Conditions on holographic entangling surfaces in higher curvature gravity

We study the extremal surfaces of functionals recently proposed for the holographic calculation of entanglement entropy in general higher curvature theories, using New Massive gravity and Gauss-Bonnet gravity as concrete examples. We show that the entropy functionals admit closed extremal surfaces, which for black hole backgrounds can encircle the event horizon of the black hole. In the examples considered, such closed surfaces correspond to a lower value of the entropy functional than expected from CFT calculations, implying a seeming mismatch between the bulk and boundary calculations. For Lorentzian settings we show that this problem can be resolved by imposing a causality constraint on the extremal surfaces. The possibility of deriving conditions from an alternative conical boundary condition method as proposed by Lewkowycz and Maldacena is explored.


Introduction
The concept of entanglement entropy has proven to be of great interest to the physics community in recent years. The search of a better understanding of quantum systems, in particular at criticality, has lead to the establishment of general analytic results for entanglement entropy in two-dimensional CFTs (see e.g. [1]), while in higher dimensions the study of entanglement entropy is less simple. Entanglement entropy also appears to be instrumental for a deeper understanding of holography [2,3].
The development of an understanding of entanglement entropy in the framework of the AdS/CFT correspondence [4][5][6] is therefore of utmost interest. A simple example of the AdS/CFT corre-spondence at play in this context is given by the Bekenstein-Hawking entropy [7,8] S BH = Area of Horizon 4G N , (1.1) which for BPS black holes has a microscopic derivation in string theory [9], implying a relation between gravitational entropy and the degeneracy of quantum field theory as its microscopic description. Important progress was made by Ryu and Takayanagi (RT) [10,11], who generalised the Bekenstein-Hawking formula above to the holographic calculation of entanglement entropy in CFTs with Einstein-Hilbert gravity duals. Here, the entanglement entropy is given by the area of a minimal surface which is anchored to the given spatial region in the CFT. This prescription for entanglement entropy was recently proven by Lewkowycz and Maldacena in [12] by translating the replica trick to the classical gravity action in the bulk, and showing that the resultant prescription for holographic entanglement entropy is equivalent to that of RT. However to date there is no proof of its covariant generalisation by Hubeny, Rangamani and Takayanagi (HRT) [13].
More recently, generalisations of the RT and HRT formulae to general higher derivative theories of gravity have been proposed [14][15][16][17]. In black hole backgrounds, for consistency a correct prescription should reduce to Wald's entropy functional [18][19][20] when evaluated on the black hole bifurcation surface B (see appendix A for notation). Despite parallels to the Einstein-Hilbert case, Wald's functional does not provide a prescription for the calculation of entanglement entropy in field theory duals to higher derivative gravity [21], and the current proposals take into account the extrinsic curvature of the surfaces upon which they are evaluated.
When using functionals to calculate entanglement entropy, they must be evaluated on a particular surface in the bulk anchored to the region of interest in the boundary CFT. One way to locate these surfaces could be to directly extremise the functionals, which has already been proposed in [16]. In this paper we investigate this method for static and partially static backgrounds (i.e. AdS and black hole backgrounds) in New Massive Gravity (NMG) and Gauss-Bonnet gravity, with a particular focus on black hole solutions and the nature of the closed extremal surfaces of the corresponding functionals. In black hole backgrounds, a particularly important closed extremal surface is the black hole bifurcation surface. When evaluated on the black hole bifurcation surface, the entanglement entropy functional yields Wald's formula (1.2) for black hole entropy. This coincides with the dual CFT entropy by standard AdS/CFT lore [22].
