aa r X i v : . [ h e p - t h ] J un Blackfolds
Roberto Emparan
Instituci´o Catalana de Recerca i Estudis Avan¸cats (ICREA)andDepartament de F´ısica Fonamental and Institut de Ci`encies del Cosmos,Universitat de Barcelona, Mart´ı i Franqu`es 1, Barcelona, Spain
Abstract
This is an introduction to the blackfold effective worldvolume theory for the dynam-ics of black branes, as well as its use as an approximate method for the constructionof new black hole solutions. We also explain how the theory is useful for the analy-sis of dynamical, non-stationary situations, in particular of the Gregory-Laflammeinstability of black branes.
The existence of black p -branes in higher-dimensional General Relativityhints at the possibility of large classes of black holes without any four-dimensional counterpart. Black rings provide a nice explicit example: in[1] they were introduced as the result of bending a black string into theshape of a circle and spinning it up to balance forces. One can naturallyexpect that this heuristic construction extends to other black branes. If theworldvolume of a black p -brane could be similarly bent into the shape ofa compact hypersurface, for instance a torus T p or a sphere S p , we wouldobtain many new geometries and topologies of black hole horizons.Unfortunately, the techniques that allow to construct exact black holesolutions in four and five dimensions have not been successfully extended tomore dimensions. Still, one may want to hold on to the intuition that a longcircular black string, or more generally a smoothly bent black brane, couldbe obtained as a perturbation of a straight one.The experience with brane-like objects in other areas of physics suggeststhat such approximate methods may be efficiently applied to this problem.Consider, for instance, the Abelian Higgs theory and its familiar string-like vortex solutions. These are first obtained in the form of static, straightstrings, but one expects that they can also bend and vibrate. It has been long Chapter of the book
Black Holes in Higher Dimensions to be published by Cambridge UniversityPress (editor: G. Horowitz)
Blackfolds recognized that, if the wavelength of these deformations is much longer thanthe thickness of the vortex, the dynamics of the full non-linear theory is wellapproximated by the simpler Nambu-Goto worldsheet action. One can thenuse it for studying loops of strings of diverse shape. Another example is pro-vided by D-branes in string theory, which are defined as surfaces where openstrings can attach their endpoints. Although the bending and vibrations ofD-branes are generally intractable in an exact manner in string theory, theyare again very efficiently captured by the Dirac-Born-Infeld worldvolumefield theory, which is applicable as long as the scale of the deformations issufficiently large that locally the brane can still be well approximated as aflat D-brane. In all these cases, the full dynamics of the brane is replaced byan effective worldvolume theory for a set of “collective coordinates”.So, similarly, we seek an effective theory for describing black branes whoseworldvolume is not exactly flat, or not in stationary equilibrium, but wherethe deviations from the flat stationary black brane occur on scales muchlonger than the brane thickness. Black branes whose worldvolume is bentinto the shape of a submanifold of a background spacetime have been named blackfolds .This chapter introduces the blackfold effective worldvolume theory for thedynamics of black branes, as well as its use as an approximate method forthe construction of new black hole solutions [2, 3, 4]. Of special interest isa class of helical black rings that provide the first example of black holesin all D ≥ p -brane withworldvolume W p +1 embedded in a D -dimensional background spacetime, wedenote n = D − p − . (1.1)Spacetime indices µ, ν run in 0 , . . . , D and the covariant derivative compat-ible with the background metric g µν ( x ) is ∇ µ . Worldvolume indices a, b runin 0 , . . . , p , and the covariant derivative compatible with the metric γ ab ( σ )induced on W p +1 is D a . Effective theory for black hole motion Above we have motivated the blackfold effective approach by drawing analo-gies between black branes and the extended brane-like solutions of othernon-linear theories. However, the general-relativistic aspects of the effectivetheory of black p -brane dynamics are better introduced by considering firstthe simpler case of p = 0: the effective dynamics of a black hole that movesin a background whose curvature radii ∼ R are large compared to the blackhole horizon radius r , r ≪ R . (2.1)This separation of scales implies the existence of two distinct regions inthe geometry. First, there is a region around the black hole where the ge-ometry is well approximated by the Schwarzschild(-Tangherlini) solution. Ifin this “near zone” we choose a coordinate r centered at the black hole, theSchwarzschild geometry is a good approximation as long as r ≪ R , i.e., ds near ) = ds (Schwarzschild) + O ( r/R ) . (2.2)The corrections to the Schwarzschild metric are the (tidal) distortions thatthe background curvature creates on the black hole.The second region is far enough from the black hole that its effect on thebackground geometry is very mild and can be treated as a small perturba-tion. This is the “far zone” where r ≫ r , and in which we can expand ds far ) = ds (background) + O ( r /r ) . (2.3)Since we are too far from the black hole to resolve its size, effectively itis a pointlike source whose gravitational effect on the background can becomputed perturbatively. To this source we can assign an effective trajec-tory X µ ( τ ), with proper time τ and with velocity ˙ X µ = ∂ τ X µ such that g µν ˙ X µ ˙ X ν = −
1. We also ascribe to it an effective stress-energy tensor thatencodes how the black hole affects the gravitational field in the far zone.Let us determine the general form of this stress-energy tensor. We cannaturally assume that the acceleration and other higher derivatives of theparticle’s velocity are small, since they must be caused by the deviationsaway from flatness of the background in the region where the black holemoves. Since these variations occur on scales ∼ R , these higher-derivativeterms must be suppressed by powers of r /R . To leading order in this ex-pansion, the stress-energy tensor of the effective source is fixed by symmetryand worldline reparametrization invariance to have the form T µν = m ˙ X µ ˙ X ν . (2.4) Blackfolds
In principle the coefficient m can also depend on τ . This tensor is understoodto be localized at x µ = X µ ( τ ).In effect, we are replacing the entire near-zone geometry with a pointlikeobject. In the language of effective field theories, we are ‘integrating out’the short-wavelength degrees of freedom of the near-zone, and replacingthem with an effective worldline theory of a point particle. The coefficient m must then be related through a matching calculation to a parameter ofthe ‘microscopic’ configuration, which in this case can only be the horizonsize r . The matching condition is that the gravitational field of the effectivesource reproduces that of the black hole in the far zone. Thanks to theseparation of scales (2.1) the near and far zones overlap in r ≪ r ≪ R (2.5)so we can match the fields there. In this region, the near-zone Schwarzschildsolution is in a weak-field regime and can be linearized around Minkowskispacetime. On the other hand, the background curvature of the far-zonegeometry can be neglected in (2.5), so the far field is the linear perturbationof Minkowski spacetime sourced by (2.4). It is now clear that these two fieldsare the same if m is equal to the ADM mass of the Schwarzschild solutionwith horizon radius r , so m = ( D − D − πG r D − , (2.6)where Ω D − is the volume of the unit ( D − r .In this matching construction, a subset of the Einstein equations canbe written as equations for r ( τ ) and X µ ( τ ). Their derivation is a well-understood but technically complicated procedure. Fortunately, if we areonly interested in the leading order equations we can use a shortcut tothem by applying a main guiding principle of effective theories: symmetry Note that m = ˙ X µ ˙ X ν T µν = T ττ is proportional, but not necessarily equal, to the massmeasured at asymptotic infinity in the background. The relation is given by the redshiftbetween the particle’s proper time and the asymptotic time of the background. Effective blackfold theory and conservation principles. In this case the principle is general covariance,which imposes that ∇ µ T µν = 0 . (2.7)This is indeed naturally required for a source of the gravitational field in thefar zone. Put a bit more fancifully, this equation ensures the consistency ofthe coupling between short- and long-wavelength degrees of freedom.The overline in the covariant derivative in (2.7) takes into account thatthis makes sense only when projected along the effective particle trajectory, ∇ µ = − ˙ X µ ˙ X ν ∇ ν . (2.8)Nevertheless the equation (2.7) has components both in directions orthogo-nal and parallel to the particle’s velocity( g ρν + ˙ X ρ ˙ X ν ) ∇ µ T µν = 0 ⇒ ma µ = 0 , (2.9)˙ X ν ∇ µ T µν = 0 ⇒ ∂ τ m ( τ ) = 0 , (2.10)with a µ = D τ ˙ X µ = ˙ X ν ∇ ν ˙ X µ being the effective particle’s acceleration. Thefirst of these equations is the geodesic equation that determines the trajec-tory of a test particle. The second equation implies that m is a constantalong the trajectory.Geodesic motion is so familiar that the effective theory for a black holebecomes of real interest only when it includes the corrections to the leadingorder equations [7]. In contrast, we will see that for black p -branes with p > Our aim is to extend the effective worldline theory of black holes to a world-volume theory that describes the collective dynamics of a black p -brane. There is a long history of deriving the geodesic equation for a small particle (not necessarily ablack hole) from the Einstein field equations, see [8] for a recent rigorous version. This can betaken as confirmation of our generic symmetry argument.
