On shape dependence of holographic entanglement entropy in AdS4/CFT3

We study the finite term of the holographic entanglement entropy of finite domains with smooth shapes and for four dimensional gravitational backgrounds. Analytic expressions depending on the unit vectors normal to the minimal area surface are obtained for both stationary and time dependent spacetimes. The special cases of AdS4, asymptotically AdS4 black holes, domain wall geometries and Vaidya-AdS backgrounds have been analysed explicitly. When the bulk spacetime is AdS4, the finite term is the Willmore energy of the minimal area surface viewed as a submanifold of the three dimensional flat Euclidean space. For the static spacetimes, some numerical checks involving spatial regions delimited by ellipses and non convex domains have been performed. In the case of AdS4, the infinite wedge has been also considered, recovering the known analytic formula for the coefficient of the logarithmic divergence.

Entanglement entropy of extended quantum systems has attracted a lot of interest during the last decades and its importance is firmly established within different areas of theoretical physics like condensed matter, quantum information and quantum gravity [1][2][3]. Given a quantum system in a state characterised by the density matrix ρ and whose Hilbert space can be written as H = H A ⊗H B , the reduced density ρ A matrix associated to H A is obtained by taking the partial trace over H B , namely ρ A = Tr B ρ. The entanglement entropy is the von Neumann entropy of ρ A , i.e. S A = −Tr A (ρ A log ρ A ). When ρ is a pure state, the entanglement entropy is a good measure of the entanglement associated to the bipartition of the Hilbert space and S A = S B . One of the most important properties of this quantity with respect to other measures of entanglement is the strong subadditivity [4,5]. Here we will consider only geometric bipartitions, i.e. cases where A is a spatial part of the whole system and B is its complement (notice that A can be the union of many disjoint regions). For a quantum field theory in a d dimensional spacetime, the spatial domain A is (d − 1) dimensional and the hypersurface ∂A = ∂B separating A and B is (d − 2) dimensional.
Among the quantum field theories, conformal field theories (CFTs) are the ones for which the entanglement entropy has been mostly studied. In general, S A can be written as a series expansion in terms of the ultraviolet cutoff ε → 0 and the leading term is S A ∝ Area(∂A)/ε d−2 + . . . , where the dots denote subleading terms. This behaviour is known as the area law of the entanglement entropy [6][7][8]. For two dimensional CFTs on the infinite line at zero temperature, when A is an interval of length the famous formula S A = (c/3) log( /ε)+const holds, where c is the central charge of the theory [9][10][11][12] (see [13] for a review). In this manuscript we will employ the holographic prescription of [14,15] to compute the entanglement entropy for quantum field theories with a gravity dual (see [16] for a review).
Extending the definition of the central charge also to non critical models, Zamolodchikov proved that the central charge decreases along a renormalization group (RG) flow going from the ultraviolet to the infrared fixed point [17]. This result can be derived also from the strong subadditivity of the entanglement entropy [18]. In higher dimensions, important results have been obtained for spherical domains [19]. In particular, in 2 + 1 dimensions, it has been found that the constant term occurring in the ε → 0 expansion of S A for a disk decreases along an RG flow (F theorem) [20][21][22][23][24][25]. Thus, in three spacetime dimensions this quantity plays a role similar to the central charge c in two dimensions.
In the context of quantum gravity, a remarkable progress in the comprehension of entanglement has been done through the AdS/CFT correspondence. An important result is the holographic formula to compute the entanglement entropy of a d dimensional CFT having a gravitational holographic dual characterised by an asymptotically AdS d+1 background. For static backgrounds it is given by [14,15] 1) where G N is the (d + 1) dimensional gravitational Newton constant and A A ≡ A[γ ε ] is the area of the (d − 1) dimensional (codimension two) hypersurfaceγ ε obtained from ∂A JHEP12(2015)037 as follows. Given the hypersurface ∂A on some constant time slice of the CFT living at the boundary of the asymptotically AdS d+1 background, one must consider all the spatial hypersurfaces γ A in the bulk such that ∂γ A = ∂A. Among these hypersurfaces, we have to find the one having minimal area, which will be denoted byγ A . Since these hypersurfaces reach the boundary of the asymptotically AdS d+1 spacetime, which is located at z = 0 in some convenient system of coordinates, their area is infinite. The regularization of this divergence is done by restricting to z ε > 0, where ε is a small quantity which coincides with the ultraviolet cutoff of the dual CFT, according to the AdS/CFT dictionary. Denoting byγ ε the restriction ofγ A to z ε, its area A[γ ε ] can be written as a series expansion for ε → 0 and the terms of this expansion can be compared with the ones occurring in the expansion of S A computed through CFT techniques. This prescription has been derived through a generalization of the usual black hole entropy in [26]. The covariant generalisation of (1.1), which allows to deal with time dependent gravitational backgrounds, has been found in [27]. In this case the formula is formally identical to (1.1) but A A is evaluated by extremizing the area functional without forcing the spatial hypersurfaces γ A to live on some constant time slice.
The formula (1.1) has passed many consistency checks (e.g. it satisfies the strong subadditivity property [28]) and nowadays it is a well established piece of information within the holographic dictionary. When the dual CFT is at finite temperature, the dual gravitational background is an asymptotically AdS d+1 black hole and (1.1) provides the corresponding holographic entanglement entropy. Let us remind that the entanglement entropy is not a measure of entanglement when the whole system is in a mixed state. It is important to remark that (1.1) holds for those regimes of the CFT parameters which are described by classical gravity through the AdS/CFT correspondence. The corrections coming from quantum effects have been discussed in [29].
The minimal area surface entering in the holographic formula (1.1) for the entanglement entropy is difficult to find analytically for domains A which are not highly symmetric because typically a partial differential equation must be solved. Numerical methods can be employed, but for non trivial domains finding a convenient parameterisation of the surface is already a non trivial task. The shape dependence of some subleading terms in the expansion of A A as ε → 0 have been studied in various papers [30][31][32][33][34][35][36][37].
The holographic formula (1.1) can be employed also when A = ∪ i A i is the union of two or more disjoint spatial domains A i . In these cases, one can construct combinations of entanglement entropies which are finite as ε → 0: the simplest case is the mutual information For two dimensional CFTs, the mutual information or its generalizations to more than two intervals encode all the CFT data of the model [38][39][40][41][42][43][44][45][46]. Some results for the mutual information are also known in 2 + 1 dimensions from the quantum field theory point of view, where the analysis is more difficult because of the non local nature of ∂A [47][48][49][50][51][52]. As for the holographic analysis for disjoint domains through (1.1), the main feature to deal with is the occurrence of two or more local extrema of the area functional [43,[53][54][55][56][57]. Thus, the holographic mutual information is zero when the two regions are distant enough (see [37] for the transition curves of domains A 1 and A 2 which are not disks).
In this paper we will consider only asymptotically AdS 4 bulk spacetimes whose boundary is the three dimensional Minkowski spacetime. Given a finite domain A delimited by a finite and smooth boundary ∂A (entangling curve), the expansion of the area of the surfacê γ ε entering in the holographic entanglement entropy (1.1) reads where P A = length(∂A) is the perimeter of the spatial region A (we set the AdS radius to one). In order to find the O(1) term F A , the whole surfaceγ A is needed. Exact analytic expressions of the F A are known only for few cases which are highly symmetric like the disk [53,74] and the annulus for AdS 4 [54,55,75]. Among the infinite domains, namely the ones elongated in one particular direction, the strip has been studied because its symmetry makes it the simplest case to address from the analytical point of view [14,15,76]. In [37] the interpolation between the disk and the elongated strip through various domains has been considered.
In this paper, we derive closed expressions for F A in terms of the unit vectors normal toγ A for both static and time dependent backgrounds. When the bulk spacetime is AdS 4 , our formula for F A becomes the Willmore energy [77][78][79][80][81] of the minimal area surfacê γ A viewed as a submanifold of R 3 , recovering the result of [82,83]. The formulas for some static backgrounds are checked numerically for regions delimited by ellipses and also for non convex domains, while for the Vaidya-AdS spacetime only disks are employed as benchmark of our results. The numerical analysis for generic entangling curves have been performed by employing Surface Evolver [84,85]. We will not consider spacetimes which are asymptotically global AdS. In these cases the homology constraint in the holographic prescription (1.1) plays a crucial role [28,[86][87][88].
The paper is organized as follows. In section 2 we find F A for generic static backgrounds which are conformally related to asymptotically flat spacetimes and then specialise the formula to the explicit examples given by AdS 4 , asymptotically AdS 4 black holes [89][90][91] and domain wall geometries [21,[92][93][94][95][96]. The latter spacetimes are simple holographic models dual to RG flows in the boundary theory. In section 3 we extend the analysis to the time dependent spacetimes, considering then the Vaidya-AdS backgrounds as special case. In section 4 some particular domains are discussed for the above backgrounds, in order to recover the known results for disks and strips and extend them through the formulas found in the previous sections. Spatial regions delimited by ellipses and also a non convex domain are considered. When the bulk geometry is AdS 4 , we also consider the infinite wedge [97], which includes also a logarithmic divergence as ε → 0 (see [98,99] for recent developments about entangling curves with corners). Some consequences for the holographic mutual information are addressed in section 5 and concluding remarks are given in section 6. In the appendices A, B, C, D and E we have collected technical details and some further discussions related to issues occurred in the main text.

