OCU-PHYS-422, KUNS 2557, YITP-15-33
Effective theory of Black Holes
in the expansion
Roberto Emparan, Tetsuya Shiromizu, Ryotaku Suzuki,
Kentaro Tanabe, Takahiro Tanaka
Institució Catalana de Recerca i Estudis Avançats (ICREA)
Passeig Lluís Companys 23, E-08010 Barcelona, Spain
Departament de Física Fonamental, Institut de Ciències del Cosmos,
Universitat de Barcelona, Martí i Franquès 1, E-08028 Barcelona, Spain
Department of Mathematics, Nagoya University, Nagoya 464-8602, Japan
Kobayashi-Maskawa Institute, Nagoya University, Nagoya 464-8602, Japan
Department of Physics, Osaka City University, Osaka 558-8585, Japan
Theory Center, Institute of Particles and Nuclear Studies, KEK,
Tsukuba, Ibaraki, 305-0801, Japan
Department of Physics, Kyoto University, Kyoto, 606-8502, Japan
Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto, 606-8502, Japan
firstname.lastname@example.org, email@example.com, firstname.lastname@example.org,
The gravitational field of a black hole is strongly localized near its horizon when the number of dimensions is very large. In this limit, we can effectively replace the black hole with a surface in a background geometry (e.g., Minkowski or Anti-deSitter space). The Einstein equations determine the effective equations that this ‘black hole surface’ (or membrane) must satisfy. We obtain them up to next-to-leading order in for static black holes of the Einstein-(A)dS theory. To leading order, and also to next order in Minkowski backgrounds, the equations of the effective theory are the same as soap-film equations, possibly up to a redshift factor. In particular, the Schwarzschild black hole is recovered as a spherical soap bubble. Less trivially, we find solutions for ‘black droplets’, i.e., black holes localized at the boundary of AdS, and for non-uniform black strings.
- 1 Introduction
- 2 Set up
- 3 Effective theory: leading order
- 4 Black droplets
- 5 Effective theory: next-to-leading order
- 6 Non-uniform black strings
- 7 Outlook
- A Curvature on sections at constant
- B Another derivation of the effective equation
- C Far-zone Schwarzschild field from the effective stress tensor
- D Black cavities?
Recently it has been demonstrated that black hole physics can be efficiently solved in an expansion around the limit of large number of dimensions, [1, 2, 3, 4, 5, 6, 7]. In this limit there appears a near-horizon region that is universal for all non-extremal neutral black holes, and which encompasses the small radial extent of the gravitational field outside of a horizon of radius . One important feature is that the quasinormal spectrum of the black hole splits into modes of high frequency, , and of low frequency [4, 6, 7, 8]. The latter are particularly interesting, since they are fully localized in the near-horizon region (where they are normalizable excitations) and are decoupled from the asymptotic ‘far zone’ to all perturbative orders in . This split into two scales makes it natural to try to integrate out the high-frequency, short-distance degrees of freedom to obtain a fully non-linear effective theory of the long-wavelength, decoupled dynamics. Such a theory should capture the physics of black holes on lengths and timescales , hence allowing for fluctuations on scales comparable to the horizon radius, , or (as appropriate for black branes [1, 2]) .
We can also motivate this effective theory in a more geometric fashion. In the limit the gravitational field of the black hole vanishes outside the near-horizon region, and thus there is a neat separation between the black hole and the background where it resides (e.g., Minkowski or (Anti-)deSitter spacetime). The black hole can then be effectively identified with a particular surface in this background. Since ultimately the properties of the black hole are dictated by Einstein’s equations, it must be possible to derive from them a set of equations that must satisfy. These constitute the effective theory of the black hole in the large limit.