Blackfolds
The geometry of a flat, static black p -brane in D spacetime dimensions is ds = − (cid:18) − r n r n (cid:19) dt + p X i =1 ( dz i ) + dr − r n r n + r d Ω n +1 . (3.1)Like in the previous example, the parameters of this solution include the‘horizon thickness’ r and the D − p − X ⊥ (making them explicit in the metric is possible butcumbersome). But now we must also include the possibility of a velocity u i along the worldvolume of the brane. A covariant form of the boosted blackbrane metric is obtained by first introducing the coordinates σ a = ( t, z i )that span the brane worldvolume with Minkowski metric η ab , and a velocity u a such that u a u b η ab = −
1. Then ds = (cid:18) η ab + r n r n u a u b (cid:19) dσ a dσ b + dr − r n r n + r d Ω n +1 . (3.2)Constant shifts of r , of u i , and of X ⊥ still give solutions to the Einsteinequations. In total, these are D zero-modes that yield D collective coordi-nates of the black brane. The long-wavelength effective theory describes fluc-tuations in which these variables change slowly on the worldvolume W p +1 ,over a large length scale R ≫ r . Typically R is set by the smallest extrinsiccurvature radius of W p +1 , or by the gradient of ln r along the worldvolume.Background curvatures may also be present but they are usually alreadyaccounted for by the extrinsic curvature they induce on W p +1 .With this variation the worldvolume metric deviates from the Minkowskimetric η ab , and the near-zone geometry is of the form ds = (cid:18) γ ab ( X µ ( σ )) + r n ( σ ) r n u a ( σ ) u b ( σ ) (cid:19) dσ a dσ b + dr − r n ( σ ) r n + r d Ω n +1 + . . . (3.3)where the dependence of γ ab on the transverse coordinates gives rise toextrinsic curvature of the worldvolume, and the dots indicate that additionalterms, of order O ( r /R ), are required for this to be a solution to Einstein’sequations. When r ≫ r this metric must match the geometry of the far-zone back-ground with metric g µν , in the region r ≪ R around the worldvolume of aninfinitely thin brane at x µ = X µ ( σ ). Thus we identify γ ab with the metric This is very similar to the long-wavelength perturbation of anti-deSitter black branes studiedin [6].
Effective blackfold theory induced on the effective brane worldvolume γ ab = g µν ∂ a X µ ∂ b X ν . (3.4)Again, we are replacing the near-zone geometry with an infinitely thin p -brane, with worldvolume W p +1 , embedded in the background geometry. The stress-energy tensor of the black brane is, like the mass m of the blackhole in the previous example, computed in the overlap region r ≪ r ≪ R where the deviations away from Minkowski spacetime are small. It canbe obtained as a generalization of the ADM mass, or equivalently, fromthe Brown-York quasilocal stress-energy tensor [10]. This is computed ona timelike surface with induced metric ˜ h µν (not to be confused with h µν below) and extrinsic curvature Θ µν , as T ( ql ) µν = 18 πG (cid:16) Θ µν − ˜ h µν Θ (cid:17) . (3.5)When measured at constant r ≫ r , the divergent contributions to thistensor, which grow with r , can be subtracted in any of the conventionalways; for instance, the method of background subtraction from Minkowskispace is enough for our purposes. Then, equivalently, this is the stress-energytensor of a domain wall that encloses empty space and creates a field outsideit equal to that of the black brane. This interpretation fits well with the ideathat we replace the black brane with an effective source.The surface at large constant r where the quasilocal T ( ql ) µν is computedhas geometry R ,p × S n +1 . We integrate it over the sphere to obtain thestress-energy tensor of the black p -brane T ab = Z S n +1 T ( ql ) ab , (3.6)with components along the worldvolume directions.For the boosted black p -brane (3.2) the result of this calculation is T ab = Ω ( n +1) πG r n (cid:16) nu a u b − η ab (cid:17) . (3.7)This is the stress-energy tensor of an isotropic perfect fluid, T ab = ( ε + P ) u a u b + P η ab , (3.8)where the energy density ε and pressure P are ε = Ω ( n +1) πG ( n + 1) r n , P = − Ω ( n +1) πG r n . (3.9) Blackfolds
In the rest frame of the fluid, and at any given point on the worldvolume, theBekenstein-Hawking identification between horizon area and entropy appliesto the black hole obtained by compactifying the p directions along the brane.Thus we identify an entropy density from the horizon area density of (3.1), s = Ω ( n +1) G r n +10 . (3.10)Locally, we also have the conventional relation between surface gravity andtemperature T = n πr , (3.11)in such a way that the first law of black hole thermodynamics applies in thelocal form dε = T ds . (3.12)In addition, the Euler-Gibbs-Duhem relation ε + P = T s (3.13)is verified.After introducing a slow variation of the collective coordinates, the stress-energy tensor becomes T ab ( σ ) = Ω ( n +1) πG r n ( σ ) (cid:16) nu a ( σ ) u b ( σ ) − γ ab ( σ ) (cid:17) + . . . (3.14)where the dots stand for terms with gradients of ln r , u a , and γ ab , respon-sible for dissipative effects that we are taking to be small. We neglect themfor now, but will return to some of them in section 6. In order to obtain the equations for the collective variables we need to recalla few notions about the geometry of worldvolume embeddings. More detailsand proofs are provided in the appendix.
