The motivation for considering higher-dimensional black holes with a cosmological constant arises from the
AdS/CFT correspondence [1]. This is an equivalence between string theory on spacetimes asymptotic to
, where
is a compact manifold, and a conformal field theory (CFT) defined on the Einstein
universe
, which is the conformal boundary of
. The best understood example is
the case of type IIB string theory on spacetimes asymptotic to
, which is dual to
super-Yang-Mills theory on
. Type IIB string theory can be replaced
by IIB supergravity in the limit of large
and strong ’t Hooft coupling in the Yang-Mills
theory.
Most studies of black holes in the AdS/CFT correspondence involve dimensional reduction on to
obtain a
-dimensional gauged supergravity theory with a negative cosmological constant. For example,
one can reduce type IIB supergravity on
to obtain
,
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
black-hole solutions that cannot be
dimensionally reduced to
dimensions. Such solutions would not be discovered using gauged
supergravity.
In this section we shall discuss asymptotically black-hole solutions of the
gauged
supergravity theories arising from the reduction of
or
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.
The simplest example of an asymptotically AdS black hole is the Schwarzschild-AdS solution [172, 250]:
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 [134
] and
Witten for
[250
]. Let
denote the horizon radius of the solution. For a small black hole,
, the thermodynamic properties are qualitatively similar to those of an asymptotically-flat
Schwarzschild black hole, i.e., the temperature decreases with increasing
so the heat capacity of the
hole is negative (as
is a monotonic function of
). However, there is an intermediate value of
at which the temperature reaches a global minimum
and then becomes an increasing
function of
. 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
at large
). Note that for
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, , there is no black-hole solution and the preferred phase is thermal
radiation in AdS. At
, black holes exists but have greater free energy than thermal
radiation. However, there is a critical temperature
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
.
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 [249, 250].
When oxidized, to ten or eleven dimensions, the radius of a small Schwarzschild-AdS black hole
will be much less than the radius of curvature (
) of the internal space
. 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
, and therefore would not admit a description in
gauged supergravity. Since the radius of curvature of
is typically
and the black hole is
much smaller than
, 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.
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 Kerr-AdS solution was constructed long ago [27]. It can be parameterized by its mass
and angular momentum
, which have been calculated (using the definitions of [4]) in [113
]. The
region of the
plane covered by these black holes is shown in Figure 13
. Note that, in
AdS, angular momentum is a central charge [108]. Hence regular vacuum solutions exhibit a
nontrivial lower bound on their mass:
. The Kerr-AdS solution never saturates this
bound.
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The Myers–Perry-AdS solution was obtained in [133] for
and for
with rotation in a
single plane. The general
solution was obtained in [111, 112]. 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
, the region of
-space covered by the Myers–Perry-AdS solution
is shown in Figure 14
.
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 [207, 173, 90].
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:
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 The dual CFT interpretation is of CFT matter in thermal equilibrium rotating around the
Einstein universe [133]. There is an interesting phase structure, generalizing that found for
Schwarzschild-AdS [133
, 13, 20
, 135
]. 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 ? Such black holes are believed to be classically unstable. It was observed
in [135
] 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
. But if
then an instability may be present. This makes sense from a
dual CFT perspective; configurations with
would correspond to CFT matter rotating faster than
light in the Einstein universe [133]. The existence of an instability was first demonstrated for small
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
[177
], i.e., precisely
where the stability argument of [135] fails. The endpoint of this classical bulk instability is not
known.
In , Figure 13
reveals (using
) that all extremal Kerr-AdS black holes have
and are, therefore, expected to be unstable. We have checked that
extremal
Myers–Perry-AdS black holes also have
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
, and
one expects it to disappear as
: 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 metric
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
or
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 supergravity on
to give
,
,
gauged supergravity [67]. This non-Abelian theory can be consistently truncated to give
,
,
gauged supergravity, whose bosonic sector is Einstein gravity coupled to four Maxwell
fields and three complex scalars [55
]. The scalar potential is negative at its global maximum. The
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
,
gauged supergravity, whose
bosonic sector is Einstein–Maxwell theory with a cosmological constant. The embedding of
minimal
,
gauged supergravity theories into
supergravity can be
given explicitly [31
], and is much simpler than the embedding of the non-Abelian
theory.
