aa r X i v : . [ h e p - t h ] N ov CPHT-RR095.1009NORDITA-2009-65
Essentials of Blackfold Dynamics
Roberto Emparan a,b , Troels Harmark c,d , Vasilis Niarchos e , Niels A. Obers ca Instituci´o Catalana de Recerca i Estudis Avan¸cats (ICREA) b Departament de F´ısica Fonamental, Universitat de Barcelona,Marti i Franqu`es 1, E-08028 Barcelona, Spain c The Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen Ø, Denmark d NORDITA, Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden e Centre de Physique Th´eorique, ´Ecole Polytechnique, 91128 Palaiseau, FranceUnit´e mixte de Recherche 7644, CNRS [email protected], [email protected], [email protected],[email protected]
Abstract
We develop and significantly generalize the effective worldvolume theory for higher-dimensional black holes recently proposed by the authors. The theory, which regardsthe black hole as a black brane curved into a submanifold of a background spacetime—a blackfold —, can be formulated in terms of an effective fluid that lives on a dynamicalworldvolume. Thus the blackfold equations split into intrinsic (fluid-dynamical) equations,and extrinsic (generalized geodesic embedding) equations. The intrinsic equations can beeasily solved for equilibrium configurations, thus providing an efficient formalism for theapproximate construction of novel stationary black holes. Furthermore, it is possibleto study time evolution. In particular, the long-wavelength component of the Gregory-Laflamme instability of black branes is obtained as a sound-mode instability of the effectivefluid. We also discuss action principles, connections to black hole thermodynamics, andother consequences and possible extensions of the approach. Finally, we outline how thefluid/AdS-gravity correspondence is related to this formalism. ontents
A.1 Extrinsic curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30A.2 Variational calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
References 33 Introduction
In recent work we have identified the origin of the rich variety of higher-dimensional blackholes in the possibility of having horizons that are much longer along some directionsthan in others [1]. In such cases there are at least two different horizon length-scales, andwe have proposed an effective theory that captures the long-distance physics when thesescales are widely separate. Focusing on the simplest, universal case of neutral, vacuumblack holes, the two length scales are associated with the mass and angular momentum , ℓ M ∼ ( GM ) D − , ℓ J ∼ JM . (1.1)In a four-dimensional black hole, by virtue of the Kerr bound J ≤ GM these lengths arealways parametrically similar, but higher-dimensional black holes (including also blackrings) with ℓ J ≫ ℓ M are known to exist in ultra-spinning regimes where the angularmomentum for a given mass can be arbitrarily high [2, 3, 4, 5]. It appears that essentiallyall their novel features, compared to their four-dimensional cousins, arise from the abilityto separate these two lengths. This suggests that higher-dimensional black holes must beorganized according to a hierarchy of scales:1. ℓ J . ℓ M : black holes behave qualitatively similarly to the four-dimensional Kerrblack hole.2. ℓ J ≈ ℓ M : threshold of new black hole dynamics.3. ℓ J ≫ ℓ M : the separation of scales suggests an effective description of long-wavelengthdynamics.The first and second regimes fully involve the non-linearities of General Relativity, butin the first case we have no hints of qualitatively new properties of black holes comparedto four-dimensional ones. In particular we conjecture that for J < J crit = α D M ( GM ) D − (with α D a yet-undetermined numerical constant of order one), the Myers-Perry black holesare dynamically stable and unique among solutions with connected regular horizons . Incontrast, as the two scales begin to diverge in the second regime, we have good evidenceof the onset of new phenomena: horizon instabilities, inhomogeneous (‘pinched’) phases,non-spherical horizon topologies, and absence of uniqueness [4, 3, 6, 7]. This regime seemshard to investigate by means of exact analytical techniques, but on the other hand thepresence of only one scale in the problem is actually convenient for numerical investigation,since one does not require high precision over widely different scales. In the third regime,which is the focus of this paper, the existence of a small parameter ℓ M /ℓ J allows the J = ( P i J i ) / aggregates the effect of all possible angular momenta. This extends a previous conjecture in [3] about stability of MP black holes to involve their uniquenesstoo. Here J crit refers to the minimum J for which either instability or non-uniqueness first appear. provides an outline for a program to investigate higher-dimensional black holes .We base the effective description of black holes in regime 3 on the idea that the limit ℓ M /ℓ J → black brane whose worldvolume spans acurved submani fold of a background spacetime — what we refer to as a blackfold . Thesimplest example is the characterization of a (thin) black ring as a circular boosted blackstring, which was worked out in detail in [6] and expanded upon in [13, 14]. But, as shownin [1], ultraspinning Myers-Perry black holes, as well as a number of other new black holes,can also be appropriately captured with this approach.Thus, the effective theory of black holes when ℓ M /ℓ J ≪ ℓ M /ℓ J the backreaction of the blackfold on thegeometry is neglected — it is a ‘test’ blackfold. Corrections to the geometry are foundby first computing the linearized gravitational backreaction on the background, whichis of order ( ℓ M /ℓ J ) D − , then this correction induces a perturbation of the near-horizongeometry, and so forth. This can be systematically pursued in the form of a matchedasymptotic expansion. For higher-dimensional black rings these first corrections havebeen computed [6]. In this paper, however, we remain at the ‘test blackfold’ level ofapproximation. This is enough to reveal new kinds of black holes, compute their physicalproperties, and also study time-dependent situations and stability.Ref. [1] gave a basic outline of the theory of blackfolds. Here we develop its concep-tual basis and considerably improve and generalize the presentation. Its application tospecific new classes of higher-dimensional black holes will be discussed in a forthcomingpublication.We draw heavily from the beautiful theory of classical brane dynamics developed byCarter in [15] . In this respect, we may say that a main part of our contribution is to The approach to extremality in a black hole introduces a long length scale transverse to the horizonand allows to decouple a different sector of the physics, namely the near-horizon region [8]. See [9, 10] for brief reviews of higher-dimensional black holes and [5] for a more extensive one. Our title deliberately highlights this. black branes, and to interpret the results in the context of higher-dimensional black hole physics according to the general considerations above. But wealso emphasize that the general classical brane dynamics, regarded as a long-wavelengtheffective theory, must take the form of the dynamics of a fluid that lives on a dynamicalworldvolume. Black branes correspond to a specific type of fluid, with a certain equation ofstate and with specific values for transport coefficients. At the leading order that we workin this paper, the fluid is a perfect one and the brane equations are the Euler equationsfor the fluid — intrinsic equations— plus a generalization of the geodesic equation for themotion of a p -brane — extrinsic equations for the worldvolume embedding. We believethat in principle it should be possible to incorporate higher-derivative corrections to theseand compute transport coefficients by performing a derivative expansion of the underlyingmicroscopic theory, in this case Einstein’s theory.Closely related precedents of a mapping of black hole dynamics to fluid dynamics arethe ‘membrane paradigm’ [16], and the more recent ‘fluid/AdS-gravity correspondence’[17]. As we shall argue near the end, the fluid/AdS-gravity correspondence can be embed-ded within the approach we advocate here — in fact it has been an important influencein developing it. The general arguments discussed above indicate that a fluid-dynamicaldescription should indeed be expected to exist for any long-wavelength fluctuations aroundan equilibrium state. From this perspective, perhaps the main qualitative novelty of ourapproach is that the existence of a hierarchy of scales in higher-dimensional black holesmakes this effective theory useful not only for studying fluctuations, but also for construct-ing and analyzing in a very general manner novel kinds of stationary ( i.e., equilibrium)black holes, including vacuum solutions.The outline of the paper is the following: Section 2 develops the conceptual basisunderlying the blackfold approach as a worldvolume theory of the dynamics of blackbranes. Section 3 presents a main result of this paper: the blackfold equations , a set ofcoupled non-linear differential equations for the collective coordinates of a neutral blackbrane. Section 4 focuses on the important case of stationary blackfolds, for which theintrinsic subset of these equations can be explicitly solved. Section 5 analyzes the issuesraised by the possible presence of boundaries of the blackfold worldvolume. In section 6we describe how to compute the physical magnitudes of a blackfold. Section 7 presentsan action principle for stationary blackfolds. This is useful for practical calculations, butalso admits a simple and appealing interpretation in terms of black hole thermodynamics.Section 8 discusses briefly the stability of blackfolds exhibiting how the approach canuncover in a remarkably simple way the Gregory-Laflamme instability of black branes.We close in Section 9 with a discussion of the relation of blackfolds to other effectivetheories of black hole dynamics, in particular the fluid/AdS-gravity correspondence. In theappendix we collect a number of technical results on the extrinsic geometry of submanifoldembeddings. 4 otation and terminology: For clarity and later reference, we summarize here some of our notation.For a blackfold of p spatial dimensions in D -dimensional spacetime it is convenient tointroduce n = D − p − . (1.2)The codimension of the blackfold worldvolume is n + 2.Spacetime (background) and worldvolume magnitudes are denoted and distinguishedas follows: • Spacetime coordinates (and embedding functions): X µ , µ, ν . . . = 0 , . . . , D − g µν .Background metric connection: Γ σµν .Background covariant derivative: ∇ µ . • Worldvolume coordinates: σ a , a, b . . . = 0 , . . . , p .(Induced) worldvolume metric: γ ab .Worldvolume metric connection: (cid:8) ab c (cid:9) .Worldvolume covariant derivative: D a .Indices µ, ν, . . . are lowered and raised with g µν , indices a, b, . . . with γ ab .We use the same letter for a background tensor tangent to the worldvolume, t µ...ν... , andfor its pullback onto the worldvolume, t a...b... (the only exception is the first fundamentalform h µν and the induced metric γ ab ).Ω ( n ) denotes the volume of the unit n -sphere. Ω i denotes the angular velocity of theblackfold in the i -th direction. V ( p ) is the volume of a spatial section of the blackfold. V i is the spatial velocity fieldon the worldvolume, and V = qP i V i .Note that we refer to ‘long-wavelength’ and not ‘low-energy’, effective theory, thereason being that in classical gravity, without ~ , such notions are not equivalent (indeedlarge energies typically imply long distances in classical gravity). We present the effective theory of blackfolds trying to highlight the similarities with thefield-theoretical effective description of other extended objects, such as cosmic strings orD-branes. The main differences with these are, first, that the short-distance degrees offreedom that are integrated out are not those of an Abelian Higgs model nor massivestring modes, but rather purely gravitational degrees of freedom. Second, the extendedobjects —curved black branes— possess black hole horizons. We obtain the equationsusing general symmetry and conservation considerations, rather than doing a detailedderivation from first principles. 5 .1 Collective coordinates for a black brane