In addition to the black hole bifurcation surface, we find there exists an extra closed extremal surface for certain parameter ranges in NMG and Gauss-Bonnet gravity. Such surfaces can encircle the event horizon of the black hole, and in the cases we examine they give a lower value when inserted into the entropy functional than the black hole bifurcation surface. A prescription for calculating entanglement entropy based on only extremising the functionals would hence lead us to incorrectly equate the dual CFT entropy with that given by the additional closed extremal surface. Our findings lead us to conclude that, for consistency and physical values for entanglement entropy, additional constraints on the extremal surfaces need to be imposed. In section 6 we show that in a covariant setting such constraints can be motivated by causality. For Euclidean settings, since causality cannot be used to derive constraints on the surfaces, in section 7 we demonstrate that at least for the closed extremal surfaces constraints can be derived by solving the equations of motion with conical boundary conditions. This paper is organised as follows: In section 2 we introduce static and covariant prescriptions for calculating entanglement entropy in both the Einstein-Hilbert and higher derivative gravity theories. For higher derivative gravity theories, we define these as the RT-like and HRT-like prescriptions, respectively. In section 3 we explain the significance of the closed extremal surfaces of holographic entanglement entropy functionals, in particular the role played by the homology constraint in the interpretation of such surfaces. In sections 4 and 5 we study the nature of the extremal surfaces of the entropy functionals in NMG and Gauss-Bonnet gravity, paying particular attention to their closed extremal surfaces in the bulk. We explain that in using the functionals to calculate entanglement entropy, constraints on the extremal surfaces additional to the homology constraint need to be imposed. In section 6, we introduce causality arguments which provide constraints in the covariant HRT-like prescription. In search of constraints for the static RT-like prescription, in section 7 we turn to an alternative method of locating extremal surfaces for entanglement entropy in static scenarios, first proposed by Lewkowycz and Maldacena [12]. In section 8 we summarise our results, and discuss their implications for functional prescriptions of entanglement entropy. Several lengthy details are relegated to the appendices: In appendix A we explain the notation used throughout the paper, and in appendix B the full form of the equation of motion studied in section 4 is given. In appendix C we briefly discuss extremal surfaces for the rotating BTZ and Lifshitz black holes of NMG.
While in the final stages of this project, we learned of a similar study [23] where it is investigated under which conditions there might be a clash between holographic prescriptions for computing relative entropy and the manifest positivity of this quantity in field theory terms.

Functional prescriptions of holographic entanglement entropy
Let us introduce the covariant and static functional prescriptions for the holographic calculation of entanglement entropy in Einstein-Hilbert gravity and general higher derivative theories. We utilise functional prescriptions for entanglement entropy throughout this paper, and only in section 7 is an alternative prescription discussed, which is based on extending the replica trick into the bulk. We refer to this alternative method as the conical boundary condition method.
For Einstein-Hilbert gravity, the Ryu-Takayanagi (RT) prescription provides a means of holo-graphically calculating entanglement entropy in the static case: When the bulk spacetime M with asymptotic boundary ∂M is static, there exists a timelike Killing vector field that induces a foliation of both M and ∂M into spacelike surfaces. For a CFT region A on such a spacelike slice of the boundary ∂M, the entanglement entropy S A associated with A is given by the area of the bulk minimal-area surface E A located on the same spacelike slice in the bulk via where G N is the gravitational constant. In this formula, E A is anchored to ∂M, such that the intersection of E A with the boundary ∂M equals the boundary ∂A of the CFT-region A (figure 1). The bulk minimal surface is also required to satisfy the homology condition: there must exist a hypersurface F in M such that the boundary of F is the union of E A and A [24]. E A is then said to be homologous to A, 4 and we will refer to such surfaces as (holographic) entangling surfaces.
