Black hole near-horizon geometries

Black hole near-horizon geometries James Lucietti Durham University Imperial College, March 5, 2008

Point of this talk: To highlight that a precise concept of a black hole near-horizon geometry can be used to learn definite things about (a given set of) black holes. 1. Supersymmetric AdS 5 black holes: hep-th/0601156, hep-th/0611351; Kunduri, JL, Reall arxiv:0708.3695; Kunduri, JL 2. Structure of general near-horizon geometries of D = 4, 5 extremal black holes arxiv:0705.4214, Kunduri, JL, Reall 3. Extremal black holes and black rings in D > 5; Work in progress Figueras, Kunduri, JL, Rangamani

Motivation Generalities Examples Applications Supersymmetric AdS 5 black holes General near-horizon geometries in D = 4, 5 Black holes in D > 5 Future directions and Open problems Conclusions

Motivation Classification of stationary black holes in flat space (D > 4) and AdS (D 4). Relevant to string theory, AdS/CFT etc. D = 4. No-hair theorem: Kerr(-Newmann) unique asymptotically flat black hole solution, uniquely parametrised by conserved charges M, J, Q. D = 5. Black hole non-uniqueness (Emparan, Reall 01): asymptotically flat vacuum black holes of topology S 3 (Myers-Perry) and S 1 S 2 (Black ring) with the same conserved charges M, J 1, J 2

Motivation Simplification - restrict to extremal black holes. These include supersymmetric black holes relevance to string theory and AdS/CFT (short multiplets protected etc.) Extremal black holes have a well defined concept of a near-horizon geometry Classification of all near-horizon geometries in a given theory provides some information on set of allowed black hole solutions Near-horizon geometries of interest in their own right AdS 2 (AdS 3 )/CFT?

Black hole horizons Stationary black hole event horizon is a Killing horizon, i.e. null surface whose normal is Killing vector V Gaussian null coords (v, r, x a ): ds 2 = r f (r, x)dv 2 +2dvdr+2r h a (r, x)dvdx a +γ ab (r, x)dx a dx b Horizon is at r = 0 where V = / v is null. x a coords on D 2 compact manifold H (spatial section of horizon). Extremal black hole surface gravity κ = 0. But κ = 1 2 f r=0 so must have f (r, x) = r F (r, x). That is g vv = O(r 2 ).

Near-horizon limit An extremal black hole admits a near-horizon limit, defined by v v/ɛ and r ɛr. The limiting metric is called the near-horizon geometry : ds 2 = r 2 F (x)dv 2 + 2dvdr + 2r h a (x)dvdx a + γ ab (x)dx a dx b with F (x) 0 (V non-spacelike) The function, one-form and metric (F, h a, γ ab ) are the near-horizon data. Defined purely on H. Symmetry enhancement. In addition to v v + c one has a dilation symmetry v λv, r r/λ, which combine to form a 2d non-abelian group G 2.

Near-horizon geometry The near-horizon geometry solves the same field equations as the full black hole metric. Finding a near-horizon geometry is a D 2 dimensional problem easier. E.g. vacuum Einstein equations imply R ab (γ) = 1 2 h ah b (a h b) Can use near-horizon geometries to rule out existence of a given class of black holes, i.e.: non-existence of near-horizon geometries of topology H = non-existence of black holes with horizon topology H

Static near-horizon geometries Stationary black holes can be static (non-rotating). The near-horizon geometry inherits this. Can prove (KLR 07): static near-horizon geometry is (locally) a warped product of either AdS 2 (F 0 > 0) or R 1,1 (F 0 = 0) and H with metric γ ab : ds 2 = e λ(x) ( F 0 r 2 dv 2 + 2dvdr) + γ ab (x)dx a dx b Symmetry: SO(2, 1) or 2d Poincare. Important: A non-static black hole can have a static near-horizon geometry: e.g. BPS black ring (F 0 = 0) locally is AdS 3 S 2.

