Bounds on the local energy density of holographic CFTs from bulk geometry

Bounds on the local energy density of holographic CFTs from bulk geometry

Sebastian Fischetti s.fischetti@imperial.ac.uk    Andrew Hickling a.hickling12@imperial.ac.uk    Toby Wiseman t.wiseman@imperial.ac.uk Theoretical Physics Group, Blackett Laboratory, Imperial College, London SW7 2AZ, UK
April 2016
Abstract

The stress tensor is a basic local operator in any field theory; in the context of AdS/CFT, it is the operator which is dual to the bulk geometry itself. Here we exploit this feature by using the bulk geometry to place constraints on the local energy density in static states of holographic -dimensional CFTs living on a closed (but otherwise generally curved) spatial geometry. We allow for the presence of a marginal scalar deformation, dual to a massless scalar field in the bulk. For certain vacuum states in which the bulk geometry is well-behaved at zero temperature, we find that the bulk equations of motion imply that the local energy density integrated over specific boundary domains is negative. In the absence of scalar deformations, we use the inverse mean curvature flow to show that if the CFT spatial geometry has spherical topology but non-constant curvature, the local energy density must be positive somewhere. This result extends to other topologies, but only for certain types of vacuum; in particular, for a generic toroidal boundary, the vacuum’s bulk dual must be the zero-temperature limit of a toroidal black hole.

I Introduction

AdS/CFT Maldacena (1999); Gubser et al. (1998); Witten (1998a) provides a fascinating way to understand the behaviour of specific strongly coupled conformal field theories. It has the potential to be a powerful tool to better understand theories of physical interest, such as QCD Casalderrey-Solana et al. (2011) or condensed matter at strongly coupled critical points Hartnoll (2009). However, since it is unclear whether or not holographic gauge theories describe our universe, an important goal is to understand what lessons may be gleaned from them that apply to non-holographic systems more directly relevant to our universe. To this end, it is interesting to focus on behaviour that is universal amongst holographic models and also universal to classes of states described by these models. Such lessons have the potential to generalise to other strongly coupled non-holographic theories, whereas detailed model or state dependent phenomena are presumably less likely to do so. It is this search for universal behaviours in theories with a gravity dual that is the motivation for this work.

One of the most basic operators in any CFT (indeed, in any QFT) is the stress tensor; in AdS/CFT, this object is intimately linked to the existence of a gravitational dual. All holographic models described by usual two-derivative semiclassical gravitational bulk theories have a “universal sector” described by pure Einstein gravity in an asymptotically locally AdS spacetime. This sector is dual to the dynamics of the CFT stress tensor, and is the reason the stress tensor plays a special role. In this work we consider a -dimensional holographic CFT and this universal sector. We will restrict to the class of states which are static and in which the CFT has been placed on a spacetime which is the ultrastatic product of time with a closed but otherwise general spatial geometry (i.e. a Riemann surface). In particular, in this static context this singles out the local energy density from the stress tensor as a basic observable of interest; this the quantity on which we will focus in this paper. We should note that there has been considerable interest in constraints on the behaviour of the local energy density of QFTs in the study of “quantum energy inequalities” (QEIs) Ford (2010); Fewster (2012) which typically bound integrals of the local energy density from above. Such bounds are hard to derive even for the simplest case of free fields111Though some progress has been made for e.g. the non-minimally coupled scalar field Fewster and Osterbrink (2008)., and obtaining analogous bounds in general strongly coupled theories on curved spacetimes directly seems an intractable challenge. This is again motivation for why constraints on the stress tensor in classes of strongly coupled theory with gravity duals are potentially so interesting Marolf et al. (2014).

To keep our results slightly more general, in the first portion of this paper we will in fact also allow deformations of the CFT by a static but otherwise general marginal CFT scalar operator, which is dual to a massless scalar in the bulk222The gravity and massless scalar sector arises as a consistent top-down reduction in the -dimensional holographic CFT setting, in which case the scalar may be taken to be the Type IIB dilaton and is associated to the existence of the exactly marginal gauge coupling. In the -dimensional setting it is less clear that such massless fields arise in consistent truncations; for example, in the case of reductions of 11-dimensional supergravity on AdS one does not necessarily obtain these Gauntlett et al. (2009). . Thus the space of deformations considered is the space of smooth functions over Riemann surfaces.

We derive two classes of results. Firstly, for static thermal states dual to Einstein-scalar gravity we consider a certain one-parameter family of regions  of the CFT geometry which are defined in terms of the curvature and scalar source. We show that the energy of these regions is bounded above by objects constructed from the geometry and scalar field on any bulk horizons. This upper bound vanishes in the limit of zero temperature, so that for static vacuum (zero-temperature) CFT states, the energy of the regions  is generically negative (and may vanish only when the CFT spatial geometry and scalar source are homogeneous). This result generalises that of Hickling and Wiseman (2016a): for pure gravity, the total energy – the Casimir energy – of the CFT is non-positive (see also the earlier result Galloway and Woolgar (2015) for fixed homogeneous boundary space). Below we will make precise how the  limit should behave; here we note only that the zero temperature bulk solution is not required to be smooth, but may have singularities of a “good” variety Gubser (2000).