Worldvolume geometry
The worldvolume W p +1 is embedded in a background with metric g µν , andits induced metric is (3.4). Indices µ , ν are raised and lowered with g µν , and a, b with γ ab . The first fundamental form of the submanifold h µν = ∂ a X µ ∂ b X ν γ ab (3.15) Effective blackfold theory acts as a projector onto W p +1 (for a worldline, h µν = − ˙ X µ ˙ X ν ), and thetensor ⊥ µν = g µν − h µν (3.16)projects along directions orthogonal to it.Background tensors t µ...ν... can be converted into worldvolume tensors t a...b... and viceversa using the pull-back map ∂ a X µ , a relevant example beingthe stress-energy tensor T µν = ∂ a X µ ∂ b X ν T ab . (3.17)The covariant differentiation of these background tensors is well definedonly along tangential directions, which we denote by an overline, ∇ µ = h µν ∇ ν . (3.18)The divergence of the stress-energy tensor, projected parallel to W p +1 , sat-isfies (see (A.9)) h ρν ∇ µ T µν = ∂ b X ρ D a T ab . (3.19)The extrinsic curvature tensor K µν ρ = h µσ ∇ ν h σρ = − h µσ ∇ ν ⊥ σρ (3.20)is tangent to W p +1 along its (symmetric) lower indices µ , ν , and orthogonalto W p +1 along ρ . Its trace is the mean curvature vector K ρ = h µν K µν ρ = ∇ µ h µρ . (3.21)A useful result is that for any two vectors s µ and t µ tangent to W p +1 , s µ t ν K µν ρ = ⊥ ρµ ∇ s t µ = ⊥ ρµ ∇ t s µ . (3.22) Blackfold equations
The classical dynamics of a generic brane has been studied by Carter in [11].The equations are formulated in terms of a stress-energy tensor supportedon, and tangent to, the p + 1-dimensional brane worldvolume W p +1 , ⊥ ρµ T µν = 0 . (3.23)As in the example of p = 0, general covariance implies that the stress-energy tensor must obey the equations ∇ µ T µρ = 0 . (3.24)This is a consequence of the underlying conservative dynamics of the full Blackfolds
General Relativity theory, but the effective worldvolume theory may be dis-sipative.The divergence in (3.24) can be written as ∇ µ T µρ = ∇ µ ( T µν h νρ ) = T µν ∇ µ h ν ρ + h ν ρ ∇ µ T µν = T µν K µν ρ + ∂ b X ρ D a T ab , (3.25)where in the last line we used (3.19) and (3.20). Thus the D equations (3.24)separate into D − p − W p +1 and p + 1equations parallel to W p +1 , T µν K µνρ = 0 (extrinsic equations) , (3.26) D a T ab = 0 (intrinsic equations) . (3.27)The extrinsic equations can be regarded as a generalization to p -branes ofthe geodesic equation (2.9) (where the acceleration is the extrinsic curvatureof the worldline, a ρ = − K ρ ). In other words, this is the generalization ofNewton’s equation “mass × acceleration= 0” to relativistic extended objects.The second set of equations, (3.27), express energy-momentum conservationon the worldvolume. For a black hole this was a rather trivial equation, butfor a p -brane we get all the complexity of the hydrodynamics of a perfectfluid.If we insert the stress-energy tensor of the black brane (3.7) and use (3.22),the extrinsic equations (3.26) become K ρ = n ⊥ ρµ ˙ u µ , (3.28)and the intrinsic equations (3.27),˙ u a + 1 n + 1 u a D b u b = ∂ a ln r . (3.29)Here ˙ u µ = u ν ∇ ν u µ and ˙ u b = u c D c u b .Eqs. (3.28) and (3.29) are the blackfold equations , a set of D equations thatdescribe the dynamics of the D collective variables of a neutral black brane,in the approximation where we neglect its backreaction on the background(‘test brane’) as well as the dissipative effects on its worldvolume. The worldvolume of the black p -brane may have boundaries specified by afunction f ( σ a ) such that f | ∂ W p +1 = 0. If the effective fluid remains withinthese boundaries, the velocity must remain parallel to them, u a ∂ a f | ∂ W p +1 = 0 . (3.30) Stationary blackfolds, action principle and thermodynamics If the boundary is ‘free’, i.e., there is no surface tension, then the Euler(force) equations for a generic perfect fluid require that the pressure vanishesat the boundary. For the black brane this is r (cid:12)(cid:12) ∂ W p +1 = 0 . (3.31)Geometrically, this means that the horizon must approach zero size at theboundary, so the horizon closes off at the edge of the blackfold. The blackfold construction puts, on any point in the worldvolume W p +1 , a(small) transverse sphere s n +1 with Schwarzschild radius r ( σ ). If B p is aspatial section of W p +1 , then the geometry of the horizon is the product of B p and s n +1 — the product is warped since the radius r ( σ ) of s n +1 variesalong B p . If r is non-zero everywhere on B p then the s n +1 are triviallyfibered on B p and the horizon topology is the product topology of B p andthe sphere.The regularity of this horizon in the perturbative expansion, in whichit is distorted by long-wavelength fluctuations, is believed to be satisfiedwhen the blackfold equations, which incorporate local thermodynamic equi-librium, are satisfied. A complete proof is still lacking, but refs. [12] and [13]provide evidence that this is true, respectively, for extrinsic and intrinsicdeformations.As we have seen, at boundaries of B p the size of s n +1 vanishes, so thehorizon topology will be different. The regularity of the horizon at theseboundaries is not fully understood yet and appears to depend on the specifictype of boundary. We will return to this issue later. Equilibrium configurations for blackfolds that remain stationary in timeare of particular interest since they correspond to stationary black holes.Requiring stationarity allows to solve explicitly the intrinsic equations for thethickness r and velocity u a , so one is left only with the extrinsic equationsfor the worldvolume embedding X µ ( σ ). For a fluid configuration to be stationary, dissipative effects must be absent.In general, this requires that the shear and expansion of its velocity field Blackfolds u vanish. One can then prove, using the fluid equations, that u must beproportional to a (worldvolume) Killing field k = k a ∂ a . That is, u = k | k | , | k | = p − γ ab k a k b (4.1)where k satisfies the worldvolume Killing equation D ( a k b ) = 0. Actually, wewill assume that this Killing vector on the worldvolume is the pull-back ofa timelike Killing vector k µ ∂ µ in the background, ∇ ( µ k ν ) = 0 , (4.2)such that k a = ∂ a X µ k µ . The existence of this timelike Killing vector fieldis in fact a necessary assumption if we intend to describe stationary blackholes.The Killing equation (4.2) implies ∇ ( µ u ν ) = − u ( µ ∇ ν ) ln | k | , (4.3)so the acceleration is ˙ u µ = ∂ µ ln | k | . (4.4)Since the expansion of u vanishes, the intrinsic equation (3.29) becomes ∂ a ln | k | = ∂ a ln r (4.5)so r | k | = constant . (4.6)Expressed in terms of the local temperature T , (3.11), this equation saysthat the worldvolume variation of the temperature is dictated by the localredshift factor | k | − , T ( σ ) = T | k | . (4.7)This result can also be derived for a general fluid using the equations offluid dynamics. The integration constant T can be interpreted, using thethermodynamic first law that we derive below, as the global temperature ofthe black hole. Equation (4.6) can be read as saying that the thickness r ( σ ) = n πT | k | (4.8)adjusts its value along the worldvolume so that T is a constant. Stationary blackfolds, action principle and thermodynamics With the intrinsic solutions (4.4) and (4.6), the extrinsic equations (3.28)reduce to K ρ = n ⊥ ρµ ∂ µ ln r = ⊥ ρµ ∂ µ ln( − P ) . (4.9)Using eq. (A.16) from the appendix, this equation can be equivalentlyfound by varying, under deformations of the brane embedding, the action I = Z W p +1 d p +1 σ √− h P . (4.10)This action, whose derivation actually need not assume any specific fluidequation of state, is a familiar one for branes with constant tension − P ,whose worldvolume must be a minimal hypersurface so K ρ = 0 (an exampleare Dirac-Born-Infeld branes with zero gauge fields on their worldvolume).More generally, this is the action of a perfect fluid on W p +1 .Assume now that the background spacetime has a timelike Killing vector ξ , canonically normalized to generate unit time translations at asymptoticinfinity, and whose norm on the worldvolume is − ξ (cid:12)(cid:12) W p +1 = R ( σ ) . (4.11)Let us further assume that ξ is hypersurface-orthogonal, so we can foliatethe blackfold in spacelike slices B p normal to ξ . The unit normal to B p is n a = 1 R ξ a . (4.12) R measures the local gravitational redshift between worldvolume time andasymptotic time. Integrations over W p +1 reduce, over an interval ∆ t of theKilling time generated by ξ , to integrals over B p with measure dV ( p ) , so I = ∆ t Z B p dV ( p ) R P . (4.13)Using (4.8) in (3.9) we get an expression for the action in terms of k that isvery practical for deriving the extrinsic equations in explicit cases,˜ I [ X µ ( σ )] = Z B p dV ( p ) R | k | n . (4.14)The tilde distinguishes it from (4.13), since we have removed an overall con-stant factor (including a sign) that is irrelevant for obtaining the equations. Blackfolds