JHEP12(2015)037 2 Static backgrounds
In this section we derive a formula for F A in (1.2) for static backgrounds which are conformally related to asymptotically flat spacetimes whose boundary is the four dimensional Minkowski space. The discussion for the general case is given in section 2.1, while in section 2.2 we specify the result to some explicit backgrounds: AdS 4 , asymptotically AdS 4 black holes and domain wall geometries.

General case
Let us consider the three dimensional Euclidean space M 3 obtained by taking a constant time slice of a static asymptotically AdS 4 background, namely Given a two dimensional surface γ embedded into M 3 , let us denote by n µ the spacelike unit vector normal to γ and by h µν = g µν −n µ n ν the metric induced on γ (first fundamental form). The trace of the induced metric is h µν g µν = h µν h µν = 2 and the tensor h ν µ allows to project all the other tensors on γ. The extrinsic curvature (second fundamental form) of γ embedded in M 3 is defined as where ∇ α is the torsionless covariant derivative compatible with g µν . We find it convenient to introduce also the following traceless tensor constructed through the extrinsic curvature An important identity to employ in our analysis is the following contracted Gauss-Codazzi relation [100] R − TrK where R is the Ricci scalar, which provides the intrinsic curvature of γ, and ⊥ R µνρσ = h α µ h β ν h γ ρ h λ σ R αβγλ is the Riemann tensor of g µν projected on γ. Performing explicitly the contractions, the r.h.s. of (2.4) reads where G µν is the Einstein tensor of g µν . At any given point of γ, two principal curvatures κ 1 and κ 2 can be introduced, which are the eigenvalues of the extrinsic curvature. Thus, the mean curvature is given by (κ 1 + κ 2 )/2 = TrK/2.
Many gravitational backgrounds occurring in the AdS/CFT correspondence are conformally related to asymptotically flat spacetimes (e.g. the asymptotic AdS 4 black holes and the domain wall geometries that will be introduced in section 2.2). Motivated by this fact, let us assume that the metric g µν of the background space is conformal tog µν , namely

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whereg µν defines an Euclidean asymptotically flat space M 3 and ϕ is a function of the coordinates. The surface γ can be also seen as embedded into M 3 and therefore we can define the induced metrich µν and the extrinsic curvature K µν characterising this embedding throughg µν as above. Denoting byñ µ the unit vector normal to the surface γ ⊂ M 3 , we have n µ = e ϕñ µ (and therefore n µ = e −ϕñµ ), and this implies that h µν = e 2ϕh µν . Considering the determinants (restricted to the tangent vectors) h andh of the induced metrics, we find that h = e 4ϕh . This leads us to conclude that the area elements dA = √ h dΣ and dÃ = h dΣ with dΣ = dσ 1 dσ 2 (we denoted by σ i some local coordinates) are related as dA = e 2ϕ dÃ. Being the metrics g µν andg µν conformally related, the corresponding extrinsic curvatures K µν and K µν obey the following relation From the transformation rules given above, it is not difficult to realise that the following combination is Weyl invariant By employing the Gauss-Codazzi relation (2.4), together with (2.5) to eliminate TrK 2 in (2.8), the Weyl invariance of the combination (2.8) can be recast as where the tilded quantities refer to the asymptotically flat metricg µν . In the left and right side of (2.9), the same surface γ is embedded either in M 3 or in M 3 respectively. The formulas for the change of R and G µν under a Weyl transformation are given respectively by where D µ is the covariant derivative constructed throughh µν and D 2 the corresponding Laplacian operator. By first plugging (2.10) and (2.11) into (2.9) (using also that n µ = e −ϕñµ and dA = e 2ϕ dÃ) and then integrating the resulting equation over γ, we find (2.12) Adding the area to both sides of this identity, it becomes We remark that (2.13) holds for a generic two dimensional surface embedded into the three dimensional Euclidean space given by (2.6). The first term is a total derivative and therefore it vanishes if γ is a closed surface without boundaries.

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When γ has a boundary (which could be made by many disjoint components), the first term in (2.13) is a boundary term (2.14) In our case the three dimensional metric g µν is asymptotically H 3 and ∂γ lies close to the boundary. The three dimensional Euclidean hyperbolic space H 3 is characterised by the metric ds 2 = z −2 (dz 2 + dx 2 ), where z > 0 and dx 2 is the space element of R 2 . Thus, let us consider a system of coordinates (z, x) in M 3 , where z > 0, the boundary of M 3 is given by z = 0 and x is the position vector in the z = 0 plane. The boundaries of the surfaces γ ε belong to the plane z = ε. Taking ϕ = − log(z) + O(z a ) with a > 1 when z → 0 in (2.6), we have that g µν is asymptotically H 3 whileg µν is asymptotically flat. Considering the surfaces γ ε and this behaviour for ϕ in (2.14), we need to knowb z at z = ε. In section A.1 we report the analysis of [83,101], which shows thatb z = −1 + o(ε) as ε → 0. We remark that the latter condition holds also for a surface intersecting orthogonally the plane z = 0 which is not necessarily minimal. Thus, from (2.14) we have that the area of the surfaces γ ε reads where the area law term comes from the boundary integral in (2.14) and the O(1) term is given by (2.16) Let us specialize (2.16) to the minimal area surfacesγ A entering in the computation of the holographic entanglement entropy [14,15]. For minimal area surfaces we have where the right side of the equivalence comes from (2.7). Thus (2.15) becomes where the first term of the integrand can be also written in terms of ϕ like the other ones through (2.17). The formula (2.19) is the main result of this section. It is worth remarking that it holds for any smooth entangling curve ∂A, including the ones made by many disjoint components.

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A two dimensional surface can be defined implicitly as a real constraint C = 0, being C a function of the three coordinates x µ . The unit vectorñ µ normal to this surface is obtained from this constraint as follows Since the global sign of C is unspecified, the orientation of the vectorñ µ is a matter of choice as well. Notice that this sign does not change (2.19) because only quadratic terms inñ µ occur.
In section B we briefly discuss the application of the method employed above to the higher dimensional case.

Some static backgrounds
In this manuscript we will consider three examples of static asymptotically AdS 4 metrics: AdS 4 , asymptotically AdS 4 black holes and some domain wall geometries. We will not consider geometries which are singular when z → ∞. For the backgrounds we are interested in, (2.6) holds with ϕ = − log(z). Hence, the first and the last term of the integrand in (2.19) become respectively where the first expression is obtained through (2.17) and Γ z µν in the second expression are some components of the Christoffel connection compatible withg µν . In the following we specify (2.19) to these three backgrounds.

AdS 4 : the Willmore energy
The simplest bulk geometry to study is AdS 4 , which is given by where the AdS radius has been set to one and dx 2 is infinitesimal spacetime interval of R 2 at z = 0. Comparing (2.22) with (2.1) and (2.6), we have that g µν is the metric of H 3 andg µν is the flat metric of R 3 . The latter fact leads to important simplifications in the general formulas given in section 2.1. Indeed, ∇ 2 ϕ − e 2ϕ = 0 and all the components of Γ z µν vanish. Thus, for a generic surfaces γ A the expression (2.16) reduces to [82,83] For the minimal area surfacesγ A , which satisfy the condition (2.17), it simplifies further to  Left: a minimal area surfaceγ A for AdS 4 whose boundary at z = 0 (entangling curve) is given by the red curve. Right: the closed surfaceγ (d) A embedded in R 3 , obtained fromγ A by attachingγ A (blue part) and its reflected copyγ (r) A (green part) along ∂A (red curve), which is an umbilic line forγ (d) A [82].
which can be found also by specifying (2.19) tog µν = δ µν . Notice that (2.24) does not depend on the choice of the coordinate system in the z = 0 plane but, for explicit computations, this coordinate system must be fixed in order to writeñ z and dÃ (see section A). Following [82], we find it convenient to introduce a closed surfaceγ (d) A embedded in R 3 obtained by "doubling"γ A . In particular, A is the surface with z < 0 obtained by reflecting the minimal surfaceγ A with respect to the plane z = 0. The entangling curve ∂A is a particular curve on the closed surfaceγ (d) A and in [82] it has been found that the two principal curvatures are equal on this curve (i.e. ∂A is an umbilic line). The set of closed oriented compact surfaces given byγ (d) A as A varies within the set of domains with smooth ∂A is strictly included into the set of the Riemann surfaces embedded in R 3 . Indeed, they are symmetric with respect to the z = 0 plane and their intersection with such plane is an umbilic closed curve. In figure 1 we show a minimal surfaceγ A and the corresponding closed surfaceγ (d) A (the red curve onγ (d) A along whichγ A andγ (r) A match is an umbilic line). It is worth remarking that already among the connected domains A one can find cases such thatγ (d) A has genus two or higher. 1 The formula in (2.24) tells us that F A is related to the Willmore energy ofγ A ⊂ R 3 . Given an oriented, smooth and closed two dimensional surface Σ g with genus g embedded in R 3 , the Willmore energy functional evaluated on Σ g is defined as [77][78][79][80][81] In terms of the principal curvatures κ 1 and κ 2 of the surface Σ g , the Willmore energy ( A is also a critical point of the functional (2.25) [82]. Among the large number of papers in the mathematical literature about the Willmore functional, let us mention [102][103][104][105][106][107][108][109][110].
Given (2.24) and (2.25), one concludes that, when the bulk geometry is AdS 4 , the term F A is the Willmore energy of the surfaceγ A embedded in R 3 [82]. The surfaceγ A lies in the part z 0 of R 3 and its boundary is at z = 0. Considering the closed surfaceγ (d) A ⊂ R 3 introduced above, it is straightforward to observe that where the bound of 2π is saturated only when A is a disk. Thus, the disk maximises the holographic entanglement entropy for AdS 4 among the domains having the same perimeter (the problem of finding the shape which maximises S A in higher dimensions has been addressed in [35]). We remark that the bound (2.26) applies also for A made by disjoint domains (see section 5).
It is interesting to observe that, considering a domain A and another one A obtained by rescaling A through a factor λ keeping the same shape, i.e. the same ratios of the various geometric parameters, we have that F A = F A . Indeed, the minimal surfaceγ A can be found by rescalingγ A through the same factor λ and in the integrand of (2.24) we have that z → λz, dÃ → λ 2 dÃ, whileñ z remains invariant. Thus, F A is obtained from F A through a straightforward change of variables. Since this result comes from the fact that (z, x, y) → λ(z, x, y) is an isometry of H 3 , it does not hold for the spacetimes occurring in section 2.2.2 and section 2.2.3, which do not have this isometry.
A generalisation of the Willmore energy functional (2.25) is the Helfrich energy functional [111], whose role is very important in the study of the cell membranes [112]. Considering the surfaces γ A intersecting the boundary z = 0 orthogonally, in section C we have briefly discussed the surfaceγ (H) A whose part restricted to z ε has an area given by (2.15) where the O(1) term of the expansion given in (2.23) is the Helfrich energy of γ (H) A as surface embedded in R 3 .