This first class of results raises the question of whether the energy density must be everywhere negative for such vacua. Our second class of results provides an answer to this question. Restricting now to the universal sector (i.e. pure Einstein gravity in the bulk with arbitrary boundary spatial geometry), we show certain integrals of the energy density are bounded below by geometric data at bulk horizons. Restricting to zero temperature static vacuum states, these bounds imply that for a boundary with spherical topology the energy cannot be negative everywhere. For a boundary with genus , we obtain the same result provided the vacuum is one such that raising it to any small but finite temperature results in the bulk ending on a horizon with the same topology as the boundary (and additionally which for  has vanishing entropy in the zero-temperature limit).

We emphasise that both sets of results derive from relatively simple geometric considerations. For the first results we use simple Riemannian methods. Our second results derive from a simple generalisation of the work of Lee and Neves Lee and Neves (2015) on Inverse Mean Curvature (IMC) flow in hyperbolic spaces. It is likely that more sophisticated geometric techniques could provide stronger constraints on physical quantities that derive from the existence of a dual gravity description.

This paper is structured as follows. In Section II we provide a detailed summary of our results. In Section III we review some basic properties of the bulk which will then be used in Section IV to derive the first class of results mentioned above. In Section V we introduce the IMC flow and use it to derive constraints on the negativity of the local energy density. We conclude in Section VI with a discussion of interesting potential generalisations of our results and future directions.

Notation: For the benefit of the reader, here we summarise the notation used in this paper. Tensorial objects in the bulk spacetime will be dressed with a left superscript  (e.g. , etc.), while tensorial objects living in the optical geometry (defined under (15) below) of a static bulk time slice will be undecorated (e.g. , etc.). Boundary quantities will be dressed with an overline; tensorial objects living on a static boundary spatial slice will be given no other decoration (e.g. ), while tensorial objects living in the full Lorentzian boundary geometry will be given a left superscript  (e.g. , etc.). Objects living on the bifurcation surface of a bulk horizon will be typeset in calligraphic font (e.g. , etc.). We will use a capital  (dressed appropriately) for covariant derivatives on these geometries. Less-frequently used notation (specifically in Section V) will be defined as it is introduced.

Index Conventions: Lower-case letters from the beginning of the Latin alphabet () will be used as abstract indices; bulk spacetime coordinates will be indexed by upper-case letters from the beginning of the Latin alphabet (); boundary spacetime coordinates will be indexed by lower-case letters from the middle of the Greek alphabet (); bulk spatial coordinates will be indexed by lower-case letters from the middle of the Latin alphabet (); and two-dimensional spatial coordinates (used for either spatial slices of the boundary or a horizon) will be indexed by lower-case letters from the beginning of the Greek alphabet ().

Ii Summary of Results

In this section we summarise our results, beginning by setting the framework in which we work. We are interested in a static state of a CFT living on a (globally static) boundary metric  whose spatial geometry  is closed but generally curved333We emphasise that here we use the notation  to denote only a static time slice of the CFT spacetime, not the entire spacetime itself.. Unless otherwise specified, we will also turn on a (static) boundary source  for the bulk scalar. Thus we assume there exists a static infilling bulk solution whose time-translation Killing vector field becomes that of the boundary444More precisely, the conformal Killing vector of the boundary conformal class. when suitably restricted. As usual, we introduce a time coordinate  adapted to the static isometry such that  is the time-translation Killing vector field of both the bulk and boundary.

We now consider two cases: either the CFT is in a thermal state with nonzero temperature , or it is in a vacuum state with zero temperature. Note that when we refer to a vacuum state from this point on it will be implicit that we are referring to zero temperature.

If the CFT is in a thermal state at (nonzero) temperature , we assume that in addition to the conformal boundary, the bulk geometry may also end on some number of regular horizons, each with the same surface gravity  (with respect to ). If there are no bulk horizons we expect the CFT is in a confined phase, and with bulk horizons in a deconfined phase Witten (1998b).

If instead the CFT is in a (zero-temperature) vacuum state, we follow Hickling and Wiseman (2016a) and assume the bulk metric is the  limit of some finite temperature solution. In fact, we will restrict to three varieties of vacuum geometries, classified according to the presence of horizons and the behaviour of their area  in the zero-temperature limit:

  1. Confined vacuum: the bulk has no ends other than the asymptotic AdS boundary.

  2. Deconfined vacuum: besides the asymptotic AdS boundary, the bulk ends on a null surface that is the limit of a black hole horizon with the same topology as the boundary.