Let k be given by a linear combination of orthogonal commuting Killingvectors of the background spacetime, k = ξ + X i Ω i χ i , (4.15)where ξ is the generator of time-translations that we introduced above, and χ i are generators of angular rotations in the background, normalized suchthat their orbits have periods 2 π . Then Ω i are the angular velocities of theblackfold along these directions.The mass and angular momenta of the blackfold are now given by theintegrals of the energy and momentum densities over B p , M = Z B p dV ( p ) T ab n a ξ b , J i = − Z B p dV ( p ) T ab n a χ bi . (4.16)The total entropy is deduced from the entropy current s a = s ( σ ) u a , S = − Z B p dV ( p ) s a n a = Z B p dV ( p ) R | k | s ( σ ) . (4.17)Let us now express the action (4.13) in terms of these quantities. Con-tracting the stress-energy tensor (3.8) with n a k b , then using (3.13) and (4.7),we find T ab n a ξ b + X i Ω i χ bi ! + T su a n a = n a k a P = − R P , (4.18)so, integrating over B p , I = − ∆ t M − T S − X i Ω i J i ! . (4.19)This is an action in real Lorentzian time, but since we are dealing withtime-independent configurations we can rotate to Euclidean time with peri-odicity 1 /T and recover the relation between the Euclidean action and thethermodynamic grand-canonical potential, I E = W [ T, Ω i ] /T .The identity (4.19) holds for any embedding, not necessarily a solution tothe extrinsic equations, so if we regard M , J i and S as functionals of the X µ ( σ ) and consider variations where T and Ω i are held constant, we have δI [ X µ ] δX µ = 0 ⇔ δMδX µ = T δSδX µ + X i Ω i δJ i δX µ . (4.20) Stationary blackfolds, action principle and thermodynamics Therefore, the extrinsic equations are equivalent to the requirement that thefirst law of black hole thermodynamics holds for variations of the embedding.Eq. (4.19), Wick-rotated to I E [ X µ ], is therefore an effective worldvolumeaction that approximates, to leading order in r /R , the Euclidean gravita-tional action of the black hole. One might have taken this thermodynamiceffective action as the starting point for the theory of stationary blackfolds,but we have preferred to work with the equations of motion. These allowto consider situations away from stationary equilibrium and they also makemore explicit the connection with worldvolume fluid dynamics.Performing manipulations similar to (4.18) one finds that( D − M − ( D − T S + X i Ω i J i ! = T tot , (4.21)where T tot = − Z B p dV ( p ) R (cid:16) γ ab + n a n b (cid:17) T ab (4.22)is the total tensional energy, obtained by integrating the local tension overthe blackfold volume.The Smarr relation for asymptotically flat vacuum black holes in D di-mensions [14], ( D − M − ( D − T S + X i Ω i J i ! = 0 (4.23)must be recovered when the extrinsic equations for equilibrium are satisfiedfor a blackfold with compact B p in a Minkowski background where R = 1.Thus, extrinsic equilibrium in Minkowski backgrounds implies T tot = 0 . (4.24)If the tensional energy did not vanish, it would imply the presence of sourcesof tension acting on the blackfold, e.g., in the form of conical or strongersingularities of the background space.Another general identity is obtained by noticing that the blackfold fluidsatisfies − P = 1 n T s . (4.25)Upon integration and using (4.19) we get M − T S − X i Ω i J i = 1 n T S , (4.26) Blackfolds or, in terms of the Euclidean action, I E = S/n .Note that while the thermodynamic form of the action (4.19) and theSmarr relation (4.23) are exactly valid for neutral black holes, eqs. (4.21)and (4.26) instead hold only to leading order in the expansion in r /R . Let us now investigate what it means, for a stationary blackfold, that thethickness vanishes at its boundary, eq. (3.31).In section 4.2 we have introduced the generators of unit time translationsat asymptotic infinity, ξ a , and on W p +1 , n a , which are related by the factor R that measures the gravitational redshift between these two locations. Onthe other hand, relative to the worldvolume time generated by n a , a fluidelement on W p +1 has a Lorentz-boost gamma factor equal to − n a u a = 1 √ − v (4.27)where v is the local fluid velocity, v = X i v i , v i = Ω i | χ i | R . (4.28)Since the velocity u a is determined by (4.1) and (4.15), then ξ a u a = ξ a ξ a / | k | = − R / | k | , and − n a u a = − R ξ a u a = R | k | . (4.29)With (4.27), this implies | k | = R p − v . (4.30)At a blackfold boundary we must have r →
0. According to (4.6), ifthe blackfold is stationary it must also be that | k | →
0. There are twopossibilities:(i) v →
1: the fluid velocity becomes null at the boundary. There is someevidence that in this case the full horizon is smooth: as we will see insection 5.1, there are blackfolds with this kind of boundary for which wecan compare to an exact black hole solution with a regular horizon.(ii) R →
0: the blackfold encounters a horizon of the background, where thegravitational redshift diverges. This boundary can be regarded as an inter-section of horizons, and the evidence from known exact solutions indicatesthat the intersection point, i.e., the blackfold boundary, is singular.
Examples of blackfold solutions Nevertheless, the evidence for these two behaviours is largely circumstan-tial and it would be desirable to have a better understanding of horizonregularity at blackfold boundaries.
Let us assume that all length scales along B p are of the same order ∼ R andthat there are no large redshifts, of gravitational or Lorentz type, over mostof the blackfold — this is naturally satisfied since the redshifts become largeonly near the boundaries. Then (setting temporarily G = 1) eqs. (4.16) and(4.17) generically give M ∼ R p r n , J ∼ R p +1 r n , S ∼ R p r n +10 . (4.31)This implies JM ∼ R , (4.32)so that neutral blackfolds are always in an ultraspinning regime, in which theangular momentum for fixed mass is very large. More precisely, in a neutralblackfold the length scale of angular-momentum (4.32) is always much largerthan the length scale of the mass M / ( D − , J/MM / ( D − ∼ (cid:16) r R (cid:17) D − p − D − ≪ . (4.33)The entropy in (4.31) scales like S ( M, J ) ∼ J − pD − p − M D − D − p − , (4.34)so, in dimension D and for fixed mass, the most entropic solution for agiven number of large angular momenta J is attained by blackfolds withthe smallest p . The intuitive reason is that, for a given mass, the horizon isthicker if p is smaller — the horizon spreads out less. A thicker horizon haslower temperature, and since T S ∼ M , the entropy is higher. In section 5.3we will find that there is always a black 1-fold for any number of angularmomenta, which therefore maximizes the entropy. This formalism can be applied easily to the explicit construction of sta-tionary black holes. Besides finding new solutions, we will show that theblackfold method correctly recovers the limit in which known exact solu-tions become similar to black branes — these are the ultraspinning regime Blackfolds of Myers-Perry black holes and the very-thin limit of the five-dimensionalblack ring, which provide non-trivial checks on the method.
