The Young Modulus of Black Strings and the Fine Structure of Blackfolds
aa r X i v : . [ h e p - t h ] J a n DCPT-11/49NORDITA-2011-91
The Young Modulus of Black Stringsand the Fine Structure of Blackfolds
Jay Armas a , Joan Camps b , Troels Harmark c and Niels A. Obers aa The Niels Bohr Institute, University of CopenhagenBlegdamsvej 17, DK-2100 Copenhagen Ø, Denmark b Centre for Particle Theory & Department of Mathematical SciencesScience Laboratories, South Road, Durham DH1 3LE, United Kingdom c NORDITARoslagstullsbacken 23, SE-106 91 Stockholm, Sweden [email protected], [email protected], [email protected], [email protected]
Abstract
We explore corrections in the blackfold approach, which is a worldvolume theory capturing the dy-namics of thin black branes. The corrections probe the fine structure of the branes, going beyond theapproximation in which they are infinitely thin, and account for the dipole moment of worldvolumestress-energy as well as the internal spin degrees of freedom. We show that the dipole correction isinduced elastically by bending a black brane. We argue that the long-wavelength linear response co-efficient capturing this effect is a relativistic generalization of the Young modulus of elastic materialsand we compute it analytically. Using this we draw predictions for black rings in dimensions greaterthan six. Furthermore, we employ our corrected blackfold equations to various multi-spinning blackhole configurations in the blackfold limit, finding perfect agreement with known analytic solutions. ontents
While four-dimensional black holes are always close to be round, higher-dimensional black holes canhave parametrically separated length scales. The blackfold approach [1, 2, 3, 4] gives an effectivedescription of such black holes in terms of black p -branes of thickness r on a world-volume with scale R ≫ r .The blackfold effective degrees of freedom are the moduli of black p − branes: their thickness r ,velocity u a , and position X µ ( σ a ). R is the shortest scale on which these parameters vary, and may bethe wavelength of the velocity and thickness fields, or the curvature of the manifold that the blackfoldwraps. Like other long-wavelength effective descriptions, the effective blackfold expansion, which isan expansion in powers of r /R , is a derivative expansion.In this paper we focus for simplicity on neutral blackfolds, which describe analytic approximatesolutions to the vacuum Einstein equations (in [5, 6, 7] it is described how to introduce charges) R µν = 0 . (1.1)1he blackfold approximation is a higher-dimensional analog of the familiar approximation of ce-lestial objects by point particles following geodesics [8] . Consider, for instance, a black hole binarysystem in which one of the black holes is much smaller than the other one. To lowest order in theratio of their radii, r /R , it is a good approximation to model the small black hole by a probe pointparticle of mass r / r /R , is that of probe branes propagating on a background spacetime. The probe brane is ideal-ized as a zero-thickness object with an effective stress-energy tensor with support on its worldvolume,which is the ADM stress-energy tensor of black p − branes. A covariant way to write a zero-thicknessstress-energy tensor is T µν = Z d p +1 x √− γB µν δ D ( x µ − X µ ( σ a )) √− g , (1.2)where σ a are coordinates on the worldvolume, and γ ab is the metric induced on it. In fact, the indicesof such a p − brane tensor are parallel to its worldvolume, B µν = u µa u νb B ab , u ρc = ∂X ρ ∂σ c , (1.3)and, for a uniform, flat, black p − brane in D = p + n + 3 spacetime dimensions with Schwarzschildradius r , it takes the form of a perfect fluid: B ab = ( ε + P ) u a u b + P γ ab , ε = − ( n + 1) P = ( n + 1) Ω ( n +1) r n πG , u a γ ab u b = − . (1.4)Because the stress-energy tensor (1.2) is coupled to the gravitational field, it is covariantly con-served, ∇ µ T µν = 0 , (1.5)which constitute the blackfold effective equations of motion. It is useful to project Eqs. (1.5) ontodirections parallel and orthogonal to the worldvolume to obtain D a B ab = 0 , B ab K abρ = 0 , (1.6)where D a is the covariant derivative on the worldvolume compatible with γ ab . The first of theseequations stands for the perfect fluid dynamics of the black brane fluid (1.4) on the submanifold.The second equation is a generalisation of the geodesic equation, and K abµ is the extrinsic curvatureof the submanifold . The blackfold approach integrates out the short scale gravitational dynamicsthat resolve the horizon of the black hole, leaving an effective description in terms of perfect fluiddynamics on a dynamical worldvolume. In the probe approximation, which applies to lowest order inthe blackfold expansion, the backreaction of the blackfold is neglected. See [9] for another review. See App. A for a summary of notation. r /R ) k . These corrections can be of two types: self-gravitation corrections and those due to the internal structure of the brane, which we denote byfine structure corrections. Self-gravitation corrections are backreaction effects of the brane on thebackground spacetime and include stress-energy losses by gravitational radiation and self-attraction.These corrections modify Eq. (1.5) since the backreaction effects give corrections to the backgroundspace-time of the brane. Instead, the fine structure corrections are those that preserve Eq. (1.5) butgo beyond (1.6) by introducing corrections to the monopole approximation of the stress-energy tensor T µν (1.2).Because self-gravitational corrections appear when the blackfold reacts to its own gravitationalfield and the range of the gravitational interaction is dimension-dependent, the order at which theseeffects become important is dimension-dependent. For a black ring of radius R and thickness r in4 + n dimensions, Newtonian considerations estimate this order to be ( r /R ) n . Thus, depending onthe co-dimension, self-gravitational interactions can be subleading with respect to other correctionsthat come in at order r /R . These r /R corrections are fine-structure corrections and are the focusof this paper. We will find that they dominate over backreaction corrections for n > B ab (1.4), and are importantin time-dependent processes.In this paper, instead, we focus on fine structure corrections to stationary configurations withnon-trivial curvatures. The prototype example to have in mind is the black ring, and the correctionswe will study are dipole contributions to the stress-energy distribution tensor (1.2). It is convenientto introduce an analogy to develop intuition about the physics of this type of corrections. Considera dielectric object with electric charge q under the influence of an electric field ~E . In the point-likeapproximation the charge density of the object is ρ ( x ) = q δ D − ( x ) and its equation of motion is m~a = q ~E . (1.7)For a real material the electric field ~E causes a charge redistribution and induces an electric dipole,which to lowest order in ~E is given by linear response ~d = κ ~E , (1.8)where κ depends on the material (and could be a matrix). The object is no longer an electric monopole, ρ ( x ) = q δ D − ( x ) − ~d · ~∂ (cid:0) δ D − ( x ) (cid:1) , (1.9)and the equations of motion for this pole-dipole object now read m~a = q ~E + ~d · ~∂ ~E . (1.10)3f the induced dipole is small with respect to the scale in which the electric field varies, the secondterm in the rhs of this equation is a small perturbation.A blackfold on a manifold with extrinsic curvature behaves analogously to this dielectric object,and in this paper we quantify this phenomenon. Eq. (1.9) is the analog of the correction that theleading approximation (1.2) receives. Sec. 2 reviews, following closely [11], the analog of Eq. (1.10),which are the equations of motion of an extended probe pole-dipole distribution of stress-energy, andgeneralize Eqs. (1.6). In the 0 − brane case, these equations were famously derived by Papapetrou[12], and describe the motion of a particle with spin in curved space. We will see that for extendedobjects the dipole contribution to the stress-energy distribution contains angular momentum degreesof freedom as well as genuine dipoles of worldvolume stress-energy. Contrary to the point particlecase, these worldvolume dipoles cannot be gauged away in p − branes.Sec. 3 develops the analog of Eq. (1.8) for bent black strings, and can be read independently of mostof Sec. 2. As in linear elasticity theory, the bending of a black string induces a dipole of worldvolumestress-energy. This is controlled by a response coefficient, the analog of κ in the dielectric example, thatgeneralizes the Young modulus in ordinary non-relativistic elasticity. We use this dipole to computefine structure corrections to certain properties of higher dimensional thin black rings.In Sec. 4 we use the formalism developed in Sec. 2 to describe ultra-spinning doubly-spinningMyers-Perry black holes, and compare them to limits of the exact solutions, finding perfect agreement.Sec. 5 concludes by discussing our results from a more general perspective, such as the AdS/CFTcorrespondence and long wavelength effective theories, and outlining a number of interesting openproblems. We further supplement this paper with three appendices. In App. A the details aboutnotation and conventions are given. In App. B it is shown how to take a refined ultra-spinning limit ofMyers-Perry black holes valid over the entire horizon, filling a gap in the literature. Finally, in App. Cwe consider spin corrections to higher-dimensional Kerr-(A)dS black holes as blackfolds. This section is dedicated to a brief review of the equations of motion for p -dimensional objects in thepole-dipole approximation. Following closely the work done in Ref. [11], it is shown how to iterativelyaccount for higher-pole deformations to the stress-energy tensor T µν while the extra symmetries thatthis object exhibits are commented upon. The equations of motion are then presented in their originalform, as derived in [11], which, when applied to black p -branes, are collectively called blackfold pole-dipole equations. In search of a clearer physical interpretation, we introduce a new set of quantitiesthat make apparent the physics involved. Towards the end of this section, we provide a characterizationof these p -branes in terms of well defined physical quantities and, in the particular case of blackfoldconstructions, of well defined thermodynamic properties.4 .1 Stress-energy tensor and extra symmetries The stress-energy tensor is a well-localized object on the brane and can be consistently expandedinto a Dirac delta function series around the embedding surface x µ = X µ ( σ a ). Schematically, theexpansion has the following form: T µν ( x α ) = Z W p +1 d p +1 σ √− γ " B µν ( σ a ) δ ( D ) ( x α − X α ) √− g − ∇ ρ B µνρ ( σ a ) δ ( D ) ( x α − X α ) √− g ! + ... . (2.1)In the context of electrodynamics, (2.1) corresponds to the usual multipole expansion of a chargedistribution. For the series (2.1) to be well defined we must require T µν to fall off exponentially tozero as we move away from the surface x µ = X µ ( σ a ), which implies that each of the coefficients B µνα ...α k must become smaller and smaller at each order k of the expansion. At order k = 0 theonly non-vanishing coefficient is B µν , resulting in T µν acquiring the form of (1.2) and, by means ofEq. (1.5), leading to the equations of motion for single-pole branes as presented in (1.6). In this paperwe are concerned with truncating the expansion (2.1) to order k = 1 and obtaining, in the same way,the equations of motion for pole-dipole branes moving in curved backgrounds. Truncation of the seriesis a covariant operation and can be done at any arbitrary order. As it stands, (2.1) is written in amanifestly invariant way both under spacetime diffeomorphisms and worldvolume reparametrizations.These are not the only gauge redundancies that T µν possesses since it is also invariant under twoother gauge transformations, which were coined by the authors of [11] as ‘extra symmetry 1’ and ‘extrasymmetry 2’. As these symmetries play an important role in understanding the physics of pole-dipolebranes, we proceed by describing their action on the B -tensors. Extra symmetry 1