The supergravity theory can also be dimensionally reduced on
to give
,
,
gauged supergravity [201, 202].
The massive IIA supergravity can be dimensionally reduced on
to give
gauged supergravity [59
]. This theory has half-maximal supersymmetry.
It is believed that the type IIB supergravity theory can be consistently reduced on
to
give
,
,
gauged supergravity, although this has been established only for a
subsector of the full theory [62]. This theory can be truncated further to give
,
,
gauged supergravity with three vectors and two scalars. Again, setting the scalars to constants and
making the vectors equal gives the minimal
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
type IIB supergravity are
known [31
, 55
].
It is sometimes possible to obtain a given lower-dimensional supergravity theory from several different
compactifications of a higher-dimensional theory. For example, minimal gauged supergravity can be
obtained by compactifying type IIB supergravity on any Sasaki-Einstein space
[18]. More generally,
if there is a supersymmetric solution of type IIB supergravity that is a warped product of
with
some compact manifold
, then type IIB supergravity can be dimensionally reduced on
to give
minimal
gauged supergravity [104
]. An analogous statement holds for compactifications of
supergravity to give minimal
,
gauged supergravity or minimal
gauged
supergravity [103, 104].
The Reissner–Nordström-AdS black hole is a solution of minimal
gauged supergravity. It
is parameterized by its mass
and electric and magnetic charges
,
. This solution is stable
against linearized perturbations within this (Einstein–Maxwell) theory [164
]. 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 [221, 185].
Static, spherically-symmetric, charged, black-hole solutions of the ,
,
gauged supergravity theory were obtained in [69]. The solutions carry only electric charges and
are parameterized by their mass
and electric charges
. 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 ,
gauged supergravity
were obtained in [9]. They are parameterized by their mass
and electric charges
. If the charges
are set equal to each other then one recovers the
Reissner–Nordström solution of minimal
gauged supergravity. The solutions never saturate a BPS bound.
A static, spherically-symmetric, charged black-hole solution of ,
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 ,
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 [31, 56, 57, 32].
These black holes exhibit an interesting instability [121, 122]. 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
The most general, known, stationary, black-hole solution of minimal ,
gauged supergravity
is the Kerr-Newman-AdS solution, which is uniquely parameterized by its mass
, angular momentum
and electric and magnetic charges
. 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:
,
,
[171, 21]. 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 gauged supergravity theories,
e.g.,
,
gauged supergravity, should also exist. Electrically charged, rotating
solutions of the
theory, with the four charges set pairwise equal, were constructed
in [36].
Charged, rotating black-hole solutions of ,
gauged supergravity have been
constructed by truncating to a
theory [40
, 43
]. 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 gauged supergravity have not yet been
constructed.
The construction of charged rotating black-hole solutions of gauged supergravity has
attracted more attention [125
, 124, 60, 61, 37, 38
, 178
, 39, 191
]. 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
, electric charge
and two angular momenta
,
. 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
, e.g., the BMPV black hole or BPS black
rings).
Analogous solutions of ,
gauged supergravity are expected to be parameterized by
the six conserved quantities
,
,
,
,
,
. However, a six-parameter
solution is not yet known. The most general known solutions are the four-parameter BPS solution
of [178
], and the five-parameter nonextremal solution of [191], which has two of the charges
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 , 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 gauged supergravity [179]. The proof involves classifying
supersymmetric near-horizon geometries (with two rotational symmetries), and showing that
topology horizons always suffer from a conical singularity, except in the limit in
which the cosmological constant vanishes. Analogous results for the
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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