4d near-horizon geometry examples Vacuum: extremal Kerr (S 2 topology), cohomogeneity-1 ds 2 = (1 + cos2 θ) 2 + 2r 2 0 sin2 θ 1 + cos 2 θ [ r 2 ] dv 2 + 2dvdr + r0 2 dθ 2 r 2 0 ( dφ + r r0 2 dv ) 2 Einstein-Maxwell: extremal Reissner-Nordstrom, AdS 2 S 2 ds 2 = r 2 L 2 dv 2 + 2dvdr + L 2 dω 2 2

5d near-horizon geometry examples Vacuum S 3 topology: extremal Myers-Perry J 1, J 2 0 S 1 S 2 topology: extremal black ring J 1, J 2 0, boosted Kerr string Note: Near-horizon of extremal black ring (J 1, J 2 0) is the same as near-horizon of a tensionless boosted Kerr string. Minimal supergravity (contains Einstein-Maxwell): extremal dipole black ring BMPV [J 1 = J 2 ] extremal Reissner-Nordstrom (AdS 2 S 3 ) supersymmetric black ring (AdS 3 S 2 ) supersymmetric AdS5 black holes. In all cases NH is a fibration of H over AdS 2. Observe: that all known solutions possess SO(2, 1) symmetry.

Supersymmetric AdS 5 black holes: Motivation AdS 5 S 5 BPS black holes dual to high energy (i.e. O(N 2 )) 1/16-BPS states in N = 4 U(N) SYM on R S 3 at large N. Open problem: derive (O(N 2 )) entropy of black holes from counting of states in N = 4 SYM Such states in CFT labelled by 5 conserved charges (J 1, J 2, Q 1, Q 2, Q 3 ). BPS: E = J 1 + J 2 + Q 1 + Q 2 + Q 3 Index does not work: O(1). Free partition function over-counts small non-zero coupling result might be enough (c.f. 1/8-BPS states, giant gravitons) (Kinney, Maldacena, Minwalla, Raju, 05)

Supersymmetric AdS 5 black holes Truncate type IIB supergravity: U(1) 3 -gauged supergravity in D = 5 explicit systematic techniques available Gauntlett, Hull, Gutowski, Pakis, Reall 02. Most general known black hole solution (Kunduri, JL, Reall 06) carries conserved charges (J 1, J 2, Q 1, Q 2, Q 3 ) plus one constraint relating these. Four parameter family of S 3 ( S 5 ) topology black holes are there more general or a different class of black holes? Focus on U(1) 3 -gauged sugra in D = 5. Would like uniqueness theorem or a complete classification of such black holes difficult! What about near-horizon geometries?

BPS black holes: Previous work The following ingredients can be sufficient to determine all near-horizon geometries in given theory (Reall 02): 1. General form of near-horizon metric 2. SUSY constraints 3. Global properties (compactness of H) First done in 5d ungauged minimal supergravity (Reall 02). Only solutions are: near-horizon of BMPV, AdS 3 S 2 (NH of black ring), R 1,1 T 3 (trivial solution?). All homogeneous spaces. BMPV proved to be unique AF BPS solution with its NH uniqueness theorem for susy (S 3 )-black holes! Method used to prove a uniqueness theorem for susy black holes in D = 4, N = 2 minimal sugra (Chrusciel, Reall, Tod 06).

Near-horizon analysis Would like to repeat classification of NH geometries in minimal 5d gauged sugra. Harder! Previous work Gutowski, Reall 04: particular solutions found previously which lead to first example of susy AdS 5 BH (J 1 = J 2 ) Kunduri, JL, Reall 06 and Kunduri, JL 07 Classification of all NH geometries of AdS 5 black holes with R U(1) 2 symmetry U(1) 2 rotational symmetry on H (cohomogeneity-1 problem). Introduce coordinates (ρ, x 1, x 2 ) adapted to symmetry: ds 2 (H) = dρ 2 + γ ij (ρ)dx i dx j

Results: minimal gauged supergravity (KLR 06) Truncation of U(1) 3 - gauged supergravity Einstein-Maxwell +CS+ negative cosmological constant. Only regular solution: near-horizon geometry of known S 3 black hole solution of (Chong,Cvetic,Lu,Pope 05) Q i = Q(J 1, J 2 ) This implies there are no BPS AdS 5 black rings in this theory with R U(1) 2 symmetry! (c.f ungauged case there are such solutions) No evidence for more general black holes with U(1) 2 symmetry.

Results: U(1) 3 -gauged supergravity (KL 07) S 3 topology: 4 parameter cohomogeneity-1 solution, S 3 fibration over AdS 2. Turns out to be near-horizon limit of most general known AdS 5 BPS black hole. Also found AdS 3 S 2 (T 2 ) solutions with horizons S 1 S 2 (and T 3 ). Have constant scalars and only exist in certain regions of moduli space. S 1 S 2 solution: radius of S 2 is O(l) does not reduce to solution in ungauged supergravity. Unlikely it corresponds to the NH of an AdS 5 black ring. No evidence for more general black holes with U(1) 2 symmetry.

