On the apparent horizon in fluid-gravity duality
This article develops a computational framework for determining the location of boundary-covariant apparent horizons in the geometry of conformal fluid-gravity duality in arbitrary dimensions. In particular, it is shown up to second order and conjectured to hold to all orders in the gradient expansion that there is a unique apparent horizon which is covariantly expressible in terms of fluid velocity, temperature and boundary metric. This leads to the first explicit example of an entropy current defined by an apparent horizon and opens the possibility that in the near-equilibrium regime there is preferred foliation of apparent horizons for black holes in asymptotically-AdS spacetimes.
pacs:11.25.Tq, 04.50.Gh, 04.20.Gz.
On leave from: ]Sołtan Institute for Nuclear Studies, Hoża 69, 00-681 Warsaw, Poland
The AdS/CFT correspondence Maldacena:1997re; Witten:1998qj; Gubser:1998bc, or more generally gauge-gravity duality, demonstrates a deep and fascinating link between black hole physics and the plasma phase dynamics of certain (holographic) strongly coupled gauge theories. Over the last 10 years the gravitational side of the correspondence has also started to be used as a tractable theoretical model of strongly interacting non-Abelian media with properties similar to quark-gluon plasma studied first at RHIC and now also at the LHC (see CasalderreySolana:2011us for the most recent review of these developments). Early efforts in the applications of gauge-gravity duality methods to hot QCD matter were motivated by hydrodynamic simulations of the expanding fireball created in heavy ion collisions and focused on obtaining transport properties of holographic plasmas by analyzing low-lying quasinormal modes and linear response theory. These results provided concrete numerical predictions for the simplest transport coefficients of strongly coupled non-Abelian media with super Yang-Mills plasma as the primary example111For an excellent review of these early developments see Son:2007vk. and have eventually led to the formulation of fluid-gravity duality Bhattacharyya:2008jc. Fluid-gravity duality is a correspondence which maps solutions of relativistic Navier-Stokes equations describing holographic liquids to long-wavelength distortions of black branes in higher dimensional geometry. The direct connection between the dynamics of black objects in higher dimensional spacetimes and solutions of nonlinear hydrodynamics provided an opportunity to understand black brane geometries and their features in terms of dual fluids, as well as to gain insights about hydrodynamics from the properties of Einstein’s equations. These perspectives, as well as the possibility of applications, have generated significant interest in the nonlinear dynamics of black brane spacetimes.
Dynamical black holes and their characterization has also been an important research theme in mathematical relativity for the last couple of decades (see Ashtekar:2004cn; Booth:2005qc; Gourgoulhon:2005ng for useful reviews of this subject). The exact characterization of a dynamical black hole has proven to be a surprisingly thorny theoretical problem for general relativity. The standard textbook definition associates black hole interiors with regions of spacetime from which no signal can ever escape Hawking:1973uf. Thus, finding the exact extent of such a region is necessarily a teleological procedure: properly defining “ever” and “escape” means that one must examine the ultimate fate of all signals from a point before ruling whether or not that point is part of the black hole. Identifying the event horizon boundary of a causal black hole is similarly teleological. Thus, even though an event horizon is a congruence of null geodesics obeying the same rules as any other congruence, its evolution can appear to be acausal. For example the area increase of an event horizon is not directly driven by infalling matter or energy; instead the actual effect of an influx through the event horizon is a decrease in its rate of expansion.
These observations are not just mathematical curiosities. The non-local nature of the event horizon is acceptable as long as one treats it as a causal boundary removing the region containing a curvature singularity from the dynamics of the rest of spacetime222Hence guaranteeing consistency of low energy description of black holes in terms of classical gravity. and does not associate any physical characteristics with it. However, this is not the only role of the event horizon – for the last four decades, one of the most celebrated results of black hole physics has been the link between the area of the event horizon and entropy. Already in the 1960s it was established that event horizons necessarily increase in area Hawking:1973uf and this has usually been interpreted as being equivalent to the second law of thermodynamics. Thus, the apparently acausal expansion of event horizons would seem to imply a similarly acausal evolution of entropy. This leads to problems since the origin of black hole entropy needs to be sought within microscopic theories underlying gravitational interactions in the sense of the holographic principle 'tHooft:1993gx; Susskind:1994vu. For asymptotically flat or asymptotically de Sitter spacetimes such theories are not known (in principle one could imagine any of these being non-local Li:2010dr), but in the anti-de Sitter context these are local quantum field theories in a suitably understood large limit. At the superficial level it is hard to judge whether acausality of the event horizon is a real problem in the first two cases, but in the AdS/CFT context it definitely is.