Myers-Perry black holes have ultraspinning regimes where the geometry nearthe rotation axis approaches that of a black brane spread along the rotationplane [15]. This suggests that these regimes may be reproduced by blackfoldconstructions, and indeed they are, in a rather non-trivial manner. Insteadof studying the most general construction (see [4]), we will illustrate it in thecase of the six-dimensional black hole rotating along a single plane, whichalready exhibits all the relevant features.Take D = 6 Minkowski spacetime as a background and a black 2-foldthat extends along a plane within it (so n = 1). The extrinsic equations aretrivially solved, and we can restrict the analysis to the blackfold plane withpolar coordinates ( r, φ ) and metric ds = − dt + dr + r dφ . (5.1)We set the blackfold in rotation along φ . Stationarity requires that the fluidrotates rigidly, eqs. (4.1) and (4.15) with ξ = ∂/∂t , χ = ∂/∂φ and angularvelocity Ω, so the velocity is u = 1 √ − Ω r (cid:18) ∂∂t + Ω ∂∂φ (cid:19) . (5.2)The intrinsic equations are solved by appropriately redshifting the temper-ature, which determines the horizon thickness as in (4.8), r ( r ) = 14 πT p − Ω r . (5.3)This implies that the extent of the blackfold along the rotation plane islimited to 0 ≤ r ≤ Ω − . (5.4)Recall that according to eq. (3.31) the condition r (Ω − ) = 0 specifies aboundary of the blackfold. The effective fluid velocity becomes lightlikethere, and would be superluminal if we tried to go beyond this radius. There-fore the blackfold worldvolume is a disk.It will be convenient to introduce two geometric parameters in place of T and Ω: the disk radius a , and the blackfold thickness r + at the rotationaxis, a = Ω − , r + = r (0) = 14 πT . (5.5) Examples of blackfold solutions What is the topology of the horizon of this blackfold? The disk is fiberedat each point with a sphere S of radius r ( r ) that shrinks to zero at thedisk’s edge. Topologically, this is S , i.e., the same as the topology of thehorizon of the Myers-Perry black hole.The mass and angular momentum of the blackfold are obtained from itsenergy and momentum densities, T ab n a ξ b = T tt = r + G − r /a p − r /a , (5.6) − T ab n a χ b = T tφ = r + G r /a p − r /a (5.7)(since n a = ξ a ). Then M = Z π dφ Z a dr r T tt = 2 π G r + a , (5.8) J = Z π dφ Z a dr r T tφ = π G r + a . (5.9)The entropy-density current s a = s u a gives − s a n a = πG r p − r /a (5.10)which integrates to S = − Z π dφ Z a dr r s a n a = 2 π G r a . (5.11)Let us now compare to the ultraspinning limit of the exact Myers-Perryblack hole. This solution is specified by a mass parameter µ and a rotationparameter a . The horizon radius r + is obtained as the largest real root of µ = ( r + a ) r + . (5.12)In terms of these, the exact mass, spin and entropy are M = 2 π G µ , J = 12 aM , S = 2 π G r + µ . (5.13)The ultraspinning regime of J → ∞ with fixed M is obtained as a → ∞ . Inthis limit eq. (5.12) becomes µ → a r + , (5.14)and the physical quantities (5.13) become precisely the same as (5.8), (5.9),(5.11), after identifying the parameters r + and a in both sides. Blackfolds
This identification of parameters is in fact geometrically meaningful. Foran ultraspinning black hole the radii of the horizon in directions transverseand parallel to the rotation plane are [15] r MP ⊥ → r + cos θ , r MP k → a , (5.15)where θ is the polar angle on the horizon. For the blackfold disk, we havealready seen that the radius in the plane parallel to the rotation is r bf k = a .The square root in (5.3) suggests to introduce a polar angle θ = arcsin( r/a )such that the thickness of the blackfold in directions transverse to the rota-tion plane is r bf ⊥ = r ( r ) = r + cos θ . (5.16)Identifying this polar angle with the one in the Myers-Perry solution, wefind a perfect match between both sides.Since the horizon of the Myers-Perry solution is smooth, this exampleyields evidence of regularity at the boundary of the blackfold where thefluid velocity becomes lightlike. We now consider stationary black 1-folds (i.e., curved black strings) in aMinkowski background ds = − dt + d x D − (5.17)with stationarity Killing vector ξ = ∂/∂t so R = 1.Stationarity requires that the black string lies along a spatial Killing direc-tion. Then the simplest way to solve the extrinsic equations is by imposingthat the tensional energy vanishes, eq. (4.24). Since the string lies alongan isometry, the integral in (4.22) is trivial and so the integrand, i.e., thetension measured relative to the frame defined by ξ , must vanish:tension = − (cid:16) γ ab + ξ a ξ b (cid:17) T ab = 0 . (5.18)Let us denote − ξ a u a = cosh σ (5.19)so σ is the rapidity that parametrizes the relative boost between the fluidand the background frame of observers along orbits of ξ . For a generic perfectfluid, the zero-tension condition (5.18) is0 = (cid:16) γ ab + ξ a ξ b (cid:17) (cid:16) ( ε + P ) u a u b + P γ ab (cid:17) = ε sinh σ + P cosh σ (5.20) Examples of blackfold solutions so at equilibrium the fluid velocity tanh σ is fixed to the valuetanh σ = − Pε . (5.21)Since − P/ε is actually the square of the velocity of transverse, elastic wavesalong the string, we see that the bending of the string can be regarded asthe effect of supporting a stationary elastic wave along itself.For the specific black string fluid (3.9), the condition (5.21) can be writtenas sinh σ = 1 n . (5.22)When n = 1, i.e., in D = 5, this is precisely the value for the boost that wasfound in [1] from the exact black ring solution in the limit where the ringis very thin and long, r /R →
0. In this case the black string lies along acircle of radius R on a plane within R .It is straightforward to calculate (4.16) and (4.17) and check that, withthis value of the boost, the blackfold construction reproduces all the phys-ical magnitudes of the exact black ring solution to leading order in r /R .Beyond this order, ref. [12] has computed the first corrections to the metric,finding again perfect agreement with the perturbative expansion of the ex-act solution in D = 5, and checking that horizon regularity is preserved inany D ≥ The previous construction did not specify yet the geometry of the curvealong which the string lies. However, as we saw, this must be along a spatialisometry of Euclidean space R D − . If we are interested in blackfolds of finiteextent, which correspond to asymptotically flat black objects, this isome-try must be compact and therefore generated by rotational Killing vectors ∂/∂φ i . This is, if we write the subspace of R D − in which the embedding ofthe string is non-trivial as ds = m X i =1 (cid:0) dr i + r i dφ i (cid:1) , (5.23)then the string lies along the curve r i = R i , φ i = n i σ , ≤ σ < π . (5.24)An upper limit m ≤ (cid:4) D − (cid:5) = (cid:4) n +32 (cid:5) is set by the rank of the spatial rotationgroup in D = n + 4 spacetime dimensions. The D − m − Blackfolds space that are not explicit in (5.23) are totally orthogonal to the string andwe will ignore them. They only play a role in providing, together with the m directions r i , the n + 2 dimensions orthogonal to the worldsheet in whichthe horizon of the black string is ‘thickened’ into a transverse s n +1 of radius r .The n i must be integers in order that the curve closes in on itself. Withoutloss of generality we assume n i ≥