AdS black rings? In a certain limit near-horizon geometries S 3 S 1 S 2 which do reduce to that of BPS ring in flat space; not regular! These are warped products of AdS 3 and S 2 with a conical singularity. E.g. in minimal supergravity: ds 2 = Γ [2dvdr + 2r C 2 dx 1 dv + C 2 (dx 1 ) 2] + l2 ΓdΓ 2 4P(Γ) + P(Γ) Γ 2 (dx 2 ) 2 Oxidising this AdS 3 solution gives a metric locally isometric to the AdS 3 M 7 geometries in IIB discrete family of these are regular Gauntlett, Mac Conamhna, Mateos, Waldram ( 06) Question: are there 10d black holes in AdS 5 S 5 with this near-horizon geometry and hence horizon topology S 1 M 7?

Extremal black holes: Attractor mechanism Attractor mechanism is phenomenon that the entropy of BH does not depend on moduli of theory (e.g. scalars at infinity) Initially observed for static BPS black holes (Ferrara, Kallosh, Strominger 1995). Extended later to extremal non-bps and then extremal rotating BH (Sen et al 05). Attractor mechanism for extremal rotating black holes relies on assumption of SO(2, 1) near-horizon symmetry Does one expect this symmetry on general grounds? For static black holes yes, but one only expects 2d symmetry in general... Under some assumptions, one can prove that this symmetry emerges dynamically (Kunduri, JL, Reall 07)

Assumptions: Rotational symmetries A stationary (non-extremal) rotating black hole must be axisymmetric: admits a rotational U(1) symmetry (Hollands, Ishibashi, Wald 06). Therefore expect near-horizon limit of extremal black hole to have G 2 U(1) symmetry. Assume this in D = 4. In D = 5 all known black holes have two rotational symmetries: thus assume G 2 U(1) 2. Assuming G 2 U(1) D 3 symmetry makes near-horizon cohomogeneity-1.

Coordinates Symmetries restrict horizon topology (Gowdy 1973): 1. D = 4: compact horizon topologies consistent with U(1) global isometry are S 2, T 2. 2. D = 5: compact horizon topologies consistent with U(1) 2 global isometry are S 1 S 2, T 3 or S 3 (and quotients) U(1) D 3 isometry allows one to introduce coords on H, x a = (ρ, x i ) where i = 1, D 3 and / x i are Killing: γ ab dx a dx b = dρ 2 + γ ij (ρ)dx i dx j in non-toroidal cases ρ belongs to finite interval with γ ij degenerate at endpoints.

General theory in D = 4, 5 Take a general two-derivative theory of gravity with (uncharged) scalars, abelian vectors and non-trivial scalar potential. This includes many theories of interest e.g. vacuum gravity (with cosmological constant), (gauged) supergravity... Consider asymptotically flat or asymptotically AdS extremal black hole solution with R U(1) D 3 symmetry. Assume matter also invariant under these symmetries. Note: can extend proof of SO(2, 1) symmetry to include higher derivative terms, as long as one has a regular horizon to lowest order (i.e. in Einstein gravity)

Proof Perform gauge transformation r Γ(ρ)r and h = Γ 1 k i (ρ)dx i Γ Γ dρ, where Γ(ρ) > 0, to rewrite near-horizon geometry in suggestive form: ds 2 = r 2 A(ρ)dv 2 + 2Γ(ρ)dvdr +γ ij (ρ)(dx i + rk i (ρ)dv)(dx j + rk j (ρ)dv) The near-horizon limit must have T ρi = T ρv = 0. R ρi = R ρv = 0 implies k i are constants and A(ρ) = A 0 Γ(ρ) so: ds 2 = Γ(ρ)[r 2 A 0 dv 2 +2dvdr]+dρ 2 +γ ij (ρ)(dx i +rk i dv)(dx j +rk j dv) where A 0 0 (i.e. / v not space-like).