The teleological nature of the event horizon is one of the main motivations for the ongoing research program to characterize black holes (quasi)locally, identifying their interiors from the presence of strong gravitational fields rather than on the basis of the causal structure of the entire spacetime. Quasilocal horizons go by such names as trapping Hayward:1993wb, isolated or dynamical horizons (the last two are both reviewed in Ashtekar:2004cn). In all cases though, these horizons can be thought of as generalizations of the classical apparent horizons Hawking:1973uf. Recall that apparent horizons are associated with foliations of spacetimes. Areas of strong gravitational field are identified with the region on each surface that is covered by trapped surfaces. The boundary of that region is an apparent horizon and it is the outermost surface for which the outgoing light front does not expand in area333It is also required that ingoing light front shrinks in area and that inside this surface there are other surfaces, such that both ingoing and outgoing light fronts emitted from them shrink in area. For a more precise definition see Section III or the review articles Booth:2005qc; Ashtekar:2004cn.. With a slight abuse of terminology the union of such surfaces over all time slices is often also referred to as an apparent horizon and it is in this sense that it will be used it here.
In the context of gauge-gravity duality characteristics of black holes with planar horizons in higher dimensional spacetimes are at the same time the quantities describing dual finite energy density or finite charge density states of local quantum field theories. In static situations, where the acausal nature of the event horizon plays no role, the entropy defined by the event horizon was identified with the thermodynamic entropy of a dual holographic field theory. Such entropy satisfies a very strong constraint: the first law of thermodynamics linking IR quantities (i.e. temperature and entropy density) with UV quantities (the energy density). Since the latter are well defined in the dual quantum field theories (energy density is the expectation value of one of the components of the energy-momentum tensor in thermal equilibrium), there are no controversies with associating thermodynamic entropy with the event horizon in time-independent situations. However in static situations (at least in the context of Kerr-Newman black holes/branes) the event horizon and one of apparent horizons coincide, so that by associating the entropy with the event horizon one actually associates it at the same time with an apparent horizon. The latter identification actually turns out to be more robust, as the example of conformal soliton flow Friess:2006kw suggests Figueras:2009iu.
Beyond equilibrium three problems arise. The first follows from the aforementioned nonlocality of the event horizon, the second comes from foliation dependence of apparent horizons, whereas the third one is related to the various ways in which one can associate points on any of horizons with points on the boundary (this is referred to as the bulk-boundary map Bhattacharyya:2008xc). The last of these is necessary to localize the entropy production in the dual field theory. To illustrate that the first issue is a serious problem quite disconnected from any ambiguities of the bulk-boundary map, one can consider the example of gravitational dynamics with a sharp distinction between a dual equilibrium regime without entropy production and a dynamical transition with dissipation. The relevant backgrounds, much in the spirit of the Vaidya solution, appeared in the context of the thermalization problem of strongly coupled non-Abelian media and describe gravitational processes in which the dual quantum field theory undergoes a transition between two equilibrium states in a finite time interval Chesler:2008hg; Chesler:2009cy. In such a situation the event horizon evolves past the bulk lightcones spanned by the transition region on the boundary. The latter patch of bulk spacetime is dual to the boundary region where the holographic field theory is in equilibrium, so that its thermodynamic entropy stays constant. This result strongly suggests that the causal boundary of a black hole is not the relevant entropy carrier regardless of any ambiguities of the bulk-boundary map Figueras:2009iu. In contrast with the area of the event horizon, the entropy defined by the unique apparent horizon respecting symmetries of 1-dimensional boundary dynamics considered in Chesler:2008hg; Chesler:2009cy was constant before the transition process and eventually in the far future agreed with the one given by the event horizon.
In less symmetric situations, apart from the choice of bulk-boundary map, the foliation dependence of apparent horizons becomes a significant issue – different foliations of spacetime lead to different apparent horizons. The most trivial and at the same time pessimistic possibility is that the notion of local entropy does not extend beyond equilibrium situations and foliation dependence, as well as the freedom of bulk-boundary mapping, signal exactly this. It might also be that on the dual field theory side there are many relevant local notions of entropy and different apparent horizons correspond to such different notions. Yet another possibility is that the field theory notion of entropy does not suffer from ambiguities of kinds introduced by foliation dependence of apparent horizons, which might be used as a guiding principle for finding preferred apparent horizon.