The Hubeny-Rangamani-Takayanagi (HRT) prescription extends RT to arbitrary time-dependent states, thus providing a covariant prescription for holographic entanglement entropy. More concretely, for a given spacelike CFT region A on the boundary ∂M of the asymptotically AdS spacetime, one searches for bulk surfaces E A anchored to ∂M which extremise the area functional, picking the surface that gives the smallest entropy: It was recently noted in [25] that in order to obtain consistent results in certain bulk spacetimes, for example those with multiple bifurcation surfaces like the Reissner-Nordström black hole, the homology condition has to be either refined or appended with an additional causality condition. The authors of [25] imposed the additional constraint that the hypersurface F is spacelike, as was already done by Wall in [26]. This seems to follow naturally from the RT approach, where F is spacelike by construction. Headrick in contrast proposed the imposition of a causality constraint on the HRT prescription in [27]. Primarily, this causality constraint demands that there should be no causal contact possible between E A and A. This argument will be explained in more detail in section 6, and plays an instrumental role in the findings of this paper. 5 We will henceforth adopt the convention that the HRT prescription is based on the variation of the area functional using a general homology condition, as originally proposed in [13]. Whenever we make use of a refined homology condition or additional causality conditions, such as in section 6, we will explicitly mention this. 4 Note that in the static RT prescription F is by construction located in the constant-time slice of M. 5 It would be an interesting question to investigate when the causality constraint and the spacelike constraint on F are equivalent. For example, it is conceivable that in a spacetime where closed timelike curves are created by identifying two spacelike slices it is impossible to satisfy the causality condition, while the demand that F should be spacelike could still be easily met. The relation between these two possible conditions will be investigated in [28]. However we are not going to deal with causally pathological examples in this work, and we assume in the discussion of section 6 that both conditions could be used equivalently. As mentioned in the introduction, for the holographic calculation of entanglement entropy in CFTs dual to higher curvature gravity theories, modified functionals have been proposed in [14][15][16][17]. In particular, for a general four-derivative theory of gravity with (Lorentzian) Lagrangian the following functional was first derived in [14]: where Σ is a spacelike co-dimension two hypersurface, with induced metric γ ij and extrinsic curvature terms k and Tr(k) 2 . Further explanation for the notation used in this formula can be found in appendix A.
In using the entropy functionals (2.4) to calculate entanglement entropy of a given CFT region A holographically, a crucial step is to locate the particular co-dimension two surface anchored to ∂A in the bulk upon which the functional should be evaluated. Since for any given A there is an infinite number of possible surfaces obeying the homology condition, a priori it is not clear which of these should be chosen. In principle it should be possible to determine the surface by solving all the equations of motion of the higher curvature theory with conical boundary conditions [12,16], and this is demonstrated for Gauss-Bonnet gravity in section 7. However for a general higher curvature theory, in practice this calculation is very involved. As noted in [16], it would be advantageous to instead be able to determine the surface by directly extremising the functional. This was argued to be true for theories of the form (2.3) by comparing the equations of motion derived from extremising the functional with certain equations derived from the conical boundary condition method in [16]. In this paper we find that this method of locating the correct co-dimension two surfaces of the functional (2.4), for consistency and physicality, needs to be supplemented by constraints additional to the homology constraint on the extremal surfaces. Our conclusions are drawn by studying the nature of the extremal surfaces of the entropy functional (2.4) for (partially) static backgrounds in New Massive gravity [29,30] and Gauss-Bonnet gravity [31,32].
Let us introduce some definitions employed throughout this paper, which translate the RT and HRT prescriptions of Einstein-Hilbert gravity to general higher curvature theories. When extremising the entropy functional (2.4) on an equal time slice of a (partially) static spacetime, we refer to this as the RT-like prescription. In the HRT-like prescription, we need not restrict extremising the functional to equal time slices of the spacetime, as they may not be uniquely defined at all (e.g. when the extremal curves lie inside a black hole horizon as in section 4.4). Whenever using causality arguments that intrinsically depend on a covariant setting this will be refered to as an HRT-like prescription amended with additional conditions.
In Einstein-Hilbert gravity, for a (partially) static setting the RT prescription and the HRT prescription with additional conditions are believed to be equivalent. However in the higher curvature case, our findings show that when taking causality conditions into account, the RTand HRT-like prescriptions do not agree at least in NMG and Gauss-Bonnet gravity and it is the HRT-like prescription amended with a causality constraint that gives the physically expected result, see section 6. We find these inconsistencies by investigating closed extremal surfaces of the entropy functionals in black hole backgrounds, whose significance we explain in the next section.

