotating solutions of the U (1)4 theory, with the four charges set pairwise equal, were constructed in [36].

Charged, rotating black-hole solutions of d = 7, SO (5) gauged supergravity have been constructed by truncating to a 2 U (1) theory [40Jump To The Next Citation Point43Jump To The Next Citation Point]. In this theory, one expects the existence of a topologically-spherical black-hole solution parameterized by its mass, three angular momenta, and two electric charges. This general solution is not yet known. However, solutions with three equal angular momenta but unequal charges have been constructed [40], as have solutions with equal charges but unequal angular momenta [43]. Both types of solution admit BPS limits.

Charged, rotating black-hole solutions of d = 6 gauged supergravity have not yet been constructed.

The construction of charged rotating black-hole solutions of d = 5 gauged supergravity has attracted more attention [125Jump To The Next Citation Point12460613738Jump To The Next Citation Point178Jump To The Next Citation Point39191Jump To The Next Citation Point]. The most general known black-hole solution of the minimal theory is that of [38]. This solution is parameterized by the conserved charges of the theory, i.e., the mass M, electric charge Q and two angular momenta J1, J2. Intuition based on results proved for asymptotically-flat solutions suggests that, for this theory, this is the most general topologically-spherical stationary black hole with two rotational symmetries. In the BPS limit, these solutions reduce to a two-parameter family of supersymmetric black holes. In other words, one loses two parameters in the BPS limit (just as for nonstatic asymptotically-flat black holes in d = 5, e.g., the BMPV black hole or BPS black rings).

Analogous solutions of d = 5, U (1)3 gauged supergravity are expected to be parameterized by the six conserved quantities M, J1, J2, Q1, Q2, Q3. However, a six-parameter solution is not yet known. The most general known solutions are the four-parameter BPS solution of [178Jump To The Next Citation Point], and the five-parameter nonextremal solution of [191], which has two of the charges Qi equal. The former is expected to be the general BPS limit of the yet to be discovered six-parameter black-hole solution (as one expects to lose two parameters in the BPS limit). The latter solution should be obtained from the general six-parameter solution by setting two of the charges equal.

Supersymmetric AdS black holes have Ωiℓ = 1, which implies that they rotate at the speed of light with respect to the conformal boundary [125]. More precisely, the co-rotating Killing field becomes null on the conformal boundary. Hence, the CFT interpretation of these black holes involves matter rotating at the speed of light in the Einstein universe. The main motivation for studying supersymmetric AdS black holes is the expectation that it should be possible to perform a microscopic CFT calculation of their entropy. The idea is to count states in weakly coupled CFT and then extrapolate to strong coupling. In doing this, one has to count only states in short BPS multiplets that do not combine into long multiplets as the coupling is increased. One way of trying to do this is to work with an index that receives vanishing contributions from states in multiplets that can combine into long multiplets. Unfortunately, such indices do not give agreement with black-hole entropy [160]. This is not a contradiction; although certain multiplets may have the right quantum numbers to combine into a long multiplet, the dynamics of the theory may prevent this from occurring, so the index undercounts BPS states.

The fact that these supersymmetric black holes have only four independent parameters is puzzling from the CFT perspective, since BPS states in the CFT carry five independent charges. Maybe there are more general black-hole solutions. It seems unlikely that one could generalize the solutions of [178] to include an extra parameter since then one would also have an extra parameter, in the corresponding non-BPS solutions, which would therefore form a seven parameter family in a theory with only six conserved charges. This seems unlikely for topologically-spherical black holes. But we know that black rings can carry nonconserved charges, so maybe this points to the existence of supersymmetric AdS black rings. However, it has been shown that such solutions do not exist in minimal d = 5 gauged supergravity [179]. The proof involves classifying supersymmetric near-horizon geometries (with two rotational symmetries), and showing that S1 × S2 topology horizons always suffer from a conical singularity, except in the limit in which the cosmological constant vanishes. Analogous results for the U(1)3 theory have also been obtained [176]. So if AdS black rings exist then they cannot be “balanced” in the BPS limit.

Maybe the resolution of the puzzle involves 10d black holes with no 5D interpretation, or 5D black holes involving non-Abelian gauge fields, or 5D black holes with only one rotational symmetry. Alternatively, perhaps we already know all the BPS black-hole solutions and the puzzle arises from a lack of understanding of the CFT. For example, maybe, at strong coupling, only a four charge subspace of BPS CFT states carries enough entropy to correspond to a macroscopic black hole.

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