The black hole is then described by a set of ‘collective coordinates’ which specify the embedding of in the background spacetime, and which vary over scales much larger than , where is a characteristic length of . The approach to obtain the effective theory employs the parametric separation between the large radial gradients and the smaller temporal and spatial gradients along the horizon, which allows to solve the radial dependence of the Einstein equations. Then, the vector-constraints in the radial direction yield the effective equations for the embedding functions. Readers familiar with other effective theories of black holes, in particular the fluid/gravity correspondence  and the blackfold approach , will recognize similarities here. They are all based on a parametric separation between the fluctuations that are transverse to the horizon and those that are parallel to it. However, since is a parameter of the theory instead of a parameter specific to some solutions, in principle the expansion allows to tackle a larger set of problems —at the expense of possibly losing accuracy at finite values of or missing phenomena which are non-perturbative in .
This effective theory is an important step in the program to understand gravity in the large limit. General Relativity in vacuum (possibly with a cosmological constant, and without compactified dimensions) is essentially a theory of black holes and gravitational waves. The large effective theory is a reformulation of the black hole sector of General Relativity in terms of membrane-like variables. The decoupling property of the near-horizon zone implies that, as long as its gradients remain much smaller than , the effective membrane can not radiate any gravitational waves to the far zone at any perturbative order in the expansion. Conversely, no gravitational waves from the far zone of frequency can interact with the effective membrane; and, while far-zone waves of frequency or larger can penetrate to the near zone, they are perfectly absorbed by the horizon on a short timescale and do not interact with the low-frequency modes of the effective membrane . So the two sectors of the theory —black holes and gravitational waves— appear to be well separated, with the coupling between them being non-perturbative in . However, there do exist black holes that when have large spatial gradients along their horizons. Moreover, in the evolution of some horizons it can occur that initially small temporal gradients grow to values . Such situations imply breakdowns of the applicability of the effective theory, and are reminiscent of the breakdown of hydrodynamics when turbulence develops.
In this article we start to develop the large effective theory of black holes by focusing on the simplest case of static configurations of neutral black holes, possibly with a cosmological constant. The equations for the embedding of in the background take a remarkably simple form: if the trace of the extrinsic curvature of is , and the redshift factor on is , then the effective equation at leading order is
where the constant gives the surface gravity of the black hole. When this is satisfied, our results provide explicitly the near-horizon black hole metric that solves the full Einstein-(A)dS equations in the leading large limit. Observe that in backgrounds like Minkowski space where there is no redshift, this is the same as the equation for a soap film. The stress-energy tensor of the effective membrane that lives on is also simple: it is a modulation along the membrane directions of the quasilocal stress tensor of the large- black hole.
We also obtain the next-to-leading order corrections to the effective theory. Although in general the form of the equations becomes a little more complicated than (1.1), in the absence of redshifts we get the same equation at the next order in the expansion.
In order to test the usefulness of the theory, we have applied it to obtain several non-trivial solutions. Staticity greatly restricts the number of possible black holes. For instance, in all dimensions the unique asymptotically flat, static black hole of Einstein’s vacuum equations is the Schwarzschild-Tangherlini solution , which is recovered from (1.1) as a spherical soap bubble — correspondingly, the spherical soap bubble is the unique closed surface of constant mean curvature in flat space in any dimension111This is widely believed among differential geometers to be the case, see e.g., . We thank Gary Gibbons for this reference.. However, we will show that the effective theory easily yields other static solutions: black holes localized at the boundary of AdS (‘black droplets’) , and non-uniform black strings in asymptotically flat space [14, 15, 16]. The construction of these solutions at specific finite values of required sophisticated numerical solution of systems of partial differential equations. In contrast, the large- equations for these problems are single ordinary differential equations that can be easily solved, when not in an analytical exact or approximate form, at least with a one-line command of NDSolve in Mathematica.
In the next section we introduce a formalism that is adequate for the resolution of the problem. Then in sec. 3 we solve the Einstein equations at leading order in and derive the effective equation for the surface . We also discuss simple examples, and obtain the stress-energy tensor for the effective membrane on . In sec. 4 the effective equation is solved to obtain black droplets in AdS. Sec. 5 contains the derivation of the effective theory at next-to-leading (NLO) order. This is then applied in sec. 6 to the construction of non-uniform black strings. Sec. 7 finishes the paper with some brief remarks. The appendices contain technical details and elaborations on asides mentioned in the main text.