Significance of closed extremal surfaces in black hole backgrounds
When calculating the entanglement entropy of a (small) subsystem of the whole boundary using the RT and HRT prescriptions, one needs to search for surfaces extremising the entropy functional that are anchored at the asymptotic boundary. But there also exist extremal surfaces which are not anchored to the boundary, and instead form closed surfaces in the bulk, see figure  2.
In [25,34], for stationary black hole backgrounds in Einstein-Hilbert gravity it was shown that these closed extremal surfaces are very important: They are homologous to the full boundary and thus determine the total entropy of the dual CFT. These surfaces play a role when larger regions of the boundary are considered [25].
In Einstein-Hilbert gravity, closed extremal surfaces of the area functional only exist in black hole backgrounds. They are always the bifurcation surface(s) of the black hole, at least assuming the weak energy condition: T µν V µ V ν ≥ 0 for any timelike vector V µ (see [27,35]). The RT prescription, being restricted to equal time slices, will then naturally equate CFT entropy with the area of the outer (if an inner one exists at all) bifurcation surface, as expected from standard AdS/CFT lore [22]. In order to reproduce this result in the HRT prescription, as mentioned in the previous section, one has to impose additional constraints such as the causality Figure 2: On the left side, we sketch a spacelike slice of a black hole spacetime with t =const., also known as an Einstein-Rosen bridge. In a conformal diagram such as figure 3, this corresponds to a straight line through the center connecting the two asymptotic boundaries. On the right hand side, we sketch the (compact) boundary with CFT 1 and the interior of the bulk between the boundary and the bifurcation surface B. This surface B is also an extremal surface. C and D sketch typical extremal surfaces anchored to the boundary, as found in [25,33], while A depicts the possibility of an additional closed extremal surface that might appear in higher derivative theories.
constraint [25]. The CFT entropy is then correctly determined by the (outer) bifurcation surface B, and is then equal to the usual Bekenstein-Hawking black hole entropy In higher curvature theories, we may argue by symmetry that black hole bifurcation surfaces will also be saddle points of any functional of the form (2.4): The bifurcation surface is defined by the vanishing of a Killing vector field ξ. By symmetry the functional, when evaluated on certain curves, has to be invariant under flows generated by this Killing field. Hence the bifurcation surface will define a saddle point of this functional in spacetimes with the typical global structure of a black hole, see figure 3. It is an important consistency check that when evaluated on the (outer) bifurcation surface of a black hole, the functional (2.4) reproduces Wald's formula [18][19][20] for black hole entropy in higher derivative theories.
However, due to the complexity of the functional (2.4) compared to the area functional of Einstein-Hilbert gravity, it is possible for other closed extremal surfaces to exist. This is sketched in figure 2. A priori it is not clear in this case whether the RT-like and HRT-like prescriptions will both correctly identify the (outer) bifurcation surface as the closed extremal curve associated to total CFT entropy. This is addressed in section 6, after investigating the existence of such surfaces in NMG and 4+1-dimensional Gauss-Bonnet gravity in the following sections. 4 Extremal curves in New Massive gravity

NMG and proposed entropy functional
The action for New Massive gravity (NMG) with cosmological parameter λ in its Lorentzian form is given by [29,30] where the mass parameter m 2 can take any sign, and σ = ±1 is the sign of the Einstein-Hilbert term. This theory has been investigated in a holographic contect for example in [36][37][38][39][40][41][42][43]. For later convenience, some important properties of this theory are detailed in the following [30]: For λ ≥ −1 the equations of motion from the above action admit maximally symmetric vacua as solutions. In particular, NMG admits AdS 3 vacua with the AdS radius ℓ (and Λ = −1/ℓ 2 ) determined by the real solutions of the equation When linearising around a maximally symmetric background with curvature Λ, negative energy gravitons (ghosts) are avoided when implying a clash between unitarity and positive energy in the bulk, and positive central charge of the boundary CFT: the ghost-free condition (4.3) and the condition c ≥ 0 are mutually exclusive. Amongst other solutions, NMG gravity admits the BTZ black hole [46,47], whose entropy is proportional to the above central charge [30,45]: r + is the radius of the (outer) event horizon. Positivity of BTZ entropy hence demands the positivity of the central charge.