Resolving these issues in the general case is very difficult if not impossible, so the only hope is to proceed example by example. In global equilibrium the foliation dependence essentially vanishes and the event and apparent horizons coincide. In the near-equilibrium regime one expects the horizons to be “close” (in a sense of Nielsen:2010h or Booth:2010eu) so that a dynamical apparent horizon behaves almost like an isolated horizon. If one takes the near-equilibrium regime as point of departure for further studies, one is immediately led to considering apparent horizons in the geometry of fluid-gravity duality. This background captures the hydrodynamic regime on the field theory side starting from a locally boosted and dilated black brane supplemented with gradient corrections Bhattacharyya:2008jc and from this perspective hydrodynamics can be regarded as the simplest (because of its universality) type of collective dynamics that quantum field theory can undergo.
The generalization of entropy to hydrodynamics is provided by the notion of an entropy current. Such a current is constructed phenomenologically in the gradient expansion by requiring that in equilibrium it reproduces thermodynamic entropy and that its divergence evaluated on solutions of the equations of hydrodynamics is non-negative. A detailed analysis of the consequences of this generalized second law of thermodynamics on the form of the entropy current in Romatschke:2009im showed that even up to second order in gradients there is an ambiguity inherent in such a definition444Part of the ambiguity is trivial and comes from a term whose divergence vanishes.. From the point of view of fluid-gravity duality it was very natural to ask what is the gravity interpretation of the coefficients appearing in the boundary hydrodynamic entropy current and what might be the bulk counterpart of the ambiguity in its definition. In the pioneering work Bhattacharyya:2008xc a candidate entropy current was obtained by mapping the area theorem on the event horizon onto the boundary along ingoing null geodesics. In general, there are infinitely many directions in which such geodesics can propagate from the boundary, but hydrodynamic covariance requires that such geodesics close to the boundary move in the direction specified (at leading order of the gradient expansion) by the local fluid velocity Bhattacharyya:2008xc. Ambiguities appearing in such procedure appear at third and higher orders in the gradient expansion and were irrelevant in the second order construction of Bhattacharyya:2008xc; Romatschke:2009im. This causal bulk-boundary map can be supplemented with suitably understood boundary diffeomorphisms and the latter turn out to capture precisely the ambiguity discussed in Romatschke:2009im.
Furthermore, in a recent paper Booth:2009ct it was shown that for a fixed bulk-boundary map the same freedom in entropy current can be understood as coming from different bulk hypersurfaces with a fixed foliation satisfying a generalized area theorem and asymptoting to the event horizon. Such surfaces were dubbed “generalized horizons” with the event horizon and the (asymptoting to it) apparent horizon being just two particular instances of the more general notion. From the perspective of the phenomenological definition of the hydrodynamic entropy current none of these hypersurfaces and none of the available bulk-boundary maps is favored over any other. However, causality of the boundary field theory seems to favor the entropy current dual to the apparent horizon – providing that it is free of the ambiguities related to foliation dependence and that the bulk-boundary map in use is causal. The task of this paper is to elaborate on the proposal Booth:2010kr by presenting a derivation of the apparent horizon in the geometry of fluid-gravity duality, its features, as well as discussing the properties of the dual entropy current.
The organization of the paper is the following. Section II presents in a self-contained fashion the geometry dual to conformal fluid dynamics in arbitrary dimensions obtained in Bhattacharyya:2008mz. Section III is the main part of the paper and provides the detailed calculation of the relevant apparent horizon in the case of conformal fluid-gravity duality up to second order in gradients. Section IV focuses on the hydrodynamic side of fluid-gravity duality and analyzes the dual entropy current using the technology introduced in Booth:2010kr. The general discussions of the results and possible future directions of research are provided in Section V. Appendix A provides some details on the Weyl-covariant derivative and Weyl-covariant hydrodynamic tensors, whereas Appendix B illustrates the methods developed in Section III by describing the construction of an apparent horizon in the Vaidya spacetime. Readers interested mostly in the general-relativistic aspects of these considerations can skip Section IV and regard the paper as an example of a perturbative calculation of an apparent horizon in a geometry governed by Einstein’s equations with negative cosmological constant.