2 Set up
In our metric ansatz we separate a radial direction , where gradients are of order , from all other spacetime directions along which variations are smaller. To this effect, we use a decomposition in ‘synchronous gauge’
where run over all the directions orthogonal to . The Einstein equations in vacuum, with a cosmological constant
can be written in terms of the intrinsic, , and extrinsic, , curvature tensors of the -dimensional constant- surfaces,
Eqs. (2.3) and (2.4) are respectively the scalar and vector constraints, while (2.5) is the ‘dynamical evolution’ equation in the radial direction. Knowing , eq. (2.6) can be integrated to obtain the metric. It is convenient to consider separately the equation for obtained from eq. (2.5),
where we define by
We take the metric to be static. In the spatial directions orthogonal to , we have non-trivial dependence on a number of spatial directions , which are orthogonal to a -dimensional space that we take to be a sphere .222This can be extended to other spaces with intrinsic curvature at large . We introduce
which can be used as the large expansion parameter instead of . Our metric ansatz is then
where , and is the metric on the unit .
The different metric functions will be assumed to scale with in specific ways. In order to get oriented, note that one solution that we intend to recover is the leading-order near-horizon geometry of the Schwarzschild black hole , which can be written in the ansatz (2.10) with as333Here is twice the one in .
Observe that in the sphere radius we are keeping terms of order : due to the large dimensionality of the sphere, such terms enter (through traces) in the leading order equations and thus must be kept at this order. The takeaway here is that, while the -dependence in appears at the leading order, instead in and it is at .
3 Effective theory: leading order
Our strategy is to first solve eqs. (2.3) and (2.5) for the radial dependence of the extrinsic curvature with regularity at the black hole horizon, and then obtain the metric from (2.6). Radial integrations leave an undetermined function of which cannot be eliminated by gauge choices: this is the one collective degree of freedom of the static black hole. The vector constraint (2.4) then yields a non-linear differential equation, on only, for this degree of freedom, which is the effective equation we seek.
3.1 Solving the leading order equations
According to our assumptions we can separate the leading-order, -independent terms as
In this case we can integrate (2.7) to find444Henceforth in this section we omit the symbol to unburden the notation. In addition, the subsequent analysis is valid only when .
The divergence at is the expected pole at the horizon, coming from (all other components of the extrinsic curvature must be regular there). The metric to leading order is invariant under , which we use to set . Similarly, by rescaling by an function of we can reach a gauge in which, to leading order,
Next, the equation for is
since can be neglected as it is of lower order. This equation is solved as
where the integration function of has been fixed again by requiring that have a pole at .
Since and are , in order to solve for them we need and . The components of the curvature in a constant- section
are readily obtained (see appendix A). The scalar curvature gets negligible contributions from the time direction, and is given at leading order by
where we abbreviate
It is easy to see that this equation is actually equivalent to the scalar constraint (2.3) at leading order.
Since the scalar curvature is dominated by the components along the sphere we have
Using the covariant derivative for the metric , the curvature along the directions, obtained from (A.3), is
Now we have all the terms needed to integrate the equations from (2.5),
Imposing regularity at we find
where we have defined the tensor
is an integration function from the -integration of . For the and components the integration functions of have been absorbed in a redefinition of and . However, cannot be absorbed in that way.
Up to this point we have solved all the radial dependence of the metric and it only remains to impose the vector constraint (2.4), whose only non-trivial component is along . To leading order it takes the form
Plugging in our previous results and using the identity
derived from (3.13), we find,
Under either of the two cases in (2.13), (2.14), the last term is subleading and can be neglected. Thus, consistently, the -dependence cancels out of the leading-order equation, which requires that be proportional to . Observe that this condition is equivalent to requiring that the surface gravity be uniform on the horizon. Indeed, the precise relation is
Defining a rescaled surface gravity
our solution of the Einstein equations, valid in the near-horizon region to leading order in , is
Although could be absorbed by rescaling the time coordinate, this may not be convenient, since the normalization of is fixed by matching to the far-zone.