0. If we want to avoid multiple coveringof the curve then the smallest of the n i (which need not be unique), say n ,must be coprime with all the n i . Thus the set of n i can be specified by m positive rational numbers n i /n ≥ n i = 1 we obtain a circular planar ring along an orbit of P i ∂ φ i withradius qP i R i . If, instead, at least two n i are non-zero and not both equalto one then we obtain helical black rings . Together with planar black rings,this exhausts all possible stationary black 1-folds in a Minkowski backgroundwith a spatially compact worldsheet.The Killing generator of the worldsheet velocity field is k = ∂∂t + X i Ω i ∂∂φ i , (5.25)where the ratios between angular velocities must be rational (cid:12)(cid:12)(cid:12)(cid:12) Ω i Ω j (cid:12)(cid:12)(cid:12)(cid:12) = n i n j ∀ i, j , (5.26)and the equilibrium condition (5.22) fixes m X i =1 Ω i R i = 1 n + 1 . (5.27)The physical properties (mass, spins, entropy etc) of helical black ringsin the approximation r /R i ≪ n i , since this makes the ring shorter and hencethicker. For a single angular momentum, the planar black ring maximizesthe entropy. Examples of blackfold solutions Maximal symmetry breaking and saturation of the rigidity theorem
A spacetime containing a helical ring has the isometry generated by X i n i ∂∂φ i (5.28)along the direction of the string. However, this string breaks in general other U (1) isometries of the background, possibly leaving (5.28) as the only spatialKilling vector of the configuration.In order to prove this point, observe that any additional U (1) symmetrymust leave the curve invariant, i.e., the curve must lie at a fixed point of theisometry. This is, the curve must be on a point in some plane in R D − , sorotations in this plane around the point leave it invariant. Let us parametrizethe most general possible helical curve in R D − as a curve in C m , with m = (cid:4) D − (cid:5) , of the form z i = R i e in i σ , (5.29)where possibly some of the n i are zero. In order to find a plane in whichrotations leave the curve invariant we must solve the equation m X i =1 a i z i = 0 (5.30)with complex a i , for all values of σ . This equation admits a non-trivialsolution only if some of the n i are equal to each other (possibly zero).Therefore, if (i) the string circles around in all of the m = (cid:4) D − (cid:5) inde-pendent rotation planes, i.e., all the possible n i are non-zero, and (ii) all the n i ’s are different from each other, then the only spatial Killing vector of theconfiguration is (5.28). In this case, we obtain an asymptotically flat helicalblack ring with only one spatial U (1) isometry.The black hole rigidity theorem of [16, 17] requires that stationary non-static black holes have at least one such isometry, and ref. [18] had conjec-tured that black holes exist with not more than this symmetry. The con-struction of helical black rings that rotate in all possible planes and haveexactly one spatial U (1) proves this conjecture in any D ≥ Blackfolds
For our final example, we describe a large family of solutions for black holesin D -dimensional flat space with horizon topology (cid:16) Y p a =odd S p a (cid:17) × s n +1 , ℓ X a =1 p a = p . (5.31)In this case, the spatial section of the blackfold worldvolume B p is a productof odd-spheres. S k +1 blackfolds We consider first a single odd-sphere S k +1 , which contains the black ringas the particular case k = 0. We embed the sphere S k +1 into a (2 k + 2)-dimensional flat subspace of R D − with metric dr + r d Ω k +1 . (5.32)The unit metric on S k +1 can be written as d Ω k +1 = k +1 X i =1 (cid:0) dµ i + µ i dφ i (cid:1) , k +1 X i =1 µ i = 1 , (5.33)where φ i are the angles that parametrize the Cartan subgroup of the rotationgroup SO (2 k + 2).In the general stationary case we would consider the blackfold embeddedas r = R ( µ , . . . , µ k ). Then the extrinsic equations give a set of differentialequations for R ( µ i ) involving k angular velocities Ω i . These are complicatedto solve, but they simplify to algebraic equations in the still non-trivialinstance that the sphere is geometrically round with constant radius r = R , and rotates with the same angular velocity Ω in all angles. Then thestationarity Killing vector is k = ∂∂t + Ω k +1 X i =1 ∂∂φ i , | k | = p − Ω R , (5.34)and the blackfold is homogeneous so the thickness r is constant over theworldvolume.The extrinsic equations for R can be easily (and consistently) obtainedfrom the stationary blackfold action (4.14)˜ I [ R ] = Ω ( p ) R p (cid:0) − R Ω (cid:1) n , p = 2 k + 1 . (5.35) Examples of blackfold solutions This is extremized when R = 1Ω r pn + p . (5.36)Equivalently, this value makes the local tension (5.18) vanish at each point.When p = 1 we recover the result for black rings. Having this equilibriumradius of round blackfolds, it is straightforward to compute their physicalproperties [4]. Products of odd-spheres
This construction can be easily extended to solutions where B p is a productof round odd-spheres, each one labeled by an index a = 1 , . . . , ℓ . Denotingthe radius of the S p a factor ( p a =odd) by R a we take the angular momentaof the a -th sphere to be all equal to Ω ( a ) .We embed the product of ℓ odd-spheres in a flat ( p + ℓ )-dimensional sub-space of R D − with metric ℓ X a =1 (cid:16) dr a + r a d Ω p a ) (cid:17) , ℓ X a =1 p a = p (5.37)and locate the blackfold at r a = R a . Note that given a value of n = D − p − ≥
1, the number ℓ of spheres in the product is limited to ℓ ≤ n + 2.With the spheres being geometrically round, the stationary blackfold ac-tion (4.14) reduces to˜ I [ R a ] = ℓ Y b =1 Ω ( p b ) R p b b − ℓ X a =1 (cid:16) R a Ω ( a ) (cid:17) ! n/ , (5.38)whose variation with respect to each of the R a ’s gives the equilibrium con-ditions R a = 1Ω ( a ) r p a n + p . (5.39)A simple case is the p -torus, where we set all p a = 1. This gives blackholes with horizon topology T p × s n +1 that rotate simultaneously along allorthogonal one-cycles of T p .It is easy to see that, like the Myers-Perry black holes and the planarblack rings, these odd-sphere solutions do not break any of the commutingisometries of the background.To finish this section we note that, except for the case of black 1-folds, ouranalysis has not been systematic enough to be complete, and further classesof black holes can be expected in D ≥
6. But already with the ones we Blackfolds have presented, one can easily see that black hole uniqueness is very badlyviolated in higher-dimensions.