  3. Degenerate vacuum: besides the asymptotic AdS boundary, the bulk ends on an extremal horizon which is the but  limit of a black hole horizon with the same topology as the boundary.

We note here that the deconfined and degenerate vacua may contain more than one horizon or null surface; the above categories only require at least one of these to have the same topology as the boundary.

While there may exist vacua which do not fall into any of these categories, all the vacua of which we are aware do. For instance, global AdS and the AdS soliton Horowitz and Myers (1998) are examples of confined vacua with spherical and toroidal boundary, respectively. Compact quotients of the extremal hyperbolic AdS-Schwarzschild black hole Emparan (1999) provide examples of degenerate vacua in which the genus  of the boundary (and extremal horizon) is greater than one. Finally, we may obtain a deconfined vacuum by quotienting the Poincaré patch of pure AdS so that the boundary geometry becomes a flat torus. As emphasised in Hickling and Wiseman (2016a), this is an important physical example: while the Poincaré patch ends on a smooth extremal horizon, when it is compactified to a spatial torus the IR geometry becomes a null singularity Kunduri and Lucietti (2013). Nevertheless, it is a “good” singularity in the sense that at any nonzero temperature the bulk solution becomes planar AdS-Schwarzschild which has a perfectly regular horizon when compactified to a torus.

We remind the reader of the basis for this terminology. In a confined vacuum, we expect that since the only end is the conformal boundary then the fluctuation spectrum will be gapped rather than continuous, with the gap representing the confinement scale which in turn is set by a geometric scale in the boundary data. In addition the geometry will not change in the supergravity approximation going to small finite temperature, and hence the entropy associated to thermal excitation about such a vacuum will be , which is characteristic of confinement in a large  gauge theory. On the other hand, in a deconfined vacuum we expect the entropy to grow as  when finite temperature is introduced (corresponding to the presence of a horizon), and the fluctuation spectrum is expected to be continuous due to the presence of the extended null surface of infinite redshift in the IR of the bulk vacuum geometry. Finally, the non-vanishing area of an extremal horizon implies that the entropy of the dual CFT state is , and thus such a vacuum state is highly degenerate.

Next, since the static Killing vector field  of is globally timelike, we may work in an ultrastatic conformal frame where . In this frame, the boundary metric takes the form

(1)

while the local energy density . Note that while we will cast our results in terms of , we can re-express them in terms of the local energy density  in any other frame using .

Now, consider the tensor

(2)

defined on a spatial slice  of the ultrastatic geometry (and recall that  is the Ricci tensor of ). Now, there must exist points on the boundary  where the trace is (globally) minimised. We define the regions containing these points and bounded by a level set  for some constant  as

(3)

Denoting the energy of such a region as

(4)

then our first result is that for any value of ,

(5)

in any of the three vacua we consider555In fact, this result holds in the deconfined and degenerate vacua even if the topology constraint is removed.. Moreover, equality is only possible if  and  are constant (which in turn implies  is constant as well); in this case,  is either empty (if ) or the entire boundary (if ). Thus it is only possible to obtain an equality in (5) on the total energy  of the CFT.

A corollary of (5) is that if  attains its global minimum at only a single point , then the local energy density at this point must be non-positive:

(6)

in addition, we show that this inequality becomes strict in the confined vacuum.

This result generalises that in Hickling and Wiseman (2016a) where the total energy of the vacuum was found to be non-positive when there was pure gravity in the bulk. Interestingly, in that case the bound was related to geometric inequalities involving the Gauss-Bonnet theorem and the renormalised bulk volume Anderson (2001). In contrast, here we have included a massless scalar, allowed deformations to its dual, and have a local version of the bound rather than just a global statement about the total energy.

Given that for any , a natural question is whether in fact must be non-positive everywhere. We will provide an answer to this question when boundary scalar deformations are turned off, which will imply that  in the bulk is constant and hence can be ignored, effectively leaving pure gravity. Then using a simple adaptation of the rigorous geometric result in Lee and Neves (2015), we come to the interesting conclusion that in many cases  cannot be everywhere negative: for a sphere topology boundary deformed away from being round, in any of the three vacua we must have  somewhere on the boundary. For a higher genus  topology boundary with non-constant scalar curvature, this result only generalises to some of the vacuum classes:

(7)

We note that for pure Einstein gravity in the bulk, we are unaware of any examples of deconfined vacua with  or degenerate vacua with . Indeed, Result 2 may be interpreted as a constraint on which classes of vacuum may exist given the positivity properties of . Specifically, the vacua listed in Result 2 cannot exist if  is everywhere non-positive. For the special case of spherical topology , this result implies that any CFT vacuum state must have  somewhere.

Finally, we note that for genus  or , Result 2 admits an extension to nonzero temperature, subject to the constraint that for  there exists a toroidal horizon in the bulk. For higher genus , we find an analogous result provided the temperature is sufficiently large, , and there exists a horizon of genus  in the bulk.