Schematically, the degrees of freedom of General Relativity are split into long and shortwavelength components, g µν = { g (long) µν , g (short) µν } . (2.1)The Einstein-Hilbert action is then approximated as I EH = 116 πG Z d D x √− gR ≈ πG Z d D x q − g (long) R (long) + I eff [ g (long) µν , φ ] , (2.2)where I eff [ g (long) µν , φ ] is an effective action obtained after integrating-out the short-wavelengthgravitational degrees of freedom (precisely what we mean by this will be made clear insec. 2.2). The coupling of these to the long-wavelength component of the gravitationalfield is captured through a set of ‘collective coordinates’ that we denote schematically by φ . Our first task is to identify these effective field variables and the length scales thatallow this splitting of degrees of freedom.The main clue to the nature of the effective theory comes from the observation that thelimit ℓ M /ℓ J → p -brane.Its geometry in D = 3 + p + n spacetime dimensions is ds p − brane = − (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 . (2.3)The coordinates σ a = ( t, z i ) span the brane worldvolume. A more general form of themetric is obtained by boosting it along the worldvolume. If the velocity field is u a , with u a u b η ab = − ds p − brane = (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 . (2.4)The parameters of this black brane solution consist of the ‘horizon thickness’ r , the p independent components of the velocity u (say, its spatial components u i ), and the D − p − X ⊥ . The D collective coordinates ofthe black brane are φ ( σ a ) = { X ⊥ ( σ a ) , r ( σ a ) , u i ( σ a ) } (2.5)and in the long-wavelength effective theory one allows ∂X ⊥ , ln r and u i to vary slowlyalong the worldvolume, W p +1 , over a length scale R much longer than the size-scale of theblack brane, R ≫ r . (2.6) For clarity of presentation, at this initial stage we consider that both long and short degrees of freedomobey vacuum gravity dynamics, R µν = 0, but this can be easily generalized, see sec. 3.5 below. R is set by the smallest intrinsic or extrinsic curvature radius of theworldvolume. Observe that we require slow variations of ∂X ⊥ , not of X ⊥ . Like thelongitudinal velocities u a , the transverse ‘velocities’ ∂X ⊥ can be arbitrary.In order to preserve manifest diffeomorphism invariance it is convenient to introducesome gauge redundancy and enlarge the set of embedding coordinates of the worldvolumeof the black brane to include all the spacetime coordinates X µ ( σ a ). From this embeddingwe can compute an induced metric γ ab = g (long) µν ∂ a X µ ∂ b X ν . (2.7)This is naturally interpreted as the geometry induced on the worldvolume of the brane.To understand what this means, regard the split between degrees of freedom as follows:the long-wavelength degrees of freedom live in a ‘far-zone’ r ≫ r , and they describe thebackground geometry in which the (thin) brane lives. Then (2.7) is the metric induced onthe brane worldvolume. The short-wavelength degrees of freedom live in the ‘near-zone’ r ≪ R . In the strict limit where R → ∞ , the near-zone solution is (2.4), but when R islarge but finite, the collective coordinates depend on σ . Also, the long and short degreesof freedom interact together in the ‘overlap’ or ‘matching-zone’ r ≪ r ≪ R , where themetrics g (long) µν and g (short) µν must match. Then the near-zone metric for the black branemust be of the form ds = (cid:18) γ ab ( 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 + . . . . (2.8)The dots here indicate that, without additional terms, in general this is not a solutionto the Einstein equations. These equations contain terms with gradients of ln r , u a and γ ab . However these terms can be seen to come multiplied by powers of r so they aresmall when r /R ≪
1. Then we can consider an expansion of the equations in derivativesand add a correction to (2.8) to find a solution to the Einstein equations to first order inthe derivative expansion. A subset of the resulting Einstein equations can be rewrittenas equations on the collective field variables φ ( σ ). An important requirement is that theperturbations preserve the regularity of the horizon, and to this effect working in a setof coordinates (Eddington-Finkelstein type) different than the ones above may be moreappropriate.The development of this line of argument, which can be regarded as a blend of the ideasfor the effective descriptions of black hole dynamics in [18, 19] (and references therein),and in [17], produces a systematic derivation of the blackfold equations. This is howevera technically involved approach that we hope to discuss elsewhere. Here we shall insteadfollow a less rigorous but quicker and physically well-motivated path, relying on generaleffective-theory-type of arguments that allow us to readily obtain the blackfold formalismvalid to lowest order in the derivative expansion. As we will see, this is the ‘perfect fluid’and ‘generalized geodesic’ approximation. The more systematic method outlined above7ould be needed to go beyond these approximations and account for dissipation and effectsof internal structure and gravitational self-force .We have considered a black brane in (2.3), (2.4) that is not rotating along the transverse( n + 1)-sphere, nor have we included any possible deformations of it. This is just asimplification and is not essential. Note first that ultraspins in this sphere, with rotationparameter a ∝ J/M ≫ r , can indeed be considered in the blackfold approach, but thenthe starting point must be the black brane limit that results when a/r → ∞ . On theother hand small spins, with a . r , can also be included easily to the order that we workin this paper. The reason is that the modifications to the blackfold dynamics introducedby these spins only enter at a higher order in the expansion in r /R . This is familiar, forinstance, in that spin effects on the worldline of a test particle enter through couplings tothe background curvature tensor: they reflect the internal structure of the particle, whichclearly is a higher-order correction. Thus the dynamics associated to internal spin andpolarization effects of the spheres of size r is effectively integrated-over without affectingthe lowest-order formalism. It must be noted, though, that higher-dimensional neutralblack holes exhibit zero-mode deformations at discrete values of the spin when n ≥ n = 1 , r than those of the mass, the computation of the effective worldvolume stresstensor in the next subsection is unaffected. The blackfold equations also apply in the formbelow in the presence of internal spins. By the phrase ‘integrating out the short-distance dynamics’ we mean that the Einsteinequations are solved at distances r ≪ R and then the effects of the solution at distances r ≫ r are encoded in a stress-energy tensor that depends only on the collective coor-dinates. The stress tensor is such that its effect on the long-wavelength field g (long) µν isthe same as that of the black brane at distances r ≫ r . For reasons that will becomeapparent as we proceed, it is both simpler and more convenient to work with an effectivestress-energy tensor rather than with an effective action. In any case, nothing is lost sincewe work at the classical level.The effective equations from (2.2) are R (long) µν − R (long) g (long) µν = 8 πGT eff µν , (2.9) Also, see [20] for an approach to the calculation of these corrections (for zero-branes) in an effective-theory framework somewhat akin to the spirit in this paper. T eff µν = − p − g (long) δI eff δg µν (long) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) W p +1 . (2.10)We now argue that the appropriate notion for this effective stress-tensor that capturesthe coupling of the short-wavelength degrees of freedom to the long-wavelength ones, isthe quasilocal stress-energy tensor introduced by Brown and York [21]. This is defined byconsidering a timelike hypersurface that lies away from the black brane and encloses itby extending along the worldvolume directions and the angular directions Ω ( n +1) , i.e., thehypersurface acts as a boundary. Actually, as we explained above, the angular directionsare integrated over in our description (and to leading order they do not play any role),so we can simplify the discussion by focusing exclusively on the worldvolume directionsof the boundary. If the boundary metric (along worldvolume directions) is γ ab then thequasilocal stress tensor is T (quasilocal) ab = − √− γ δI cl δγ ab , (2.11)where I cl is the classical on-shell action of the solution. For our purposes, this is the actionwhere the short-distance gravitational degrees of freedom, r ≪ R , are integrated and soit must be the same function of the collective variables as I eff . Together with the relation(2.7) this implies that we can identify (2.10) with (2.11).It is shown in [21] that the Einstein equations with an index orthogonal to the boundaryare first-order equations equivalent to the equation of conservation of the quasilocal stresstensor, D a T ab (quasilocal) = 0 , (2.12)where D a is the covariant derivative associated to the boundary metric γ ab . Hence, solvingthe equations (2.12) is equivalent to solving (a subset of) the Einstein equations.Since we identify the stress tensors (2.10) and (2.11), henceforth we drop the super-scripts from them. We also drop the superscript (long) from the background metric g µν .The effective stress tensor is computed in the zone r ≪ r ≪ R , where the gravitationalfield is weak and the quasilocal stress tensor T ab is, to leading order in r /R , the same asthe ADM stress tensor. For the boosted black p -brane (2.4) one can readily compute itand find T ab = Ω ( n +1) πG r n (cid:16) nu a u b − η ab (cid:17) . (2.13)After introducing a slow variation of the collective coordinates the stress 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) + . . . (2.14)where the dots stand for terms with gradients of ln r , u a , and γ ab , which we are takingto be small and are neglected in this paper. 9 .3 General branes: fluid perspective On general grounds, the long-wavelength effective theory for any kind of brane will takethe form of a derivative expansion for an effective stress-energy tensor that satisfies theconservation equations (2.12). This is the dynamics of an effective fluid that lives on theworldvolume spanned by the brane. If the worldvolume theory is isotropic, then to lowestderivative order the stress tensor is that of an isotropic perfect fluid, T ab = ( ε + P ) u a u b + P γ ab , (2.15)with energy density ε , pressure P and velocity u a satisfying u a u b γ ab = − . (2.16)Thermodynamics provides the universal macroscopic description of equilibrium configura-tions, and fluid dynamics is the general long-wavelength description of fluctuations underthe assumption of local equilibrium. So in general there will be an equation of state, whichwe write in the form P ( ε ), and the system will obey locally the laws of thermodynamics dε = T ds (2.17)and Euler-Gibbs-Duhem relation ε + P = T s (2.18)where T is the local temperature and s the entropy density of the fluid in its rest frame.The fluid may also carry additional conserved charges, but we do not consider these inthis paper.For a black brane, (2.14) tells us that the effective fluid has ε = Ω ( n +1) πG ( n + 1) r n , P = − n + 1 ε . (2.19)Moreover, in the rest frame of the fluid the Bekenstein-Hawking identification betweenhorizon area and entropy s = Ω ( n +1) G r n +10 (2.20)and between surface gravity and temperature T = n πr (2.21)is well known to reproduce the correct thermodynamic relations (2.17), (2.18).Going beyond the perfect fluid approximation (2.15), the stress tensor will acquiredissipative terms proportional to gradients of ln r , u a , γ ab . As discussed above, these areneglected in this paper. At any rate their effects are absent for stationary configurations.10 Blackfold dynamics
We have argued that the general effective theory of classical brane dynamics can be formu-lated as a theory of a fluid on a dynamical worldvolume. The fluid variables must satisfythe intrinsic equations (2.12), and they will be coupled to the ‘extrinsic’ equations for thedynamics of the worldvolume geometry, which we still have to determine. To this effect,in the next subsection we introduce a few notions about the geometry of worldvolumeembeddings. More details and proofs are provided in the appendix.