Black holes
The asymptotically AdS 4 charged black hole (Reissner-Nordström-AdS black hole) [89][90][91] is given by Figure 2. Minimal area surfaceγ A for a Schwarzschild-AdS black hole. The entangling curve ∂A is an ellipse with semi-major axis R 1 and semi-minor axis R 2 (the red curve is plotted at z = ε).
Here ε = 0.01 and the grey plane corresponds to the horizon at z h = 1. Only half of the surface is shown in order to highlight a section of the surface (green curve) which reaches the highest value z * < z h of the coordinate z for the whole surface. In this case z * is the intersection between the green curve and the z axis.
where M is the mass and Q is the charge of the black hole. The Hawking temperature of this black hole vanishes in the extremal case, for which the two horizons coincide and the emblacking function becomes f (z) = 1 − 4(z/z h ) 3 + 3(z/z h ) 4 . The Schwarzschild-AdS black hole corresponds to the uncharged case Q = 0 and for this geometry the horizon is z h = 1/ 3 √ M . Comparing (2.6) and (2.29), we have that ϕ = − log(z) andg µν is provided by the metric within the parenthesis in (2.29). In this case all the terms occurring in (2.19) are non trivial. In particular, we get where we recall thatñ z = f (z)ñ z . Combining these results with the expression for (Tr K) 2 in (2.21) we find that (2.19) becomes The choice of the system of coordinates in the z = 0 plane enters in the explicit expressions ofñ z and dÃ (see section A). In figure 2 we show a minimal area surfaceγ A for which the entangling curve ∂A is an ellipse and the bulk geometry is the Schwarzschild-AdS black hole. Denoting by z * the highest value of the coordinate z reached by the points ofγ A , for a static asymptotically AdS black hole we have that z * < z h , i.e. the minimal surface does not penetrate the horizon [31,113]. As first consistency check of (2.31), we observe that for f (z) = 1 identically the expression (2.24) for AdS 4 is recovered, as expected.
When the domain A is very large, we expect a minimal area surfaceγ A close to a cylindrical surfaceγ cyl A whose horizontal cross section is ∂A and having only one base at JHEP12(2015)037 Figure 3. Minimal area surfaceγ A for the domain wall geometry (2.33) with α = 2 and γ = 1. The yellow plane corresponds to z RG = 1. The entangling curve ∂A is an ellipse whose semi axis R 2 < R 1 (the red curve is plotted at z = ε and here ε = 0.01). The green curve is a section whose intersection with the z axis provides the highest value z * for the coordinate z on the surface. When the domain A is very large z * z RG and the deep IR region is probed, where the asymptotic geometry is AdS 4 with radius L IR = 1/(1 + γα).
constant z = z * z h . Hence, we expect that F A is also close to the integral in (2.31) evaluated onγ cyl A , that will be denoted by F cyl A . The latter quantity is the sum of two contributions: the integral over the base and the one over the vertical part of the cylinder, whose height is z * z h . As for the former term, whose integration domain is horizontal, we haveñ z = 1/ f (z * ) and therefore the integral turns out to be proportional to the area of A. Instead, on the vertical part ofγ cyl A we haveñ z = 0 and the corresponding integral is proportional to P A . The sum of these terms reads where the term containing Area(A) dominates when A is large. Since z * is close to the horizon, one easily finds that the leading term in (2.32) is F cyl A = − Area(A)/z 2 h + . . . (see e.g. [56,96,113]).

Domain wall geometries
Asymptotically AdS 4 static backgrounds have been introduced also to provide a holographic dual description of a RG flow of the boundary theory [92][93][94]. The holographic entanglement entropy for these geometries has been already studied in [21,24,61,95,96], mainly for the infinite strip and for the disk.
The example that we are going to consider is given by the following four dimensional bulk metric 2 where z > 0 and α > 0 to guarantee a well defined z → 0 behaviour. The background (2.33) has a crossover scale z RG separating the ultraviolet (UV) region z z RG from the infrared 2 We are grateful to Rob Myers for addressing our attention to this metric and for useful discussions about it.

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(IR) region z z RG , where the metric (2.33) asymptotes to AdS 4 with different radii. Indeed, when z/z RG 1 we easily recover AdS 4 with unit radius L UV = 1, while for z/z RG 1, by introducing the variable u/L IR = z 1+γα /(z γα RG L UV ), we get AdS 4 with radius L IR = 1/(1 + γα). The null energy condition for the four dimensional metric g M N in (2.33) specified to null vector M = (− p(z), 1, 0, 0) provides the condition p[z p +p ]−z(p ) 2 0, which tells us that γ > 0, once the explicit expression for p(z) in (2.33) is substituted. Thus, since γα > 0, we have that L IR < L UV . Plotting the Ricci scalar of (2.33) normalized by its value at large z/z RG in terms of z/z RG , one observes that the smooth transition between the two asymptotic AdS 4 is faster as α increases for a given γ.
The metric (2.33) can be written also as [24,96]), where ζ = z p(z) and P (ζ) = (1 + z p (z)/[2p(z)]) 2 . In terms of this holographic coordinate the above null energy condition becomes simply P (ζ) 0. Notice that we cannot write z = z(ζ) analytically for generic values of the parameters.
Denoting by z * the highest value of the coordinate z for the minimal area surfaceγ A , we have thatγ A probes the UV regime when z * z RG and the IR regime when z * z RG . As for the term F A of the holographic entanglement entropy given in (2.19) for this gravitational background, by specifying (2.21) for the metric (2.33) we find Notice that a coordinate system must be chosen to evaluate Γ z µν = − 1 2 ∂ zgµν and to implement the normalisation condition for the vectorñ z . Nevertheless, the expressions we give here hold for both cartesian and polar coordinate systems in the z = 0 plane. By employing (2.34), the formula (2.19) for (2.33) becomes Let us restrict to α > 1 to guarantee the finiteness of (2.35). When p(z) = 1 identically (2.35) reduces to (2.24) for AdS 4 , as expected. In figure 3 we show a minimal surfacê γ A whose entangling curve ∂A is an ellipse (the same one of figure 2) and for which the bulk spacetime is the domain wall geometry (2.33). The parameters of the ellipse and the scale z RG are such that z * > z RG .

Time dependent backgrounds
The holographic entanglement entropy can be computed also for asymptotically AdS time dependent backgrounds by employing the prescription given in [27]. In these cases, the area functional to extremize must be evaluated on a class of two dimensional surfaces γ A (i.e. such that ∂γ A = ∂A) which is larger than the one occurring in the static case. Indeed, the covariance of the proposal removes the restriction to the constant time slice, that is natural in the static case. Thus, for the time dependent backgrounds the surfaces γ A to consider in the extremization process are embedded into the whole four dimensional Lorentzian spacetime.
In this section we extend the analysis performed in section 2.1 to four dimensional time dependent bulk spacetimes.