Iii The Dual Bulk Geometry

In this Section, we outline some properties of the bulk that will be useful in the derivation of our results. Let the -dimensional bulk metric be  and the massless bulk scalar be . Then the full bulk equations of motion are

(8a)
(8b)

where the AdS scale  is related to the CFT “effective central charge” as , with the bulk Newton constant.

iii.0.1 Near-Horizon

Now, consider some (connected) bulk Killing horizon component . Locally near  we can introduce a normal coordinate  to write the metric as

(9)

where the horizon is at  and  is its (constant) surface gravity (this implies that we have normalised ). Regularity then demands that , and the components  be smooth functions of . The bulk equations locally determine these smooth functions in terms of the spatial geometry of the horizon (given by the 2-metric ) and the scalar profile on the horizon. Expanding in , the equations yield the expansions

(10a)
(10b)
(10c)

where the terms in the expansion above are written covariantly in the 2-dimensional geometry . Note that as before, we have defined the tensor

(11)

where is the Ricci tensor of the horizon spatial metric . In the semiclassical limit, the entropy of the horizon component is determined by its area  as  and is summed with that of other horizon components to give the total CFT entropy at temperature .

iii.0.2 Near-Boundary

Next, consider the asymptotically locally AdS boundary. Recall that in an asymptotically locally AdS spacetime with metric , the conformally rescaled metric  induces a regular metric on the boundary for any  satisfying  and  there Fischetti et al. (2012). This function  is referred to as a defining function, as it defines a conformal frame: that is, it determines a boundary metric which is a representative of the conformal class of the boundary. Of course, the choice of  is not unique; for any positive definite function , the choice  is a valid defining function as well.

Having chosen a conformal frame, we may then expand the bulk metric (and any bulk fields) near the boundary in Fefferman-Graham (FG) coordinates adapted to this frame; note in particular that the FG radial coordinate  obeys . Working in the conformal frame in which the boundary metric takes the ultrastatic form (1), we have666Recall that the time coordinate  is adapted to the static isometry, so no  cross-terms may appear.

(12)

where Skenderis (2002)

(13a)
(13b)
(13c)

Note that the terms at each order in  are written covariantly in the spatial boundary metric  of the ultrastatic frame, and recall that the tensor  was defined in (2). Per the AdS/CFT dictionary, we have also written the expansion in terms of the one-point functions of scalar operators:  is the one-point function of the CFT operator  dual to the bulk scalar, while  and  are the CFT energy density and spatial stress tensor, respectively. Note that the static bulk equations guarantee that the latter obeys the sourced conservation equation

(14)

Iv Local Energy Bounds

We now analyse the geometry governing static bulk solutions to deduce bounds on the local energy density. Following Hickling and Wiseman (2016b, a) we find it is convenient to write the bulk solutions in the “optical frame”,

(15)

where is a Riemannian 3-geometry which we term the “optical geometry”. Then may be regarded as a function over .

We choose  to be a defining function for the bulk conformal boundary, so that at the bulk conformal boundary with . Then the optical geometry has a smooth boundary, , that corresponds to the spacetime asymptotic region. In particular our defining function  is chosen so the corresponding representative of the conformal class of the boundary is the ultrastatic metric (1), whose spatial metric is given by the metric induced from on .

It is again convenient to define a generalisation of the optical Ricci tensor to include the bulk scalar:

(16)

Then the static bulk equations, including the massless scalar, can be written covariantly in the optical geometry as

(17a)
(17b)
(17c)

Let us now consider the behaviour of the optical geometry near the spacetime conformal boundary and near a horizon. Near the spacetime boundary, we may read off the metric components  and defining function  of the optical geometry from the FG expansion (13). More relevant to our purposes will be the expansions of ,

(18)

as well as of the trace ,

(19)

Thus the boundary quantity is given (up to a factor of three) by the bulk quantity restricted to the boundary:

(20)

In addition the normal gradient of at the boundary of the optical geometry determines the local energy density as

(21)

where is the outer-directed unit normal vector (in the optical geometry) to the boundary , given by at .

To study the behaviour of the optical geometry near a horizon, we first note that a smooth finite temperature bulk horizon with surface gravity (with respect to ) corresponds to an asymptotic region in the optical metric where . In fact, it is a conformal boundary. Translating our previous expansion near the horizon from equations (10), we find the behaviour

(22a)
(22b)
(22c)

where the scalar is still as given in (10a). We see the spatial 2-geometry of the horizon determines the conformal class of the conformal boundary of the optical metric.

Using this expansion we find the quantity  has the behaviour

(23)

near the horizon. Note then that at a horizon,  has the nice properties that it is constant, negative and simply determined by the temperature:

(24)

Then the normal derivative of at the horizon conformal boundary, weighted by the local volume element, takes the form

(25)

where is the outwards-pointing unit normal (in the optical geometry), given as as . We pause here to note that when the scalar field is constant, the right-hand side of (25) is a combination of the Euler density and volume element of the horizon; we will discuss this observation more later.