Ii The geometry of fluid-gravity duality
The geometry of fluid-gravity duality in arbitrary dimensions Bhattacharyya:2008mz is a solution to Einstein gravity with negative cosmological constant
where is the -dimensional Newton’s constant and the AdS radius is set to 1. The action (1) arises in the context of string theory (for and , see Haack:2008cp for details) and describes a sector of decoupled dynamics of the one-point function of the energy-momentum tensor operator in planar strongly coupled holographic conformal field theories Bhattacharyya:2008mz. The equations of motion derived from (1) support a -parameter family of exact, static black hole solutions with planar horizons obtained by boosting and dilating the AdS-Schwarzschild black brane solution
Here denotes components of the flat Minkowski metric on the conformal boundary of the asymptotically AdS spacetime (2). The boost parameter is a -component velocity in the directions, normalized so that in the sense of the boundary metric . The lines of constant in (2) are ingoing null geodesic, for large propagating in the direction set by , and the radial coordinate parametrizes them in an affine way Bhattacharyya:2008xc. The geometry (2) may be regarded as a stack of constant- -dimensional planes, starting from the boundary at (which is -dimensional Minkowski spacetime) right down to the curvature singularity at . The latter is shielded by the event horizon at , which is at the same time an (isolated) apparent horizon. The dilation parameter appearing in (2) is related to the Hawking temperature of the event horizon by
Unlike black holes in asymptotically flat spacetime, the metric (2) supports perturbations varying much slower within the transverse planes than within the radial direction. The parameter controlling the scale of variations in the radial direction is .
If , and are allowed to vary slowly compared to the scale set by , the metric (2) should be an approximate solution of nonlinear Einstein’s equations with corrections organized in an expansion in the number of gradients in the directions. Direct calculations Bhattacharyya:2008jc have shown that proceeding in this way is a systematic way of solving Einstein’s equations, provided that the dual energy-momentum tensor Bianchi:2001kw; deHaro:2000xn depending on and is conserved. For the metric (2) the dual energy-momentum tensor is that of a relativistic perfect fluid with a conformal equation of state
is the projector operator onto the space transverse to ,
and is some weakly curved metric in which the fluid lives.
Both the metric (2) and the energy-momentum tensor (4) receive gradient corrections. The separation of scales mentioned earlier implies that corrections to the metric (2) will be tensorial quantities made of -derivatives of , and with scalar functions of and as coefficients. The relevance of these terms is suppressed by the number of gradients and for practical reasons the expansion is terminated at the 2-derivative level. The tensorial quantities in question are scalars , transverse () vectors and transverse () traceless symmetric rank 2 tensors 555It needs to be borne in mind, that all these quantities are also required to be independent when evaluated on lower order solutions of hydrodynamics.. A priori one should consider all possible terms, as was done originally in Bhattacharyya:2008jc. This task can however be greatly simplified by utilizing the underlying conformal symmetry and seeking Weyl-invariant solutions of Einstein’s equations, i.e. solutions invariant under simultaneous rescalings of
where depends on the coordinates Bhattacharyya:2008mz. The leading order metric (2) is Weyl-invariant, but due to the presence of it does not retain its form at higher orders. It can however be written in a manifestly Weyl-invariant form upon introducing a vector field defined by Loganayagam:2008is
This quantity is of order one in the gradient expansion and transforms as a connection under Weyl-transformations
The Weyl-invariant form of the metric (2) reads
This metric is a leading order approximation to a spacetime whose metric is of the form
with the condition completely fixing the gauge freedom. The subleading corrections to (10) need to be Weyl-invariant and the simplest way to construct them is by summing individual Weyl-invariant contributions order by order in the gradient expansion. A single Weyl-invariant contribution to (11) can be represented as a scalar function of the Weyl-invariant combination multiplying a Weyl-covariant (i.e. transforming homogeneously under Weyl transformations of , and , see Appendix A) tensor of a given weight supplemented with a factor of .
A powerful tool in generating Weyl-covariant gradient terms is the Weyl-covariant derivative , which uses the connection (8) to compensate for derivatives of the Weyl factor coming from derivatives of Weyl-covariant tensors. It has the property that a Weyl-covariant derivative of a Weyl-covariant expression is itself Weyl-covariant with the same weight (see Appendix A or the original publications Loganayagam:2008is; Bhattacharyya:2008xc; Bhattacharyya:2008mz for details).
At first order in gradients there is only a single Weyl-covariant term available, which is the shear tensor of the fluid . It reads
and transforms with Weyl-weight . At second order in gradients, there are in total 10 Weyl-covariant terms: 3 scalars, 2 transverse vectors and 5 transverse traceless symmetric rank 2 tensors. For convenience these objects can be defined with appropriate powers of to render them Weyl-invariant. The scalar contributions read
where is the vorticity of the flow and is the Weyl-covariant curvature tensor and curvature scalar (see Appendix A for details). The Weyl-invariant transverse vectors are
Finally, the Weyl-invariant tensors read
where is a Weyl-covariantized curvature tensor (consult Appendix A for its detailed form).