General case
Consider a two dimensional spacelike surface γ embedded in a four dimensional Lorentzian spacetime M 4 characterized by the metric g M N . Given two unit vectors n (i) (with i ∈ {1, 2}) normal to γ and orthogonal between them, the induced metric on γ reads For each unit normal vector n (i) , we can compute the corresponding extrinsic curvature and the associated traceless combination, which are respectively 2}. In this case we need to consider the following Gauss-Codazzi equation [100] h and, following the analysis done in section 2.1 for the static case, let us take the contraction given by By employing (3.1), the r.h.s. of (3.4) can be expanded in terms of the orthogonal vectors n (1) e n (2) , finding where, in order to avoid a proliferation of indices, we have adopted the notation such that a scalar quantity with parenthesis stands for the contraction of the corresponding tensor with the vectors within the parenthesis in the specified order. Let us rewrite the r.h.s. of (3.5) by replacing the contraction involving the Riemann tensor with the same contraction of the Weyl tensor according to the following formula 3 We recall that, for a q 4 dimensional spacetime (in our case q = 4), the Weyl tensor is defined as [114]

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The reason to prefer the Weyl tensor to the Riemann tensor in our analysis is that the former one changes in a nice way under conformal transformations [114]. Thus, (3.5) becomes where we have also employed the definition of the Einstein tensor G M N of the metric g M N .
In order to follow the procedure discussed in section 2.1 for the static case, we need to construct a Weyl invariant expression suggested by the contracted Gauss-Codazzi equation (3.4). From (2.7), we have that Tr(K (i) ) 2 dA is Weyl invariant. Hence, in this case we need to consider where in the last step we have eliminated the i i Tr( K (i) ) 2 by means of the contracted Gauss-Codazzi equation (3.4). By employing (3.7), the Weyl invariant expression in (3.8b) can be written as follows Let us first write explicitly the Weyl invariance of (3.9) and then integrate the resulting equation on a surface γ. Given the transformation property of the Weyl tensor, the two terms containing it cancel in the equation provided by the Weyl invariance of (3.9). Then, we need the following transformation rules for the four dimensional Ricci scalar and Einstein tensor respectively where D M is covariant derivative compatible withg M N . By employing (2.10), (3.10) and (3.11) into the equation for the Weyl invariance of (3.9), one finds that At this point, one adds the area A[γ] to both sides of (3.12). Then, by specialising the resulting expression to the class of surfaces given by γ ε and using the divergence theorem

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(see also [83]) we find again the expansion We find it useful to give F A also in terms of the energy-momentum tensor T M N of the bulk metric g M N . By employing the traceless tensors K where the l.h.s. is Weyl invariant, once multiplied by the area element dA. The Einstein equations with negative cosmological constant for the bulk metric g M N relate its Einstein tensor and the corresponding energy-momentum tensor as follows where we have absorbed the factor 8πG N into the definition of the bulk energy-momentum tensor. Taking the proper contractions of the Einstein equations (3.15), one finds that the combination involving the Einstein tensor and the Ricci scalar occurring in the r.h.s. of (3.14) can be written as where T is the trace of the energy-momentum tensor. By plugging (3.16) into (3.14), integrating the resulting expression on a surface γ and then exploiting the Weyl invariance of the terms coming from the l.h.s. of (3.14), we find where A[γ] originates from the first term in the r.h.s. of (3.16).

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When γ has a boundary, the Gauss-Bonnet theorem allows us to simplify the first two terms in the r.h.s. of (3.17) as follows where in the last step we have employed the transformation law for the geodesic curvature under Weyl transformations, which reads Restricting our analysis to the class of surfaces given by γ ε , we can easily adapt to the time dependent case the steps followed in the static backgrounds to obtain (2.15) from (2.13), as done also above to write (3.13). The final result is (2.15) with the O(1) term given by By using (3.16), (3.10) and (3.11), it is not difficult to check that (3.13) is recovered from (3.20).
It is worth recalling that (3.13) and (3.20) hold for a generic surface γ A ending orthogonally on the boundary at z = 0. For a given domain A, the extremal area surfaceγ A is the solution of the following equations where the second expression comes from (2.7) properly adapted to the case we are considering. Specifying (3.13) and (3.20) to extremal area surfaces we find respectively In explicit computations, the vectors n (i) must be chosen. Taking n (1) timelike and n (2) spacelike, i.e. 1 = −1 and 2 = 1, the sums in (3.13) and (3.20) become differences of two terms. Further simplifications occur if the following lightlike vectors are introduced Indeed, T (n (1) , n (1) )−T (n (2) , n (2) ) /2 = T ( (−) , (+) ) and a similar expression holds for the terms involving the Einstein tensor. By employing that K (±) M N / √ 2 are the extrinsic curvatures defined through the null vectors in (3.23), one finds that (3.22) becomes (3.24) In order to check the consistency of (3.13), let us recover the formula (2.16) for static backgrounds.
A generic static asymptotically AdS 4 spacetime is given by 25) where N and g µν are functions of the space coordinates x µ = (z, x), being x the position vector in the z = 0 plane. The three dimensional Euclidean metric g µν is conformally related tog µν as in (2.6). In this case, the timelike and spacelike unit vectors mentioned above are n A direct computation tells us that K M N = 0 identically, which implies that the minimality equation for n (1) M is trivially satisfied. Since ϕ is independent of time, we have K As for the Laplacian term, notice that D 2 ϕ specified to the static metric (3.25) provides ∇ 2 ϕ plus an extra term which is canceled by the remaining term containing n (1)M , namely The spacelike vectorñ (2) provides all the other terms in (2.16). Indeed, the terms n (2)M n (2)N D M ϕ D N ϕ and n (2)M n (2)N D M D N ϕ in (3.13) for the static metric (3.25) become (ñ λ ∂ λ ϕ) 2 andñ µñν ∇ µ ∇ ν ϕ respectively. Finally, it is immediate to see that in K (2) M N only the spatial part K (2) µν is non vanishing and therefore (TrK (2) ) 2 reduces to (TrK) 2 (the same observation holds for K (2) M N ).

Vaidya-AdS backgrounds
In order to test the result of the section section 3.1, let us consider the dynamical background given by the Vaidya-AdS metric [58,59]. In Poincaré coordinates, it reads where v is the outgoing Eddington-Filkenstein coordinate which becomes the time coordinate t of the boundary theory at z = 0. The metric (3.27) is a solution of the Einstein equations (3.15) with an energy-momentum tensor T M N having only one non vanishing component The holographic entanglement entropy for the Vaidya-AdS backgrounds (3.27) must be computed through the covariant prescription of [27]. The result depends also on the boundary time coordinate t. Keeping the entangling curve ∂A fixed, the expansion ( In order to specify the result of section 3 for F A to this case, we find it more convenient to consider the expression (3.24). Since g vv = 0, the trace of the energy-momentum tensor vanishes. Moreover, R = 6zM (v), while for the Einstein tensor we need to choose a coordinate system in the z = 0 plane. For example, the only non vanishing components of in polar coordinates. In order to simplify the term of (3.24) containing the extrinsic curvatures, it is useful to employ first the extremal surface conditions (3.21) and then the vectors (3.23). The final result reads where the parameter v 0 determines the steepness of the transition between the two asymptotic regimes of AdS 4 (when v → −∞) and Schwarzschild-AdS 4 black hole with mass M (when v → +∞). Indeed, it parameterises the thickness of the shell falling along v = 0. The holographic entanglement entropy for the Vaidya-AdS background (3.27) with the mass profile (3.30) has been largely studied during the last years (see e.g. [60][61][62][63][64][65][66][67][68][69][70][71][72][73]). In section 4.3.3, considering circular domains A, we check numerically that (3.29) reproduces the same results already found by subtracting the most divergent term from the area of the extremal surface (see figure 9). It would be interesting to find some analytic result from (3.29) in the thin shell limit v 0 → 0, along the lines of [62,63,69,70].

Some particular domains
After a brief explanation of the numerical methods employed in this manuscript, in this section we test the formulas for F A given above by first considering some simple cases of simply connected domains A which have been largely studied in the literature: the infinite strip and the disk. Then, we extend the numerical analysis to the case of the elliptical entangling curves. These domains belong to a large class of spatial regions A such that the corresponding minimal surfaceγ A can be parameterised by z = z(x), where x ∈ A. In

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this section we will also study F A for domains which do not belong to this class, since on the correspondingγ A one can find pairs of distinct points having the same projection on the z = 0 plane.

Numerical methods
The crucial numerical tool employed in this manuscript to study minimal surfacesγ A for finite domains A different from disks is Surface Evolver [84,85], a multipurpose shape optimization program created by Ken Brakke [84] to address generic problems on energy minimizing surfaces. In the context of AdS/CFT, it has been first employed in [37] to get some numerical results about the shape dependence of the holographic mutual information in AdS 4 . Here we extend its application to other backgrounds. In Surface Evolver, a surface is implemented as a union of triangles (see e.g. figures 1, 2, 3 and 10). Given the background metric g µν , the boundary curve ∂A in the plane z = ε and an initial trial triangulated surface, the program evolves the surface towards a local minimum of the area functional by employing a gradient descent method (see the appendix B of [37] for a very brief discussion). The final stage of the evolution is a triangulated surface close toγ A . The approximation improves as the number of triangles increases. For any triangulated surface, one can read off both the area of the whole surface and all the unit normal vectors. Let us denote byγ SE A the best approximation of the minimal surfacê γ A found with Surface Evolver. Given the corresponding area A SE and unit vectors n SE µ , we can numerically compute the following two quantities whereF SE A is obtained from the analytic expression (2.19) evaluated on the triangulated surfaceγ SE A through its unit normal vectorñ SE µ . Both these expressions are finite in the limit ε → 0. Verifying that both the quantities in (4.1) give the same values provides a strong check of the analytic formula (2.19). Indeed, from the series expansion of the holographic entanglement entropy, we expect that | F SE A −F SE A | = o(1) as ε → 0. Examples will be provided involving both the black hole and the domain wall geometry introduced in section 2.2 (see [37] for AdS 4 ).
We perform the numerical analysis through Surface Evolver whenever the partial differential equation definingγ A cannot be simplified (e.g. for the elliptic domains in figures 4, 5, 6, 7 and for the non convex domains of figure 11). For highly symmetric regions A, the corresponding minimal area equation simplifies to an ordinary differential equation in one variable. This happens for the infinite strip (section 4.2), the disk (section 4.3) and the annulus (section E). For these domains, more standard softwares (e.g. Mathematica) can be employed to study numerically the corresponding ordinary differential equations.