The behaviour of the optical geometry, and in particular the quantity both near the boundary and any horizon ends, allows us to derive a bound on the local energy density. The key relation is

(26)

where we have defined the traceless part of as

(27)

It is a straightforward computation to confirm that (26) follows from the intermediate result (obtained using the Bianchi identity)

(28)

The important consequence of (26) is that

(29)

This result is remarkably simple and will prove crucial in the derivation of Result 1, but a deeper understanding of its physical origin eludes us. Understanding it better may be useful for formulating bounds in other contexts.

Let us briefly note when the inequality (29) may be saturated. We see from (26) that vanishing  requires both  and  to vanish; via (17c) and (28), these imply that

(30a)
(30b)

Via the expansions of and near the boundary (equations (18) and (19)), these conditions in turn imply

(31a)
(31b)

Since the boundary is closed the first condition above implies is constant, in which case the second implies  is constant as well. Hence equality everywhere in equation (29) is only possible if we choose a constant source and boundary metric with a constant curvature (which will be determined by the boundary topology by Gauss-Bonnet). Any smooth deformation of either the boundary scalar source or metric implies (29) becomes a strict inequality somewhere in .

iv.1 Simple Bound for Confining Vacua

We may quickly derive a simple bound on the local energy density in the case that the optical metric has no boundary other than the one corresponding to the spacetime conformal boundary. In this case the bulk falls into our confining vacuum class (though we note it also may be interpreted as having finite temperature, the effect of which is invisible at the level of semiclassical bulk gravity).

Then implies a minimum principle for . Since in this case is a smooth bounded geometry we may apply the Hopf minimum principle Gilbarg and Trudinger () which implies that unless is constant, then the minimum of must occur at the boundary of the optical geometry. Furthermore, the outer normal gradient is strictly negative there, so .

Now since , the minimum of occurs precisely on the boundary where is minimised – these are the points we labelled in Section II above. Then for non-constant , the strictly negative outer normal gradient there determines that

(32)

directly from our boundary expansion (21). Note that this is a strict inequality, and it holds even when there is more than one point  at which  attains its global minimum; thus this result is slightly stronger than (6).

While this is a simple bound, it is highly non-trivial from a field theoretic perspective. Simply knowing the spatial metric and scalar source (and hence ) for our CFT allows us to identify where the energy density is guaranteed to be negative, no matter what the behaviour of the expectation value of the stress tensor does elsewhere.

iv.2 Improved Bound

We wish to improve on this simple bound in two regards. First, it requires a confining vacuum, so there are no other boundaries or ends to the optical geometry - i.e. no horizons or singularities in the full bulk spacetime. However as we have emphasised, for toroidal boundaries null singularities are required already in the canonical case of Poincaré-AdS suitably compactified. This is certainly expected to persist when the boundary metric is deformed and/or a scalar source is turned on. Second, we have a statement about the energy density only at the special points .

Let us define a non-negative function which is monotonically decreasing and suitably smooth so that

(33)

Then we have

(34)

which due to our constraints on the function and equation (29) yields

(35)

Provided we have not chosen a pathological such that vanishes everywhere in the bulk, then we may only have equality everywhere in the bound above if (and thus ) is constant (and hence  and  are independently constant as well).

Integrating and using the divergence theorem we obtain the inequality

(36)

where represents the contribution from asymptotic ends of the optical geometry corresponding to bulk horizons , and we remind the reader that  is the Hodge star.

Let us firstly examine the surface term at . Using our expansion (21) of  near this boundary, we obtain

(37)

where is the outer unit normal, and we have defined for convenience

(38)

so that obviously shares the same properties (33) as . Hence this boundary term is simply an integral of the energy density weighted by the function evaluated on the boundary quantity .

The surface term at the horizon can be evaluated remembering that is constant (and equal to ) when restricted to a horizon. Using the expansion (25) near the horizon, we have

(39)

where is the outer unit normal to a constant surface.

Thus denoting as the area of the horizon component we obtain the inequality

(40)

for any non-negative decreasing . Recall that , so consists of a contribution from the Euler characteristic of the horizon and a term involving the gradient of the scalar.

In particular, suppose we choose to be a Heaviside function:

(41)

Then we obtain

(42)

where the regions  and corresponding energies  were defined in (3) and (4), respectively. Note that whether the contribution from the horizons is included depends on their temperature and the value of . In Figure 1, we show a generic example (in the case of a single bulk horizon component) where  and : in this case, the horizon contributes to  (this will be the case for e.g. small deformations of global AdS-Schwarzschild). The other cases ( and ) are shown in Figures 1 and 1.