Given the induced metric on W p +1 , (2.7), the first fundamental form of the submanifoldis h µν = ∂ a X µ ∂ b X ν γ ab . (3.1)Indices µ , ν are raised and lowered with g µν , and a, b with γ ab . Defining ⊥ µν = g µν − h µν (3.2)it is easy to see that the tensor h µν acts as a projector onto W p +1 , and ⊥ µν along directionsorthogonal to W p +1 .Background tensors t µ...ν... with support on W p +1 can be converted into worldvolumetensors t a...b... and viceversa using ∂ a X µ . For instance, the velocity field u µ = ∂ a X µ u a , (3.3)preserves its negative-unit norm under this mapping.The covariant differentiation of tensors that live in the worldvolume is well definedonly along tangential directions, which we denote by an overbar, ∇ µ = h µν ∇ ν . (3.4)Note that in general ∇ ρ t µ...ν... has both orthogonal and tangential components. Thetangentially projected part is essentially the same as the worldvolume covariant derivative D c t a...b... for the metric γ ab , both tensors being related via the pull-back map ∂ a X µ . Inparticular, the divergence of the stress-energy tensor T µν = ∂ a X µ ∂ b X ν T ab (3.5)satisfies (see (A.21)) h ρν ∇ µ T µν = ∂ b X ρ D a T ab . (3.6)The extrinsic curvature tensor K µν ρ = h µσ ∇ ν h σρ (3.7)11s tangent to W p +1 along its (symmetric) lower indices µ , ν , and orthogonal to W p +1 along ρ . Its trace is the mean curvature vector K ρ = h µν K µν ρ = ∇ µ h µρ . (3.8)Explicit expressions for the extrinsic curvature tensor in terms of the embedding functions X µ ( σ a ) can be found in the appendix. The general extrinsic dynamics of a brane has been analyzed by Carter in [15]. Theequations are formulated in terms of a stress-energy tensor with support on the p + 1-dimensional worldvolume W p +1 satisfying the tangentiality condition ⊥ ρµ T µν = 0 . (3.9)The basic assumptions are that (i) this effective stress-energy tensor derives from anunderlying conservative dynamics (in our case, General Relativity), even if the macroscopic(=long-wavelength) dynamics may be dissipative; and that (ii) spacetime diffeomorphisminvariance holds, or equivalently, the worldvolume theory can be consistently coupled tothe long-wavelength gravitational field g µν . Under these assumptions, the stress tensormust obey the conservation equations ∇ µ T µρ = 0 . (3.10)These are in fact the generic equations of motion for the entire set of worldvolume fieldvariables φ ( σ a ), both intrinsic and extrinsic: we can decompose (3.10) along directionsparallel and orthogonal to W p +1 as ∇ µ T µρ = ∇ µ ( T µν h ν ρ ) = T µν ∇ µ h ν ρ + h ν ρ ∇ µ T µν = T µν h ν σ ∇ µ h σρ + h νρ ∇ µ T µν = T µν K µν ρ + ∂ b X ρ D a T ab (3.11)where in the last line we used (3.6) and (3.7). Thus the D equations (3.10) separate into D − p − W p +1 and p + 1 equations parallel to W p +1 , T µν K µν ρ = 0 ( extrinsic equations ) , (3.12) D a T ab = 0 ( intrinsic equations ) . (3.13)Let us now apply the equations (3.10) onto the generic stress-energy tensor of a perfectfluid on the worldvolume T µν = ( ε + P ) u µ u ν + P h µν . (3.14)We find u µ u ν ∇ ν ε + ( ε + P )( ˙ u µ + u µ ∇ ν u ν ) + ( h µν + u µ u ν ) ∇ ν P + P K µ = 0 , (3.15)12here ˙ u = u ν ∇ ν u (3.16)is the acceleration of u µ . These are the general equations, to leading order in the derivativeexpansion, for the dynamics of a classical brane with worldvolume spatial isotropy, possiblysupplemented by conservation equations for charges, if present. A familiar example is aNambu-Goto-Dirac brane, with T µν = −| P | h µν and ∇ P = ∇ ε = ∇ u = 0, for which theextrinsic equations, K ρ = 0, require that the worldvolume be a minimal submanifold. Butany classical brane will satisfy equations of this form.As usual the projection of (3.15) onto u is the continuity equation for the energy ofthe fluid, u ν ∇ ν ε + ( ε + P ) ∇ µ u µ = 0 , (3.17)while the projections orthogonal to u ( ε + P ) ˙ u µ = − ( h µν + u µ u ν ) ∇ ν P − P K µ (3.18)say that the force that accelerates an element of the fluid is given along worldvolume di-rections by pressure gradients (Euler equation) and in directions transverse to the world-volume by the extrinsic curvature.For the specific stress tensor of a neutral black brane, (2.19), the equations (3.15)become, after a little manipulation,˙ u µ + 1 n + 1 u µ ∇ ν u ν = 1 n K µ + ∇ µ ln r . (3.19)These blackfold equations describe the general collective dynamics of a neutral black brane.Again, we can decompose them into different projections. In directions orthogonal tothe worldvolume we have K ρ = n ⊥ ρµ ˙ u µ . (3.20)The equivalence of this equation to (3.12) follows by using (A.12).The equations parallel to the worldvolume are h µν ˙ u ν + 1 n + 1 u µ ∇ ν u ν = ∇ µ ln r , (3.21)which we can also write using worldvolume indices and derivatives,˙ u a + 1 n + 1 u a D b u b = ∂ a ln r , (3.22)with ˙ u b = u c D c u b . Thus the temporal and spatial worldvolume gradients of r determinethe worldvolume acceleration and expansion of u , respectively.Although we have emphasized the fluid-dynamical interpretation of the equations, it isinteresting to observe that the extrinsic equations (3.12), when written explicitly in termsof the embedding X µ ( σ a ) become T ab ⊥ σρ (cid:0) ∂ a ∂ b X σ + Γ σµν ∂ a X µ ∂ b X ν (cid:1) = 0 , (3.23)13r alternatively T ab (cid:0) D a ∂ b X ρ + Γ ρµν ∂ a X µ ∂ b X ν (cid:1) = 0 (3.24)(see eqs. (A.28) and (A.30)). These can be regarded as generalizations to p -branes of thegeodesic equation for free particles, or more simply, of “mass × acceleration= 0”.Blackfolds differ from other branes in that they represent objects with black hole hori-zons. In the long-distance effective theory we lose sight of the horizon, since its thicknessis of the order of the scale r that we integrate out. But the presence of the horizon isreflected in the effective theory in the existence of an entropy and in the local thermo-dynamic equilibrium of the effective fluid. Indeed, we shall assume that the regularityof the event horizon under long-wavelength perturbations —including those that bendthe worldvolume away from the flat geometry or that excite the effective fluid away fromequilibrium— is satisfied when the blackfold equations, which incorporate in particularlocal thermodynamic equilibrium, are satisfied. A proof of this statement requires the rig-orous derivation, outlined at the end of sec. 2.1, of the blackfold equations in the derivativeexpansion of Einstein’s equations, which is outside the scope of this paper. However, thereis already significant evidence that horizon regularity is preserved for blackfold solutions.First, analyses of the perturbations of black strings that bend them into a circle [6] (andextensions thereof to branes curved into tori) show that the extrinsic equations (3.12)are equivalent to demanding absence of singularities on or outside the horizon. Second,the intrinsic, hydrodynamical perturbations of a black brane in AdS have been studiedin detail in [17] and shown to be consistent with horizon regularity. Note however thatthe regular solution to higher orders may not preserve the same symmetries as the lowest-order solution. In particular, in some cases horizon regularity may require to abandonstationarity of the configuration at higher orders. In most instances it appears easy to de-cide from the physics of the problem whether such an effect is expected (specific exampleswill be discussed elsewhere), but it would be good if precise conditions could be stated ingenerality. Under the splitting in (2.2), the set of field variables in the system are the collectiveworldvolume fields, intrinsic and extrinsic, and the background gravitational field g µν .The complete set of equations are the extrinsic equations (3.12), intrinsic equations (3.13),and backreaction equations (2.9). Since they are a consequence of general symmetry andconservation principles, these equations retain their form at any perturbative order. The The example studied in [22] appears to fall outside the remit of our approach since the blackfold (asmall black hole) is not a small perturbation of the background spacetime. This, however, is a somewhat formal statement due to the appearance of gravitational self-force di-vergences on the worldvolume that must be dealt with carefully [18]. Ref. [23] shows how the equation ofstress tensor conservation can be used as the basis to obtain these corrections to particle motion. T µν can be consistently coupled to the long-wavelength gravitational field.These are just the intrinsic and extrinsic equations, and backreaction is neglected. Theexplicit blackfold equations (3.19) that result are valid only for test branes.From the point of view of effective field theory one is interested only in quantitiesthat are measured in the long-wavelength regime. The short-wavelength dynamics entersonly to determine the coefficients in the effective stress tensor, which can be computed by, e.g., matching the calculations of some observables [20]. However, in General Relativityone is often interested in also having an explicit solution, even if an approximate one, forthe geometry at all scales, including near the horizon of the black hole. The systematicway to construct an explicit metric in an expansion in r /R is through the method ofmatched asymptotic expansions [18]. In the context of blackfolds this was discussed in [6](following [24, 25]), and explicitly applied to the construction of higher-dimensional blackrings. We review it briefly here.As described in sec. 2.1, the full geometry splits into near- and far-zones, that share acommon overlap-zone. To zeroth order in r /R the near-zone metric is (2.4) and the far-zone metric is the background metric g µν . This is as far as we go in this paper in terms ofproviding explicit solutions to the Einstein equations: this is the test-brane approximation.The next order involves the gravitational backreaction of the brane: the equations (2.9)are linearized around the background and solved with the distributional worldvolumesource T µν . The blackfold equations would appear here among the Einstein equationsas constraints (first-order equations), but we can assume they have already been solvedat zeroth order. The solution of the linearized equations with