Strip
When A is an elongated strip with sides having lengths and L with L, it is convenient to adopt cartesian coordinates {z, x, y} which can be always chosen such that A = {(x, y) ; |x| /2 , |y| L/2}. Since L , we can assume that z = z(x) and therefore z y = 0 andñ y = 0. Moreover, the symmetry of the domain with respect to the the y JHEP12(2015)037 allows us to consider 0 x /2 only. In [37] a numerical analysis through Surface Evolver has been done where the elongated strip is approximated through various smooth domains.

Black holes
Let us first address the case of the black holes characterised by the metric (2.29), which includes AdS 4 as special case when f (z) = 1 identically.
The area functional evaluated for the class of surfaces given by γ ε reads [14,15] A[γ ε ] = 2Lˆ where f (z) is the emblacking factor in (2.29) and the parameter ω is defined by z( /2−ω) = ε. Since the integrand of (4.2) does not depend on x explicitly, we can simplify the problem of finding the extremum of (4.2) by writing the following first integral being z * the highest value reached from the minimal surface along the holographic direction.
The expression (4.3) is a first order ordinary differential equation and therefore much easier to solve with respect to the equation of motion coming from (4.2). By isolating z x in (4.3) (we recall that z x < 0), the first order differential equation becomes which can be solved through separation of the variables, getting the relation between and z * , namely dz . (4.5) As for the finite term F A of the holographic entanglement entropy for the strip in this black hole background, it is obtained simply by specifying (2.31) to this case. By using (A.13) and (A.16) for the vectorñ z and the area element dÃ respectively, one finds where (4.6b) and (4.6c) has been obtained by employing (4.3) and (4.4) respectively. Thus F A /L is a complicated function of that could be found by first performing the integral (4.6c) explicitly and then by finding z * ( ) from (4.5).

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A major simplification occurs for AdS 4 . Indeed, when f (z) = 1 identically the integrand in (4.6b) becomes equal to 1. Moreover, also the integral (4.5) can be performed explicitly in this case. Thus, for AdS 4 we have that which is the result of [15,116,117].

Domain wall geometries
When the bulk geometry is (2.33) and the domain A in the boundary is the elongated strip described above, the area functional for the class of surfaces γ ε reads where ω has been already introduced below (4.2). Since the metric (2.33) on a constant time slice can be written like a black hole metric at t = const with a proper f (z), one could employ the results of section 4.2.1. Nevertheless, we find instructive to provide explicitly the analysis also in this coordinates.
Since the integrand in (4.8) does not depend explicitly on x, we can write the following conserved quantity which allows us to find z x (we recall that z x < 0) (4.10) By separating the variables in this first order differential equation, one finds the relation between and z * 2 =ˆz * 0 z 2 p(z) The finite term F A in the holographic entanglement entropy for these domain wall geometries with A given by the elongated strip is obtained by specializing (2.35) to this domain. By employing (A.18) and (A.21) for the vectorñ z and the area element dÃ respectively, one gets where (4.12b) and (4.12c) have been found through (4.9) and (4.10) respectively. From (4.12b) it is straightforward to check that the AdS 4 result for F A in (4.7) is recovered when p(z) = 1 identically.

Vaidya-AdS backgrounds
Let us consider the elongated strip and the gravitational background in the bulk given by the Vaidya-AdS metric (3.27). Choosing the cartesian coordinate system in the boundary as explained in the beginning of section 4.2, the profile of the surfaces γ A can be described by the two functions z(x) and v(x). In this case, the area functional to extremize reads In order to apply the formula (3.29), we need the vectors and the area element discussed in section A.3. Considering the explicit expression f (v, z) = 1 − M (v)z 3 , the formula (3.29) becomes (4.14) where M = M (v). As consistency check of (4.14), we notice that (4.6a) can be recovered in the special case of M (v) constant.

Disk
When A is a disk of radius R, it is convenient to adopt the cylindrical coordinates {z, ρ, θ}, with the origin of the polar coordinates {ρ, θ} in the z = 0 plane given by the center of the disk. The symmetry of the domain tells us that z = z(ρ). This means that z θ = 0 andñ θ = 0. The disk is more complicated than the strip considered in section 4.2 because the coordinate ρ is not cyclic and therefore the ordinary differential equation to study is a second order one.

Black holes
Let us consider the black hole metric (2.29) at constant time slice with polar coordinates {ρ, θ} in the z = 0 plane. Given the ansatz z = z(ρ), the area functional for the surfaces γ ε reads where ω is defined by the condition z(R − ω) = ε. As already remarked, in this case the integrand explicitly depends on ρ and therefore we cannot write a first integral as done for the strip. The equation of motion is a second order ordinary differential equation and its analytic solution is not known for a non trivial f (z). As for the finite term F A in the holographic entanglement entropy, by employing the proper expressions in (A.13) and (A.16) for the vectorñ z and the area element dÃ respectively, (2.31) becomes This expression holds for both the Schwarzschild-AdS black hole and the charged black hole. It can be employed only once the solution z(ρ) of the extremal area equation is known. In (4.16) the profile z(ρ) satisfies the boundary condition z(R) = 0. Since the second order ordinary differential equation providing z(ρ) is quite complicated for non trivial f (z), we have to rely on numerical methods. An important special case of (4.16) is AdS 4 , for which f (z) = 1 identically. In this case the profile z(ρ) is known analytically and it is given by the hemisphere. By simplifying (4.16) first and then employing the explicit solution for the profile, the result of [14,15] is recovered, namely Let us restrict to the Schwarzschild-AdS black hole, i.e. f (z) = 1 − (z/z h ) 3 , where z h the position of the event horizon, and perform the following rescalinĝ In terms ofρ andẑ, we have f (z) = 1 −ẑ 3 ≡f (ẑ) and z f (z) =ẑf (ẑ), wheref (ẑ) ≡ ∂ẑf (ẑ). Moreover, z ρ =ẑρ and, denoting by L the integrand of (4.15), we have that It is straightforward to observe that the equation of motion d dρ ∂L ∂zρ = ∂L ∂z can be written as the equation of motion forL, i.e. d dρ ∂L ∂ẑρ = ∂L ∂ẑ . Thus, the profile of the minimal area surface is given byẑ =ẑ(ρ). As for the boundary conditions for this differential equation, from z(R) = 0 and (4.18) one finds thatẑ(R/z h ) = 0. By employing these observations and performing the rescaling (4.18), we can conclude that (4.16) for the Schwarzschild-AdS black hole can be written as (4.20) From this expression we read that F A = F A (R/z h ), which is given by the bottom curve in figure 4.
For the extremal black hole, where f (z) = 1 − 4(z/z h ) 3 + 3(z/z h ) 4 and the inner and outer horizons coincide, one can repeat the same reasoning finding again that F A = F A (R/z h ) given by (4.20) withf (ẑ) = 1 − 4ẑ 3 + 3ẑ 4 (see the bottom curve in figure 5). In the non extremal case the analysis can be done in the same way but the outcome is slightly  different because of the occurrence of two independent parameters. Indeed, by performing the rescalingρ = 3 √ M ρ andẑ = 3 √ M z, and repeating the steps explained above, one finds that F A = F A (R 3 √ M , Q 3 /M 2 ), whose explicit expression is given by (4.20) properly adapted to the rescaling entering in this case.

Domain wall geometries
Given a disk A in the z = 0 plane with radius R, in this subsection we consider the background (2.33). Since z = z(ρ), the area functional evaluated on the class of surfaces γ ε associated to the disk is given by As already remarked above, also in this case we can observe that, since the integrand depends explicitly on ρ, we cannot write a first integral. The equation of motion to solve remains an ordinary differential equation of the second order and its analytic solution is not known for a non trivial p(z). The finite term F A for the holographic entanglement entropy of a disk can be obtained from (2.35). Indeed, by employing the proper expressions for the vectorñ z and the area JHEP12(2015)037 This expression needs the explicit form of z(ρ), which can be found by solving numerically the second order ordinary differential equation coming from the variation of (4.21). In order to check the consistency of this expression, notice that for R/z RG 1 we have that p(z) → 1 (i.e. p (z) → 0) and in this limit (4.22) becomes (4.17) for AdS 4 , as expected.

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The equation of motion for L can be written as the equation of motion forL and the boundary condition isẑ(R/z RG ) = 0, as one can see from z(R) = 0 and (4.23). These observations allow us to write (4.22) in terms of (4.23), finding The bottom curves in figures 6 and 7 provide a check of the expressions (4.22) and (4.25) against numerical results obtained through Surface Evolver (coloured lines) and Mathematica (black line). Further observations can be made from these curves. In particular, an interesting quantity to compute is C = −(1−R ∂ R )S A when A is a disk of radius R because for 2 + 1 dimensional field theories it plays a role similar to the one of the central charge in 1+1 dimensions [24,25]. It is straightforward to observe that the leading term proportional to R giving the area law in (1.2) does not contribute to C and therefore we have When R RG 1 the minimal surface probes AdS 4 with radius equal to one and therefore 4G N C UV = 2π. In order to probe the IR regime very large values of R RG must be considered. In figure 8 we have performed a numerical analysis of F A and of the C function (4.26) in terms of R RG (reported in the top panel and in the bottom panel respectively) by taking values of R RG much larger than the ones explored in figures 6 and 7, finding that the latter ones do not allow us to capture the correct IR behaviour. Indeed, in the IR regime a linear behaviour occurs F A = aR RG +2π/(1+αγ) 2 +. . . , where the dots correspond to subleading terms [96]. Thus 4G N C IR = 2π/(1 + αγ) 2 = 2πL 2 IR and therefore C IR < C UV . Let us stress that, despite the fact that already for the values of R RG in figures 6 and 7 a linear behaviour seems to arise, it is not enough to get the expected value for C IR , as one can appreciate by means of a comparison with the plot shown in the bottom panel of figure 8. While C IR depends only on the product αγ (the asymptotic values are highlighted by the horizontal dashed lines in the bottom panel of figure 8), the slope a of the linear behaviour in the IR regime depends on these parameters separately, as one can observe from the top panel of figure 8.