The quantities are functions of introduced in Bhattacharyya:2008mz and read
The metric given above is a solution of Einstein equations with negative cosmological constant up to second order in gradients, provided that and satisfy the equations of dual hydrodynamics, i.e. the equations of covariant conservation of the energy-momentum tensor obtained from (16) by holographic renormalization
The geometric picture emerging is that of spacetime locally approximated by tubes of uniform black branes spanned along ingoing null geodesics given by lines of constant . The dilation and boost parameters and , as well as the boundary metric vary from tube to tube, but, as anticipated, the scales of these variations are small compared to variations of the bulk metric along the radial null direction Bhattacharyya:2008xc. Due to this tubewise approximation, the leading order geometry of fluid-gravity duality inherits the causal structure of static black brane, i.e. the event horizon located at , now with depending on Bhattacharyya:2008xc. It is to be expected, and is confirmed by direct calculation further in the text, that the event horizon of (2) with slowly varying , and is at the same time an apparent horizon. Such an apparent horizon is called isolated and does not lead to entropy production. The isolated apparent horizon at is expected to become dynamical once corrections to (2) are included and its position will also be modified. The dynamics of this almost isolated apparent horizon can be described in a gradient expansion much in the spirit of the framework of slowly evolving horizons Booth:2003ji; Booth:2006bn; Kavanagh:2006qe.
Iii Locating the apparent horizon in the geometry of fluid-gravity duality
This section is devoted to identifying an apparent horizon for the spacetimes defined by the metric (11). This search will be based on two criteria: 1) from the isolated horizon contained in the unperturbed geometry (2) it is natural to expect the apparent horizon to be a perturbation of the hypersurface and 2) to ensure compatibility with the dual conformal fluid solution those perturbations are required to be manifestly Weyl-invariant.
Apparent horizons are defined in terms of trapped and marginally trapped surfaces. In both cases the term “surface” means a codimension-two hypersurface embedded in a larger spacetime. The normal space to such a surface is spanned at any point by a pair of null vectors and . The following considerations apply to spacetimes where it makes sense to specify that both of these are future-oriented and respectively outwards and inwards pointing. It is convenient to cross-normalize them so that (this leaves a degree of scaling freedom).
The induced metric on can be written as
while the outward and inward null expansions of are
or, more generally, for an arbitrary normal vector
Now is said to be outer trapped if , trapped if and and untrapped if and . It is outer marginally trapped if and marginally trapped if and . Trapped surfaces are indicative of black hole regions, with well-known theorems linking them to both singularities and the existence of event horizons Hawking:1973uf. As recalled in the introduction they are also used to define apparent horizons Hawking:1973uf. Given a foliation of spacetime into spacelike hypersurfaces (“instants” of time) one can define the total trapped region on each as the union of all the outer trapped surfaces. Then (up to some technicalities which will be ignored here) the boundary of each of those regions is outer marginally trapped and known as the apparent horizon. A common abuse of terminology (adopted in the following) also uses the term apparent horizon to refer to the hypersurface defined by the evolving (that is the union of the ).
In practical calculations, this definition of an apparent horizon is not very usable and instead one just searches directly for hypersurfaces foliated by outer marginally trapped surfaces. This is common practice in numerical relativity (see, for example, Gourgoulhon:2005ng and references therein). More generally, the teleological nature of classical black holes and their event horizons has lead many to search for a (quasi)local and properly causal replacement. Horizons foliated by (outer) marginally trapped surfaces which (hopefully) bound regions of trapped surfaces are the most obvious and mathematically tractable candidates.
For example, the boundaries of stationary black holes (or branes) are taken to be weakly isolated horizons: codimension-one hypersurfaces that are foliated by outer marginally trapped surfaces or isolated horizons if their extrinsic geometry is also invariant (see for example the discussions in Ashtekar:2000hw; Ashtekar:2001jb or review articles such as Ashtekar:2004cn; Booth:2005qc; Gourgoulhon:2005ng). These are closely related (though more general than) Killing horizons and under many circumstances do a good job of characterizing a stationary black hole boundary without reference to causal structure or infinities. This is particularly so if one adds extra conditions to ensure that there are fully trapped surfaces “just inside” the horizon. For Hayward’s Hayward:1993wb future outer trapping horizons (FOTHs) one assumes that
That is, the inward expansion is negative and under a small inwards deformation the outward expansion also becomes negative. The black branes considered in this paper are examples of FOTHs.