Figure 1: The behaviour of  for the step function (41);  is unity in the grey regions and zero in the white regions. The boundary region  (defined in (3)) corresponds to the portion of the boundary intersected by the grey region. For simplicity, here we show a static time slice of the bulk in the case where the boundary has spherical topology and there is one bulk horizon  with surface gravity . \subrefsubfig:LH: the case where  and , so  contains a contribution from . \subrefsubfig:LnoH: the case where , so  contains no contribution from . \subrefsubfig:LHnoboundary: the case where , so the boundary region  is trivial. Then one obtains the bound (IV.2) relating the area and topology of the horizon to the scalar field on it.

We can rewrite (42) in terms of the temperature and entropies , the latter of which sum to give the total entropy . We thus obtain a one-parameter set of bounds parameterised by :

(43)

with equality possible only for constant boundary  and .

We have now derived a bound on the energy associated to the compact regions on the boundary. If  has an isolated global minimum at , then when the bulk contains no horizons we can reproduce a slightly weaker version of (32): we simply take the limit  and obtain . More generally, however, our bound allows for bulk horizons and applies not just to the energy density at this point.

We now make some remarks. If there are no bulk horizons then we obtain simply for any region . However, we emphasise that in Section V we will show that it is not necessarily the case that will be everywhere non-positive.

If there are bulk horizons, but , then we can choose , as shown in Figure 1. In such a case, there are non-trivial regions where .

If the boundary satisfies (as in, for instance, global or planar AdS-Schwarzschild), then we can always choose a such that the region  is trivial and so the only contribution comes from the horizons; this is illustrated in Figure 1. Then the bound implies the condition

(44)

with the Euler characteristic777Recall that for closed, orientable surfaces like , the Euler characteristic is related to the genus  as . of the horizon component , and we note that since is not constant over (since ) the inequality above is strict. In fact, this bound holds horizon component-by-component. To see this, first note that (29) implies that has no minima in the interior of . Then assuming , from (20) we have , and so  must obtain its minimum on the bulk horizons (since from (24), ). Hence the outer normal on the horizons must be non-positive, , implying from (25) that  on all horizons. Integrating this condition over just one horizon component we thus conclude that

(45)

Without a scalar source this bound is trivial unless the horizons have negative Euler characteristic (genus greater than one), in which case it provides a lower bound on the area, and hence entropy, associated to a black hole state. With a non-trivial boundary scalar source, the bound is non-trivial for any horizon topologies and can be viewed as bounding the magnitude of scalar variations on the horizons in terms of their area.

Now let us conclude this section by considering the implication of our bound for a (zero temperature) vacuum state. First, consider a confined vacuum state (in which the bulk contains no boundaries besides the AdS boundary); then from (43) we immediately find that  for all . Next, consider a deconfined or degenerate vacuum. Then provided that  is finite in the limit  (which we would require for a regular horizon), we have that and . Thus again from (43) we find  for all . We have therefore obtained Result 1 (in fact, a slightly stronger version: Result 1 holds in the deconfined and degenerate vacua even if we remove the constraint requiring the topology of the bulk horizon to match that of the boundary).

V Inverse Mean Curvature Flow

Having used simple Riemannian methods to consider bounds on the local energy density, we now consider the use of the inverse mean curvature (IMC) flow. This flow was originally introduced by Geroch Geroch (1973) and extended by Jang and Wald Jang and Wald (1977) as a way of deriving Penrose inequalities in flat space (that is, a lower bound on the ADM mass of an asymptotically flat spacetime in terms of the area of any horizons), and was made rigorous in Huisken and Ilmanen (2001). Interestingly, the flow itself was generalised to an asymptotically hyperbolic setting by Lee and Neves Lee and Neves (2015) (though its use in proving Penrose-like inequalities is subject to limitations, as shown in Neves (2010)).

For the benefit of the reader we will review the construction of the flow, though we refer to Huisken and Ilmanen (2001); Lee and Neves (2015) for details and a rigorous construction and to Bray (2002) for an introduction and overview. In order to more closely connect to the results of the previous section, we will initially continue to restrict to a static bulk spacetime. However, as we will discuss briefly later, the IMC flow methods reviewed here may be applied in a much more general (i.e. dynamical) context.

We will require the (massless) bulk scalar to be constant. This follows necessarily for a static bulk if there is a constant boundary source for the scalar: then since is harmonic in the bulk and has vanishing normal derivative at any horizons, it must be constant everywhere. Thus for now we consider only pure gravity in the bulk - i.e. the universal sector of AdS/CFT.

v.1 Review of IMC Flow

Consider a static time slice  of the bulk. Using the Gauss-Codazzi equations, the staticity of this slice implies that its Ricci scalar is

(46)

Let us foliate this slice by a one-parameter family of compact surfaces . We construct this foliation by requiring that flowing along  (the “flow time”) moves the surfaces along their normals with speed equal to their inverse mean curvature; that is, defining  to be the unit normal vector field to the  and  to be their mean curvature, we require

(47)

This implies that the rate of change of a quantity with respect to  can be computed by a Lie derivative along ; we will denote such derivatives with dots. Thus denoting the induced metric on the surfaces  as , we have that the rate of change of  with respect to  is

(48)

where  is the extrinsic curvature of the  in .