Vaidya-AdS backgrounds
When the bulk background is the Vaidya-AdS metric (3.27) and A is a disk of radius R, the rotational symmetry allows to describe the profile of γ A in terms of two functions, z(ρ) and v(ρ), once the polar coordinates (t, ρ, θ) have been chosen for the Minkowski space at z = 0.
The area functional for γ ε in this case reads

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Considering the explicit expression f (v, z) = 1 − M (v)z 3 and by employing the results discussed in section A.3 for the unit vectors and the area element, the formula (3.29) for the finite term becomes Choosing the mass profile (3.30), in figure 9 we plot F A found in two ways: through our formula (4.28) (solid coloured lines) or through the usual method of subtracting the divergence from the area of the extremal surface. The good agreement of these results provides an important check for (4.28).

Other domains
In the previous discussions we have considered domains A which are highly symmetric because their symmetry usually allows to treat the problem of the minimal area surface analytically up to some point.
In order to study analytically the minimal area surfaceγ A associated to a generic domain A, the first problem to address is the parameterisation of the class of surfaces γ A . Then, one has to solve the differential equation coming from the extremal area condition to getγ A and finally compute the area ofγ ε . For simply connected domains A with smooth boundary which do not have any particular symmetry, already the first step could be very difficult (see e.g. figure 10). Assuming that a convenient parameterisation for the surface has been found, the differential equation coming from the extremal area condition is usually a second order partial differential equation very difficult to solve. The main simplification introduced by highly symmetric domains (e.g. strips, disks and annuli) is that this differential equation reduces to an ordinary differential equation. The latter one could be difficult to solve anyway (e.g. for the black holes or for the domain wall geometries), but ordinary differential equations are much easier to study than partial differential equations, even from the numerical point of view.
The formulas for F A discussed in section 2.1 and section 3.1 hold for a generic domain A with smooth boundary, including the ones made by disjoint components. In the latter case two or more local minima occur and the holographic prescription (1.1) requires to choose the global minimum, as we will discuss in section 5 for the case of two regions. Nevertheless, the formulas for F A discussed in section 2.1 and section 3.1 involve the unit normal vectorñ and therefore one should know the analytic solution forγ A in order to find it. For instance, whenγ A can be parameterised as z = z(x, y), the expression for Tr K contains all possible first and second order partial derivatives in a complicated way that we do not find interesting to report here.
The big advantage of the numerical analysis with Surface Evolver [84,85] is that the minimal area surface is obtained without going through this procedure of finding the convenient parameterisation first and then solving the differential equation (see section 4.1). Moreover, as already remarked in section 4.1, besides the area of the surface, also its unit normal vectorñ can be found and this allows us to check the formulas found in section 2.2 for non trivial domains.
Besides the cases of disks and strips discussed in section 4.2 and section 4.3, we have considered F A also for more complicated simply connected domains, both convex and non convex. In particular we have studied regions A delimited by ellipses for all the static backgrounds of section 2.2. For the domain wall geometries, we have considered also the non convex domains delimited by the blue and the red curves in figure 10. Once the shape and all the relative ratios between the various geometrical parameters have been fixed, we have computed F A changing the total size of the region A. The numerical analysis has been done as explained in section 4.1. The area A A for domains A delimited by ellipses as small perturbations of circumferences has been already considered through the standard approach e.g. in [31,36] and by employing the interesting method of [82,[118][119][120] (which is based on the solution of the cosh-Gordon equation in terms of algebraic curves) in [121].  Figure 11. The quantity F A for the domain wall geometry (2.33) with α = 2 and γ = 1. The entangling curves are the blue and the red ones in the bottom right part of the plot, which are obtained by joining arcs of circumferences whose centers provide an opening angle given by π and 1.54π respectively. The radius of the external circumference is R and the radius of the internal one is R/3 (see figure 10 for two examples of minimal surfacesγ A anchored to these entangling curves).

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The numerical analysis has been done with Surface Evolver by taking ε = 0.03, R = 3 and moving z RG in the interval (0.5, 70). Solid and dashed lines correspond respectively to the two ways to find F A given in (4.1). In the inset we show z * /z RG in terms of R/z RG corresponding to all the points in the main plot.
When the bulk geometry is AdS 4 , this rescaling of A does not change F A because the Willmore energy is invariant, as already discussed in section 2.2.1. On the other hand, for asymptotically AdS 4 black holes and domain wall geometries this invariance is broken and a non trivial behaviour is found under rescaling of A.
In figure 4 and figure 5 we study this rescaling for the Schwarzschild-AdS black hole and the extremal Reissner-Nordström-AdS black hole respectively by employing both the formula (2.31) and the usual way to get F A by subtracting the area law divergence, as explained in section 4.1. We show F A for A given by disks or domains delimited by ellipses with semi-axis R 1 R 2 having two different eccentricity. Let us remind that the perimeter P A of an ellipse with semi-axis , where E is the complete elliptic integral of the second kind, 4 and its area is Area(A) = πR 1 R 2 . For the disks we have employed also the simpler formula (4.16), which can be evaluated numerically by using Mathematica. The plots in figure 4 and figure 5 show that F A is a function of R 1 /z h for a given eccentricity. It would be helpful to have data for large ellipses in order to check the behaviour F A = − Area(A)/z 2 h + . . . expected from (2.32).

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As for the domain wall geometry (2.33), in figures 6 and figure 7 we have considered the domains A just mentioned having elliptical entangling curves for two different sets of parameters for the background ((α, γ) = (2, 1) and (α, γ) = (4, 1) respectively). The expression of F A for the domain wall geometry is (2.35) and the numerical analysis has been done as mentioned above and explained in section 4.1. As the size of A changes, the qualitative behaviour of F A for the domains delimited by ellipses is the same one found for the disks. In particular, F A has a finite limit when A is very small (z * z RG ). Nevertheless, we remark that the values of R RG explored in figures 6 and figure 7 are too small to capture the correct IR behaviour, as we have seen in figure 8 for the disks.
The similarity between F A for the disk and the ones corresponding to domains delimited by ellipses observed in figures 6 and figure 7 motivated us to explore also the case of non convex domains. In particular, we have considered the non convex domains delimited by the red and blue curves in figure 10. For both these cases the domains A have the same shape and only one geometrical parameter (the opening angle) distinguishes them. It is worth remarking that, for these domains, finding a parameterisation for γ A is not easy, as one can immediately understand from figure 10, where the minimal area surfacesγ A obtained with Surface Evolver are shown.
In figure 11 we show the results for F A corresponding to these non convex domains. Interestingly, the qualitative behaviour of F A is again the same one observed for the disk, which led to the definition (4.26) of the C function. This suggests that it could be worth generalising the definition (4.26) introduced for the disks by interpreting R as a global parameter of A and exploring better whether proper C functions can be defined from domains which are not disks [24,96]. We remark that in our numerical analysis for domains which are not disks we have not considered domains large enough to capture the IR behaviour of F A . Indeed, from the case of the disk, whose relevant plots are shown in figure 8, we have learned that the values of R RG explored in figures 6, 7 and 11 are too small to probe the deep IR regime. We should push our numerical analysis to much higher values of R RG but, unfortunately, our present code is numerically unstable. We hope to overcome this technical obstacle in the near future.
As for the time dependent backgrounds, it would be very interesting to check (3.29) by considering finite regions with smooth non circular boundaries. This would be helpful to have a better understanding of the tsunami picture introduced in [69,70].

Infinite wedge
An important class of domains A to study is given by the ones whose boundary ∂A contains some corners. In these cases, the entanglement entropy has also a logarithmic divergence besides the area law term. The simplest example to address is the infinite wedge for the gravitational background given by AdS 4 , whose corresponding minimal area surface has been first studied in [97] and its area has been computed. The aim of this section is to show how the analytic result of [97] can be recovered through the formula (2.24). Here we give only the main expressions to understand the result, but all the technical details of this computation have been reported in section D). It is worth recalling that (2.24) has been obtained by assuming smooth entangling curves for ∂A and, under this hypothesis, it JHEP12(2015)037 provides a finite result as ε → 0. Nevertheless, we find it interesting and non trivial that the Willmore energy (2.24) provides the expected logarithmic divergence for non smooth entangling curves.
Choosing polar coordinates {ρ, θ} in the z = 0 plane such that the origin coincides with the tip of the wedge, the domain that we are going to consider is A = {(ρ, θ), |θ| Ω/2, ρ L}, where Ω is the opening angle of the wedge and L 1 its length along the edges. Since L 1, we can employ the following ansatz for the minimal surfaceγ A [97] z = ρ q(θ) , (4.29) where the function q(θ) can be found by imposing the extremal area condition.
Considering the boundary ofγ ε , part of this curve lies at z = ε, which will be denoted by ∂γ ε , and the remaining part ∂γ ⊥ ε belongs to a vertical cylinder. From the projection of ∂γ ε on the z = 0 plane, one finds that the range of the radial coordinate ofγ ε is being ω ∼ 0 + the angle between the edge of A and the straight line connecting the tip of the wedge to the intersection point between the circumference ρ = ρ max and the projection of ∂γ ε on the z = 0 plane. In section D we show that for the wedge we are considering the formula (2.24) gives where b(Ω) is the function found in [97] b(Ω) =ˆ∞ (4.32) beingq 2 0 ≡ q 2 0 /(1 + 2q 2 0 ) ∈ [0, 1/2] and the functions K and E are the complete elliptic integrals of the first and second kind respectively.
As for the contribution of ∂γ ε to the holographic entanglement entropy, it is given by the contour integral in the r.h.s. of (2.14), whose integrand isb z /z in our case. Such contour integral is the sum of two contributions: the line integral over ∂γ ε and the line integral over ∂γ ⊥ ε . Along ∂γ ε we haveb z ∼ −1 for the parts of the curve close to the edges of A, while it significantly deviates from −1 for the part of ∂γ ε close to the tip of the wedge (i.e. at ρ ∼ ρ min ) without becoming infinitesimal. Instead, along ∂γ ⊥ ε we haveb z ∼ 0 close to the boundary and becomes finite around θ = 0, which is also the point ofγ A with the highest value of z. Considering these two contributions together and using that L 1, one finds thatˆ∂γ Thus, while the area term for the infinite wedge comes from the boundary integral (4.33), the subleading logarithmic divergence (4.31) is encoded into the Willmore energy (2.24) with the expected coefficient.