This additional gauge freedom arises naturally in the expansion (2.1) due to the p + 1 δ -functionsand p + 1 integrations that were introduced solely with the purpose of making the full expressioncovariant. Specifically, derivatives along the worldvolume directions are integrated out, implyingthat there are redundant components of B µνρ . Physically, this is a consequence of the fact that themultipole expansion is an expansion in derivatives transverse to the brane, rather than longitudinal.The invariance of the stress-energy tensor under this symmetry is defined by its action on the B µν and B µνρ tensors as δ B µν = −∇ a ǫ µνa , δ B µνρ = ǫ µνa u ρa , (2.2)with ǫ µνa = ǫ νµa being free parameters except at the boundary of the worldvolume where they arerequired to obey, ˆ n a ǫ µνa | ∂ W p +1 = 0 , (2.3)where ˆ n a is the unit normal vector to the brane boundary (see App. A for details). Using the trans-formation laws (2.2) one can easily check that the purely tangential components to the worldvolume5f B µνρ are in fact a gauge artifact, δ ( B µνρ u aρ ) = ǫ µνa . (2.4)Hence, the components B µνa can be gauged away everywhere except at the boundary where theparameters ǫ µνa cannot be freely chosen. This implies that there are degrees of freedom that liveexclusively on the boundary of the worldvolume, for which a physical interpretation will be given inthe next section. Extra symmetry 2
The stress-energy tensor T µν has been expanded around the surface x µ = X µ ( σ a ) as in (2.1) but sincewe are dealing with objects of finite thickness there is freedom in choosing a different worldvolume. Inphysical terms, the finite thickness of the brane allows for different choices of worldvolume surfaces .This redundancy is an exact symmetry of the full series expansion (2.1) but only an approximate one ofthe truncated series to order k = 1. This is because neglecting higher order terms in the expansion (2.1)already constrains the allowed choices of worldvolumes. In particular, choosing the surface X α ( σ a ) tolie outside the localized matter would require a non-zero contribution from the higher order B -tensors.Therefore, we choose the surface to lie within the localized matter and assume the following hierarchyof scales: B µν = O , B µνρ = O , B µνρλ = O , ... . (2.5)In this way, we can define the action of ‘extra symmetry 2’ as X ′ α ( σ a ) = X α ( σ a ) + ǫ α ( σ a ) , (2.6)where ǫ α is constrained by the requirement that the transformed B -tensors obey B k +1 = O k +1 . Inboth the single-pole and pole-dipole cases this implies ǫ α = O . The action of (2.6) to order k = 1demands the following transformation rule for the B -tensors: δ B µν = − B µν u aρ ∇ a ǫ ρ − B λ ( µ Γ ν ) λρ ǫ ρ , δ B µνρ = − B µν ǫ ρ , (2.7)where we have ignored contributions of O and higher. In the single-pole approximation we find δ X α = 0 and δ B µν = 0, emphasizing the fact that there is no freedom in choosing the worldvolumesurface for the object as they are infinitely thin. The equations of motion (EOMs) for probe pole-dipole branes moving in a curved background space-time can be obtained by solving Eq. (1.5) using the stress-energy tensor given in (2.1) truncated to In the particle case there is a natural choice of reference frame which is the centre of mass. k = 1. The derivation of these equations is somewhat involved and we refer to [11] for anextensive detailed analysis.It is convenient to decompose the objects B µν and B µνρ into tangential and orthogonal componentsto the worldvolume, the latter being subjected to the constraint equation [11] ⊥ νλ ⊥ σρ B µ ( λρ ) = 0 , (2.8)while the former is not altogether independent and bares a relation with B µνρ which will be describedbelow. This suggests the following decomposition : B µν = B µν ⊥ + 2 u ( µb B ν ) b ⊥ + u µa u νb B ab , B µνρ = 2 u ( µb B ν ) ρb ⊥ + u µa u νb B ρab ⊥ + u ρa B µνa , (2.9)that can be shown to obey the properties B ( µν ) a ⊥ = B µ [ ab ] ⊥ = B [ µν ] a = 0. Due to ‘extra symmetry 1’,described in the previous section, the last components in the decomposition of B µνρ are left neitherparallel nor perpendicular to the worldvolume, as B µνa can be gauged away in the bulk of the brane.A convenient form of the EOMs can be obtained by defining a new set of tensors S µνa = B µνa ⊥ + u [ µb B ν ] ba ⊥ , N µνa = B µνa + u ( µb B ν ) ba ⊥ , (2.10)which are, respectively, anti-symmetric and symmetric in the first two indices µ, ν . In terms of theseit is straightforward to check that B µνρ can be recast as B µνρ = 2 u ( ua S ν ) ρa + N µνa u ρa . (2.11)Furthermore, the interdependence between the orthogonal components of B µν and the quantities S µνa and N µνa is expressed through the relations B µν ⊥ = ⊥ µλ ⊥ νρ ∇ a N λρa , B µa ⊥ = u aλ ⊥ µρ ∇ b (cid:16) S λρb + N λρb (cid:17) , (2.12)while the tangential components B ab describe the monopole contribution to the intrinsic stress-energytensor of the brane.Parametrizing the EOMs using (2.11)-(2.12) yields two sets of bulk equations: a partial conserva-tion equation of the brane worldvolume currents S µνa , ⊥ µλ ⊥ νρ ∇ a S λρa = 0 , (2.13)and the equation that describes the motion of the pole-dipole brane ∇ b (cid:16) m ab u µa − u bλ ∇ a S µλa + u µc u cρ u bλ ∇ a S ρλa (cid:17) − u νa S λρa R µνλρ = 0 , (2.14)where we have defined, for later convenience, the worldvolume tensor m ab through the formula m ab = B ab − u aρ u bλ ∇ c N ρλc . (2.15) A subindex ⊥ on a tensor indicates that all µ, ν type of indices are orthogonal, e.g., B aµ ⊥ = ⊥ µν B aν ⊥ . Details can befound in App. A. p = 0. In order to highlight the physical meaning of Eq. (2.14), we project it along the tangentialand orthogonal directions to the worldvolume. This operation leads to the intrinsic and extrinsicworldvolume equations: ∇ b m ab = 2 ∇ b (cid:16) u [ bρ K a ] cλ S ρλc (cid:17) − u bλ K abρ ∇ c S ρλc − u aµ u νc S ρλc R µνλρ , (2.16) m ab K abρ = (cid:16) ⊥ ρλ K bbν + u cν u bλ K bcρ (cid:17) ∇ a S νλa + 2 u bν ⊥ ρλ ∇ b ∇ a S νλa − u νa ⊥ ρ µ S σλa R µνλσ . (2.17)In this way, it is clear that Eq. (2.16) can be interpreted as an equation for the conservation of theintrinsic monopole stress-energy tensor B ab , which can be violated due to the higher order dipolecontributions , while Eq. (2.17) is the generalized geodesic equation for a pole-dipole p -dimensionalobject, in contrast with the single-pole case (1.6).In turn, the EOMs that govern the brane dynamics (2.13)-(2.14) everywhere inside W p +1 aresupplemented by well defined boundary conditions derived from solving Eq. (1.5), S µνa ˆ n a ˆ n ν | ∂ W p +1 = 0 ⊥ µλ ⊥ νρ S λρa ˆ n a | ∂ W p +1 = 0 h ∇ ˆ i (cid:16) N ˆ i ˆ j v µ ˆ j + 2 S µνa ˆ n a v ˆ iν (cid:17) − ˆ n b (cid:16) m ab u µa − u bλ ∇ a S µλa + u µc u cρ u bλ ∇ a S ρλa (cid:17)i | ∂ W p +1 = 0 , (2.18)where v ˆ jν are the boundary coordinate vectors (see App. A for details) and we have defined N ˆ i ˆ j = N µνa ˆ n a v ˆ iµ v ˆ jν . These quantities appear only in the boundary conditions and nowhere else and, asmentioned in the previous section while discussing ‘extra symmetry 1’, contain the degrees of freedomthat live exclusively on the boundary. In full generality, the tensors m ab , S µνa and N ˆ i ˆ j characterizethe internal structure of the brane and play a crucial role in describing its dynamics. We will proceedby analyzing their physical meaning and of the resulting EOMs. Physical interpretation
Ref. [11] introduced the S µνa and m ab quantities, in terms of which the equations of motion of pole-dipole branes simplify considerably. In the following we give an interpretation of S µνa , and see that itcontains two types of contributions: the genuine intrinsic transverse angular momenta, and the dipolemoment of the distribution of worldvolume stress-energy . Thus, besides generalising Papapetrou’sequations, Eqs. (2.16), (2.25) include dipole interactions analogous to those in (1.10).We begin by analyzing which components of S µνa are involved in the description of the intrinsicangular momenta. To this end, we assume to be working in flat spacetime written in Cartesian Even though the conservation of the monopole stress-energy tensor is not necessarily guaranteed, in the cases studiedhere we always find that B ab is conserved. In the case of the 0 − brane the dipole can be gauged away and, as demonstrated in the original work of Corinaldesiand Papapetrou [12, 13], S µνa describes only spin degrees of freedom. p -branes extended along the x , ..., x p directions. Evaluating thetotal angular momentum on the transverse plane labeled by the indices µ, ν leads to J µν ⊥ = Z Σ d D − x (cid:0) T µ x ν − T ν x µ (cid:1) = Z B p d p σ √− γ (cid:16) B µν ⊥ (cid:17) + boundary terms , (2.19)where Σ is a constant time slice in the bulk spacetime. At this point, we ignore the boundary terms,which only depend on the components B µνa , but we will consider them towards the end of this section.From (2.19), we can see that the monopole contribution to the intrinsic stress-energy tensor B ab doesnot play a role in (2.19) and hence a p -brane when treated in the single-pole approximation cannever carry intrinsic angular momenta. Furthermore, only the components B aµν ⊥ contain informationabout the spin of the object. This suggests the introduction of a current density of transverse angularmomenta as j aµν = 2 u aρ ⊥ µσ ⊥ νλ B ρ [ σλ ] = 2 B aµν ⊥ , (2.20)where both indices µ, ν are orthogonal to the worldvolume.On the other hand, there is another source of B µνρ which is of a different nature than transverseangular momenta. It arises from the fact that, since we are probing the finite thickness of the brane,we need to take into account corrections to the intrinsic stress-energy tensor T ab due to the dipole-typeeffects. This is characterized by the integral , D abρ = Z Σ d D − xT ab x ρ = Z Σ d D − xT µν u aµ u bν x ρ = Z B p d p σ √− γB ρab ⊥ + boundary terms , (2.21)where x ρ is an orthogonal coordinate to the worldvolume. D abρ captures the dipole moment of thedistribution of worldvolume stress-energy. As in the case of intrinsic angular momenta, we introducea current density that describes such deformations to the intrinsic stress-energy tensor by d abρ = u aµ u bν ⊥ ρλ B µνλ = B ρab ⊥ , (2.22)where the index ρ is orthogonal to the worldvolume W p +1 .Our aim now is to recast the EOMs, including the boundary conditions, in terms of these newlydefined quantities. Using the definitions of the current densities (2.20) and (2.22), we can rewrite thetensors introduced in (2.10) as S µνa = 12 j aµν − d ab [ µ u ν ] b , N µνa = B µνa + d ab ( µ u ν ) b . (2.23) As a matter of a fact, this is the usual notion of an electric induced dipole. In electrostatics, given a density of charge ρ ( x ), the dipole can computed as, ~D = Z Σ d D − x~xρ ( x ) .