For the classical definition, it is clear that time-evolved apparent horizons are foliation dependent: different foliations will sample a different set of trapped surfaces and so give rise to a different “time-evolved” horizon. Alternatively, focusing on the time-evolved horizon itself, it can be shown (see, for example Ashtekar:2005ez; Booth:2006bn) that a hypersurface foliated by outer marginally trapped surfaces is not rigid and may be deformed while maintaining its properties. The non-uniqueness of apparent horizons has been explicitly demonstrated in several papers Schnetter:2006; Nielsen:2010.
iii.2 Finding the horizon: strategy
Problems with uniqueness are somewhat alleviated in the present calculation by the xxrequirement that perturbations of the horizon be manifestly Weyl covariant. Then, the time-evolved apparent horizon should be specified as the level set of a scalar function
where is a Weyl-invariant scalar defined by
where denotes a linear combination of all Weyl-invariant scalars at order in the gradient expansion. There are no Weyl-invariant scalars at order 1, and 3 at order 2, so one expects to find
where the are the 3 independent Weyl-invariant scalars (13) and the constants will be determined by solving and the conditions (30). Once this is done, the expression for the position of the apparent horizon will take the form
This is a strong constraint, but a reasonable one to impose in a perturbative regime where physical considerations suggest that the horizon should be given by a Weyl covariant structure. Testing these surfaces as potential horizons means that one must consider their possible foliations and find out whether any of them satisfy and the conditions (30). Again however, one can lean on the Weyl covariance to simplify the calculation. Specifically, the outer marginally trapped surfaces of will have their own (in ) normal . This vector is required to be expressible as a sum of Weyl invariant terms and further that it be surface forming
Though this only really needs to apply on the horizon itself, it turns out to be computationally much easier to check this condition for specified not only on the putative horizon but also in some neighbourhood. Thus, in practice one should look for Weyl-covariant one-form fields that are surface forming in some neighbourhood of .
Thus the search domain will not be arbitrarily large, but rather be restricted to potential horizons and foliations that are essentially Weyl-covariant perturbations of the unperturbed boosted black brane solution (2). Marginally outer trapped surfaces are to be sought among intersections of these classes. It will be shown in the following that for the geometry of fluid-gravity duality, up to second order in the gradient expansion the conditions and (35) determine (as well as the in (34)) uniquely.
iii.3 Finding the horizon: hypersurfaces and intersections
The program outlined above can be implemented as follows. The normal covector to a surface of the form (31) is
which up to second order in the gradient expansion is
The function does not contribute above, since the leading term involves , which is of third order in gradients. It is convenient to write the normal in terms of the Weyl-covariant derivative, which acting on (Weyl weight ) is
One then has666Recall now that the geometry found in Bhattacharyya:2008mz satisfies Einstein equations provided that the equations of hydrodynamics are satisfied by the quantities , (in terms of which also is expressed). These equations imply Bhattacharyya:2008mz that is of the second order in gradients.
Raising the index using the metric (11) one gets (using the formula for the inverse given in Bhattacharyya:2008mz), up to second order
Next one must consider potential foliations of . As noted earlier, the foliation of the apparent horizon can be specified by a vector field which is tangent to and but otherwise normal to the leaves. Given a parametrization of the horizon so that , this means that
In terms of this coordinate system the tangent vectors are, of course, which push-forward into the full space time as
and applying (42) one finds
By construction these vectors all satisfy (as they should).
It is very convenient to choose the coordinates on the horizon . This should be reasonable as long as the horizon does not “fold over”; this should be the case in this perturbative, gradient expansion limit. This choice also has the advantage of making the bulk-boundary map trivial (as discussed in Section IV). Then the tangent vectors to can be written as
and a general vector field tangent to the horizon is given by
In terms of the coordinate basis in the bulk one then has
Requiring that the vector be Weyl covariant fixes (up to second order)
where , , , and are some constants. It is computationally convenient to normalize so that
in which case the coefficients of the longitudinal terms in (48) vanish
The remaining coefficients () appearing in are also not arbitrary. As discussed earlier, to ensure that the vector defines a foliation one has to impose the Frobenius condition (35). There are two types of terms, which turn out to be given by
up to terms of higher order in the gradient expansion. This determines the coefficients and
This way one finds that the foliation vector is completely determined once is fixed
Since the Frobenius condition was imposed for the full spacetime (rather than just on the horizon), this vector actually gives rise to foliation of the full spacetime, at least in a neighborhood of the horizon.