JHEP12(2015)037 5 Holographic mutual information
In this section we briefly discuss the holographic mutual information and, in the case of AdS 4 , some straightforward consequences of the formula (2.24). Given two disjoint spatial domains A 1 and A 2 in the boundary, one can consider the entanglement entropy S A 1 ∪A 2 , which measures the entanglement between A 1 ∪ A 2 and its complement. A very useful quantity to introduce is the mutual information Since the divergent terms of the entanglement entropy S A depend on the entangling surface ∂A, they cancel in this combination and the mutual information (5.1) is UV finite. For the two disjoint domains A 1 and A 2 , the subadditivity property of the entanglement entropy reads which means that the mutual information (5.1) is non negative I A 1 ,A 2 0. When A 1 and A 2 have a non vanishing intersection, the strong subadditivity property of the entanglement entropy holds [4,5] which tells us the mutual information increases as one of the two disjoint domains is enlarged The holographic entanglement entropy formula (1.1) of [14,15] can be applied also for disjoint domains and the strong subadditivity property for the holographic prescription has been proven in [28]. For two disjoint regions A 1 and A 2 , let us introduce I A 1 ,A 2 as follows As already remarked above, the mutual information is UV finite and, from (1.2), we have that where F A 1 ∪A 2 is found by taking the global minimumγ A 1 ∪A 2 of the area functional among all the surfaces γ A 1 ∪A 2 such that ∂γ In this computation, it is well known that typically two local minima occur: a connected surfaceγ con joining ∂A 1 and ∂A 2 through the bulk and a disconnected configurationγ A 1 ∪γ A 2 made by the two disjoint surfaces found for the holographic entanglement entropy of A 1 and A 2 separately [43,53,54,56]. Sinceγ con A 1 ,A 2 andγ A 1 ∪γ A 2 have the same boundary, from (1.1) and (1.2) one gets

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where F A 1 ,A 2 is defined as the O(1) term in (1.2) when the global minimum is provided by the connected surfaceγ con A 1 ,A 2 . In (5.7) the max occurs because F A enters with a minus sign in the expansion of S A .
When the two disjoint domains A 1 and A 2 are very close, the connected surfaceγ con is the global minimum and I A 1 ,A 2 > 0, while, when the distance between the domains is large enough, the configuration made by the union of the two disconnected surfacesγ A 1 andγ A 2 becomes the global minimum and I A 1 ,A 2 = 0. The transition between these two regimes occurs when Keeping the shapes of ∂A 1 and ∂A 2 and their relative orientation fixed, one can change the relative distance and find where (3.24) holds. This is difficult for shapes which are not highly symmetric like disks or infinite strips (see [37] for a numerical analysis). One could employ the expressions discussed in the previous sections to find some further results for the solution of (5.8) in terms of the shapes of the domains. It is worth remarking that, while the disconnected configurationγ A 1 ∪γ A 2 can be found for every distance between ∂A 1 and ∂A 2 , the connected one does not exist for distances larger than a critical one, which is obviously bigger than the distance defined by (5.8) [37,54,55,122].
Given the disjoint domains A 1 and A 2 , let us enlarge A 1 getting A 1 ∪ A 0 and consider the corresponding extremal area surfaces occurring in the computation of the holographic mutual information. Plugging the holographic formula (5.5) into the strong subadditivity property (5.4), one finds that I A 1 ∪A 0 ,A 2 I A 1 ,A 2 and, from (5.6), this tells us that Notice that, while in the l.h.s. of this inequality disjoint domains occur and therefore the maximisation in (5.7) must be performed for both terms, in the r.h.s. only connected domains are involved. When A 1 ∪A 0 and A 1 are sufficiently far from A 2 , i.e. their distances are such that I A 1 ∪A 0 ,A 2 = 0 and I A 1 ,A 2 = 0, in (5.9) we have F A 1 ∪A 0 ∪A 2 = F A 1 ∪A 0 + F A 2 and F A 1 ∪A 2 = F A 1 + F A 2 and the inequality is trivially saturated. Let us restrict to the part of the space of configurations where I A 1 ∪A 0 ,A 2 > 0 and I A 1 ,A 2 > 0, whereγ con A 1 ∪A 0 ,A 2 andγ con A 1 ,A 2 are the global minima to consider for the holographic entanglement entropy S A 1 ∪A 0 ∪A 2 and S A 1 ∪A 2 respectively. In this case (5.9) becomes Considering a domain A 0 very small with respect to A 1 , we can interpret the enlarging of A 1 by A 0 as a small perturbation of A 1 . For this case, (5.10) tells us that the variation of F under such perturbation is bigger when A 2 occurs. The inequality (5.10) is a non trivial property of the formulas for F A discussed in the previous sections.

AdS 4
When the gravitational background is AdS 4 , we have that 2F A is the Willmore energy of the closed surfaceγ (d) A embedded in R 3 , as stated in (2.27 In the former case we will denote the corresponding closed surface byγ con,(d) A 1 ,A 2 , while in the latter case the two surfacesγ (d) A 1 ∪γ (d) A 2 occur. When I A 1 ,A 2 > 0, the genus ofγ con,(d) A 1 ,A 2 is g 1, depending on the shape of the entangling curve.
For domains A 1 and A 2 such thatγ (d) A has genus one, we can apply the fact that for any g = 1 closed surface embedded in R 3 , we have (see Theorem 7.2.4 in [81]) where the bound is saturated by a regular torus whose ratio between its radii is √ 2, which is known as the Clifford torus. This claim has been conjectured by Willmore [79,80] and proved only recently [108].
Considering two disjoint disks for A 1 and A 2 , ifγ con,(d) A 1 ,A 2 were the Clifford torus, then the holographic mutual information would be F A 1 ∪A 2 − F A 1 − F A 2 = π 2 − 4π < 0. Thus, the Clifford torus does not occur among the genus one closed surfacesγ con,(d) A 1 ,A 2 providing the holographic mutual information of some configuration of two disks, which is always non negative. Nevertheless, it is reasonable to ask whether the Clifford torus occurs anyway as local minimum of the area functional which is not a global one. For two disjoint disks it has been found that [37,54,55,122]. Thus, half of the Clifford torus never occurs among the surfaces γ A ⊂ H 3 which are extremal points of the area functional. This happens because not all the genus one surfaces can be spanned by considering γ (d) A with varying A, but only those ones which are symmetric with respect to the plane z = 0 and such that the curve ∂A is umbilic. For regular tori, i.e. the ones obtained from two circumferences at fixed radii (and the Clifford torus is among them), the latter condition is not satisfied.
An interesting observation about AdS 4 that we find it worth remarking here concerns the strong subadditivity condition (5.9) for the holographic prescription. Choosing A 0 such that A 1 ∪ A 0 has the same shape of A 1 , namely A 1 ∪ A 0 is a rescaling of A 1 by a factor greater than one, by employing the observation made in the last paragraph of section 2.2.1, we have that the r.h.s. of (5.9) vanishes. This does not happen for the black holes and the domain wall geometries, where the invariance under scale transformations is broken by the occurrence of a scale.
Finding the minimal area surfaceγ A such that A is made by two equal disjoint disks is equivalent to obtainγ A when A is an annulus [37,122]. In section E we consider the latter domain, showing that the formula (2.24) specified to this case provides the analytic expression already found in through a direct computation of the area [54,55,75].