9e note that we have not been concerned so far with giving a physical interpretation to the components B µνa . This is because, due to ‘extra symmetry 1’, we can gauge them away everywhere in the bulkwhile on the boundary we will have to deal with N ˆ i ˆ j as we will see below. In turn, the currentconservation equation (2.13) becomes:12 ⊥ µλ ⊥ νρ ∇ a j aρλ + d ab [ ν K ab µ ] = 0 . (2.24)This equation can be interpreted as the balance between orbital angular momentum and intrinsicangular momentum. Nevertheless, as it will be argued in Sec. 3, for blackfold-type objects, the dipolecurrent d abρ is induced by the extrinsic curvature. In all such situations, the second term in Eq. (2.24)vanishes, leading to a conserved spin current which can be naturally interpreted as a 0-brane particlecurrent on the worldvolume.We now turn our attention to the intrinsic and extrinsic Eqs. (2.16)-(2.17), which can be rewrittenusing (2.20) and (2.22) as D a m ab = K cbµ (cid:16) K acλ j aµλ + ∇ a d acµ (cid:17) + ∇ c K a [ b µ d c ] aµ − u bµ u νa (cid:18) j aλρ − d ab [ λ u ρ ] b (cid:19) R µνλρ , (2.25) m ab K abρ = ∇ b (cid:16) j aλρ K baλ (cid:17) + u ρc K abλ K caσ j bλσ − K acρ K ( cbλ d a ) bλ − ⊥ ρσ ∇ b ∇ a d abσ − u νa ⊥ ρµ (cid:18) j aλσ − d ab [ λ u σ ] b (cid:19) R µνλσ . (2.26)These equations provide the gravitational analog of Eq. (1.10) for p -branes. Written in this way itis apparent that, besides spin interactions, we also have couplings to the dipole current d abρ . Forblackfold objects these interactions can be interpreted as elastic forces, for which a derivation in termsof high-pressure elasticity theory [14] can in principle be accomplished and will be presented elsewhere.This point will be further motivated in Sec. 3. Finally, the boundary conditions (2.18) take the form: d abµ ˆ n a ˆ n b | ∂ W p +1 = 0 j aµν ˆ n a | ∂ W p +1 = 0 h ∇ ˆ i (cid:16) N ˆ i ˆ j v µ ˆ j − d abµ ˆ n a v ˆ ib (cid:17) − ˆ n b (cid:16) m ab u µa − j aµλ K abλ − K a ( c λ d b ) aλ u µc − ∇ a d abµ (cid:17)i | ∂ W p +1 = 0 , (2.27)where N ˆ i ˆ j v µ ˆ j is in fact N ˆ i ˆ j v µ ˆ j = B λνa ˆ n a v ˆ iλ v ˆ jν v µ ˆ j . (2.28)Eqs. (2.24)-(2.27) when applied to black p -branes constitute the blackfold equations in the pole-dipoleapproximation and will be analyzed in detail in particular cases throughout the course of this work.We end this section by briefly commenting on the physical interpretation of the coefficients N ˆ i ˆ j .As remarked in [11], N ˆ i ˆ j characterizes the tangential components to the worldvolume of the branethickness. The reason why these components drop out of the bulk equations is because thickening the10rane along tangential directions does not affect the brane interior but if the brane has a boundarythen it will be affected by such process. Essentially, N ˆ i ˆ j is a correction to the intrinsic stress-energytensor of the brane boundary T ˆ i ˆ j and can be read off from an analytic solution by evaluating thestress-energy tensor on the boundary of the brane surface. Since for all the cases analyzed in thispaper B µνa vanishes everywhere, including at the boundary, we assume B µνa = 0 from hereon. Branes in the pole-dipole approximation can carry different conserved charges and this section isdevoted to describing them. In the case of blackfolds, where these charges acquire a thermodynamicinterpretation, we present a method for obtaining the remaining thermodynamic quantities involved.As mentioned in Sec. 1, fine structure corrections do not violate stress-energy conservation (1.5).Hence, since we are working in the probe approximation, associated with any background Killingvector field k µ , there exists a conserved current given by T µν k ν and thus the conserved charges canbe obtained in the usual way. In general, we have a charge Q k given by | Q k | = Z B p dV ( p ) B µν n µ k ν + Z B p dV ( p ) B µνρ ∇ ρ ( n µ k ν ) , (2.29)where n µ is the normal vector to a constant time slice of the background spacetime and is defined as n µ = ξ ν R , R = p − ξ | W p +1 . (2.30)Here, ξ ν is the Killing vector field associated with asymptotic time translations and R is the redshiftfactor. In writing Eq. (2.29) we have assumed that ξ ν is hypersurface orthogonal with respect to thebackground spacetime and that ξ ν is parallel to the worldvolume timelike Killing vector field, whichis also hypersurface orthogonal with respect to the worldvolume metric, i.e., ξ µ = u µa ξ a .Using the decompositions (2.9) and Eqs. (2.20),(2.22) we can write down general expressions forthe physical quantities in terms of the spin and dipole currents. The total mass, associated with ξ ν ,reads M = Z B p dV ( p ) B ab n a ξ b + Z B p dV ( p ) d abρ ξ a u µb ∂ ρ n µ . (2.31)The first term above describes the contribution to the total mass arising from a monopole source ofstress-energy, while the second is the extra contribution coming from a dipole distribution. Similarly,the total angular momentum along a worldvolume spatial direction i associated with the rotationalKilling vector field χ i takes the form J i = − Z B p dV ( p ) B ab n a χ ib − Z B p dV ( p ) d abρ u µb ξ a ∂ ρ χ iµ R + χ ia ∂ ρ n µ ! . (2.32) Here we have used the Killing equation ∇ ( µ k µ ) = 0 to exchange covariant derivatives for partial derivatives. χ α ⊥ isgiven by J α ⊥ = − Z B p dV ( p ) (cid:18) ∇ a j aλρ + u λb ∇ a d abρ (cid:19) n λ χ ⊥ ρ − Z B p dV ( p ) j aµρ u νa (cid:18) ξ ν ∂ ρ χ α ⊥ µ R + χ α ⊥ µ ∂ ρ n ν (cid:19) . (2.33)We see that the spin current j aµρ only plays a role in the transverse angular momenta. On the otherhand, the dipole current d abρ does influence all charges, including J α ⊥ (which corresponds to orbitalangular momentum). This is because, in general, j aµρ is not a conserved current due to Eq. (2.24).In the cases considered in this paper, the first term in Eq. (2.33) always vanishes, as the spin currentis always conserved. Thermodynamic quantities
When considering the dynamics of black p -branes, the conserved charges have a thermodynamicinterpretation. Furthermore, a local temperature and entropy density can be assigned to the p -dimensional object from which one can then compute a global temperature and total area. As wewill be mainly concerned with blackfolds constructed from wrapped Schwarzschild and Myers-Perry p -branes on curved submanifolds, the knowledge of these quantities amounts to perform a near-horizoncomputation which is beyond the scope of this paper. Instead, we present here an alternative methodto obtain the on-shell temperature and entropy.Focusing for the moment on asymptotically flat spacetimes, the method consists of using the firstlaw of black hole thermodynamics dM = X i Ω i dJ i + X α Ω α dJ α ⊥ + T dS , (2.34)together with the Smarr relation( n + p ) M = ( n + p + 1) X i Ω i J i + X α Ω α J α ⊥ + T S ! , (2.35)in order to determine the two unknown quantities T and S . Since, in general, all physical quantitiesdepend on a set of intrinsic parameters, such as r and R , which we collectively call by Φ w , then,by means of Eq. (2.35) the product T S can be determined, as all other quantities involved can beevaluated using the expressions (2.31)-(2.33). Inserting this result into Eq. (2.34) leads to a set ofequations, one for each Φ w , which take the following form:1 T ∂T∂ Φ w = 1 T S ∂T S∂ Φ w + X i Ω i ∂J i ∂ Φ w + X α Ω α ∂J α ⊥ ∂ Φ w − ∂M∂ Φ w ! . (2.36)Solving this set of equations yields the temperature of the black object and hence also the entropy byEq. (2.35). The method just described provides the knowledge of T and S up to a constant, which12an later be fixed by demanding the correct behavior in the infinitely thin limit, or in the single-poleapproximation, where all quantities can be unambiguously determined using the results of Ref. [3].We conclude this section by briefly considering blackfolds in (A)dS backgrounds in situations wherethe dipole current d abρ vanishes. In such cases the total integrated tension T , given by T = − Z B p dV ( p ) R B µν ( h µν − n µ n ν ) − Z B p dV ( p ) R B µνρ ∇ ρ ( h µν − n µ n ν ) , (2.37)plays a role in the thermodynamics and must be added to the rhs of Eq. (2.35). These considerationswill be used later in Sec. (4) and App. (C) in order to make contact with the thermodynamic propertiesof doubly-spinning Myers-Perry and higher-dimensional Kerr-(A)dS black holes. The purpose of this section is to study how a Schwarzschild black string reacts to its bending. This isdone using an explicit solution describing a black string of thickness r bent on a circle of radius R , tofirst order in r /R . Very similarly to what happens to elastic rods, the effective blackfold stress-energytensor distribution acquires a dipole contribution, exhibiting an effective elastic behavior. This section uses the solutions constructed in [1] to compute the dipole contribution to the effectivestress-energy distribution induced by the bending of strings. This reference computes, to first orderin r /R , the spacetime of a bent Schwarzschild string, where r is the thickness of the string and R isthe radius of curvature of the circle on which the string is bent. In Ref. [1] this geometry is obtainedby solving Einstein’s equations using a matched asymptotic expansion (MAE), in which there are twodifferent coordinate patches. These two zones are the zone near the horizon r ≪ R , and the far zone r ≫ r , where the weak field approximation applies. The applicability of a MAE requires that thereis an overlap zone, and this happens when r ≪ R , which is why this technique is amenable in thisregime of parameters.In the near zone, the geometry is a perturbation of the Schwarzschild black string, and in the farregion the gravitational field is well described by the linear approximation sourced by the appropriateblackfold effective probe distribution of stress-energy. In MAE each zone feeds the other zone withboundary conditions.Let us summarize the order-by-order logic at each zone. We denote by k th (near/far) thegeometry to order ( r /R ) k in each of the zones. The first few orders go as follows: • The geometry is that of the straight boosted Schwarzschild