It is interesting that one gets a unique result. It seems plausible that this will also be the case at higher orders in the gradient expansion. To see this, note that at a given order , is entirely specified in terms of its components, and its component does not depend on the -th order contribution to . In complete analogy with the second order, at order will be a linear combination of all available transverse and longitudinal vectors. The hypersurface is specified as the level set of a scalar function (see (31)), which at order contains all the available hydrodynamic scalars of order . The vector normal to is defined by , so the construction does not introduce any further coefficients to be determined. Now consider the normalization condition (50) and expand the contribution
In order to evaluate the -th order contribution to (55) from it is sufficient to take the zeroth order metric. Note however that since at leading order is proportional to and is transverse, the first term on the left hand side of (55) vanishes for all . Since does not receive corrections from the -th order , the only term which depends on this is actually . But since is also proportional to at leading order, the whole left hand side of (55) at order depends only on the longitudinal contributions to at this order. If so, formula (55) fixes them uniquely and does not constrain the transverse contributions. What is left at order are scalar contributions to and transverse contributions to . But at a given order, transverse and longitudinal quantities are independent, so the scalar condition at order fixes all the contributions to . The transverse components of , relevant for the foliation of , are likely to be fixed by the Frobenius condition (35) in analogy with what happens at second order. It can be checked that the contributions in question will appear in the Frobenius conditions, but it seems difficult to show that by choosing the transverse parts appropriately one can satisfy Frobenius conditions at any order. One argument that this is indeed the case is that such a condition must be satisfied for on the event horizon at arbitrary order and in this case is fixed and given by . It would certainly be interesting to make these statements more precise.
Dynamical quasilocal horizons are spacelike and so should be timelike and spacelike. Without loss of generality one can assume that is future pointing and is outward pointing. Then the null normals to the surfaces of constant and are
where the scalar is called the evolution parameter Booth:2003ji; Booth:2006bn. In this case
The sign of the evolution parameter indicates whether is spacelike or timelike (or null if ). The signs of the coefficients in (III.4) have been chosen to ensure that both and are future-pointing, and is outward-pointing while is inward-pointing.
The null normals are then
As a check on the results obtained so far one can determine the location of the event horizon and compare with Bhattacharyya:2008mz. Using the explicit form of the evolution parameter (obtained from (57))
which matches the results of Bhattacharyya:2008mz.
Note that because the event horizon is null it must be the case that is proportional to . Using (III.4) and the explicit form of and , it may be checked directly that this is indeed the case.
To determine the position of the apparent horizon one needs to calculate the null expansions from the forms (28)
where the metric induced on the foliation slices is calculated from (27). Using the results of the previous section one finds (up to second order)777This computation is fairly lengthy.
Note that the results are manifestly Weyl-invariant. In particular, there is no correction at first order (as required by Weyl invariance). With these results in hand, it is straightforward to determine the location of the apparent horizon by solving . One again finds (34) with
Only differs from the result for the event horizon Bhattacharyya:2008mz.
explicitly shows that the apparent horizon lies within (or coincides with) the event horizon in the sense that an ingoing radial null geodesic will cross first the event horizon and only then the apparent horizon, since is an affine parameter on such geodesics. It is also easy to check that the apparent horizon is spacelike or null
Iv The hydrodynamic entropy current defined by the apparent horizon
In hydrodynamics the entropy current is a phenomenological notion constructed order-by-order in the gradient expansion starting from the term describing the flow of thermodynamic entropy. Subleading contributions are given as a sum of all available hydrodynamic vectors (not necessarily transverse) chosen in such a way that the divergence of the current is non-negative when evaluated on the solutions of equations of hydrodynamics. In the conformal case, up to second order in gradients, there are in total 5 available contributions consisting of the 3 hydrodynamic Weyl-invariant scalars (13) multiplied by the velocity and 2 Weyl-invariant transverse vectors (14)
The overall factor of in (70) comes from the holographic expression for thermodynamic entropy. The on-shell divergence888In the sense of conservation of the energy-momentum tensor given by (25) of the current (70) was evaluated in reference Bhattacharyya:2008mz and reads, up to third order in gradients,
As understood in Romatschke:2009kr for , this expression makes it possible to constrain some of the coefficients appearing in (70). These arguments are based on the observation that local non-negativity should hold both when the shear tensor vanishes at a given point, as well as when it is arbitrary small (if it is large enough, then dominates over other contributions and there are no further constraints). The first condition automatically implies that
whereas the second sets to zero all contributions which spoil non-negativity for very small , i.e.