For NMG, the functional (2.4) for the holographic calculation of entanglement entropy reduces to [48] whose evaluation on a particular extremal surface (or curve, since we are in 2+1 dimensions) is proposed to compute entanglement entropy in the dual boundary theory. The integral is performed along a curve parametrised by τ , with the induced metric √ g τ τ = g µν dx µ dτ dx ν dτ . In Einstein-Hilbert gravity, the entropy functional in 2+1 dimensions just computes the length of a path, and its extremisation produces the geodesic equations of motion. However in NMG, the evaluation of functional (4.7) no longer has the interpretation of length, due to the presence of additional curvature terms.
In the following we investigate the nature of the extremal curves corresponding to (4.7) in (partially) static backgrounds by deriving and solving the equations of motion. In the light of section 3 we pay particular attention to closed extremal curves in black hole backgrounds.

Equations of motion of NMG entropy functional
As we mentioned in section 2, in using the functionals (2.4) to calculate entanglement entropy holographically, first the correct co-dimension two surface upon which the functional is evaluated needs to be found. It is hoped that one way of locating these surfaces is by directly extremising the functionals [16], and in light of this we investigate the possible curves which extremise (4.7), considering global AdS 3 and non-rotating BTZ black hole backgrounds. In 2+1 dimensions, these metrics take the form: in Schwarzschild coordinates. The global AdS 3 metric is obtained by setting the BTZ black hole mass M = −1. We obtain the curves which extremise (4.7) as follows: By considering curves that lie in a constant time slice of t = 0, 6 we may choose the parameterisation r = f (φ), i.e. the progression of the curve into the bulk spacetime is given as a function of the (boundary) coordinate φ. This can be inserted into (4.7), giving with the extrinsic curvature term The corresponding fourth order Euler-Lagrange equation of motion for f (φ) is quite involved, and we relegate the explicit expression to equation (B.3) of appendix B.
In following we discuss this equation and its possible solutions, which was in part already done in [42]. We begin with solutions anchored to the boundary.

Curves anchored at the boundary
The fourth order nature of equation (B.3) arises from the presence of the extrinsic curvature term k 2 in (4.7), and therefore to find a unique solution when solving this equation we have to specify initial conditions up to third order in derivatives, i.e. values f However it is first interesting to note that the geodesics of background (4.8), which are used to calculate entanglement entropy holographically in Einstein-Hilbert gravity, solve (B.3) independently of the NMG parameter m 2 . For global AdS 3 this was already noted in [42]. Setting ℓ = 1, these geodesics are [25,33] for global AdS 3 , and for a non-rotating BTZ background, with event horizon radius r + = √ M .