The blackfold approach must capture the perturbative dynamics of a blackhole when the perturbation along the horizon has long wavelength λ , λ ≫ r . (6.1)This includes in particular intrinsic fluctuations of the black brane in whichthe worldvolume geometry remains flat but r and u a are allowed to vary.A variation of the thickness of the brane, δr , is a variation of the pressureand density of the effective fluid. Then, for small fluctuations we expect torecover sound waves along the brane. These turn out to be unstable in aninteresting way. Sound waves are easily derived for a generic perfect fluid. Introduce smallperturbations in an initial uniform state at rest, ε → ε + δε , P → P + dPdε δε , u a = (1 , . . . ) → (1 , δu i ) . (6.2)To linear order in the perturbations the intrinsic fluid equations (3.27) give (cid:18) ∂ t − dPdε ∂ i (cid:19) δε = 0 , (6.3)so longitudinal, sound-mode oscillations of the fluid propagate with squaredspeed v s = dPdε . (6.4)Neutral blackfolds have imaginary speed of sound v s = − n + 1 , (6.5)which implies that sound waves along the effective black brane fluid are un-stable: under a density perturbation the fluid evolves to become more andmore inhomogenous. Thus the black brane horizon itself becomes inhomo-geneous, with the brane thickness r varying along the brane as δr ∼ e Ω t + ik i z i , (6.6) Gregory-Laflamme instability and black brane viscosity with Ω = k √ n + 1 + O ( k ) . (6.7)( k = √ k i k i ). Unstable oscillations of the form (6.6) are the type of blackbrane instability discovered by Gregory and Laflamme (see [5]). Using theblackfold effective theory, we have derived it in the regime of long wave-lengths, kr ≪
1. Many studies of the Gregory-Laflamme instability focuson the threshold mode, with Ω = 0 at k = k GL = 0, which has ‘small’wavelength 2 π/k GL ∼ r and typically needs numerical work to determine.The blackfold approach instead reveals that the hydrodynamic modes, whichhave vanishing frequency as k →
0, are much simpler to study. The slope ofthe curve Ω( k ) near k = 0 is exactly determined using only the equation ofstate P ( ε ) of the unpertubed, static black brane. It is conventional in fluid dynamics to express the speed of sound in termsof thermodynamic quantities. Using the Gibbs-Duhem relation dP = sd T one finds dPdε = s d T dε = sc v , (6.8)where c v is the isovolumetric specific heat. So the black brane is dynamicallyunstable to long wavelength, hydrodynamical perturbations, if and only ifit is locally thermodynamically unstable, c v <
0. The ‘correlated stabilityconjecture’ of Gubser and Mitra [19] posits precisely this type of connectionbetween dynamical and thermodynamic stability. The blackfold method notonly shows very simply why it holds for hydrodynamic modes, but it alsogives a quantitative expression for the unstable frequency in terms of localthermodynamics as Ω = r s − c v k + O ( k ) . (6.9) The previous analysis of the sound-wave instability employed the stress-energy tensor of eq. (3.7), which gives the perfect fluid approximation to theintrinsic dynamics of the black brane. This tensor was obtained from thestationary metric of the black brane. If we perturb this brane, it will vibrate Blackfolds in its quasinormal modes, with damped oscillations. The stress-energy ten-sor measured at large distance r ≫ r from the black p -brane will reflectthis damping through the appearance of dissipative terms, proportional toderivatives of u a (the derivatives of r are proportional to these), so that T ab = T ( perfect ) ab − ζθP ab − ησ ab + O ( D ) . (6.10)Here the orthogonal projector, expansion and shear of the velocity congru-ence are P ab = η ab + u a u b , θ = D a u a , (6.11) σ ab = P ac P bd (cid:18) D ( c u d ) − p θP cd (cid:19) , (6.12)and the coefficients η and ζ are the effective shear and bulk viscosity of theblack brane. They can be computed from a perturbative calculation verysimilar to those in the context of the fluid/AdS-gravity correspondence of[6]. For the neutral black brane in asymptotically flat space this calculationhas been carried out in [13], with the result that η = s π , ζ = s π (cid:18) p − v s (cid:19) , (6.13)where s is the entropy density of the black brane (3.10) and v s the speed ofsound (6.5).Now we can solve the fluid equations (3.27) for linearized sound-modeperturbations of the viscous fluid, and obtain the leading corrections to thedispersion relation (6.7) at order k . For the black brane fluid the result isΩ = k √ n + 1 (cid:18) − n + 2 n √ n + 1 kr (cid:19) , (6.14)which is valid up to corrections O ( k ). We see that viscosity has the expectedeffect of damping the sound waves. Figure 1. compares this dispersion rela-tion to the numerical results obtained by solving the linearized perturbationsof a black string.Eq. (6.14) gives excellent agreement to the numerical data for small kr ,but it also shows a remarkable overall resemblance to them even when kr is of order one, which is beyond the expected range of validity of the ap-proximation. The quantitative agreement improves with increasing n : in anexpansion in 1 /n , eq. (6.14) obtains the exact leading-order value for thethreshold wavenumber k GL → √ n/r . Ref. [13] suggests to explain this sur-prising agreement by noting that the thermal wavelength λ T = 1 /T ∼ r /n Gregory-Laflamme instability and black brane viscosity kr W r n = n = kr W r n = n = n = Figure 1. Ω( k ) for unstable modes of black p -branes, in units of 1 /r (theresults depend only on n in (1.1)). The continuous curves are the analyticalapproximation (6.14), the dots are the numerical solution of the perturba-tions of black branes. The top diagram shows results for n = 2 , . . . , k . The bottom diagram shows the curves for n = 6 , ,
100 at all k . For large n , eq. (6.14) underestimates the wavenumber of the thresholdmode by only 1 /n . shrinks to zero as n → ∞ for fixed r . Quite plausibly, this effect extendsthe range of wavelengths that fall under the remit of fluid dynamics.Let us emphasize how little has gone into the derivation of (6.14): onlythe equation of state P ( ε ) and the viscosities η and ζ — actually, for a blackstring there is only ζ . The determination of these coefficients requires a studyof perturbations of the black string, but this can be carried out analytically Blackfolds for all n and p and needs to be done only once. Furthermore, the result for η is known to be universal for black holes, and the value of ζ saturates aproposed bound [20] which may plausibly be proven in generality. If thereexists such a general argument for the value of ζ for a black brane, then theentire expression for the curve (6.14) can be obtained, using simple algebra,from knowledge of only dP/dε .Thus, the effective viscous fluid of the blackfold approach seems to capturein a strikingly simple manner some of the most characteristic features ofblack brane dynamics. This is a significant simplification of the complexityof Einstein’s equations. Although in section 5 we have only considered Minkowski backgrounds, theeffective theory of blackfolds can be readily applied to the construction ofblack holes in curved backgrounds, such as deSitter or anti-deSitter space-time with cosmological