The geometry (3.29) can be interpreted as a modulation along of a near-horizon Schwarzschild black hole solution. The functions , and vary slowly compared to radial gradients, and one of them, say , can be regarded as the single collective degree of freedom for the black hole. Indeed, it is possible to give an alternative derivation of the effective theory following these ideas, as described in appendix B.
3.2 Effective equation
The metric (3.29) in the near-horizon geometry must be matched to the far-zone background in the common ‘overlap zone’, which corresponds to in (3.29). The matching must be such that the metric induced on a constant- surface there, namely,
Therefore eq. (3.27) can be written in the simple form
with constant .
That is, if in a given background spacetime we find a surface that satisfies (3.32), then we can ‘resolve’ this surface by replacing it and its interior with the static black hole with geometry (3.29), whose surface gravity is . Let us remark that in the limit to the horizon is, by definition, equal to . The vector constraint is independent of , but it is non-trivial that takes the same value on the horizon and on .
When there is only one coordinate, , this equation takes the form
This is the version that we will employ in the examples in this paper.
Finally, observe that in our construction of the black hole solution, is positive when the radial normal points outwards of the horizon. This resolves any possible ambiguity about which side of in the background spacetime corresponds to the exterior of the black hole: if , the exterior of the black hole lies in the direction of the normal to .
3.3 Simple solutions
We verify that the effective theory correctly reproduces known exact solutions.
Schwarzschild-(A)dS black holes.
In global AdS spacetime (extending to Minkowski and deSitter when ),
take a surface at , so that
It is immediate to see that a spherical surface
solves (3.34) with the correct surface gravity
The static planar and hyperbolic black holes in AdS are similarly easy to obtain.
In a Minkowski background, , we can also find the same solution starting from
Setting , so that and , eq. (3.34) is solved by with . Thus the Schwarzschild solution when is obtained as a spherical soap bubble with mean curvature
Both in the Minkowski background (3.39) and in Poincaré AdS,
the black string is just , with .
Intersecting black hole and deSitter horizons.
Let us describe a less simple, lesser-known solution in the deSitter background,555We thank Jorge Santos for discussions that led to this example. with ,
We change coordinates as
so that (3.42) becomes
In this case,
is a solution of (3.34) (where and ), with
This black hole is actually known exactly in all : it is the metric
which arises in the Kerr-deSitter family of black holes in the limit in which the equator of the rotating black hole touches the deSitter horizon and the configuration becomes static. The two horizons correspond to the cosmological and black hole horizons of the submetric , which extend separately along the coordinate until they meet at the equator . It is a simple exact instance of a horizon intersection. By the same reasoning as used in , in the large limit the black hole of (3.47) becomes a ‘hole’ at in the geometry (3.44).
3.4 Effective stress tensor
In order to match the near-zone solution to the far zone, they must share the metric (3.30) at their common boundary (in the overlap zone), and also their extrinsic curvatures. In this way the full geometry is smoothly glued between the two zones. This construction is equivalent to substituting the near-zone by a membrane with geometry (3.30) and with a stress-energy tensor given by the quasilocal stress-energy tensor of the near-horizon geometry measured in the overlap zone. Having this membrane stress-energy tensor as a source for the gravitational field in the background, ensures that the matching of the near- and far-zone geometries is .
The overlap zone is the asymptotic region of the near-zone. Asymptotically at large , and neglecting terms , the metric (3.29) becomes
with the condition (3.27). When the scalar constraint (3.13) is satisfied, this is in fact a solution at all : it corresponds to empty space (Minkowski or AdS) at large . Therefore (3.13) is not specific to black holes. Instead, it pertains to the definition of near-zone asymptotics of large gravity.