Other solutions can be found numerically, solving for curves with a turning point f ′ (φ 0 ) = 0 at some initial radius f (φ 0 ) in the bulk. This allows us by symmetry to set f ′′′ (φ 0 ) = 0, leaving the freedom to specify f ′′ (φ 0 ). In global AdS 3 , we find non-geodesic curves anchored to the boundary when the collective NMG parameter σℓ 2 m 2 is negative. In fact, for a given finite interval A on the boundary, for this parameter range there seems to exist an infinite number of different curves attached to ∂A. For the non-rotating BTZ black hole background, we again appear to find an infinite number of curves anchored to a given boundary region for σℓ 2 m 2 > 0, figure (4). The freedom to specify an initial condition on f ′′ (φ 0 ) in the above appears to be behind the emergence of the multiplicity of extremal curves associated to a given boundary interval. Notably, there also appears to be an infinite subset of these curves which satisfy the homology constraint. We postpone an in-depth discussion on which of these curves would be chosen by the RT-and HRT-like prescriptions until section 6. The fact that the analytically known geodesic solutions (4.10) and (4.11) still solve the equation of motion (B.3) in this higher derivative context a priori is quite surprising. Before studying the closed extremal curves of (4.7), we comment on this result in the following: For an AdS background, in [49] Casini, Huerta and Myers showed that the entanglement entropy of spherical regions on the boundary can be calculated by a conformal mapping of the region to hyperbolic space. The gravity dual of this hyperbolic space then contains a topological black hole, and evaluating the Wald entropy of this black hole gives the entanglement entropy of the spherical CFT region. As discussed in [49], these topological black holes are related to certain foliations of AdS space, and hence only depend on the geometry of the locally AdS background and not on the bulk-gravity Lagrangian. Therefore, in calculating entanglement entropy for spherical regions on the boundary, the corresponding entangling surface in the bulk turns out to be independent of higher curvature corrections, as found in [48]. These findings are especially interesting in the light of NMG, as here every connected boundary region is necessarily an interval, i.e. a 1-ball. Hence one might think that, when working with global AdS-space or the BTZ metric (which is locally AdS), the results of [49] apply to any such region. In this way, they would give us the corresponding entanglement entropy by simply evaluating (4.7) on the geodesic that one would use to calculate entanglement entropy in the Einstein-Hilbert case. This would make it unnecessary to solve the full equation (B. 3) at all. The fact that the geodesics indeed solve (B.3) in AdS and BTZ backgrounds independently of m 2 seems to lend credibility to this idea, however there is a small caveat: In [49] it was explicitly stated that the calculations made there implicitly assume that the boundary CFT is unitary, which is not necessarily the case in NMG -see section 4.1. Independently of this argument due to Casini et al., in this paper we are interested in which extremal surfaces are permitted by the RT-and HRT-like prescriptions for entanglement entropy, and therefore consider the full range of solutions to the equations of motion (B.3).
In the next section we consider the possible closed extremal curves, which we are to study analytically.

Closed extremal curves in NMG
For the study of closed extremal curves, we restrict our attention to black hole backgrounds. In particular, for this section we focus on the non-rotating BTZ black hole while in appendix C we comment on the rotating case, and Lifshitz black holes. In section 3 the significance of closed extremal curves of holographic entropy functionals in black hole backgrounds was explained.
While the Schwarzschild coordinates (4.8) do not cover the BTZ black hole event horizon, following the discussion of section 3 the BTZ black hole bifurcation surface r + = ℓ √ M is a closed extremal surface of entropy functional (4.7). The corresponding value of the entropy functional is equal to the entropy of the black hole (4.6): For the same range that the multiplicity of extremal curves anchored to the boundary appears in the non-rotating BTZ background (σℓ 2 m 2 > 0), we find an additional closed extremal curve  from the centre. Notice that r a ≥ r + is equivalent to the BF bound (4.4). Our results for these additional closed bulk extremal curves are summarised in figures 5 and 6. The value of the entropy functional corresponding to r a is (4.14) Comparing this value with the entropy of the BTZ black hole S BT Z , we find |S BT Z | ≥ |S EEa | with equality for σℓ 2 m 2 = 1/2 where r a = r + , i.e. where the BF bound is saturated. For positive central charges (and hence positive BTZ entropy), the additional extremal surfaces will correspond to a lower entropy than the Wald entropy of the black hole. Hence in both RTand HRT-like prescriptions, naively invoking only the homology constraint and that the entropy is minimised would lead one to incorrectly identify S EEa with the entropy of the dual CFT for certain choices of σℓ 2 m 2 . We elaborate on how this changes in the HRT-like prescription when causality constraints are imposed in section 6.