We now outline monotonicity results for the IMC flow. First, note that the area of the surfaces  obeys

(49)

so the area grows exponentially in flow time, .

Next, define the Hawking mass Hawking (1968) of the surfaces  to be

(50)

where  is the Ricci scalar of  and we used the Gauss-Bonnet theorem to replace its integral with the Euler characteristic . To obtain the behaviour of the Hawking mass along the flow, we first compute

(51)

Assuming the topology of the surfaces doesn’t change,  is constant along the flow. Then combining (51) with (46) and using (49), we find that the Hawking mass increases monotonically along the flow:

(52)

Note that if the IMC flow surfaces are homogeneous so that  (as they will be in the case of e.g. spherical symmetry), then the inequality above is saturated, and the Hawking mass is constant along the flow.

This monotonicity can be used to bound the asymptotic () behaviour of the Hawking mass in terms of its value on some initial surface. To that end, consider starting the IMC flow at the bifurcation surface  of a Killing horizon (for now, we furthermore assume that there is at most one such surface). Since  is a minimal surface (), its Hawking mass is

(53)

Alternatively, in the absence of any Killing horizons, the flow may be started on the surface of a small geodesic ball centred on a point, in which case  vanishes in the limit the ball is taken very small. In both cases these surfaces provide good initial conditions for the IMC flow at some finite starting flow time, say .

In asymptotically flat space, the asymptotic value of  is precisely the ADM mass, which by monotonicity of  is bounded from below by (53) (with ): this yields a Penrose inequality. In the asymptotically hyperbolic case, it was suggested in Gibbons (1999) that the asymptotic value of  should again be the total mass of the spacetime, and thus still yield a Penrose inequality. Interestingly, Neves (2010) showed that this is not the case: an asymptotically hyperbolic space generally does not have sufficient convergence properties to guarantee that the IMC flow asymptotes to its total mass.

Figure 2: Surfaces used to study the asymptotics of the IMC flow. The flow surfaces foliate the asymptotic region of a bulk static time slice  (shaded grey); note that here we show just one surface  for simplicity. We foliate the asymptotic region of the full bulk spacetime with timelike hypersurfaces  which intersect  on the flow surfaces . The surfaces  are adapted to the time-translation isometry of the spacetime, so that the unit timelike vector field  normal to  is tangent to the .

To illustrate why this is the case, let us study the asymptotics of the IMC flow in more detail. Assume that the flow surfaces  eventually approach the asymptotic boundary  (in the precise way described in Appendix A); then in any FG coordinate system, as the flow asymptotes to

(54)

for some smooth positive definite function of the boundary coordinates . Thus a flow that asymptotes to the AdS boundary defines a “flow frame”, whose defining function we may take to be  near the conformal boundary, such that the flow asymptotes to constant coordinate surfaces in the associated FG coordinates.

The important point is that flows starting with different initial conditions in the interior of the bulk will generally have asymptotics that yield different flow frames. This is nicely illustrated with a simple example. Take (a time slice of) global AdS and consider flows that start at a point in the interior. In usual global AdS coordinates, a flow starting at the origin of course respects the spherical symmetry of the coordinates. As it asymptotes to the boundary on constant radial coordinate slices, picking out the flow frame which is the natural one where the boundary metric is the ultrastatic product of time with a round sphere. However, we show in Appendix B that starting the flow “off centre” results in a flow frame which is not the ultrastatic one.

Given that there is no preferred frame to which flows asymptote, then the question is whether the asymptotic Hawking mass is a conformal invariant or not, i.e. if it is independent of the frame. As we now show, this is not the case, and while the Hawking mass does indeed depend on the energy density in the ultrastatic conformal frame, it also depends on the data specifying the frame to which the flow asymptotes.

In the asymptotic region let us introduce a foliation of timelike hypersurfaces  whose intersections with the bulk time slice  are just the IMC flow surfaces , as shown in Figure 2. We take these hypersurfaces to be adapted to the time-translation isometry of the spacetime, i.e.  is tangent to the . Defining  and  to be the induced metric and Einstein tensor of the , their extrinsic curvature  is related to the Balasubramian-Kraus boundary stress tensor Balasubramanian and Kraus (1999) as888Some signs here differ from the original reference Balasubramanian and Kraus (1999) due to differing curvature conventions.

(55)

We remind the reader that  is defined as a bulk object, and in particular makes no use of a choice of conformal frame. However, once a conformal frame  is chosen, the boundary stress tensor in this frame is given by .