The geometry (3.52) is then the reference metric required to define the effective stress tensor. In general there can be terms in the metric at large of the type , but all the functions that appear are fully determined, up to gauge, by the Einstein equations in the overlap zone in terms of and and their derivatives. In this sense, they are analogous to the first terms in the Fefferman-Graham (FG) expansion in AdS, which are fixed by the boundary metric. The terms at large that behave like , …, correspond to normalizable perturbations in the near-zone and are not determined by the boundary metric. The leading terms will give the quasilocal stress-energy tensor.
To leading order at large , the quasilocal stress-energy tensor from (3.52) is the same for empty space and for the black hole. Thus this stress tensor cannot give any non-trivial gravitational effect on the background. Such effects come from the difference between the quasilocal stress energy tensors of empty space and of the black hole, i.e.,
where the subtraction in is from empty space with the same asymptotic boundary metric. The subtraction removes the terms at large , and leaves those that decay like (or faster). Typically, in the far zone we have , which is the fall-off of the gravitational field away from a localized source.
For the black hole solution (3.29), expanding at large , neglecting terms , and subtracting the background values we find
It is easy to check that
since this indeed follows from the vector constraint. It implies the conservation of the stress-energy tensor (3.53), which is given by
with the constraint that
That is, at this order the values of and are not fully determined, since terms of in , which remain undetermined at this order, can make a contribution to these components. However, the combination in (3.60) does not suffer from this indeterminacy and this equation is required for the conservation of the stress-energy tensor at leading order. It is not surprising that the tensions (or pressures) play a subleading role: it was already observed in  that the pressure of black branes is suppressed relative to their energy density when .
This effective stress tensor is enough to obtain the backreaction of the black hole on the background, by computing the linearized gravitational field created by this source. The specific values of and are not important at this order, so one may choose them arbitrarily as long as (3.60) (which is necessary for a consistent coupling of the source to the gravitational field) is satisfied. While these components may be determined at the next order in the expansion, one way of choosing them is by requiring that, as is the case in all known black hole solutions, the ‘hoop stresses’ vanish. Under this assumption we have
In the case we can solve (3.61) to find
3.4.1 Effective membrane source
Since the stress tensor is homogeneous on , we can integrate it over the angular directions. At large this yields a factor . Then the integrated energy density is
and the stresses are, as mentioned above, and therefore subleading but must satisfy
When we can explicitly solve this to obtain
The energy density (3.64) admits a very simple physical interpretation: it is the energy density of a black brane of radius , redshifted by the local redshift factor . The tension (3.66) is also the tension for that type of black brane, with an additional correction that accounts for the bending tension.
The backreaction of the black hole on the background is obtained by taking a membrane with this stress-energy tensor, extended along the directions in the background, and at the origin of the : by homogeneity in the sphere direction, in linearized gravity and at large , we need not consider that the membrane extends on a sphere of finite radius but rather we can collapse it to zero radius.
4 Black droplets
Now we construct solutions of (3.34) in the AdS spacetime (3.41) that are rather more complicated than in sec. 3.3. We seek surfaces that end at the boundary at on a sphere , and extend a finite distance into the bulk at until the shrinks to zero. They correspond to black holes localized at the AdS boundary. These were investigated in the context of cutoff-AdS/CFT (the holographic duality for the Randall-Sundrum-II braneworld) for the purpose of studying Hawking radiation of the dual CFT at planar (leading large ) order, and its backreaction on the black hole [18, 19]. In  they were reconsidered without the UV cutoff, in which case they are dual to the CFT in a fixed black hole background. After several years of controversy, the existence of such regular static black hole solutions seems to be settled after the numerical construction in . Nevertheless, a simpler solution to the problem, such as afforded by the large expansion, may be desirable.
We seek solutions for surfaces embedded in (3.41) in the form
Choosing a normal that near the boundary points outwards from in the direction of increasing , a direct computation of the mean curvature to leading order in gives
In this case eq. (3.32) becomes
Note that disappears from this equation, as it must, without having set it to .