black string: ds = dr − r n r n + r (cid:16) dθ + sin θd Ω n ) (cid:17) − (cid:18) − r n r n (cid:19) (cosh α dt + sinh α dz ) + (cosh α dz + sinh α dt ) , (3.1)13nd the far field is well described by the linearized approximation, ✷ ¯ h µν = − πGT µν , sourcedby the ADM stress-energy tensor of the string, which is localized on its effective worldvolume(1.2), T µν = B µν δ n +2 ( r ): B tt = Ω ( n +1) r n πG (cid:0) n cosh α + 1 (cid:1) , (3.2a) B tz = Ω ( n +1) r n πG n cosh α sinh α , (3.2b) B zz = Ω ( n +1) r n πG (cid:0) n sinh α − (cid:1) . (3.2c) • The far field is sourced by the blackfold effective stress-energy tensor (3.2), now alonga bent manifold with curvature 1 /R . Only sources satisfying ∇ µ T µν = 0 can be coupled to thegravitational field, and this conservation equation constitutes the blackfold effective equationsof motion. For the case of the ring they reduce to constancy of r and B zz = 0, which sets therapidity to sinh α = 1 n . (3.3)This is interpreted as the value at which the centrifugal repulsion compensates the tension. • The geometry in the near zone is a 1 /R perturbation to the Schwarzschild blackstring, that is found using as a boundary condition. One can compute, from the farfield of the corrected near zone solution, a corrected effective stress-energy source in the far zone. • One would use the corrected effective distribution of stress-energy measured in to source the far field. The corrected stress-energy tensor would need to satisfy acorrected effective equation of motion coming from ∇ µ T µν , which would modify (3.3). • One would use the result of , as a boundary condition to solve the neargeometry as a 1 /R perturbation of the black string. From this near solution, one would extractyet another correction to the source of the far field which would modify the effective equation ofmotion...and so on. [1] went to the step . In this section we push that analysis to byreading the correction to the stress-energy source, which is of dipole type, and which will have theequations derived in Sec. 2 as effective equation of motion.The solution of a bent black string in flat space at was written down in [1] as g tt = − n + 1 n r n r n + cos θR a ( r ) , (3.4a)14 tz = √ n + 1 n (cid:20) r n r n + cos θR b ( r ) (cid:21) , (3.4b) g zz = 1 + 1 n r n r n + cos θR c ( r ) , (3.4c) g rr = (cid:18) − r n r n (cid:19) − (cid:20) θR f ( r ) (cid:21) , (3.4d) g ij = ˆ g ij (cid:20) θR g ( r ) (cid:21) , (3.4e)where ˆ g ij is the metric of a round n + 1 sphere of radius r in polar coordinates:ˆ g ij dx i dx j = r d Ω n +1) = r (cid:16) dθ + sin θ d Ω n ) (cid:17) . (3.5)As expected, the limit R → ∞ corresponds to a black string boosted in the z direction with rapiditysinh α = 1 / √ n , according to (3.3).The regular solution to the Einstein equations found in [1] has the following large r asymptotics a ( r ) = [ k (1 + n ) − n (4 + 3 n ) ξ ( n )] r n +20 r n +1 + O (cid:16) r − ( n +2) (cid:17) , (3.6a) b ( r ) = r n r n − + (cid:2) k n − n ξ ( n ) (cid:3) r n +20 r n +1 + O (cid:16) r − ( n +2) (cid:17) , (3.6b) c ( r ) = 2 r + 1 n r n r n − + k r n +20 r n +1 − n + 2 n − n ( n − r n r n − + k r n +20 r n +1 + O (cid:16) r − (2 n +2) (cid:17) , (3.6c) f ( r ) = − n r + 1 n r n r n − + (cid:2) ( k + 2 k ) n − (4 n + 7 n + 4) ξ ( n ) (cid:3) r n +20 r n +1 + O (cid:16) r − ( n +2) (cid:17) , (3.6d) g ( r ) = − n r + 1 n r n r n − + (cid:20) k − k n )1 + n + ( n − ξ ( n ) (cid:21) r n +20 r n +1 + O (cid:16) r − ( n +2) (cid:17) , (3.6e)where we defined ξ ( n ) = − − nn n + 1 Γ (cid:0) n +1 n (cid:1) Γ (cid:0) − n +22 n (cid:1) Γ (cid:0) − n +1 n (cid:1) Γ (cid:0) n +22 n (cid:1) , (3.7)which is zero for n = 1 and divergent for n = 2. In the latter case this is because the correspondingterms in the expansion should have logarithm contributions on top the polynomial ones. The interestof this paper is in the n > can be interpretedin the probe approximation, as discussed in the introduction. In what follows we assume n > i.e., black rings in more than six dimensions.It is worth mentioning various things about expressions (3.6). First, note that all functions areexpanded to order r − ( n +2) except for c ( r ), which is expanded to order r − (2 n +2) .15econd, there is residual gauge freedom in the coordinate system in which the metric reads (3.4)and, because of that, c ( r ) is essentially unfixed by the Einstein equations. This shows up in the unfixed k and k parameters, which parametrize gauge ambiguities. c ( r ) is not completely free because wedemand that at large r the metric goes to a known metric in a familiar coordinate system. Thatknown metric is the one that one recovers if one keeps only terms in this expansions up to order r − n in(3.4). The geometry to order r − n is just the linearized gravitational field of the blackfold distributionalstress-energy tensor sitting on a circle of radius R , to first order in 1 /R .To be more specific with this interpretation, let us decompose the metric (3.4) in the followingway: g µν = η µν + h (M) µν + h (D) µν + O (cid:16) r − ( n +2) (cid:17) . (3.8)Here η µν is flat space in cylindrical coordinates η µν dx µ dx ν = − dt + dz + dr + r ( dθ + sin θd Ω n ) ) , (3.9)and h (M) µν , which is the correction to the flat metric in the (1st far) step, includes two types ofcontributions; There is a 1 /R change of coordinates of flat space such that r = 0 corresponds to acircle, which are the terms not multiplying any r in (3.6). The other type of terms, which are the onesmultiplied by r n , correspond to the linearized gravitational field sourced by the monopole blackfoldstress-energy tensor. h (D) µν are the order r − ( n +1) terms in the expansion. All these terms have h (D) µν ∝ r n +20 R cos θr n +1 , (3.10)which is the field of a dipole source located at r = 0 in the spacetime (3.9) , namely∆ h (D) µν ∝ r n +20 R ∂ ( r cos θ ) δ n +2 ( r ) , (3.11)where ∂ ( r cos θ ) points outwards from the worldvolume, such that ∂ ( r cos θ ) r = cos θ . The metric (3.4)at order r − ( n +1) is the linearized solution of G µν [ g ] = 8 πG (cid:16) B µν δ n +2 ( r ) − d µν ( r cos θ ) ∂ ( r cos θ ) δ n +2 ( r ) (cid:17) . (3.12)The goal of this section is to determine d µν ( r cos θ ) , which is the dipole source to the far field, andbecause it is proportional to 1 /R , it is induced by the bending of the monopole source. To find it, wenote that in TT gauge, the Einstein equations for h (D) µν linearize to∆¯ h (D) µν = 16 πG d µν ( r cos θ ) ∂ ( r cos θ ) δ n +2 ( r ) , (3.13) Note that the field of the dipole is insensitive to the fact that r = 0 is now a circle of curvature 1 /R . This is becausethe metric (3.4) is valid only up to 1 /R corrections. Being the dipole source already a 1 /R effect, accounting for thefact that the dipole source sits on a circle of curvature 1 /R is a 1 /R effect. h (D) µν = h (D) µν − h (D) η µν , h (D) = h (D) µν η µν , (3.14)such that ¯ h (D) µν = 16 πG Ω n +1 cos θr n +1 d µν ( r cos θ ) . (3.15)It has been argued in Sec. 2 that d µν ( r cos θ ) has µν indices parallel to the worldvolume, which is locatedat r = 0. µ and ν can thus only be t or z . This implies that in TT gauge h (D) rr = h (D)ΩΩ r , h (D) tt − h (D) zz = n h (D) rr , (3.16)which is satisfied for k = 2 n − n ( k − n + 1) ξ ( n )) , (3.17)which, of course, is a gauge choice. After this gauge fixing, it is straightforward to read d µν ( r cos θ ) from ¯ h (D) µν . For the sake of interpretation it is convenient to first redefine , k = ( n + 3) ξ ( n ) + ( n − n ˜ k . (3.18)The dipole contribution is found to be d tt ( r cos θ ) = − Ω ( n +1) r n πG r R ( n + 3 n + 4) ξ ( n ) − ˜ k r R B tt , (3.19a) d tz ( r cos θ ) = − ˜ k r R B tz , (3.19b) d zz ( r cos θ ) = Ω ( n +1) r n πG r R (3 n + 4) ξ ( n ) , (3.19c)where B ab in these expressions are (3.2) and are evaluated at equilibrium (3.3). We remind the readerthat these expressions are valid for n > k . This is the expected ‘extrasymmetry 2’ ambiguity in the dipole under changes of the representative worldvolume surface, seeSec. 2. Indeed, ˜ k → ˜ k + δ picks a worldvolume outwards by δ r /R . d zz ( r cos θ ) being unambiguouslydefined fits nicely with the fact that the equilibrium condition for the ring is precisely B zz = 0, whichrenders this symmetry (2.7) trivial for this component of the dipole. We used that in D spacetime dimensions ∆ ( D − r − ( D − = − ( D − ( D − δ D − ( r ) . See Eq. (3.23) for a motivation for this. .2 More general backgrounds The calculation of Sec. 3.1 can be generalized to black strings lying on flat submanifolds with a moregeneral extrinsic curvature than just non-vanishing K zz ( r cos θ ) . Ref. [15] studied the gravitational fieldof a black string lying on the r = 0 submanifold of ds = − (cid:18) C t r cos θR (cid:19) dt + (cid:18) C z r cos θR (cid:19) dz + (cid:18) − C t + C z n r cos θR (cid:19) h dr + r ( dθ + sin θ d Ω n ) ) i + O (cid:0) ( r cos θ ) (cid:1) . (3.20)Note that the metric induced on the worldvolume is flat, and the non-vanishing components of itsextrinsic curvature read K tt ( r cos θ ) = C t R , K zz ( r cos θ ) = − C z R . (3.21)This family of worldvolumes can be used to perturbatively study black rings in (A)dS space or inSchwarzschild-Tangherlini black hole backgrounds (in which case one is studying Black Saturns in aperturbative regime).The equilibrium condition for the boosted string lying on r = 0 solving (1.6) readssinh α = C z + ( n + 1) C t n ( C z − C t ) , (3.22)and the linear blackfold stress-energy tensor takes the form (3.2) with the boost (3.22). A calculationvery similar to the one explained in Sec. 3.1 on the approximate solution found in [15] reveals theinduced dipole induced by such extrinsic curvature. To write it down in a compact way it is useful tofirst consider K abρ d abρ = Ω ( n +1) r n πG r R (cid:2) − (3 n + 4)( C t + C z ) − n + 3 n + 4) C t C z (cid:3) ξ ( n ) , (3.23)which is ˜ k − invariant thanks to the leading extrinsic equation of motion, K abρ B ab = 0. The symmetryof this expression under C t − C z exchange motivates a parametrization of the gauge ambiguity suchthat this symmetry is apparent in the d abµ object, k = [( n + 1) C t + ( n + 3) C z ] (2 C z + ( n + 2) C t )2( C z − C t ) ξ ( n ) + ( n −