appropriate asymptotics( e.g., asymptotic flatness) produces a corrected far-zone metric with corrections of order( r /R ) n . Its value in the overlap-zone provides new asymptotic conditions for the near-zone solution. The next step is to perturb the metric (2.4) linearly, with the boundaryconditions that the horizon remains regular and that in the overlap zone the metric matchesthe corrected far-zone solution. In this manner we produce a new, corrected solution, atall scales. This process can then be iterated to higher orders. The formalism developed in this paper can easily be applied to other branes once theireffective equation of state is known. In particular, there are other neutral black branesin vacuum gravity than the ‘smooth’ black branes of (2.3): refs. [26, 27] have shown that‘lumpy’ black branes exist, branching off at the threshold of the Gregory-Laflamme (GL)instability [28]. Their horizons are inhomogeneous on a scale ∼ r , so this small-scaleinhomogeneity is averaged over in our effective description. However, there is an effect onthe effective stress tensor measured at large distance from the brane, since the equation of15tate (which is known only perturbatively near the GL threshold, or possibly numerically)is in general different than the one for smooth branes (2.19).We can use these lumpy branes as the basis for the construction of lumpy blackfolds.It should be clear that their worldvolume is smooth, but the horizons of these blackfoldsare inhomogeneous on the scale r . The simplest example would be a lumpy black ring,built by bending a lumpy black string into a circular shape. This example serves also toillustrate a feature of lumpy blackfolds: the lumps in a rotating lumpy black ring will emitgravitational radiation, so the ring will lose mass and angular momentum as it evolves intime. In more generality, if the lumps on a blackfold extend along a direction in whichthe fluid velocity is non-zero, and if this direction is not an isometry, then the lumpsmoving along these orbits will give rise to a varying quadrupole and hence to gravitationalradiation.Once this effect is taken into account, lumpy blackfolds are generically expected toevolve in time and not remain stationary. However, the time-scale for this evolution willbe very long. The effect is only visible when the small scale is resolved, so it is suppressedby a power of r /R . It will be further suppressed by the fact that gravitational radiationcouples to a higher-multipole (quadrupole) and therefore is rather inefficient. In [29] thistime scale was estimated for five-dimensional rings, and extending this estimate to blackrings in D = 4 + n dimensions we find the time to be of order T gw ∼ R ( R/r ) n , longer bya factor ( R/r ) n +1 than the short time-scale r . Lumpy black branes may also be affectedby GL-like instabilities, but these have not been investigated yet. In (2.1) and (2.2) we assumed that the full dynamics at all wavelengths is described byvacuum General Relativity, i.e., the Einstein-Hilbert action with no matter nor cosmolog-ical constant. However, this is not actually necessary for our derivation of the equations ofmotion and the effective stress tensor. The only part of the field that is actually requiredto be governed by the Einstein-Hilbert action is the sector of short-wavelength degreesof freedom that we integrate out in order to obtain the effective stress tensor (2.13). Forthe long-wavelength components we only require diffeomorphism invariance, which impliesthe equations of motion (3.10). Thus, the blackfold equations (3.19) are enough to de-scribe neutral blackfolds in any configuration that, at small distances, is dominated by theEinstein-Hilbert term. For instance, this will be the case for blackfolds in the presence ofa cosmological constant as long as r ≪ | Λ | − / (3.25)(see [13] for an explicit application), or for blackfolds in an external background gaugefield as long as the typical length scale of the background field around the blackfold ismuch larger than r . No restriction on R other than R ≫ r needs to be imposed.16 different situation arises for charged blackfolds, since then the gauge field has short-wavelength components. The effective fluid is then charged, and additional current conser-vation equations must be added. This extension of our analysis will be discussed elsewhere. Equilibrium configurations that remain stationary in time are of particular interest. For ablackfold, they correspond to stationary black holes. In this case it is possible to solve theblackfold equations explicitly for the worldvolume variables, namely the thickness r andvelocity u , so one is left only with the extrinsic equations for the worldvolume embedding X µ ( σ ).We employ a general result proven in [30] for stationary fluid configurations: if dissi-pative effects must be absent, then the fluid (intrinsic) equations require that the velocityfield be proportional to a worldvolume Killing field k = k a ∂ a . That is, u = k / | k | (4.1)where | k | = p − γ ab k a k b (4.2)and k satisfies the worldvolume Killing equation D ( a k b ) = 0 . (4.3)This does not necessarily mean that k is a Killing field of the background away from W p +1 , as the worldvolume could be at a locus of enhanced symmetry. However, this is notgeneric, and indeed when the blackfold thickness is small but non-zero, k should satisfythe Killing equations on a finite region around the worldvolume. Thus we assume theexistence of a timelike Killing vector k µ ∂ µ in the background, ∇ ( µ k ν ) = 0 , (4.4)whose pull-back onto the worldvolume determines the velocity field as in (4.1). Theexistence of a timelike Killing vector field is in fact a necessary assumption if we intendto describe stationary black holes.The contraction of the Killing equation (4.4) with k µ k ν implies k µ ∂ µ | k | = 0, and usingthis it follows easily that ˙ u µ = ∂ µ ln | k | . (4.5) For a stationary conformal fluid this is relaxed to the conformal Killing equation D ( a k b ) = λγ ab .Conformal fluids do not dissipate the expansion of u since the bulk viscosity vanishes. In [30] it is shown that the worldvolume fluid equations directly imply u b D b u a = ∂ a ln | k | , whichimplies the worldvolume projection of (4.5), but not the orthogonal component of it, which will be usedlater below in eq. (7.1). u vanishes, the intrinsic blackfold equation (3.22) becomes ∂ a ln | k | = ∂ a ln r (4.6)so r | k | = constant . (4.7)In order to fix the proportionality constant in (4.7), we turn to the fact that the stationaryblackfold describes a black brane with a Killing horizon.Above we have defined k as a Killing vector in the background. Its norm k = −| k | computed with the worldvolume metric γ ab is negative, i.e., it is a timelike vector. Butrecall that we regard the background metric as only one part, the far-zone metric at r ≫ r , of the full geometry. For r ≪ R the geometry is instead well approximated bythe near-zone metric (2.8), which matches with the far-zone metric in r ≪ r ≪ R . It isnatural to extend k as a Killing vector to all the geometry, including the near-zone (2.8).Using the metric g (short) µν in this region the norm of k is g (short) µν k µ k ν = (cid:18) γ ab + r n r n u a u b (cid:19) k a k b = − (cid:18) − r n r n (cid:19) | k | (4.8)where we used (4.1) and (4.2). Thus k becomes null as we approach the horizon r → r and indeed it is the null Killing generator of the horizon. Its surface gravity is easilyobtained as κ = n | k | r . (4.9)Equation (4.7) tells us that κ is a constant over the worldvolume of the blackfold. That is,the surface gravity is uniform not only over the ( n + 1)-sphere but over the entire horizon.Eqs. (4.1) and (4.9) provide the general solution to the intrinsic equations for stationaryblackfolds. To give them a more explicit expression it is convenient to choose a preferred setof orthogonal commuting vectors of the background geometry, ξ timelike and χ i spacelike,such that the Killing vector k is a linear combination of them k = ξ + X i Ω i χ i , (4.10)with constant Ω i . Clearly i runs at most up to p . A convenient (but not necessary)choice is to take for ξ the generator of asymptotic time translations, and for χ i the Cartangenerators of asymptotic rotations with closed orbits of periodicity 2 π . They will often beKilling vectors themselves, but in principle only the linear combination in k need be so.Introduce a set of worldvolume functions R a ( σ ) via their norms on the worldvolume, R = p − ξ (cid:12)(cid:12)(cid:12) W p +1 , R i = q χ i (cid:12)(cid:12)(cid:12) W p +1 . (4.11)The R a must be regarded as part of the embedding coordinates X µ ( σ ). If ξ is thecanonically-normalized generator of asymptotic time translations then R is a redshift18actor between infinity and the blackfold worldvolume. The R i are the proper radii of theorbits generated by χ i along the worldvolume. The Ω i are the horizon angular velocitiesrelative to observers that follow orbits of ξ .The vectors ∂∂t = 1 R ξ , ∂∂z i = 1 R i χ i (4.12)(no sum in i ) are orthonormal with respect to the metric γ ab on the worldvolume. It willalso be useful to regard them as vectors that extend into all of (2.8).Let us introduce the worldvolume spatial velocity field V i ( σ ) = u · ∂ z i − u · ∂ t = Ω i R i ( σ ) R ( σ ) (4.13)so that k = R ∂∂t + X i V i ∂∂z i ! (4.14)and | k | = − ξ − X i Ω i χ i ! / = R p − V , (4.15)where V = X i V i = 1 R X i Ω i R i . (4.16)Thus | k | can be regarded as the relativistic Lorentz factor at a point in W p +1 , with apossible local redshift, all relative to the reference frame of ξ -static observers. Plugging(4.15) into (4.9) we obtain that for given κ and Ω i the thickness r is solved in terms ofthe R a as r ( σ ) = nR ( σ )2 κ p − V ( σ ) . (4.17)The conditions that κ and Ω i must remain uniform over the blackfold worldvolumewere referred to in [1] as the blackness conditions and were imposed, invoking generaltheorems for stationary black holes — zeroth law of black hole mechanics and horizonrigidity [31, 32, 33] —, as necessary conditions for the regularity of the black hole horizon.Here instead we have derived them as general consequences of stationary fluid dynamics,where κ and Ω i appear as integration