The existence of additional closed extremal curves is not restricted to the non-rotating BTZ background: they also exist for the rotating BTZ and Lifshitz black hole backgrounds, see appendix C. Furthermore these curves are not exclusive to NMG, and in the following section we study the existence of closed extremal surfaces in Gauss-Bonnet gravity.

Closed extremal surfaces in Gauss-Bonnet gravity
In this section we comment on the existence of closed extremal surfaces of the entropy functional in Gauss-Bonnet gravity, which is a special case of Lovelock gravity [31] (see [32] for a review) and has been extensively studied before in the holographic context, see for example [21,50] and [51,52] for reviews. While NMG is afflicted with problems concerning unitarity in the bulk and on the boundary, Gauss-Bonnet gravity is believed to be much better behaved and understood in a holographic framework. It would therefore be interesting to examine whether additional extremal surfaces appear in this context. For simplicity we will restrict our discussion to five bulk dimensions, in which the (Lorentzian) action reads adopting the conventions used for example in [50]. The conjectured boundary theory is causal for [53][54][55][56] and it is possible to choose an AdS vacuum such that the bulk theory is ghost free [57] and the dual CFT is unitary (see e.g. [51,52]). The entropy functional (2.4) takes the form where the Gauss-Codazzi equations have been used to absorb the extrinsic curvature terms into the intrinsic scalar curvature R of Σ. This functional, known as Jacobson-Myers functional, has already been derived in [58] in the context of black hole entropy. It was later proposed for the holographic calculation of entanglement entropy in [14,21,24,50].
At the level of the entropy functional it is possible to conclude that spherically symmetric spacetimes in Gauss-Bonnet gravity also admit additional closed extremal surfaces: Suppose we are working with a stationary spherically symmetric black hole background of Gauss-Bonnet gravity (see e.g [59][60][61][62]), given in Schwarzschild-like coordinates t, r, θ, φ, ψ. Similar to section 4.4, we assume that the spacelike surface Σ is adopted from the spherical symmetry of the spacetime, i.e. that Σ is a 3-sphere parametrized by t =const., r =const., with θ, φ, ψ arbitrary. It then follows that Σ √ γ = Area(Σ) = 2π 2 r 3 and R = 6/r 2 , such that the functional (5.3) takes the form For negative λ this has a minimum at finite r = √ −2λL, implying the existence of a closed extremal surface at this radius. Interestingly, the case λ < 0 is often neglected in the literature as one motivation for studying the model (5.1) is that it might appear as correction to Einstein-Hilbert gravity in a string-theory expansion, where λ > 0 [57].
Note that the above argument for closed extremal surfaces is independent of the topology of the bulk spacetime, however in a stationary spherically symmetric black hole spacetime, in addition to the surface found above, the black hole bifurcation surface would also be a closed extremal surface. The bifurcation surface would not appear as an extremum in the above analysis as Schwarzschild-like coordinates are not well defined there, though it was explained in section 3 that bifurcation surfaces are always saddle points of the entropy functionals.
Of course, the additional extremal surfaces found above for λ < 0 are unphysical: They correspond to a negative value of the entropy S EE , while in the CFT the entropy is defined via the density matrix ρ as Von Neumann entropy which is manifestly non-negative 9 . One might therefore doubt that there exists a sensible holographic picture of the theory (5.1) for λ < 0. Nevertheless, the purpose of this section was merely to show that additional closed extremal surfaces are not a special feature of NMG, but might appear in a wide variety of higher curvature theories. In particular, we expect that an analysis similar to the one performed above could be done for general Lovelock theories in higher dimensions. It might then very well happen that additional closed extremal surfaces appear even in more physical situations.