1) [ C z + ( n + 1) C t ]2 n ( C z − C t ) ˜ k . (3.24)The induced dipole of a black string on this submanifold reads d tt ( r cos θ ) = Ω ( n +1) r n πG r R (cid:2) − C t (3 n + 4) − C z ( n + 3 n + 4) (cid:3) ξ ( n ) − ˜ k r R B tt , (3.25a) d tz ( r cos θ ) = − ˜ k r R B tz , (3.25b) Here, as in Eq. (3.6c), k is defined as the coefficient of the r n +20 /r n +1 term in the large r expansion of the c ( r )function, appearing in Eq. (B.15) in [15]. zz ( r cos θ ) = Ω ( n +1) r n πG r R (cid:2) C z (3 n + 4) + C t ( n + 3 n + 4) (cid:3) ξ ( n ) − ˜ k r R B zz . (3.25c) The study of how strains induce stresses is the subject of elasticity theory. It is well known fromelementary elasticity theory [16] that the bending of an elastic rod induces a stress on it that hasopposite signs on the inner and outer side. This is so because under bending, the inner side iscompressed and the outer side is stretched. Thus, in classical elasticity theory bent rods developdipoles of stress.The geometric object capturing how strain varies on directions transverse to a bent rod is theextrinsic curvature since it can be written K µνρ = − γ µκ γ ν σ £ n ρ γ κσ , (3.26)where γ µν = u aµ u bν γ ab projects on the worldvolume.Black strings also develop dipoles when bent, exhibiting elastic behavior. In the linear (Hookean)regime, the response coefficient relating stress and strain in non-relativistic elasticity theory is theYoung modulus. The black strings we have studied are in such linear regime, as their dipole is a smalldeformation, of order 1 /R .To characterize the elastic behavior of black strings we need to introduce a relativistic generalizationof the Young modulus. Let us start by going back to classical elasticity theory in flat space, in whicha bent rod develops, in the case C t = 0 and C z = 1, the stress T zz = Y r cos θR , (3.27)where Y is the Young modulus. If the rod has, to first approximation, a circular cross-section of radius r , the dipole of stress reads d zz ( r cos θ ) = Z T zz r cos θ (cid:0) r n +1 sin n θ dr dθ d Ω ( n ) (cid:1) = r n +40 ( n + 2)( n + 4) Ω ( n +1) Ω ( n ) (cid:18) YR (cid:19) . (3.28)Motivated by this we now define the Young modulus of black strings through the formula d abρ = (cid:20) − r n +40 ( n + 2)( n + 4) Ω ( n +1) Ω ( n ) (cid:21) Y abcd K cdρ . (3.29)For relativistic matter we need to use a tensor, accounting for necessary anisotropy of the worldvolumedirections: one of the a, b directions is timelike and the rest are spacelike.It is not the purpose of this work to carry out a deep study of the properties of the relativisticYoung modulus that we just defined, Y abcd . We note, however, that it should display the ‘extrasymmetry 2’ ambiguity that d abρ enjoys. By construction, for the result (3.25) ˜ k = 0 yields Y ttzz = Y zztt (cid:12)(cid:12) ˜ k =0 , (3.30)19hich is a desirable property of such tensors in non-relativistic anisotropic media.We close this section by collecting the measured components of Y abcd from (3.25) at ˜ k = 0: Y tttt = Y zzzz = Ω ( n ) ( n + 2)( n + 4)16 πG r (3 n + 4) ξ ( n ) , Y tztt = Y tzzz = 0 ,Y ttzz = Y zztt = − Ω ( n ) ( n + 2)( n + 4)16 πG r ( n + 3 n + 4) ξ ( n ) . (3.31)The addition of flat directions to (3.4) or the corresponding case of (3.20) does not threaten thesolutions to Einstein equations we have been considering in this section, and does not change thefinal result (3.31). Thus, much like the equation of state ε = − ( n + 1) P is a property only of thecodimension of black p − branes, so is the Young modulus, and expressions (3.31) are also valid for p >
1. Also, because calculations in this section are linear in 1 /R , one can include the effect of a moregeneral extrinsic curvature by just adding the effects of different components up; Considerations inthis section are also valid, for example, for black tori [4], in which the ρ index in K abρ may not bealigned in all ab components of the extrinsic curvature. One of the main applications of the blackfold approach has been the construction of approximateblack hole solutions. The dipole corrections to the stress-energy tensor of black branes that we havefound modify the effective blackfold equations of motion, and can be used to compute next to leadingorder contributions to these approximate solutions. In this subsection we consider corrections to theapproximate construction of thin black rings in flat space.The relevant blackfold equation for stationary black rings is the extrinsic equation, which is theprojection of (1.5) onto directions orthogonal to the worldvolume. It reads B ab K abρ = 0 . (3.32)In flat space, where in the thin limit black rings live on flat submanifolds with K zzρ , these equationsbecome, as discussed around Eq. (3.3), B zz = 0 (3.33)We shall consider corrections to homogeneous, stationary configurations in backgrounds of thetype (3.20). In these backgrounds, the only non-vanishing Christoffel symbols at r = 0 (apart fromthe usual coordinate pathology on the transverse sphere) have two parallel indices to the worldvolume: u νa u ρb Γ µνρ = K abµ , and u aµ u νb Γ µνρ = − K abρ . (3.34)These, together with homogeneity, imply ∇ c d abµ = − u µd K cdρ d abρ , ∇ c j aµν = − u µd K cdρ j aρν − u νd K cdρ j aµρ . (3.35)20s has been stressed, this includes black rings in (A)dS and Black Saturns [17] with a static centralblack hole, but excludes other cases, as that of Black Rings in Taub-NUT [18].Under these assumptions, the only non-trivial equation of motion is the extrinsic one (2.17), whichreduces to m ab K abµ − ⊥ µσ u νa S λρa R σνλρ = 0 . (3.36)In terms of B ab , j aµν and d abρ these become (cid:16) B ab + d caρ K cbρ (cid:17) K abµ − ⊥ µσ u νa (cid:18) j aλρ − d abλ u ρb (cid:19) R σνλρ = 0 . (3.37)We will now use this formula to draw some predictions, but it is worth noticing that we will not havecomplete predictability. The reason is that, because the dipole is an induced effect, the d abρ correctionsthat are derived from it are two orders away from the leading order in r /R , instead of just one, and acomplete accounting of effects at this order would require extending the calculations of [1, 15] to oneorder beyond.For rings in flat space or (A)dS there are no intrinsic angular momentum corrections to (3.32).This is expected, as these should be insensitive to the orientation of the intrinsic angular momentum(which is in a plane transverse to that of the ring), and thus should be a j contribution. When j aµν = 0, the leading correction for rings in flat space to (3.32) is the dipole correction B zz = − d zz ( r cos θ ) K zz ( r cos θ ) , (3.38)which implies that the critical boost issinh α = 1 n + r R n + 4 n ξ ( n ) . (3.39)In the large n limit this becomessinh α = 1 n + r R (cid:18) n + . . . (cid:19) + . . . , (3.40)highlighting the fact that, at large n , the corrections to the single pole account of this type of blackholes are further suppressed in 1 /n . Arguments along these lines have been given in the past [19]stating that this supression is due to the shorter range of the gravitational interaction at large n .The correction term in (3.39) is two powers in r /R away from the leading contribution. Thisprevents us from computing the conserved charges and thermodynamic properties of these blackrings, as we do not have enough data to carry this out; a naive calculation of these properties to theorder at which the result (3.39) is relevant, that is to ( r /R ) , gives ˜ k − dependent charges, whichis unphysical. Only the next order computation in the MAE will cancel the gauge dependence byintroducing the unknown ambiguous part in B ab , which by Eq. (2.7) will be B ab → B ab + (cid:0) B ab K ccρ + 2 B c ( a K b ) cρ (cid:1) ǫ ρ , (3.41)21nd is of order ( r /R ) .Note, however, that the leading equilibrium condition of black rings in flat space, B zz = 0, implies δ B zz = 0. One then expects Eq. (3.39) to hold to order ( r /R ) , and we trust this equation tothe order we have written it. This is not the case for rings in the more general backgrounds (3.20),and this is why we do not write down a corrected equilibrium condition for (3.22). In conclusion, wecannot at this point predict corrections to the conserved charges of the black rings at order ( r /R ) .However, we would like to stress that at order r /R we do have a prediction, namely that the blackrings do not receive corrections for n > Myers-Perry (MP) black holes exhibit ultra-spinning regimes where the horizon pancakes along one ofthe planes of rotation [20], a behavior which was later realized also to be shown by higher-dimensionalKerr-(A)dS black holes [21]. Both of these cases were recently captured within the blackfold framework[4],[21]. The analysis of references [20] and [21] consists in focusing near the axis of rotation and takinga limit such that the horizon looks like a boosted Schwarzschild membrane. On the other hand, havingbeen able to describe this particular limit using the blackfold approach implies that the entire horizonmust locally have the geometry of a boosted Schwarzschild membrane. This has not been consideredin the literature so far and in App. B we fill this gap by showing precisely how a regular ultra-spinninglimit can be taken everywhere over the horizon.This section begins with a demonstration of the existence of a similar regime for MP black holeswith one non-zero transverse angular momentum. By taking the limit in which the horizon flattensout along one of the planes of rotation and approaching any point on the horizon it is shown thatthe horizon geometry is locally that of a boosted MP membrane. The extended blackfold formalismpresented in Sec. 2 should be able to capture this behavior and, in fact, by reducing Eqs. (2.24)-(2.27)to the case under consideration together with a detailed analysis of the thermodynamic properties ofsuch blackfold geometry, this is shown to be the case. The same type of analysis can be carried outfor higher dimensional Kerr-(A)dS black holes and it is presented in App. C.