constants.As a matter of fact it is also possible to work entirely within the framework of theeffective theory and avoid any reference to the short-wavelength geometry of the horizon.Using only the fluid and thermodynamics equations one can derive, like in [30], that thevariation of the local temperature T (2.21) along the worldvolume is dictated by the localredshift T = TR √ − V . (4.18)19he integration constant T can then be interpreted, using the thermodynamic first lawthat we derive below, as the overall temperature of the black hole. As expected, T = κ/ π .However, we think it is instructive to see how the concepts of black hole physics, like nullhorizon generators and surface gravity, are recovered in this scheme. Typically branes (such as D-branes or cosmic strings) acquire boundaries when they in-tersect or end on other branes. This effect is accounted for by adding boundary terms tothe effective stress tensor. However, black branes (and other fluid branes) may also have‘free’ boundaries without any boundary stresses.Consider the case where the worldvolume of the brane has a timelike boundary ∂ W p +1 .Assume there is a smooth (but otherwise arbitrary) extension f W p +1 of W p +1 across theboundary. Introduce a level-set function f ( σ a ) such that f > W p +1 and f < f W p +1 − W p +1 , with f ( σ a ) = 0 on the boundary ∂ W p +1 . Then − ∂ a f is a one-form normalto ∂ W p +1 pointing away from the fluid. The stress tensor is T ab = [( ε + P ) u a u b + P γ ab ] Θ( f ) , (5.1)where Θ is the step function.If the fluid is to remain within its bounds then the boundary must be advected withthe fluid, i.e., it must be Lie-dragged by u , £ u | ∂ W p +1 f = 0 (5.2)or equivalently, the velocity must remain parallel to the boundary, u a ∂ a f (cid:12)(cid:12) ∂ W p +1 = 0 . (5.3)At the boundary, the stress-energy conservation equation (3.13) becomes(( ε + P ) u a u b + P γ ab ) ∂ a f (cid:12)(cid:12) ∂ W p +1 = 0 . (5.4)Imposing (5.3) we find that the pressure must approach zero at the boundary, P (cid:12)(cid:12) ∂ W p +1 = 0 . (5.5)This is simply the Young-Laplace equation for a bounded fluid when there is no surfacetension that could balance the fluid pressure at the boundary. That is, we do not introduceany such boundary stresses, although, as we mentioned they are of interest when onestudies brane intersections.For a neutral blackfold, vanishing pressure at the boundary means r (cid:12)(cid:12) ∂ W p +1 = 0 , (5.6)20hich has a nice geometric interpretation: the thickness of the horizon must approachzero size at the boundary, so the horizon closes off at the edge of the blackfold.If the blackfold is stationary, the condition (5.6) means that | k | → R →
0, or perhaps more commonly, because the fluidapproaches the speed of light at the boundary, V (cid:12)(cid:12) ∂ W p +1 = 1 . (5.7)An example of this was presented in [1], where a rigidly-rotating blackfold disk was con-structed and shown to reproduce accurately the properties of MP black holes with oneultraspin.The extrinsic equations (3.20) must also be satisfied and they impose further con-straints on the worldvolume variables. In the stationary case, using (4.5) and (4.7) we seethat if the extrinsic curvature of the worldvolume remains finite at ∂ W p +1 , then not only r → ⊥ ρµ ∂ µ r must vanish there at least as quickly as r . The blackfold construction puts, on any point in the spatial section B p of W p +1 , a (small)transverse sphere s n +1 with Schwarzschild radius r ( σ ). Thus the blackfold represents ablack hole with a horizon geometry that is a product of B p and s n +1 — the product iswarped since the radius of the s n +1 varies along B p . The null generators of the horizonare proportional to the velocity field u .If r is non-zero everywhere on B p then the s n +1 are trivially fibered on B p and thehorizon topology is (topology of B p ) × s n +1 . (6.1)However, we have seen that if B p has boundaries then r will shrink to zero size at them,resulting in a non-trivial fibration and different topology. A simple but very relevantinstance of this happens when B p is a topological p -ball. Then the horizon topology caneasily be seen to be S p + n +1 = S D − .To analyze the horizon geometry we go to the metric (2.4) that locally describes thegeometry of the blackfold to lowest order in r /R in the region r ≪ R . There we canchoose a local orthonormal frame ( ∂ t , ∂ z i ) on the worldvolume, such that ∂ t coincides inthe overlap zone r ≪ r ≪ R with the timelike unit normal n µ to B p , n µ = ( ∂ t ) µ . (6.2)To lowest order in r /R the worldvolume metric is Minkowski and the spatial metric onthe horizon at r = r is ds H = ( δ ij + u i u j ) dz i dz j + r d Ω n +1) (6.3)21ith u i = u · ∂ z i . Then the local area density of the horizon a H at a given point in B p is a H = Ω ( n +1) r n +10 p δ ij u i u j . (6.4)For a stationary blackfold the choice for ∂ t and ∂ z i was made in (4.12), so n µ = 1 R ξ µ , (6.5)and u i = Ω i R i R √ − V (6.6)so the area density is a H = Ω ( n +1) r n +10 √ − V = Ω ( n +1) (cid:16) n κ (cid:17) n +1 R n +10 ( σ a )(1 − V ( σ a )) n . (6.7)The total area of the horizon is then A H = Z B p dV ( p ) a H ( σ a ) , (6.8)where dV ( p ) denotes the volume form in B p .Again, we could have avoided any reference to the geometry of the horizon and workedinstead exclusively with quantities defined in the effective fluid theory. The entropy densityof the blackfold fluid is given in (2.20) and after taking the relativistic Lorentz factor √ − V into account, the total entropy is S = Z B p dV ( p ) s √ − V = Ω ( n +1) G (cid:16) n κ (cid:17) n +1 Z B p dV ( p ) R n +10 ( σ a )(1 − V ( σ a )) n = A H G , (6.9)in agreement with the geometric area computed from (6.8). The geometric interpretationinvolves short-wavelength physics, but is useful for exhibiting how the blackfold construc-tion gives precise information about the entire horizon geometry, including the size of the s n +1 .The mass and angular momenta are conjugate to the generators of asymptotic timetranslations and rotations, which we assume are the vectors ξ and χ i that we introducedin sec. 4. Then M = Z B p dV ( p ) T µν n µ ξ ν , J i = − Z B p dV ( p ) T µν n µ χ νi . (6.10)Plugging here (6.5) and the results from sec. 4 we obtain M = Ω ( n +1) πG (cid:16) n κ (cid:17) n Z B p dV ( p ) R n +10 (1 − V ) n − (cid:0) n + 1 − V (cid:1) , (6.11) In a more general case the relation may involve linear combinations, but this, although straightforward,comes at the expense of more cumbersome expressions. J i = Ω ( n +1) πG (cid:16) n κ (cid:17) n n Ω i Z B p dV ( p ) R n − (1 − V ) n − R i . (6.12)It is easy to extract some interesting consequences of these results. Let us assumethat all length scales along B p are ∼ R and that the velocities and redshift are moderate( i.e., − V and R of order one) over almost all the blackfold. Then the two black holelength scales introduced in (1.1) are ℓ M ∼ ( r n R p ) D − ℓ J ∼ R , (6.13)and the small expansion parameter for the effective theory is (cid:18) ℓ M ℓ J (cid:19) D − ∼ (cid:16) r R (cid:17) n . (6.14)It is interesting to compare the areas ( i.e., entropies) of different blackfolds in a givendimension D . In [6] we introduced the dimensionless angular momentum j and dimen-sionless area a for a given mass j ∼ JM ( GM ) / ( D − ∼ ℓ J ℓ M , a ∼ A H ℓ D − M . (6.15)The blackfold approximation requires j ≫
1. Since A H ∼ r n +10 R p we find that a ( j ) ∼ j − pD − − p . (6.16)This shows that for a given number of large non-zero angular momenta, the blackfoldwith smallest p is entropically preferred at fixed mass. This is just like we observed forsingle-spin MP black holes and black rings in [6]: for a given mass, the smaller p , thethicker the horizon, thus the cooler the black hole, and (since κA H ∼ GM i.e., constantfor fixed mass) the higher its entropy. We have presented in sec. 4 the general solution to the intrinsic equations for a stationaryblackfold. Given a Killing vector field k , eqs. (4.1), (4.15) and (4.17) allow to eliminatethe variables r ( σ ) and u a ( σ ) in terms of the embedding functions X µ ( σ ). The remainingextrinsic equation (3.20) that determines the embedding can be written, using (4.5), as K ρ = ⊥ ρµ ∂ µ ln | k | n . (7.1) One should not confuse the dimensionless total area for fixed mass a with the blackfold area density a H . I [ X µ ( σ )] = Z W p +1 d p +1 σ √− γ | k | n (7.2)by considering variations of X µ in directions transverse to the worldvolume.We can write it in a form that is particularly practical for obtaining the blackfoldequations in specific calculations. First observe that the asymptotic time, conjugate tothe vector ξ , is related to proper time t on the worldvolume by a factor of R . If we takethe interval for the (trivial) integration over asymptotic time to have finite length β , then Z W p +1 d p +1 σ √− γ | k | n = β Z B p dV ( p ) R | k | n . (7.3)Using now (4.15) we find I [ X µ ( σ )] = β Z B p dV ( p ) R n +10 (1 − V ) n = β Z B p dV ( p ) R ( σ ) R ( σ ) − X i Ω i R i ( σ ) ! n . (7.4)We emphasize again that R a are among the worldvolume field variables X µ ( σ ). Of coursethese enter as well through dV ( p ) . Using eqs. (6.8), (6.11), (6.12), it is straightforward to check that the action (7.4) is I = β M − X i Ω i J i − κ πG A H ! . (7.5)This identity holds for any embedding, not necessarily a solution to the extrinsic equations.Thus, if we regard M , J i and A H as functionals of X µ ( σ ), and consider variations at fixedsurface gravity and angular velocities, we have δIδX µ = 0 ⇔ δMδX µ − X i Ω i δJ i δX µ − κ πG δA H δX µ = 0 . (7.6)Hence, solutions of the blackfold equations satisfy the ‘equilibrium state’ version of thefirst law of black hole mechanics. Conversely, the blackfold equations for stationary con-figurations can be obtained as the requirement that the first law be satisfied. If we regard κ and Ω i as Lagrange multipliers we may also say that blackfolds extremize the horizonarea for given mass and angular momenta. In [1] it was asserted that the blackfold equations can be derived from the action R W p +1 d p +1 σ √− γT ab γ ab . This is not true in general, but is correct in the stationary case since T ab γ ab ∝ r n ∝ | k | n .