The asymptotic value of the Hawking mass can be expressed as an integral of . To do so, first we note that the extrinsic curvature  of  in  is just the projection of  onto . Then the mean curvatures are related by

(56)

where  is the unit timelike normal vector field to  (and which by construction is therefore tangent to ). We may then use (55) to express  in terms of boundary quantities:

(57)

Next, note that since the time slice  is static, its extrinsic curvature vanishes. The Gauss-Codazzi equations then yield

(58)

so that plugging this and (57) into (V.1), we obtain that asymptotically,

(59)

where we have used the fact that near the boundary and for arbitrary conformal factor .

Neither of the two integrals that appear in  (recall that  is an integral over ) can be written as conformal invariants of boundary data. This demonstrates our previous claim, that the asymptotic limit of the Hawking mass is not a conformal invariant of boundary quantities (and thus need not yield the total energy of the CFT). Hence the details of the specific flow will be important through the particular flow frame it picks out.

We may re-express the asymptotic behaviour of the Hawking mass in terms of boundary data in a particular frame. Consider, for instance, working in the flow frame defined by ; then one finds

(60)

where  and  are the spatial boundary metric and local energy density in the flow frame, respectively. However, this frame will generically not be the same as the ultrastatic conformal frame with defining function ; the function describes the relation between these. Note that is positive definite and will depend on the details of the flow, which in turn depend on its initial conditions and the bulk geometry. Converting to this ultrastatic frame, we find

(61)

Now consider starting the IMC flow on some surface  at . Then let us assume a smooth flow exists that asymptotes to the boundary. Note that a necessary condition for the flow to be smooth is that the topology of the surface is the same as that of the boundary . Then monotonicity of Hawking mass under the IMC flow implies the highly non-trivial result

(62)

v.2 A Rigorous Bound

This result (62) assumed the existence of a smooth IMC flow from an initial surface  to the asymptotic boundary. Certainly for static bulks with homogeneous spatial symmetry one would expect such flows to exist. The question in our context is: does such a smooth IMC flow exist for a generic (static) bulk? The answer is no: the surfaces  can e.g. cusp or encounter “obstacles” like black hole horizons. In such cases, the above analysis breaks down. However, Huisken and Ilmanen (2001); Lee and Neves (2015) rigorously showed that monotonicity of the Hawking mass is still preserved by a so-called weak solution to the IMC flow. Roughly speaking, instead of thinking of  as a parameter along the family of surfaces , one takes  to be a scalar on the time slice  and defines the surfaces  as level sets of  whose asymptotic behaviour is that described in Appendix A. These level sets are then allowed to “jump” discontinuously to avoid singularities of the flow while still preserving monotonicity of the Hawking mass. However in order to obtain the bound (62) for this weak flow we still require the topology of the starting surface and the boundary to be the same Lee and Neves (2015).

An important technical point is that Lee and Neves (2015) assumed that the boundary  was a constant-curvature space (i.e. a round sphere, flat torus, etc…). In the static AdS/CFT setting, for constant-curvature boundary we know the exact bulk solution (AdS-Schwarzschild), whereas here we are precisely interested in the case that the boundary metric is non-trivially curved so that the bulk solution is unknown. To fill in this small gap, in Appendix A we generalise Lemma 3.1 of Lee and Neves (2015), which is where the details of the asymptotics appear, in order to use the Weak Existence Theorem 3.1 of Huisken and Ilmanen (2001) to guarantee that the weak IMC flow asymptotes to the asymptotic boundary . Then the proof presented in Lee and Neves (2015) applies to our time slice  as well.

Thus the monotonicity result (62) above can be made mathematically rigorous and holds even when a smooth IMC flow does not exist. Moreover, we may apply it even when there exists more than one Killing horizon, in which case we may start the flow at any bifurcation surface . We refer to Lee and Neves (2015) for details of this construction.

Although not immediately relevant for the purpose of this paper, we should also pause to note that the IMC flow can be applied more generally than just to static time slices of static spacetimes. Indeed, the analysis presented above will go through unchanged if  is only taken to be a moment of time symmetry (rather than a static time slice). In particular, equations (46) and (58) will be unchanged, while  must be taken to be an apparent horizon rather than the bifurcation surface of a Killing horizon. In fact, all that is really needed for the monotonicity result to go through is for ; this provides considerable freedom in both the choice of time slicing and the matter content of the spacetime.

v.3 Derivation of Result 2

Despite the fact that the asymptotic behaviour of the Hawking mass is not a conformal invariant, we may nevertheless use its monotonicity to derive constraints on the local energy density of the CFT. To that end, consider first the case where the boundary has spherical topology, . Starting the IMC flow at flow time at some bulk point, we find . Second, consider the case where the boundary has toroidal topology, , and the bulk contains a (bifurcation surface of a) nonzero-temperature horizon  of the same topology. Starting the flow on , we note that since  we have . Thus in both these cases monotonicity of the IMC flow implies from (61) and (62) that