Consider the metric of a MP black hole with two angular momenta in n + 5 dimensions [22] ds = − dt + X i =1 h a i dµ i + ( r + a i ) µ i dφ i i + µr n +2 Π F ( dt − X i =1 a i µ i dφ i ) + Π F Π − µr n +2 dr + r h dθ + cos θ ( dψ + cos ψd Ω n − ) i , (4.1)where, µ = sin θ, µ = cos θ sin ψ , (4.2)22 = Y i =1 (1 + a i r ) , F = 1 − X i =1 a i µ i r + a i . (4.3)The event horizon is located at r = r + with r + being the largest positive real root of r n +2 Π − µ = 0.For clarity of notation, in what follows we label a and a as a ≡ a and a ≡ b . The aim of this sectionis to show that there exists an allowed region of parameters where, near the horizon, the metric (4.1)looks locally like a boosted MP membrane. To this end, we take the ultra-spinning limit in the firstangular momentum parametrized by a , that is, a ≫ r + . Since we want to capture the dynamics onthe transverse plane b is kept finite such that a ≫ b . Within such restricted phase space the blackfoldlooks much like the one encountered in the singly-spinning case : a disc with center at θ = 0, radius a and boundary at θ = π/ r ≪ a cos θ is needed. Under this assumption we findΠ ≃ a r ( r + b ) , F ≃ − µ − b µ r + b , g rr ≃ (1 − µ )( r + b ) − b µ r + b − ˜ µa r n − ,µr n +2 Π F ≃ µa r n − [(1 − µ )( r + b ) − b µ ] . Now we introduce the coordinate ρ as ρ = a sin θ , (4.4)which can be seen as the radius on the disc. Furthermore, assuming also that b ≪ a cos θ we obtain X i =1 a i dµ i + r dθ + r cos θdψ ≃ dρ + cos θ ( r + b cos ψ ) dψ . (4.5)As we are after a description of the local geometry of the horizon we need to approach it at any point,thus we consider the metric near a fixed angle θ ∗ (i.e., at a given radius of the disc ρ = ρ ∗ ). Thenecessary requirements become r ≪ a cos θ ∗ and b ≪ a cos θ ∗ . To make contact with the metric of aMP membrane written in its usual form, it is convenient to define˜ r = r cos θ ∗ , ˜ b = b cos θ ∗ , ˜ r n = µ (cos θ ∗ ) n a , Σ = ˜ r + ˜ b cos ψ, ∆ = ˜ r + ˜ b − ˜ r n ˜ r n − . (4.6)Using these definitions we obtain g rr dr ≃ Σ∆ d ˜ r , µr n +2 Π F ≃ ˜ r n cos θ ∗ ˜ r n − Σ . (4.7) See App. B for details. z by z = ρ ∗ φ = a sin θ ∗ φ , (4.8)which parametrizes the angular direction on the disc. The metric (4.1) near the horizon is now seento become: ds = − dt + dρ + dz + ˜ r n Σ˜ r n − (cid:16) dt cos θ ∗ − tan θ ∗ dz − ˜ b sin ψdφ (cid:17) + Σ∆ d ˜ r + Σ dψ +(˜ r + ˜ b ) sin ψdφ + ˜ r cos ψd Ω n − . (4.9)This geometry corresponds to the Myers-Perry membrane, ds = − dt + dρ + dz + ˜ r n Σ˜ r n − ( dt − ˜ b sin ψdφ ) + Σ∆ d ˜ r + Σ dψ +(˜ r + ˜ b ) sin ψdφ + ˜ r cos ψd Ω n − , (4.10)boosted along the z direction with the boost tz ! = θ ∗ − tan θ ∗ − tan θ ∗ θ ∗ ! ˜ t ˜ z ! . (4.11)Applying the boost (4.11) to (4.10) and removing the tildes from ˜ t and ˜ z leads to the metric (4.9). Inturn, (4.11) corresponds to the Lorentz boost V = sin θ ∗ = ρ a , ˜ γ = 1 √ − V = 1cos θ ∗ , ˜ γV = tan θ ∗ . (4.12)In the regime under consideration the MP black hole given in (4.1) has angular velocity along the φ direction, Ω ≃ a . (4.13)Thus, in the blackfold description, the ultra-spinning MP black hole with transverse angular momen-tum is a MP membrane which is rigidly rotating with constant angular velocity, V = ρ Ω , (4.14)as in the singly-spinning case. As seen above, there exists a limit in which the MP black hole can be locally seen as a black membrane,hence, it should be possible to describe it using the formalism of Sec. 2. Here, we construct suchblackfold geometry by solving Eqs.(2.24)-(2.27) for a disc-like topology with internal spin current. The relation between θ ∗ and the rapidity η is tan θ ∗ = sinh η . n + 5-dimensional flat spacetime with metric ds = − dt + dρ + ρ dφ + ds ⊥ + n X i =1 dx i , (4.15)where ds ⊥ is the metric on the transverse 2-plane written in the form ds ⊥ = dρ + ρ dφ . (4.16)In this we embed a membrane as t = σ , ρ = σ , φ = σ , ρ = 0 , x i = 0 , (4.17)which gives rise to the induced metric γ ab dσ a dσ b = − dt + dρ + ρ dφ . (4.18)The stress-energy tensor of the boosted MP membrane can be read off from (4.10) and takes theperfect fluid form (1.3), with energy density and pressure given by ε = Ω ( n +1) πG ( n + 1)˜ r n , P = − Ω ( n +1) πG ˜ r n . (4.19)Since we are trying to construct a black hole solution with a horizon Killing vector field of the form k = ∂∂t + Ω ∂∂φ + Ω ∂∂φ , (4.20)with constant Ω and Ω , then, the fluid velocity must be of the form u a = k a / | k | , with non-vanishingcomponents u t = ˜ γ, u ρ = 0 , u φ = ˜ γ Ω , ˜ γ = 1 p − ρ Ω . (4.21)Moreover, the boosted MP membrane spin current can also be obtained from (4.10) and in thiscoordinate system simply reads j aνρ = Ω ( n +1) πG ˜ b ˜ r n u a ρ δ νρ δ ρφ . (4.22)Since the embedding (4.17) is completely flat we must have K abρ = 0. According to the considerationsof Sec. 3 this immediately implies d abρ = 0. Physically, this is so because there is no bending takingplace and hence no elastic forces playing a role. Also, there is no extra contribution to the boundarystress-energy tensor from (4.10) leading to a vanishing B µνa everywhere. This is most likely due tothe fact that the blackfold boundary is regular, containing no extra stress-energy sources. Finally, we The constancy of Ω and Ω need not be imposed. In fact, it is a consequence of the requirement of stationarity. Aderivation of these conditions can be accomplished and it will be presented elsewhere. as a function of ˜ r n and ˜ b . This can be seen as anadditional equation of state and is given byΩ (˜ r n , ˜ b ) = 1˜ γ ˜ b ˜ r + (˜ r n , ˜ b ) + ˜ b , (4.23)where ˜ r + (˜ µ, b ) is found as the highest real root of ∆(˜ r ), see (4.6), thus,˜ r + ˜ b = ˜ r n ˜ r n − . (4.24)With this in hand, the pole-dipole blackfold equations (2.24)-(2.27), reduce in this case to: B ab K abρ = 0 , D a B ab = 0 , ⊥ µλ ⊥ µρ ∇ a j aνρ = 0 , B ab u µa n b | ∂ W = 0 , j aµν n a | ∂ W = 0 . (4.25)The first and last equations in (4.25) are trivially satisfied due to the vanishing of the extrinsiccurvature tensor and to the vanishing of the ρ component of the fluid velocity u a respectively. Theboundary condition B ab u µa n b | ∂ W = 0 requires that, at the boundary, the fluid must be moving withthe speed of light, i.e., ρ | ∂ W = Ω − .Furthermore, assuming that ǫ and P only depend on ρ , we find from the conservation of thestress-energy tensor D a B ab = ( ǫ + P ) ˙ u b + ∂ b P , (4.26)which is solved by, ˜ r ( ρ ) = ˜ r (0) q − ρ Ω . (4.27)Since j aνρ is independent of t and φ , the conservation equation ⊥ µλ ⊥ µρ ∇ a j aνρ = 0 is trivially obeyed, i.e. , it does not constrain ˜ b as a function of ρ . According to (4.20) we now use the constancy of Ω over the blackfold. As a function of ρ we see from Eq. (4.27) that ˜ r n is proportional to γ ( ρ ) − n .Using Eq. (4.23) and Eq. (4.24) we see that both ˜ b and ˜ r n should be proportional to γ ( ρ ) − . Thus,we find ˜ b ( ρ ) = ˜ b (0) q − ρ Ω . (4.28)In order to show that this blackfold solution corresponds indeed to the limit taken above in thedoubly-spinning MP metric (4.1) we proceed by computing its thermodynamic properties. The thermodynamic properties for the analytic solution given in (4.1) in the ultra-spinning regimecan be obtained from reference [23] and read, M = Ω ( n +3) πG µ ( n + 3) , J = Ω ( n +3) πG µa, J = Ω ( n +3) πG µb , (4.29)with angular velocities Ω = 1 a , Ω = br + b , (4.30)26hile the temperature and entropy are given by, S = Ω ( n +3) G a µr + , T = 14 πr + (cid:18) n − b r + b (cid:19) . (4.31)We want to show that this is correctly reproduced from the blackfold description. To this aim, we usethe expressions (2.31)-(2.33) together with (1.3),(4.22) and find, M = Ω ( n +3) πG ˜ r n (0)Ω ( n + 3) , J = Ω ( n +3) πG ˜ r n (0)Ω , J ⊥ = Ω ( n +3) πG ˜ r n (0)Ω ˜ b (0) . (4.32)We see that we find perfect agreement between the above quantities and those presented in (4.29)-(4.30) if we identify ˜ r n (0) / Ω = µ , Ω = a − and ˜ b (0) = b . To compute the entropy and temperaturewe use the method described in Sec. 2.3. We first start by computing the product T S using the Smarrrelation (2.35), yielding,
T S = Ω ( n +3) πG ˜ r n (0)Ω n − b (0) r + ˜ b (0) ! . (4.33)All the quantities given in (4.32) and the product above can be parametrized in terms of r + and b (0)using Eq. (4.24). According to Eq. (2.36) we obtain a set of two equations, for which the solution is, T = ˜ Cr + n − b (0) r + ˜ b (0) ! . (4.34)The constant ˜ C can be fixed by requiring the right result in the infinitely thin limit, i.e., when ˜ b (0) = 0.This implies ˜ C = 1 / π . It is easy to check that this matches with the temperature given in (4.31). Theentropy then follows by using Eq. (4.33). In this way, we have indeed shown that the ultra-spinningMP black hole with one transverse angular momentum is accurately described within the blackfoldframework in the pole-dipole approximation. This paper has studied corrections to the leading blackfold approach that appear as dipole contribu-tions to the effective blackfold stress-energy tensor. The consideration of these corrections required theintroduction of a new response coefficient, the Young modulus. The physics captured by the Youngmodulus is of the same type as that of the Love numbers of black holes . The calculation of theYoung modulus presented here is similar in spirit to the calculations of transport coefficients in thecontext of the AdS/CFT correspondence . Because the Young modulus is active when there is bend-ing, introducing this type of response coefficient in AdS/CFT would require considering the bending See Ref. [24], that appeared when this paper was being finished, and with which our Sec. 3 tangentially overlaps. As a matter of fact, many aspects of the blackfold derivative expansion are reminiscent of the Fluid/Gravity corre-spondence [25, 26, 27].