Proof A 0 = 0 (which implies k = 0) has 2d Poincare symmetry. Can be ruled out assuming non-toroidal topology and strong energy condition. So A 0 < 0. Isometries of AdS 2 act like rdv rdv + dφ so can compensate by x i x i k i φ(v, r). Thus near-horizon must have SO(2, 1) U(1) D 3 symmetry! Near-horizon geometry: fibration of horizon H over AdS 2. Actually one has a line or circle bundle over AdS 2 if k 0 then SO(2, 1) has 3d orbits Can show matter fields are also invariant under SO(2, 1).

AdS 3 near-horizon geometries U(1) D 3 -symmetric near-horizon geometries are static when either: 1. k = 0: warped products of AdS 2 and H. 2. k i k i = A 0 Γ and Γ 1 k i constant. The NH geometry is: ds 2 = Γ(ρ)[ g 2 r 2 dv 2 +2dvdr +(dx 1 +grdv) 2 ]+dρ 2 +γ AB (ρ)dx A dx B Warped products (locally) AdS 3 and M D 3. H = S 1 M D 3. SO(2, 2) SO(2, 1) U(1) since x 1 periodic. Includes black rings AdS 3 S 2 Note: a warped AdS 2 or AdS 3 geometry can be interpreted as a static near-horizon geometry of an extremal black hole

Analytic continuations Analytically continue non-static near-horizon geometry so AdS 2 S 2 and SO(2, 1) SU(2) or SO(3). Orbit is circle bundle over S 2 this can be S 3! In D = 5 one has SO(2, 1) U(1) 2 with 4d orbits. Possibility of having new time not acted upon by SO(2, 1). Then, SO(2, 1) U(1) 2 -symmetric near-horizon geometries SU(2) U(1) R-symmetric S 3 black hole solutions (new time generates R) E.g. NH geometry of D = 5 Myers-Perry (J 1 J 2 ) non-extremal Myers-Perry black hole solution with J 1 = J 2. Maison s classification of SO(3)-symmetric KK black holes all solutions (implicit) for vacuum near-horizon geometries.

Black holes in D > 5 Vacuum gravity: black holes have conserved charges M and [(D 1)/2] angular momenta J i (in D = 5 J 1, J 2 ). Only (vacuum) exact solution is Myers-Perry which has S D 2 topology. Presumably non-spherical topology horizon can occur: higher dimensional rings S 1 S D 3? No systematic techniques for constructing solutions due to less symmetry. Can we use near-horizon geometries to learn things?

Near-horizon geometries in D > 5 Work in progress: Figueras, Kunduri, JL, Rangamani Does SO(2, 1) symmetry persist in D > 5? Difficult to prove even with U(1) [(D 1)/2] rotational symmetry. We have proved it in special cases where rotational symmetry enhanced. Checked: near-horizon geometry of Myers-Perry has SO(2, 1) U(1) [(D 1)/2] symmetry. We have constructed families of SO(2, 1)-near-horizon geometries with S 1 S D 3 horizon topology. We argue that certain special cases are the near-horizon limits of yet to be found asymptotically flat extremal black rings.

Open problems Given a near-horizon geometry and prescribed asymptotics (e.g. flat space, AdS) what are the conditions for: 1. existence of an interpolating black hole solution 2. uniqueness of an interpolating black hole solution This is an important problem: could be used as a technique for proving uniqueness theorems for extremal (and supersymmetric) black holes. Counter-example to 1.? NH geometry of tension-ful boosted Kerr-string probably does not correspond to an asymptotically flat black ring.

Open problems Is U(1) [(D 1)/2] rotational symmetry necessary? Black holes with less symmetry? Only R U(1) has been proved. 1. Complete classification of supersymmetric AdS 5 black hole near-horizon geometries (i.e. remove U(1) 2 assumption) 2. Proof of SO(2, 1) symmetry in D = 5 with only U(1) spatial symmetry. SO(2, 1) in D > 5? Classify vacuum near-horizon geometries. NH geometries of extremal branes?

Conclusions Have discussed the concept of a near-horizon geometry from a general point of view: a tool for uncovering aspects of higher dimensional extremal black holes Application 1: classified all near-horizon geometries of supersymmetric AdS 5 black holes with R U(1) 2 symmetry (in U(1) 3 -gauged supergravity) Application 2: have explained SO(2, 1) symmetry enhancement in near-horizon limit in a fairly general setting. Application 3: constructing near-horizon geometries of solutions believed to exist, e.g. black rings in D > 5. Important open problem: integrating away from horizon = uniqueness theorems for supersymmetric black holes?