constant Λ. In this case it is necessary that the thick-ness r be much smaller than the curvature radius | Λ | − / , so the vacuumblack brane solution can be a good approximation in the near-zone. Blackrings, odd-spheres, and other blackfolds with characteristic radii R that canbe larger or smaller than | Λ | − / , are easy to construct in these spacetimes[21, 22], as well as in other non-trivial backgrounds such as Kaluza-Kleinmonopoles [23].Black p -branes can also carry on their worldvolumes the charges of q -branes, 0 ≤ q ≤ p , which are sources of ( q +2)-form gauge field strengths F q +2 [24, 25]. Then the worldvolume fluid includes a conserved q -brane numbercurrent. For q = 0 this is a theory of an isotropic fluid with a conserved par-ticle number, but when q ≥ F q +2 . When q = 0they carry a conserved charge of a Maxwell field. When q ≥ Extensions the case that the D-brane worldvolume theory has a thermal population ofexcitations. The blackfold gives a gravitational description of this thermallyexcited worldvolume, with a horizon that on short scales is like that of thestraight black D-brane. Like in the AdS/CFT correspondence, this gravita-tional description of the worldvolume theory is appropriate when there isa stack of a large number of D-branes (although not so large as to cause astrong backreaction on the background) and the theory is strongly coupled.Ref. [26] develops these methods to study a thermal version of the D3-branebion. Acknowledgments
I am indebted to my collaborators in the development of the blackfoldapproach: Marco Caldarelli, Joan Camps, Nidal Haddad, Troels Harmark,Vasilis Niarchos, Niels Obers, Mar´ıa J. Rodr´ıguez. I also thank Pau Figuerasfor the numerical data used in figure 1. Work supported by MEC FPA2010-20807-C02-02, AGAUR 2009-SGR-168 and CPAN CSD2007-00042 Consolider-Ingenio 2010 Blackfolds
Appendix: Geometry of submanifolds
A.1 Extrinsic curvature
For a submanifold W embedded as X µ ( σ a ), the pull-back of the spacetimemetric onto W , γ ab , is (3.4) and the first fundamental form of the surface, h µν , is obtained as eq. (3.15). It satisfies h µν ∂ a X ν = ∂ a X µ , h µν h ν ρ = h µρ , (A.1)so h µν projects tensors onto directions tangent to W . ⊥ µν , introduced in(3.16), projects onto orthogonal directions, ⊥ µν ∂ a X µ = 0 , ⊥ µν ⊥ νρ = ⊥ µρ . (A.2)The shape of the embedding of the submanifold W is captured by the secondfundamental tensor, or extrinsic curvature tensor, (3.20). Symmetry of thefirst two indices of K µνρ is equivalent to the integrability of the subspacesorthogonal to ⊥ µν . To see this, let s and t be any two vectors in this subspace, ⊥ µν t ν = 0 , ⊥ µν s ν = 0 . (A.3)Then one can easily prove from the definition of K µν ρ that s µ t ν K µν ρ = ⊥ ρµ ∇ s t µ , (A.4)so K [ µν ] ρ = 0 ⇔ ⊥ ρµ ( ∇ s t µ − ∇ t s µ ) = ⊥ ρµ [ s, t ] µ . (A.5)The vanishing of the last commutator is equivalent, though Frobenius’ the-orem, to the integrability of the subspace orthogonal to ⊥ µν . Therefore theextrinsic curvature tensor of the submanifold W satisfies K [ µν ] ρ = 0.Now let N be any vector orthogonal to W . Then N ρ K µν ρ = N ρ h ν σ ∇ µ h σρ = − h νρ ∇ µ N ρ . (A.6)Background tensors t µ µ ...ν ν ... can be pulled-back onto worldvolume ten-sors t a a ...b b ... using ∂ a X µ as t a a ...b b ... = ∂ a X µ ∂ a X µ · · · ∂ b X ν ∂ b X ν · · · t µ µ ...ν ν ... , (A.7)where ∂ b X ν = γ bc h νρ ∂ c X ρ . (A.8)Even when t µ µ ...ν ν ... has all indices parallel to W , in general ∇ µ t µ µ ...ν ν ... has both parallel and orthogonal components. The parallel projection alongall indices is related to the worldvolume covariant derivative D a t a a ...b b ... ppendix: Geometry of submanifolds as in (A.7). Then, the divergences of background and worldvolume tensorsare related as h ν µ · · · ∇ ρ t ρµ ... = ∂ a X ν · · · D c t ca ... . (A.9) A.2 Variational calculus
Consider a congruence of curves with tangent vector N , that intersect W orthogonally N µ h µν = 0 , N µ ⊥ µν = N ν , (A.10)and Lie-drag W along these curves. The congruence is arbitrary, other thanrequiring it to be smooth in a finite neighbourhood of W , so this realizesarbitrary smooth deformations of the worldvolume X µ → X µ + N µ .Consider now the Lie derivative of h µν along N . In general, £ N h µν = N ρ ∇ ρ h µν + h ρν ∇ µ N ρ + h µρ ∇ ν N ρ . (A.11)Using (A.6) one can derive N ρ K µν ρ = − h µλ h ν σ £ N h λσ . (A.12)Taking the trace, N ρ K ρ = − h µν £ N h µν = − p | h | £ N p | h | , (A.13)where h = det h µν . These equations generalize well-known expressions forthe extrinsic curvature of a codimension-1 surface.Consider now a functional of the embedding of the form I = Z W p | h | Φ (A.14)where Φ is a worldvolume function. Then δ N I = £ N (cid:16)p | h | Φ (cid:17) = p | h | ( − N ρ K ρ Φ + N ρ ∂ ρ Φ) . (A.15)Since N is an arbitrary orthogonal vector we have δ N I = 0 ⇔ K ρ = ⊥ ρµ ∂ µ ln Φ . (A.16)If Φ is constant then we recover the equation K ρ = 0 for minimal-volumesubmanifolds. eferences [1] R. Emparan and H. S. Reall, “Black rings,” chapter of this book.[2] R. Emparan, T. Harmark, V. Niarchos and N. A. Obers, “World-volume ef-fective theory for higher-dimensional black holes. (Blackfolds),” Phys. Rev.Lett. , 191301 (2009) [arXiv:0902.0427 [hep-th]].[3] R. Emparan, T. Harmark, V. Niarchos and N. A. Obers, “Essentials of Black-fold Dynamics,” JHEP (2010) 063 [arXiv:0910.1601 [hep-th]].[4] R. Emparan, T. Harmark, V. Niarchos and N. A. Obers, “New Horizons forBlack Holes and Branes,” JHEP (2010) 046 [arXiv:0912.2352 [hep-th]].[5] R. Gregory, “Gregory-Laflamme instability,” chapter of this book.[6] V. Hubeny, M. Rangamani, S. Minwalla, “Gravity/fluid correspondence,”chapter of this book.[7] E. Poisson, “The motion of point particles in curved spacetime,” Living Rev.Rel. , 6 (2004) [arXiv:gr-qc/0306052].[8] S. E. Gralla and R. M. Wald, “A Rigorous Derivation of Gravitational Self-force,” Class. Quant. Grav. (2008) 205009 [arXiv:0806.3293 [gr-qc]].[9] W. D. Goldberger and I. Z. Rothstein, “An effective field theory of gravity forextended objects,” Phys. Rev. D , 104029 (2006) [arXiv:hep-th/0409156].[10] J. D. Brown and J. W. York, “Quasilocal energy and conserved chargesderived from the gravitational action,” Phys. Rev. D , 1407 (1993)[arXiv:gr-qc/9209012].[11] B. Carter, “Essentials of classical brane dynamics,” Int. J. Theor. Phys. ,2099 (2001) [arXiv:gr-qc/0012036].[12] R. Emparan, T. Harmark, V. Niarchos, N. A. Obers and M. J. Rodr´ıguez,“The Phase Structure of Higher-Dimensional Black Rings and Black Holes,”JHEP , 110 (2007) [arXiv:0708.2181 [hep-th]].[13] J. Camps, R. Emparan and N. Haddad, “Black Brane Viscosity and theGregory-Laflamme Instability,” JHEP (2010) 042 [arXiv:1003.3636 [hep-th]].[14] T. Harmark, N. A. Obers, “New phase diagram for black holes and strings oncylinders,” Class. Quant. Grav. (2004) 1709. [hep-th/0309116].[15] R. Emparan and R. C. Myers, “Instability of ultra-spinning black holes,”JHEP (2003) 025 [arXiv:hep-th/0308056].[16] S. Hollands, A. Ishibashi and R. M. Wald, “A Higher Dimensional StationaryRotating Black Hole Must be Axisymmetric,” Commun. Math. Phys. ,699 (2007) [arXiv:gr-qc/0605106]. Chapter of the book
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