24n the Euclidean quantum gravity approach to black hole thermodynamics it is naturalto take β to be the period of Euclidean time, β = 1 /T . Using κA H / πG = T S and eq. (7.5)we see that I is equal, up to a factor, to the Gibbs free energy G , β − I = G = M − X i Ω i J i − T S . (7.7)Therefore (7.2) can be identified as the effective action that approximates, in the blackfoldregime r /R ≪
1, the gravitational Euclidean action of the black hole [34].It is also possible to find action functionals for general, possibly time-dependent black-folds by adapting the action principles developed for perfect fluids [35]. However, theusefulness of these actions, which must deal with constraints such as u = −
1, appears tobe somewhat limited so we do not dwell on them.
The blackfold approach must capture the perturbative dynamics of a black hole when theperturbation wavelength λ is long, λ ≫ r . (8.1)These perturbations can be either intrinsic variations in the thickness r and local velocity u , or extrinsic variations in the worldvolume embedding geometry X . In general, these twokinds of perturbations are coupled. A detailed analysis of the perturbations of solutionsto the blackfold equations and their stability will be presented elsewhere. Here we extractsome simple but important consequences for perturbations with wavelength r ≪ λ ≪ R , (8.2) i.e., perturbations for which the worldvolume looks essentially flat, K µν ρ ≈
0. In this caseit is easy to see that the intrinsic and extrinsic perturbations decouple.It is instructive to perform the analysis for a general perfect fluid (2.15), and thenparticularize to the neutral blackfold fluid (2.19). For simplicity we consider a fluid initiallyat rest u a = (1 , . . . ), with uniform equilibrium energy density ε and pressure P . Theflat worldvolume metric is parametrized, in ‘static gauge’, using orthonormal coordinates X = t , X i = z i , i = 1 , . . . p and the transverse coordinates X m are held at constantvalues. Introduce small perturbations δε , δP = dPdε δε , δu a = (0 , v i ) , δX m = ξ m , (8.3)and work to linearized order in them. The perturbed stress tensor is T tt = ε + δε , T ti = ( ε + P ) v i , T ii = P + dPdε δε , (8.4)25nd the extrinsic curvature δK abm = ∂ a ∂ b ξ m . (8.5)The extrinsic equations (3.12) then become (cid:0) ε∂ t + P ∂ i (cid:1) ξ m = 0 . (8.6)Thus transverse, elastic oscillations of the brane propagate with speed c T = − Pε . (8.7)The intrinsic equations (3.13) are ∂ t T tt + ∂ i T it = 0 , ∂ t T ti + ∂ j T ji = 0 , (8.8)which can be combined into ∂ t T tt − ∂ ij T ij = 0 . (8.9)For (8.4) we find (cid:18) ∂ t − dPdε ∂ i (cid:19) δε = 0 , (8.10)so longitudinal, sound-mode oscillations of the fluid propagate with speed c L = dPdε . (8.11)These derivations of eqs. (8.7) and (8.11) are hardly new: they are conventional ways toobtain the speeds of elastic and sound waves — in fact they have been obtained in [36, 37]for brane dynamics. They have a remarkable consequence: a brane with a worldvolumefluid equation of state such that Pε dPdε > c L c T < P = wε with constant w ,where the interpretation is easy (we assume ε > tension is required for elasticstability, but positive pressure is needed to prevent that the fluid clumps under any densityperturbation.Neutral blackfolds have c L = − c T = − n + 1 . (8.14)and therefore are generically unstable to longitudinal sound-mode oscillations and stableto elastic oscillations in the range of wavelengths (8.2).26his instability is not unexpected. Black branes suffer from the Gregory-Laflammeinstability [28], which makes the horizon radius vary as δr ∼ e Ω t + ik i z i . (8.15)Here Ω is positive real and thus the frequency is imaginary. The threshold mode for theinstability, with Ω = 0 and k = √ k i k i = 0, has ‘small’ wavelength λ = 2 π/k ∼ r andtherefore cannot be seen in the blackfold approximation. But the GL instability extendsto arbitrarily small k , i.e., arbitrarily long wavelengths, and when k is very small it shouldbe captured by the blackfold dynamics.The sound-mode instability corresponds precisely to this long-wavelength part, Ω , k →
0, of the GL instability. Observe that sound waves in a blackfold produce δε ∼ δP ∼ δr i.e., variations in the horizon thickness. Eq. (8.14) tells us that these are unstable, of theform (8.15) with dispersion relation Ω = 1 √ n + 1 k . (8.16)A simple inspection of the slope at the origin in figure 1 of [28] shows good numericalagreement with (8.16). We leave a more precise derivation of this equation from a detailedGL-type analysis to future work. Note also that the dispersion relation (8.16) indicatesthat the collective coordinate r is a ghost ( i.e., its effective Lagrangian − c − L ( ∂ t ln r ) +( ∂ z i ln r ) has the ‘wrong sign’ for the kinetic term). Moreover, observe that the Gibbs-Duhem relation dP = sd T , from (2.17) and (2.18),implies in general that dPdε = s d T dε = sc v (8.17)where c v is the isovolumetric specific heat. Thus the black brane is dynamically unstable(to long-wavelength GL modes) if and only if it is locally thermodynamically unstable, c v <
0. This is precisely the content of the ‘correlated stability conjecture’ of Gubser andMitra [38]. In fact our method gives a quantitative expression for the unstable dynamicalfrequency in terms of local thermodynamics asΩ = r s | c v | k , (8.18)which as far as we know is a new result.The ordinary derivation of the GL instability involves a complicated analysis of lin-earized gravitational perturbations of a black brane and the numerical resolution of aboundary value problem for a differential equation (which is moreover compounded atsmall k since larger grids are required to avoid finite-size problems). Here we have shownthat the long-wavelength component of the instability, (8.16), can be obtained by an almost The GL mode at threshold is instead tachyonic, since its dispersion relation has imaginary mass. . In addition, the correlation betweendynamical and thermodynamical stability follows as an elementary consequence of thethermodynamics of the effective fluid. In our opinion these results are striking evidenceof the power of the blackfold approach. The formalism we have developed bears relation to two different earlier effective descrip-tions of black hole dynamics. The extrinsic part is a generalization to p -branes of theeffective worldline formalism for small black holes, whose size r is much smaller than thelength scale R = 1 / (acceleration) of their trajectories or the wavelength of the gravita-tional radiation they emit [18, 19, 20]. The intrinsic part is similar to other fluid-dynamicalformalisms for horizon fluctuations, such as the membrane paradigm and the fluid/AdS-gravity correspondence.With respect to the first one, note that our formalism allows to consider time-dependentsituations, which typically involve the emission of gravitational waves. This can be ob-tained by coupling the blackfold effective stress tensor to the quadrupole formula forgravitational radiation. This is also common in studies of gravitational wave emissionfrom small black holes and from cosmic strings. However, accounting for the backreactionof this radiation on the blackfold requires going beyond the generalized-geodesic approx-imation and dealing with the notoriously subtle problem of the gravitational self-force[39].To relate the fluid/AdS-gravity correspondence to our approach take, instead of aneutral black brane, a near-extremal D3-brane. The blackfold formalism can be appliedto it, too . The small scale corresponds to the charge-radius r q of the D3-brane (whichis much larger than the non-extremality length r ), and in the overlap-zone r q ≪ r ≪ R where the blackfold effective stress tensor is computed, the metric is flat up to smallcorrections in r q /R . The blackfold method here could be regarded as an extension of theDBI approach to describe thermally excited worldvolumes. The difference with respectto neutral branes is that there is a region near the horizon where excitations with longwavelengths ≫ r q are localized. One can take a limit, Maldacena’s decoupling limit,to decouple all the far-zone effects from them. This region is asymptotic to AdS × S with radius r q , and far-zone effects are absent since they would give rise to non-normalizable modes in AdS and change the boundary geometry. Integrating the degrees offreedom in the asymptotically AdS region one gets only intrinsic, purely hydrodynamicalcollective modes. The effective stress tensor thus obtained is again of quasilocal typeand in fact is the holographic stress tensor in AdS [40]. This is in principle different Observe that this is not a Jeans instability of the fluid (which has sometimes been suggested as possiblyrelated to the GL instability) since gravitational forces within the fluid are entirely absent in our analysis. Charged blackfolds will be discussed elsewhere.