Eq. (4.3) is a non-linear equation that requires numerical integration, but it is convenient to first analyze its main properties. The equation simplifies for to
which is solved by a circular profile,
The extrinsic curvature of this solution vanishes: the curvature of the sphere is exactly cancelled by the cosmological curvature. Other simple solutions are:
the AdS black brane, with and ;
the black string solution, with .
Eq. (4.3) is invariant under , and it requires that vanish at the boundary . Therefore, the surface always meets the boundary orthogonally.
When extended into the bulk, we want to cap off smoothly at a finite value where . The smoothness is guaranteed by eq. (4.3): for small and large , it becomes
which shows that if closes off away from the boundary, , it does it smoothly.
The solutions of (4.3) are completely determined once the boundary radius
is specified. Therefore, the solutions are parametrized by the surface gravity, i.e., by , and by the radius at the boundary . These parameters can be varied independently, but only their product
is invariant under the scaling symmetry of the background. Therefore we obtain a one-parameter family of inequivalent solutions labeled by .
Observe that, even if the near-horizon geometries for different values of are all locally equivalent, they will match to different far-zone geometries. For instance, if we fix , then as varies we get boundary black hole geometries with the same horizon radius but with different radial dependence in . Ref.  proposed the existence of a one-parameter family of black droplets of this kind. Our construction realizes it in a natural way.
When is finite and non-zero we can rescale such that we effectively set
in which case the solutions are parametrized by . This choice is slightly convenient. With it, the two branches of solutions for in eq. (4.3) are
Consider first the sign solution. Substituting in (4.2) one finds that
Given our choice of normal in (4.2), since the criterion discussed at the end of sec. 3.2 tells us that the black hole exterior is in the direction of increasing . Thus, the sign in (4.10) gives black droplets.
On the other hand, for the sign in (4.10) we obtain666To obtain this we assume that , which is appropriate since we want solutions connected to the boundary.
so this gives solutions where the exterior is inverted relative to the previous case. Therefore these are not black droplets. It is less clear whether these solutions are relevant. We discuss them briefly in appendix D.
We see that as grows with fixed, the droplets extend more into the bulk. The parameter ranges between and , and as the solutions approach (4.5). Eq. (4.4) is actually recovered from (4.3) in the limit in which both and go to zero at the same rate. At the opposite end, when , the solutions approach the black string. Near the black string limit, with , the solution at small is close to a black string,
Close to the cap it is instead described by (4.6).
It is not fully clear from this study which of these black droplets should correspond to the solution whose boundary geometry is the Schwarzschild black hole, although a plausible candidate is the solution (4.5). This, as well as the calculation of the holographic stress-energy tensor of these solutions, requires a more detailed analysis which we postpone to the future.
Black droplets with black branes, and black funnels.
In AdS one expects other classes of static black hole solutions that are related to droplets [13, 21]. Black droplets are often considered in conjunction with a black brane of infinite extent in directions parallel to the boundary. The two horizons are separated in the bulk, and in the large limit the interaction between them is suppressed exponentiallly in . Then we can obtain these configurations by simply superimposing a flat black brane and one of our black droplets. Their interaction through graviton exchange across the bulk is non-perturbative in but in principle it is possible to incorporate such effects by computing the gravitational attraction between the two membranes with the effective stress-energy tensor of sec. 3.4.1.
‘Black funnels’ can be regarded as the merger of a black brane and a black string (or a long enough black droplet) that hangs from boundary. Our effective equations do not seem to allow to obtain these solutions. The apparent reason is that the ‘shoulder’ at which the black string and the black brane are joined involves large gradients along the horizon, of order , which fall outside the remit of the effective theory.
5 Effective theory: next-to-leading order
At the next order we must take into account the -dependence in in (2.8). It is convenient to separate it in the form
Here is different than the leading-order function in (3.4), since it also contains contributions. Since we will want to keep the horizon at , we fix the ambiguity in this split by demanding that