Note that appears in the divergence only in the combination , so that shifting and keeping the sum constant does not change the divergence. This ambiguity comes from the freedom of modifying the entropy current by adding a multiple of the divergence-free term Bhattacharyya:2008mz; Bhattacharyya:2008xc; Booth:2010kr and does not affect the local rate of entropy production. At third order there are no further constraints available so that remains the only unspecified parameter affecting the divergence (71). Results of Bhattacharyya:2008xc; Bhattacharyya:2008mz and Booth:2010kr make it clear that is not fixed by some higher order argument – dual gravitational constructions, which all guarantee non-negativity of the divergence, lead to different values of . Thus, if the notion of local entropy production in the near-equilibrium regime makes sense, there must be some further constraints on the form of the hydrodynamic entropy current. This paper argues that one such constraint might be causality, which leads to considering the holographic entropy current based on the apparent horizon in the dual gravity description.
The problem of constructing a candidate hydrodynamic entropy current on the gravity side of the correspondence was first solved in Bhattacharyya:2008xc and then generalized to weakly curved boundary Bhattacharyya:2008ji and to arbitrary dimensions Bhattacharyya:2008mz. These articles relied on using the bulk-boundary map defined by ingoing null geodesics supplemented with boundary diffeomorphisms999The bulk-boundary map along ingoing null geodesics associates points on the apparent horizon, the event horizon or any other “generalized horizon” with boundary points lying on the same null geodesics moving close to the boundary in a direction specified by a given vector field. This vector field is taken to be proportional to in the leading order with subleading corrections modifying dual entropy current at orders higher than 2. As anticipated in section III, in the gauge (11) this bulk-boundary map acts trivially and maps points of the same position. Any such bulk-boundary map may be supplemented with boundary diffeomorphisms, which are generated by another vector field specified on the boundary. Such a vector field, if non-zero at leading order of the gradient expansion, must be also proportional to , which modifies the dual entropy current at second and higher orders. The only parameter in (70) shifted by boundary diffeomorphisms of such form is . For a detailed discussion of bulk-boundary maps see Bhattacharyya:2008xc. to map the area form of the black brane event horizon satisfying the area theorem onto a dual current of non-negative divergence. The main motivation for mapping bulk data along ingoing null geodesics was causality. Note however that such a constraint on the bulk-boundary map is self-consistent only when the bulk entropy carrier is causal101010Relaxing the assumption of causality of bulk-boundary maps has so far not been explored. Note at this point that although the mapping along ingoing null geodesics seems (at least superficially) to be causal, boundary diffeomorphisms composed with a given bulk-boundary map might lead to causality violations (see Section V for a discussion of this point).. Such a notion is provided by an apparent horizon, which along with the event horizon provides an example of a “generalized horizon” introduced in Booth:2010kr.
The geometric setup described in Section III contains a distinguished vector field tangent to the horizon . As anticipated in Booth:2010kr in the context of “generalized horizons” one motivation for introducing is that the change of the area form on the horizon sections can be written in terms of the expansion along
where is the determinant of the induced metric on the section. The generalized second law of thermodynamics is then the statement that the area of the leaves is non-decreasing under the above flow
On the apparent horizon this area law is guaranteed by , and . The boundary entropy current is obtained from by means of rewriting the left hand side of (75) within a chosen bulk-boundary map as a divergence of boundary current. This current is interpreted as a candidate boundary current and is given by Booth:2010kr
where the prefactor involving has been introduced to reproduce thermodynamic entropy at leading order and the AdS radius has been set to 1, as in (1). The technical assumptions used to derive (76) match those in Section III. In particular, the formula (76) is valid for a trivial bulk-boundary map, i.e. along null geodesics, which in the vicinity of the boundary move in the direction defined by . In the conformal case this direction can be modified only by second and higher order terms which change the entropy current at third order, and thus are beyond the scope of this article. The bulk-boundary map used here is not supplemented with boundary diffeomorphisms partly due to causality reasons (see Section V for more details). Because of this, the formula (76) leads to the unique causal second order entropy current.
As discussed earlier, the vector is completely fixed by the self-consistency of the bulk construction; the second order result (54) reads