Acknowledgments
We thank Joan Camps for many useful discussions during collaboration on another strandof this program. TH also thanks Shiraz Minwalla for useful discussions. RE, TH and NOare grateful to the Benasque Center for Science for hospitality and a stimulating environ-ment during the Gravity Workshop in July 2009, and they thank the participants therefor very useful feedback and discussions. RE was supported by DURSI 2005 SGR 00082and 2009 SGR 168, MEC FPA 2007-66665-C02 and CPAN CSD2007-00042 Consolider-Ingenio 2010. TH was supported by the Carlsberg foundation. VN was supported byan Individual Marie Curie Intra-European Fellowship and by ANR-05-BLAN-0079-02 andMRTN-CT-2004-503369, and CNRS PICS
A Geometry of embedded submanifolds
We collect here several relevant definitions and results on the geometry of submanifoldembeddings. Some aspects are more extensively discussed in [42].29 .1 Extrinsic curvature
Assume the submanifold W is embedded as X µ ( σ a ). The pull-back of the spacetime metriconto W is γ ab = g µν ∂ a X µ ∂ b X ν . (A.1)The first fundamental tensor of the surface is then h µν = γ ab ∂ a X µ ∂ b X ν . (A.2)It follows easily that h µν ∂ a X ν = ∂ a X µ , (A.3)and h µν h νρ = h µρ (A.4)so h µν projects tensors onto directions tangent to W . Decomposing the metric as g µν = h µν + ⊥ µν , (A.5)we obtain the orthogonal projection tensor ⊥ µν , ⊥ µν ∂ a X µ = 0 , ⊥ µν ⊥ νρ = ⊥ µρ . (A.6)The extrinsic curvature tensor can be defined as K µν ρ = h λµ h σν ∇ λ h ρσ = − h λµ h σν ∇ λ ⊥ ρσ . (A.7)The tangentiality of the first two indices and orthogonality of the last, ⊥ µν K σν ρ = ⊥ σν K νµρ = h ν ρ K σµν = 0 (A.8)follows easily from this definition and the projector property (A.4).Following [15], it is convenient to introduce the tangential covariant derivative ∇ µ = h µν ∇ ν , (A.9)so K µν ρ = h νσ ∇ µ h σρ . (A.10)Applying the tangential derivative on (A.4) one obtains2 K µ ( νρ ) = ∇ µ h νρ = −∇ µ ⊥ νρ . (A.11)Let v be any vector tangent to W . Then [42] v µ v ν K µν ρ = − v µ v ν ∇ ν ⊥ µρ = − v ν ∇ ν ( v µ ⊥ µρ ) + ⊥ ρµ v ν ∇ ν v µ = ⊥ ρµ ˙ v µ (A.12)30here ˙ v µ = v ν ∇ ν v µ . (A.13)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.14)The symmetry K [ µν ] ρ = 0 (A.15)follows from the integrability of the subspaces orthogonal to ⊥ µν . To prove this, assumethat the latter is true, namely that there is a submanifold W defined by a set of equations f ( i ) ( X ) = 0 such that df ( i ) are a basis of one-forms normal to the submanifold. Any one-form normal to W is a linear combination of them so the subspace orthogonal to it is alsointegrable. It is always possible to choose a one-form N orthogonal to this subspace suchthat N µ = ∂ µ f ( X ) = ∇ µ f ( X ) , (A.16)so, using (A.14), N ρ K µν ρ = − h νσ h µρ ∇ σ ∇ ρ f , (A.17)which is manifestly symmetric under µ ↔ ν . The converse statement that (A.15) impliesthe integrability of the orthogonal subspace, can also be proven by a straightforwardapplication of Frobenius’s theorem [42].Background tensors t µ µ ...ν ν ... can be pulled-back onto worldvolume tensors 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.18)where ∂ b X ν = γ bc h νρ ∂ c X ρ . (A.19)Observe that even when t µ µ ...ν ν ... is a background tensor with indices parallel to W ,in general ∇ µ t µ µ ...ν ν ... has both parallel and orthogonal components. The parallel pro-jection along all indices is related to the worldvolume covariant derivative D a t a a ...b b ... asin (A.18). This can be shown by using the equation that relates the connection coefficientsfor each metric, Γ ρµν and (cid:8) ca b (cid:9) , ∂ a X µ ∂ b X ν h σρ Γ ρµν = ∂ c X σ (cid:8) ca b (cid:9) − h σρ ∂ a ∂ b X ρ , (A.20)which can be proven by direct substitution of the definitions of each term involved.In particular, the divergences of tensors are related as h ν µ · · · ∇ ρ t ρµ ... = ∂ a X ν · · · D c t ca ... . (A.21)Such relations allow to dispense with the use of worldvolume coordinate tensors and deriva-tives in most formal manipulations. However, worldvolume coordinates are very practical31or explicit calculations and also allow us to highlight the distinction between intrinsic(parallel to W ) and extrinsic (orthogonal to W ) equations.Let us now consider the divergence of a totally antisymmetric tensor J (such as acurrent associated to a gauge form field) parallel to the worldvolume. It is easy to showthat ⊥ ρµ ∇ µ J µµ ... = 0 (A.22)holds as an identity. This implies that the conservation equation ∇ µ J µµ ... = 0 (A.23)is equivalent to the worldvolume conservation equation D a J aa ... = 0 , (A.24) i.e., the orthogonal component of the current conservation equation (A.23) does not yieldany additional equation. In particular, for a 1-form current one has ∇ µ J µ = D a J a , (A.25)and continuity of charge is only meaningful as an intrinsic equation. This is in contrastto the conservation of the stress energy tensor, where the orthogonal component givesindependent extrinsic equations (3.12).Let us now obtain more explicit expressions for the pull-back of the extrinsic curvaturetensor onto W in terms of X µ ( σ ), K abρ = ∂ a X µ ∂ b X ν K µν ρ = − ∂ a X µ ∂ b X ν ∇ µ ⊥ νρ . (A.26)The property (A.6) implies0 = ∂ b X ν ∇ ν ( ⊥ σρ ∂ a X σ ) = − K abρ + ⊥ σρ ∂ b X ν ∇ ν ( ∂ a X σ ) . (A.27)Expanding the covariant derivative in the last term and using ∂ b X ν ∂ ν = ∂ b we find K abρ = ⊥ σρ (cid:0) ∂ a ∂ b X σ + Γ σµν ∂ a X µ ∂ b X ν (cid:1) , (A.28)which is reminiscent of the expression for the acceleration (deviation from self-paralleltransport) of a curve — indeed (A.12) makes this even more explicit. An alternativeexpression with this same feature can be obtained by performing some manipulations: ⊥ σρ ∂ a ∂ b X σ = ∂ a ∂ b X ρ − h ρσ ∂ a ∂ b X σ = ∂ a ∂ b X ρ − (cid:8) ca b (cid:9) ∂ c X ρ + ∂ a X µ ∂ b X ν h ρσ Γ σµν = D a ∂ b X ρ + ∂ a X µ ∂ b X ν h ρσ Γ σµν . (A.29)In the second line we have used (A.20). Inserting the last expression into (A.28) we find K abρ = D a ∂ b X ρ + Γ ρµν ∂ a X µ ∂ b X ν . (A.30)32 .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.31)and Lie-drag W along these curves. The congruence is arbitrary, other than requiring it tobe smooth in a finite neighbourhood of W , so this realizes arbitrary smooth deformationsof 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.32)Then h µλ h ν σ £ N h λσ = h µλ h ν σ N ρ ∇ ρ h λσ + h ρν ∇ µ N ρ + h µρ ∇ ν N ρ . (A.33)The first term in the rhs is zero: h µλ h ν σ N ρ ∇ ρ h λσ = − h µλ h ν σ N ρ ∇ ρ ⊥ λσ = h µλ ⊥ λσ N ρ ∇ ρ h ν σ = 0 , (A.34)and the other two terms can be rewritten using (A.14), so N ρ K µν ρ = − h µλ h νσ £ N h λσ . (A.35)This implies N ρ K ρ = − h µν £ N h µν = − p | h | £ N p | h | , (A.36)where h = det h µν . These equations generalize well-known expressions for the extrinsiccurvature of a codimension-1 surface. The last one allows to derive the equation for aminimal-volume submanifold: Vol = Z W p | h | ⇒ δ N Vol = − p | h | N ρ K ρ (A.37) i.e., for variations along an arbitrary orthogonal direction N , minimal (actually, extremal)volume requires K ρ = 0. This is of course the variational principle for Nambu-Goto-Diracbranes.Consider now a more general functional I = Z W p | h | Φ (A.38)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.39)Since N is an arbitrary orthogonal vector we have δ N I = 0 ⇔ K ρ = ⊥ ρµ ∂ µ ln Φ . (A.40)33 eferences [1] R. Emparan, T. Harmark, V. Niarchos and N. A. Obers, “World-volume effectivetheory for higher-dimensional black holes. (Blackfolds),” Phys. Rev. Lett. , 191301(2009) [arXiv:0902.0427 [hep-th]].