27f the brane. There is more to this than non-normalizable fluctuations of AdS; Non-normalizablemodes of AdS change the intrinsic curvatures in the boundary, whereas the bending of a brane man-ifests itself through the extrinsic curvature. This is seen, to leading order, as a dipole deformationof the transverse sphere; for the AdS/CFT of D3 branes bending corresponds to deforming the S ofAdS × S , which is dual to sourcing the scalars of N = 4 transforming in the fundamental of the SO (6) R-symmetry group.In the blackfold approach black holes are seen as probe branes characterized by fluid dynamics andmaterial science concepts: they have an energy density, pressure, viscosity [10], Young modulus, etc.This should not come as a surprise. We have argued that the blackfold approach is a long wavelengtheffective theory, and so are material science and fluid dynamics. In these theories, the small scale isan interatomic distance or a thermal wavelength, and in the blackfold approach it is the radius of thetransverse sphere. Material science and fluid dynamics are universal long wavelength theories and isnatural we recover them in the blackfold limit. Because the fluid behavior enters in corrections to theintrinsic stress-energy tensor ( B ab ) and the material behavior shows up as extrinsic dipole corrections( d abρ ), we can say that blackfolds behave as liquids in worldvolume directions and solids in transversedirections .As argued in [11], the dipole contribution to the stress-energy effective tensor suffers an ambiguity,that we parametrized by ˜ k . The origin of this ambiguity is the freedom to choose a representativesurface inside the finitely thick black brane as the worldvolume surface. ˜ k shifts the representativeworldvolume by ǫ ρ ∂ ρ = ˜ k r R ∂ ( r cos θ ) . Although the black brane has thickness r we are only free tomove the representative surface by a much smaller amount, of about r /R . This is so because shiftsby larger amounts threaten the hierarchy B µνρ /B µν = r /R = O , which is crucial for the consistencyof the pole-dipole approximation.In the point particle case there is a natural way to fix this ambiguity, namely by requiring the massdipole to vanish: This natural representatitve is the center of mass, whose coordinates are defined as X µ = 1 m Z Σ √− g d D − x T x µ . (5.1)For p − branes this choice is not well defined in general, simply because there is more than one world-volume dipole, d abρ . While the vanishing of one particular component may be achieved by gaugefixing, the others will not vanish in general.The monopole contribution to the effective stress-energy tensor also suffers from this ambiguity(2.7), but for the monopole this is a ( r /R ) contribution (3.41), and this is why we have not seen itat the order we have worked in the MAE. Doing the MAE to next order would allow us to see thiscontribution to B ab . A natural way of fixing this symmetry would be demanding that the equation ofstate of B ab remains ε = − ( n + 1) P .It will be very interesting to use the analysis we have carried out here to study other blackfoldconfigurations, like those in which one will see a non-zero spin-spin interaction, as in Black Saturns We thank Roberto Emparan for suggesting this. p − form fields, blackfolds willbehave dielectrically and paramagnetically, developing charge and magnetic dipoles. In this case, theresponse coefficients are polarizabilities and susceptibilities.Finally, it would be interesting to study time-dependent embeddings, from which one could measurea viscosity that would give the time scale of the damping of extrinsic oscillations of black strings. Thisviscosity is different from those measured in [10]. Using a mixed analysis, in which both types ofviscosities (intrinsic and extrinsic) would be active, one could address the very important problemof the stability of thin black rings in a controlled approximation, to an order accounting for theGregory-Laflamme long-wavelength cutoff. We plan to address this problem in the near future. Acknowledgements
We thank Marco Caldarelli, Roberto Emparan, Veronika Hubeny, Barak Kol, David Mateos, ShirazMinwalla, Mukund Rangamani and Simon Ross for discussions. Jay would like to thank the TataInstitute for Fundamental Research for hospitality during the completion of a part of this work andto FCT Portugal for the grant SFRH/BD/45893/2008. JC is supported by the STFC Rolling grantST/G000433/1. TH thanks the Niels Bohr Institute for hospitality. JC, TH and NO would like tothank the Centro de Ciencias de Benasque Pedro Pascual, and JC would also like to thank the NBI andICTP for hospitality during various stages of this project. JC acknowledges support of GEOMAPSduring his stay at the NBI. The work of NO is supported in part by the Danish National ResearchFoundation project “Black holes and their role in quantum gravity”.
A Notation
In this appendix we collect some conventions about notation. This paper deals with submanifolds W p +1 of a ( D = p + n + 3) − dimensional spacetime, described by the mapping functions X µ ( σ a ) fromthe worldvolume, parametrized by the coordinates σ a , to the ambient spacetime, with coordinates x µ . g µν is the background metric while W p +1 inherits the metric γ ab = u µa g µν u νb , u µa = ∂ a X µ . (A.1)Here, µ, ν indices are raised and lowered with g µν and its inverse g µν , and a, b indices with γ ab and itsinverse γ ab .To project any spacetime tensor along tangential directions to the worldvolume we can use u aµ ,while for directions orthogonal to W p +1 we define the projector, ⊥ µν = g µν − u aµ γ ab u bν . (A.2)29 subindex ⊥ on a tensor indicates that all µ, ν type of indices are orthogonal, e.g., B aµ ⊥ = ⊥ µν B aν ⊥ . (A.3) j aµν and d abµ defined in (2.20) and (2.22) are ⊥ -objects, but we do not write the ⊥ subindex on themin order to avoid cluttering. ∇ µ is the covariant derivative on the ambient space compatible with g µν and Γ ρµν are its Christoffelsymbols. D a is the intrinsic covariant derivative on W p +1 compatible with γ ab and { ab c } are itsChristoffel symbols. Our convention for the Riemann curvature tensor is R µλνρ = Γ µλρ,ν − Γ µλν,ρ + Γ µσν Γ σλρ − Γ µσρ Γ σλν . (A.4)The operator ∇ a is defined to be compatible both with γ ab and g µν such that, for instance, ∇ a v µνbc = u ρa ∂ ρ v µνbc − u ρa Γ σρµ v σν bc + u ρa Γ νρσ v µσbc − (cid:8) da b (cid:9) v µνdc + { ca d } v µνbd . (A.5)For a submanifold tensor, ∇ c t a...b... = D c t a...b... . Moreover, the extrinsic curvature of W p +1 can bewritten as ∇ a u ρb = K abρ , (A.6)with K abρ being also a ⊥ − object.The boundary of the submanifold is described by σ a = ζ a ( λ ) and normal vector ˆ n µ = u µa ˆ n a withunit norm. We introduce the coordinate vectors v a ˆ i as v a ˆ i = ∂ζ a ∂λ ˆ i , (A.7)satisfying the properties v µ ˆ i = u µa v a ˆ i and ˆ n a v a ˆ i = 0 such that the induced metric on the boundary takesthe form ˆ h ˆ i ˆ j = γ ab ( ζ ) v a ˆ i v b ˆ j . ∇ ˆ i is the boundary covariant derivative compatible with the metric ˆ h ˆ i ˆ j . B Ultra-spinning Myers-Perry black holes revisited