Conclusions
In this paper we have studied the holographic entanglement entropy (1.1) in the context of AdS 4 /CFT 3 for domains A having generic shapes. When the entangling curve is smooth, the first non trivial term in the expansion ε → 0 of the holographic entanglement entropy JHEP12(2015)037 is the constant term F A (see (1.2)). This term is interesting because it depends on the whole minimal surface and, therefore, it allows to probe the IR part of the geometry when the corresponding domain A is sufficiently large.
Our main results are (2.19) and (3.22), where F A is given in terms of the unit vectors normal to the extremal area surfaceγ A respectively for static and time dependent backgrounds which are conformally related to asymptotically flat spacetimes. These formulas has been applied for explicit backgrounds: among the static ones we have considered AdS 4 , asymptotically AdS 4 black holes and domain wall geometries. The latter ones provide an example of holographic RG flow. In the simplest case of AdS 4 one finds that F A is given by the Willmore energy ofγ A viewed as surface embedded in R 3 [82,83]. This allows us to easily prove that the disk maximises S A among the domains with the same perimeter. Among the time dependent spacetimes, we have considered the Vaidya-AdS metrics.
We have checked that our results reproduce the well known ones for highly symmetric domains like strips, disks and annuli. As for less symmetric domains A, which are more difficult to treat (e.g. the ones delimited by ellipses or some non convex domains), our formulas have been tested numerically by employing Surface Evolver [84,85]. An interesting outcome is obtained from the domain wall geometries. Indeed, from the holographic analysis of F A for the domains different from the disk, we have observed the same qualitative behaviour of F A for the disk, which provides the holographic C function. Unfortunately our numerics does not allow to probe the deep IR regime and therefore we cannot give conclusive statements. We hope that our analysis will be improved in the near future.
Among other open issues that would be interesting to address in the future, let us mention the higher dimensional case, where the expansion of the entanglement entropy as ε → 0 has more divergent terms whose coefficients depend on the geometry of ∂A (see [35,123] for recent papers where the properties of the Willmore energy of ∂A in d = 4 have been employed to get some insights on entanglement entropy).
As for the time evolution of the holographic entanglement entropy through the Vaidya-AdS backgrounds, the result found here could lead to some deeper understanding of the entanglement tsunami picture [69,70]. It would be also interesting to perform a numerical study of this time evolution for finite domains which are not disks, like the ones considered in this manuscript for static backgrounds.

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where κ(σ) is the geodesic curvature of the entangling curve ∂A, namely Given the tangent vectors (A.3) and (A.3), the determinant of induced metric h µν reads Another consistency condition to impose is the requirement that deth at z = 0 provides the square of the line element of the entangling curve, i.e. (deth)| z=0 = x 2 + y 2 . By employing (A.6) and u(σ, 0) = u σ (σ, 0) = 0, this condition implies that which tells us that γ A intersects orthogonally the z = 0 plane. Notice thatm (1) ·m (2) = u σ u z = 0 for z = 0. Considering the surface γ ε obtained by restricting γ A to z ε > 0, sincem (1)z = 0 the vectorm (1)µ belongs to the plane z = ε. Thus, the vectorb µ introduced in section 2.1 can be constructed as the linear combination ofm (1)µ andm (2)µ which is orthogonal tõ where where u = u 2 (σ) z 2 /2 + O(z 3 ) is the first term of the expansion of u as z → 0. The vector (A.9) has unit norm up to O(ε 2 ) terms. In particular,b µ → (−1, 0, 0) when ε → 0. Taking the vector product ofb µ andm (1)µ , we can easily find the unit vector normal to γ ε at z = ε, namelyñ µ = ε u 2 , − y which tells us thatñ z = O(ε) when u 2 is non vanishing. From (A.6) it is straightforward to write the differential equation providing the extremal area condition, which turns out to be quite complicated. Nevertheless, by plugging the expansion u = u 2 (σ) z 2 /2 + O(z 3 ) into it and expanding the result as z → 0, the first non trivial order leads to u 2 (σ) = κ(σ) . (A.11) As discussed in [82,83], this condition tells us that ∂γ A is un umbilic line, i.e. for any of its points the two principal curvatures coincide and therefore, locally, the surface looks like a sphere.

A.2 Black holes and domain wall geometries
In this subsection we give explicit expressions for the unit vectors and for the area elements that are needed in the computation of F A for specific domains. For static backgrounds, let us consider surfaces parameterised either by z = z(x, y) if cartesian coordinates {x, y} have been chosen for the z = 0 plane or by z = z(ρ, θ) for polar coordinates {ρ, θ} in the z = 0 plane.
Black holes. Let us consider first the black hole metric (2.29), which includes the special case of AdS 4 when f (z) = 1 identically. Choosing the order {z, x, y} or {z, ρ, θ}, by employing (2.20), for the unit normal vector we havẽ (A.12) and, raising the index, the corresponding vectors read As for the induced metric on Σ, it is given by Computing the determinant coming from induced metric ds 2 | Σ , one gets the area element (A.17) and we find it useful also to give the same unit vectors obtained by raising the index, namelyñ µ = ñ z , p(z)ñ x , p(z)ñ y ,ñ µ = ñ z , p(z)ñ ρ , p(z)ñ θ /ρ 2 . (A.18)

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The two dimensional metric induced on the surface Σ reads 20) and the corresponding area elements are given respectively by The above expressions (A.18) and (A.21) have been used in section 4.2.2 and section 4.3.2 to specify F A for the strips and the disks starting from the general formulas given in section 2.2; but they can be employed also for other domains.

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Then, considering the determinants h andh of the induced metrics, they are related as h = e 2(d−1)ϕh . This implies that for the area elements dA d−1 = √ h dΣ d−1 and Thus, from (B.2) and the transformation rule of the area element, we can easily construct Weyl invariant expressions as follows where the case n i = 1 is excluded because TrK = 0. Notice that (B.4) are defined only for d 3.
When d = 3 only the pair (n, a) = (2, 1) is allowed and, similarly, when d = 4 one finds only the pair (n, a) = (3, 1). Instead, for d = 5 we can construct two terms of the form (B.4) with a single term in the product: one having (n, a) = (4, 1) and (n, a) = (2, 2). Any linear combination of these two terms is Weyl invariant but let us mention that also other Weyl invariant terms different from (B.4) can be constructed [124].

C A comment from the Helfrich energy
The holographic entanglement entropy (1.1) for a two dimensional spatial domain A is given by the area of the surfaceγ A which minimises the area functional within the class of surfaces γ A such that ∂γ A = ∂A, once the cutoff z ε > 0 has been introduced. In section 2 it has been shown that, for smooth entangling curves and when the bulk spacetime is AdS 4 , the O(1) term in the ε → 0 expansion is given by the Willmore energy of the surfaceγ A embedded in R 3 (see (2.24)) [82,83].
Given an oriented, smooth and closed surface Σ g ⊂ R 3 of genus g, an interesting generalization of the Willmore functional is the Helfrich functional, which is defined as follows [111] where H 0 andλ are two constants. The functional (C.1) plays a very important role in the study of the cell membranes [112]. The last term in (C.1) is topological and the Gauss-Bonnet theorem tells us that it is proportional to (1 − g). In section 2.1 it has been shown that, when the bulk geometry is AdS 4 and considering the surfaces γ A intersecting orthogonally the boundary z = 0, the area of γ A restricted to z ε is (2.15), where the O(1) term is given by (2.23). A natural question to ask is whether exists a surfaceγ (H) ε within this class of surfaces whose part having z ε (denoted byγ (H) ε ) has an area given by ( A in the half space z 0, as explained in section 2.2.1 (see figure 1 an example of this construction involving the minimal area surfaceγ A ).
By employing the transformation properties of the extrinsic curvature and of the Ricci scalar introduced in section 3.1, from the integrands in (C.1) and (2.23) we find thatγ (H) A is defined by the following equation which is written through the curvature ofγ (H) A embedded in R 3 . In terms of the curvature ofγ (H) A as surface in H 3 , it reads As a simple consistency check, one observes that, by setting H 0 = 0 andλ = 0 in (C.5), the minimal area condition TrK = 0 is recovered. Thus, the surfaceγ (H) A ⊂ H 3 , which is characterised by the parameters H 0 andλ, reduces to the minimal area surfaceγ A occurring in the holographic entanglement entropy formula when H 0 =λ = 0.
It would be interesting to find a CFT quantity related in some way to the surfaceγ (H) A . Such quantity should depend on the parameters H 0 andλ, and reduce to the entanglement entropy when they both vanish. Moreover, it should have the same leading divergence of the entanglement entropy as ε → 0, as it can be seen from (C.2). Thus, the Rényi entropies are excluded.

D Some technical details for the infinite wedge
In this appendix we discuss the computations leading to the results presented in section 4.5 for the holographic entanglement entropy of the infinite wedge when the bulk geometry is AdS 4 .

E Annulus
In this appendix we apply (2.24) for the annulus, recovering the minimal surfaceγ A discussed in [54,55,75].
When A is an annulus delimited by two concentric circumferences with radii R − < R + and the gravitational background is AdS 4 , the global minimum of the area functional among the surfaces γ A which provides the holographic entanglement entropy depends on the ratio η ≡ R − /R + ∈ (0, 1).
In particular, for η 0.367 there are two topologically different local minima of the area functional: one is the union of the two disjoint hemispheresγ A 1 ∪γ A 2 , while the other one is a surfaceγ con A 1 ,A 2 connecting the two boundaries of the annulus through the bulk (there are two of them having the same η, but we consider only the one having minimal area). For a thin annulus η ∼ 1 andγ con A 1 ,A 2 is the global minimum. At η c = 0.419 the transition occurs and for η < η c the global minimum is given by the two disjoint hemispheres. For η < η * the solutionγ con A 1 ,A 2 does not exist and onlyγ A 1 ∪γ A 2 remains as extremal area surface. Choosing polar coordinates (ρ, θ) in the z = 0 plane centered in the origin, the expression forγ con A 1 ,A 2 can be written in a parametric form as the union of two branches z ± (t) = R ± t e −f ± (t) , ρ ± (t) = R ± e −f ± (t) , t ∈ 0, t max , (E.1)

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Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.