[2] R. C. Myers and M. J. Perry, “Black Holes In Higher Dimensional Space-Times,”Annals Phys. (1986) 304.[3] R. Emparan and R. C. Myers, “Instability of ultra-spinning black holes,” JHEP (2003) 025 [arXiv:hep-th/0308056].[4] R. Emparan and H. S. Reall, “A rotating black ring in five dimensions,” Phys. Rev.Lett. (2002) 101101 [arXiv:hep-th/0110260].[5] R. Emparan and H. S. Reall, “Black Holes in Higher Dimensions,” Living Rev. Rel. , 6 (2008) [arXiv:0801.3471 [hep-th]].[6] R. Emparan, T. Harmark, V. Niarchos, N. A. Obers and M. J. Rodr´ıguez, “ThePhase Structure of Higher-Dimensional Black Rings and Black Holes,” JHEP ,110 (2007) [arXiv:0708.2181 [hep-th]].[7] O. J. C. Dias, P. Figueras, R. Monteiro, J. E. Santos and R. Emparan, “Instabilityand new phases of higher-dimensional rotating black holes,” arXiv:0907.2248 [hep-th].[8] H. K. Kunduri, J. Lucietti and H. S. Reall, “Near-horizon symmetries of extremalblack holes,” Class. Quant. Grav. , 4169 (2007) [arXiv:0705.4214 [hep-th]].[9] N. A. Obers, “Black Holes in Higher-Dimensional Gravity,” Lect. Notes Phys. (2009) 211 [arXiv:0802.0519 [hep-th]].[10] V. Niarchos, “Phases of Higher Dimensional Black Holes,” Mod. Phys. Lett. A ,2625 (2008) [arXiv:0808.2776 [hep-th]].[11] H. Elvang and R. Emparan, “Black rings, supertubes, and a stringy resolution ofblack hole non-uniqueness,” JHEP , 035 (2003) [arXiv:hep-th/0310008].[12] R. Emparan and H. S. Reall, “Black rings,” Class. Quant. Grav. (2006) R169[arXiv:hep-th/0608012].[13] M. M. Caldarelli, R. Emparan and M. J. Rodr´ıguez, “Black Rings in (Anti)-deSitterspace,” JHEP (2008) 011 [arXiv:0806.1954 [hep-th]].[14] J. Camps, R. Emparan, P. Figueras, S. Giusto and A. Saxena, “Black Rings in Taub-NUT and D0-D6 interactions,” JHEP (2009) 021 [arXiv:0811.2088 [hep-th]].3415] B. Carter, “Essentials of classical brane dynamics,” Int. J. Theor. Phys. , 2099(2001) [arXiv:gr-qc/0012036].[16] R. S. Hanni and R. Ruffini, “Lines of Force of a Point Charge near a SchwarzschildBlack Hole,” Phys. Rev. D , 3259 (1973).T. Damour, “Black Hole Eddy Currents,” Phys. Rev. D , 3598 (1978).R.L. Znajek, “The electric and magnetic conductivity of a Kerr hole,” Mon. Not. R.Astr. Soc. 185, 833 (1978).T. Damour, Th`ese de Doctorat dEtat, Universit´e Pierre et Marie Curie, Paris VI,1979; “Surface Effects in Black Hole Physics”, Proceedings of the Second MarcelGrossmann Meeting on General Relativity, (edited by R. Ruffini, North Holland,1982) p. 587.R. H. Price and K. S. Thorne, “Membrane viewpoint on black holes: Properties andevolution of the stretched horizon,” Phys. Rev. D (1986) 915.K. S. Thorne, R. H. Price and D. A. Macdonald, “Black Holes: The MembraneParadigm” , Yale Univ. Press, New Haven, USA (1986).[17] S. Bhattacharyya, V. E. Hubeny, S. Minwalla and M. Rangamani, “Nonlinear FluidDynamics from Gravity,” JHEP , 045 (2008) [arXiv:0712.2456 [hep-th]].[18] E. Poisson, “The motion of point particles in curved spacetime,” Living Rev. Rel. ,6 (2004) [arXiv:gr-qc/0306052].[19] S. E. Gralla and R. M. Wald, “A Rigorous Derivation of Gravitational Self-force,”Class. Quant. Grav. (2008) 205009 [arXiv:0806.3293 [gr-qc]].[20] W. D. Goldberger, “Les Houches lectures on effective field theories and gravitationalradiation,” arXiv:hep-ph/0701129.W. D. Goldberger and I. Z. Rothstein, “An effective field theory of gravity for ex-tended objects,” Phys. Rev. D , 104029 (2006) [arXiv:hep-th/0409156].B. Kol and M. Smolkin, “Classical Effective Field Theory and Caged Black Holes,”Phys. Rev. D , 064033 (2008) [arXiv:0712.2822 [hep-th]].[21] J. D. Brown and J. W. York, “Quasilocal energy and conserved charges derived fromthe gravitational action,” Phys. Rev. D , 1407 (1993) [arXiv:gr-qc/9209012].[22] J. Le Witt and S. F. Ross, “Black holes and black strings in plane waves,”arXiv:0910.4332 [hep-th].[23] Y. Mino, M. Sasaki and T. Tanaka, “Gravitational radiation reaction to a particlemotion,” Phys. Rev. D , 3457 (1997) [arXiv:gr-qc/9606018].3524] T. Harmark, “Small black holes on cylinders,” Phys. Rev. D (2004) 104015[arXiv:hep-th/0310259].[25] D. Gorbonos and B. Kol, “A dialogue of multipoles: Matched asymptotic expansionfor caged black holes,” JHEP , 053 (2004) [arXiv:hep-th/0406002].[26] S. S. Gubser, “On non-uniform black branes,” Class. Quant. Grav. (2002) 4825[arXiv:hep-th/0110193].[27] T. Wiseman, “Static axisymmetric vacuum solutions and non-uniform black strings,”Class. Quant. Grav. (2003) 1137 [arXiv:hep-th/0209051].[28] R. Gregory and R. Laflamme, “Black strings and p-branes are unstable,” Phys. Rev.Lett. , 2837 (1993) [arXiv:hep-th/9301052].For a review, see T. Harmark, V. Niarchos and N. A. Obers, “Instabilities of blackstrings and branes,” Class. Quant. Grav. (2007) R1 [arXiv:hep-th/0701022].[29] H. Elvang, R. Emparan and A. Virmani, “Dynamics and stability of black rings,”JHEP , 074 (2006) [arXiv:hep-th/0608076].[30] M. M. Caldarelli, O. J. C. Dias, R. Emparan and D. Klemm, “Black Holes as Lumpsof Fluid,” JHEP (2009) 024 [arXiv:0811.2381 [hep-th]].[31] S. W. Hawking, “Black holes in general relativity,” Commun. Math. Phys. , 152(1972).[32] B. S. Kay and R. M. Wald, “Theorems on the Uniqueness and Thermal Propertiesof Stationary, Nonsingular, Quasifree States on Space-Times with a Bifurcate KillingHorizon,” Phys. Rept. , 49 (1991).[33] S. Hollands, A. Ishibashi and R. M. Wald, “A Higher Dimensional Stationary RotatingBlack Hole Must be Axisymmetric,” Commun. Math. Phys. , 699 (2007) [arXiv:gr-qc/0605106].[34] G. W. Gibbons and S. W. Hawking, “Action Integrals And Partition Functions InQuantum Gravity,” Phys. Rev. D (1977) 2752.[35] See e.g., J. D. Brown, “Action functionals for relativistic perfect fluids,” Class. Quant.Grav. , 1579 (1993) [arXiv:gr-qc/9304026], and references therein.[36] B. Carter, “Stability And Characteristic Propagation Speeds In Superconducting Cos-mic And Other String Models,” Phys. Lett. B (1989) 466.[37] B. Carter, “Perturbation Dynamics For Membranes And Strings Governed By DiracGoto Nambu Action In Curved Space,” Phys. Rev. D (1993) 4835.3638] S. S. Gubser and I. Mitra, “The evolution of unstable black holes in anti-de Sitterspace,” JHEP (2001) 018 [arXiv:hep-th/0011127].[39] In addition to [18], some entries to the literature in the specific context of strings andbranes are:R. A. Battye and E. P. S. Shellard, “String radiative back reaction,” Phys. Rev. Lett. , 4354 (1995) [arXiv:astro-ph/9408078].A. Buonanno and T. Damour, “Gravitational, dilatonic and axionic radiative dampingof cosmic strings,” Phys. Rev. D (1999) 023517 [arXiv:gr-qc/9801105].R. A. Battye, B. Carter and A. Mennim, “Linearized self-forces for branes,” Phys.Rev. D (2005) 104026 [arXiv:hep-th/0412053].[40] V. Balasubramanian and P. Kraus, “A stress tensor for anti-de Sitter gravity,” Com-mun. Math. Phys. , 413 (1999) [arXiv:hep-th/9902121].[41] See e.g., M. Parikh and F. Wilczek, “An action for black hole membranes,” Phys.Rev. D , 064011 (1998) [arXiv:gr-qc/9712077].C. Eling and Y. Oz, “Relativistic CFT Hydrodynamics from the MembraneParadigm,” arXiv:0906.4999 [hep-th].[42] B. Carter, “Outer Curvature And Conformal Geometry Of An Imbedding,” J. Geom.Phys.8