The purpose of this section is to instructively show how to take the blackfold limit for singly-spinningMP black holes as it was done in Sec. 4.1 for its doubly-spinning counterpart. Bearing this in mind,we consider the MP metric with a single angular momentum in n + 5 dimensions ds = − dt + µr n Σ ( dt − a sin θdφ ) + Σ∆ dr + Σ dθ + ( r + a ) sin θdφ + r cos θd Ω n +1) , (B.1)with Σ = r + a cos θ, ∆ = r + a − µr n . (B.2)30he horizon radius r + is given by the largest positive real root of ∆( r ) = 0. The ultra-spinning limitis attained when r + ≪ a [20]. From the definition of ∆ above we see that in this limit r n + ≃ µ/a .From the n + 1-sphere metric in (B.1) we see that the radius of the ( n + 1)-sphere is r = r + cos θ = (cid:16) µa (cid:17) n cos θ . (B.3)Hence, the blackfold is a rotating disc of radius a with its center and boundary located at θ = 0 and θ = π/ r ≪ a . However, this is not sufficiently close everywhere on the blackfold. We thusintroduce the coordinate ρ = a sin θ . (B.4)This can be seen as the radius on the disc. In terms of this we can write the thickness as r ( ρ ) = r + r − ρ a . (B.5)For a given point on the disc we should require that the distance scale over which the thickness changeis much larger than the thickness of the disc. Thus, we should require r ≪ | r ′′ | . (B.6)In terms of the horizon radius and rotation parameter, this requirement becomes: r + ≪ a r − ρ a . (B.7)This tells us, for each point on the disc, how widely separated the scales must be for the blackfoldapproximation to be valid. To see the boosted black membrane from the metric (B.1), given a radius ρ , we need r ≪ a r − ρ a . (B.8)This implies r ≪ a cos θ . Now, consider a given point with radius ρ = ρ ∗ at the disc, correspondingto the angle θ ∗ with ρ ∗ = a sin θ ∗ . Hence, we require r ≪ a cos θ ∗ . In order to make a more clearcontact with the metric of a Schwarzschild membrane we define r ∗ = r + cos θ ∗ , ˜ r = r cos θ ∗ , z = ρ ∗ φ . (B.9)In the approximation r ≪ a cos θ ∗ the metric near ρ = ρ ∗ becomes: ds = − dt + dρ + dz + (1 − f ) (cid:18) dt cos θ ∗ − tan θ ∗ dz (cid:19) + d ˜ r f + ˜ r d Ω n +1) , (B.10)with f ≡ − r n ∗ ˜ r n . (B.11)31his corresponds to boosting the static black Schwarzschild membrane ds = − dt + dz + (1 − f ) dt + dρ + d ˜ r f + ˜ r d Ω n +1) , (B.12)along z with the boost (4.11) with corresponding Lorentz boost parameter (4.12). The angular velocityin this ultra-spinning regime is also given by (4.13) and hence we see that in the blackfold descriptionthe ultra-spinning MP black hole is a black membrane which is rigidly rotating with constant angularvelocity given by V = ρ Ω . Blackfold equations with zero transverse angular momentum
To facilitate comparison with the doubly-spinning case of Sec. (4.2) we present here a brief analysisof the blackfold equations in the single-pole approximation.We begin by embedding the disc in the background (4.15) using the mapping functions (4.17),leading to the induced metric (4.18). Next, we read off the static black membrane stress-energy tensorfrom the metric (B.10), which has the form of (1.3). The fluid velocity is given by the pullback of thebackground Killing vector field k = ∂∂t + Ω ∂∂φ , (B.13)which gives rise to the same non-vanishing components as those given in (4.21).Now, we analyze the blackfold equations (2.24)-(2.27) adapted to the current situation. As theembedding is flat, the extrinsic curvature K abµ vanishes and thus the extrinsic equation T ab K abµ = 0is trivially satisfied. The remaining non-trivial equations read D a B ab = 0 , B ab u µa n b | ∂ W = 0 . (B.14)The conservation of the intrinsic stress-energy tensor, assuming that ǫ and P only depend on ρ ,implies r ( ρ ) = r (0) q − ρ Ω . (B.15)Furthermore the boundary condition B ab u µa n b | ∂ W = 0 is again satisfied provided ρ | ∂ W = Ω − , hencethe disc has a radius of ρ = Ω − and is moving at the speed of light there. This is the result obtainedin [2, 4] and matches the thickness (B.5) upon the identification r (0) = ( µ/a ) n and Ω = a − . C Spin corrections for higher-dimensional Kerr-(A)dS black holes
Here we study the same limiting behavior of higher-dimensional Kerr-(A)dS black holes as in Sec. 4 forMP black holes and show that it can be described using the blackfold formalism. Due to the similaritybetween both cases we refer to Sec. (4) for a more extensive analysis.32n spheroidal coordinates, the metric of the Kerr-AdS black hole with two angular momenta ineven dimensions is given by [28, 29] ds = − W (cid:18) r L (cid:19) dt + µU dt − X i =1 a i µ i Ξ i dφ i ! + UV − µ dr + X i =1 r + a i Ξ i dµ i + µ i (cid:18) dφ i + √ α i L dt (cid:19) ! + r ( n +5) / X i =3 dµ i + N X i =3 µ i dφ i + 1 L W (1 + r L ) X i =1 r + a i Ξ i µ i dµ i + r n +5) / X i =3 µ i dµ i , (C.1)where W = 1 + X i α i µ i Ξ i , U = r n − − X i =1 a i µ i r + a i ! Y j =1 ( r + a j ) ,V = r n − (1 + r L ) Y i =1 ( r + a i ) , (C.2)with µ and µ as given in Eq. (4.2) and UV − µ = (cid:16) − P i =1 a i µ i r + a i (cid:17) (1 + r L ) − µr n − Q j =1 ( r + a j ) , α i = a i L , Ξ i = 1 − α i . (C.3)The horizon r + is given by the largest positive real root of V ( r ) − µ = 0. For clarity of notation weset a ≡ a and a ≡ b . The ultra-spinning limit is attained when r + , b ≪ a, L with 0 ≤ α < r ≪ a, L with r finite , in a similar fashion as in the singly-spinning case [21]. However, for the metricto look locally like a boosted MP membrane, we furthermore need r ≪ a √ Ξ cos θ and b ≪ a √ Ξ cos θ .Under these assumptions, it is easy to show that the last term in (C.1) is subleading while thefunctions V, U, W reduce to W → , U → r n − a (cid:18) − µ − a i µ i r + a i (cid:19) ( r + b ) , V → r n − a ( r + b ) . (C.4)In order to parametrize the membrane in a convenient way, we introduce the coordinates ρ = a √ Ξ sin θ, z = ρ ∗ φ . (C.5)Then, near a fixed angle θ ∗ , the metric (C.1) is seen to reduce again to that of a MP membrane (4.10)but with Lorentz boost, V = sin θ ∗ = p Ξ Ω ρ , ˜ γ = 1 p − Ξ ρ Ω , Ω = 1 a . (C.6) The same analysis also holds in the case of odd dimensions. In the case of higher-dimensional Kerr-dS black holes, the parameter α is instead constrained by α i ≥ lackfold pole-dipole equations with non-zero transverse angular momentum We now want to describe the above limiting behavior of the higher-dimensional Kerr-(A)dS blackholes using the blackfold approach. It is convenient, in order to highlight the existent 2-planes of thebackground spacetime, to write the metric of AdS in conformally flat coordinates ds = − F ( ρ ) dt + H ( ρ ) − dρ + ρ dφ + ds ⊥ + n X i =1 dx i ! , ρ = ρ + ρ + n X i =1 x i , (C.7)where ds ⊥ is given by (4.16) and F ( ρ ) = ρ L − ρ +4 L ! , H ( ρ ) = 1 − ρ L . (C.8)Note that this coordinate system differs from the one used in (C.1). To see how to translate from onecoordinate system to the other see for example [21]. To embed the membrane in this background wechose the embedding coordinates (4.17), which give rise to the induced metric γ ab dσ a dσ b = − F ( ρ ) dt + H ( ρ ) − (cid:0) dρ + ρ dφ (cid:1) . (C.9)Again, all extrinsic curvature components vanish since the embedding is flat. The stress-energy tensoris still given by (1.3) but now with boost velocities, u t = ˜ γ, u ρ = 0 , u φ = ˜ γ Ω , ˜ γ = 1 s − Ξ ρ − ρ L Ω . (C.10)Furthermore the spin current in this coordinate system reads j aνρ = Ω ( n +1) πG ˜ b ˜ r n H ( ρ ) ρ u a δ νρ δ ρφ , (C.11)while the dipole current and B µνa components vanish. Due to the vanishing of the extrinsic curvatureand of all the contractions involving the Riemann tensor in Eqs. (2.24)-(2.27), the blackfold equationsreduce to Eqs. (4.25) as in the flat space case. Solving the bulk equations requires:˜ r ( ρ ) = ˜ r (0) vuut − Ξ ρ − ρ L Ω , ˜ b ( ρ ) = ˜ b (0) vuut − Ξ ρ − ρ L Ω . (C.12)Moreover the boundary condition B ab u µa n b | ∂ W = 0 implies that the disc has a maximum radius givenby ρ | ∂ W = 2 L ( L Ω − p L Ω − b (0) = 0 thethickness ˜ r obtained in [21] coincides with the one given in (C.12) and agrees with the analyticsolution (C.1) upon the identification ˜ r n (0) / Ω = µ and Ω − = a .34 hermodynamic quantities The thermodynamical quantities of the analytic solution (C.1) in the ultra-spinning regime can beobtained from [23] and read M = Ω ( n +3) πG µ Ξ (2 + Ξ ( n + 1)) , J = Ω ( n +3) πG µ Ξ a, J ⊥ = Ω ( n +3) πG µ Ξ b , (C.13)with Ω = 1 a , Ω = br + b , (C.14)while the entropy and temperature are given by S = Ω ( n +3) G a µ Ξ r + , T = 14 πr + (cid:18) n − b r + b (cid:19) . (C.15)Evaluating the conserved charges using Eqs. (2.31)-(2.33) results in M = Ω ( n +3) πG ˜ r n (0)Ξ Ω (2 + Ξ ( n + 3)) , J = Ω ( n +3) πG ˜ r n (0)Ξ Ω , J ⊥ = Ω n +3) πG ˜ r n (0)Ξ Ω ˜ b (0) . (C.16)We can straightforwardly check that these results agree with the ones presented in (C.13)-(C.14) uponthe identification ˜ r n (0) / Ω = µ , Ω − = a , and ˜ b (0) = b . Moreover, in the AdS case the tension (2.37)is non-zero and reads T = − α Ω ( n +3) πG ˜ r n (0)Ξ Ω . (C.17)Using the Smarr relation (2.35) to compute the product T S and then the first law of black holethermodynamics we can exactly reproduce the temperature and entropy as given in (C.15) in thesame way we did for MP black holes. This leads one to conclude that higher-dimensional Kerr-(A)dSblack holes in the ultra-spinning regime are correctly captured by the blackfold pole-dipole equations.
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