It was recently shown [48,[64][65][66], that the equations of motion originating from functionals that depend only on intrinsic curvature terms (such as (5.3)), take the general form ij describes the extrinsic curvature projected onto Σ. For Gauss-Bonnet gravity, we have Assuming the extremal surface to be a hypersphere with R ij = γ ij R/3 and R = 6/r 2 as above, it is then easy to show that the additional closed bulk extremal surface at r = √ −2λL exactly solves (5.6), as then X ij = 0. See also section 7 for a discussion of these equations in the context of the alternative conical boundary condition method.

Closed bulk extremal surfaces and the causal influence argument
In this section we will explain that the presence of the additional extremal surfaces found in this paper, in particular the additional closed extremal surfaces, indicates a subtle difference between RT-and HRT-like prescriptions, when causality conditions are taken into account. 10 More specifically, we will show that in HRT-like prescriptions, an argument based on causality can be used to argue that these surfaces cannot be used to calculate entanglement entropy.
In section 2 we explained how the area functional can be employed to calculate holographic entanglement entropy in Einstein-Hilbert gravity, with subtle differences between the RT and HRT prescriptions that could be resolved by imposing causality conditions on the latter, see also [25,27]. Spacetimes for which such differences were found in [25] are the Reissner-Nordström black hole and the bag-of-gold spacetime, where an eternal black hole is matched to a compact spacetime bubble along a static shell of matter [67,68].
As was furthermore explained in section 2, for higher curvature theories such as (2.3), functionals (2.4) generalising the area functional have been proposed to calculate entanglement entropy holographically. As these functionals where derived in Euclidean settings in [14][15][16][17], a priori these functionals should be seen to be most reliable in an RT-like setting. However, as all the spacetime backgrounds that are considered in the present work are at least partially (i.e. not necessarily globally) static, one would naively expect RT-and HRT-like prescription in using these functionals to agree. Here we will now consider how the imposition of a causality argument due to Headrick [27,28], which we refer to as the causal influence argument, affects this expectation in our cases: The principle behind the causal influence argument is to avoid causality paradoxes similar to the well known grandfather paradox. Suppose one wanted to calculate the entanglement entropy of a certain boundary region A (at boundary time t = 0) using a bulk co-dimension two surface E A . If E A were to lie in the future of A, i.e. that there were to be a future pointing timelike curve from A to (at least one point on) E A , the following paradox might arise: An observer living on (or near) the boundary might immediately after t = 0 send some energy into the bulk in such a way that the geometry around the part of E A in the future of A is changed. This would also affect the surface E A itself, hence potentially the associated entropy. This has the implication that it would be possible -after the entanglement entropy was fixed from the point of view of the CFT -to alter the result of the holographic entanglement entropy calculation in the bulk, leading to an obvious paradox. We therefore demand that there should be no timelike curve from A to E A , and by time inversion symmetry also not from E A to A. Similarly, the same holds for E A andĀ, the complement of A. Hence, causality implies that E A should be required to lie in the causal shadows of the boundary regions A andĀ, i.e. there should be no timelike curves connecting E A to one of the two regions 11 . For reasons that will soon become clear, we refer to this as the weak form of the causal influence argument. The explanation presented here may of course be at most a motivation and not a stringent derivation of the necessity to impose this causality condition on the extremal surfaces, however we will find this condition to be very useful in the discussion below.
This condition can still be strengthened, and to explain how let us describe the way in which this argument comes into play in RT-and HRT-like prescriptions. As explained in section 2, the RT-like prescription assumes a static bulk spacetime, in which due to the presence of a timelike Killing vector field one can unambiguously define a foliation of the bulk spacetime (as well as of the conformal boundary) into spacelike slices. Both the boundary region A and the co-dimension two surface E A are then embedded in one of these spacelike slices by construction and the weak form of the causal influence argument is therefore trivially satisfied, see the left side of figure 7.
On the other hand, the HRT prescription is intended to work for general spacetimes, and therefore the existence of a uniquely defined foliation of the bulk spacetime cannot be assumed. The holographic entangling surface belonging to A lies by construction on a similarly defined equal time slice of the bulk spacetime, and hence in the causal shadow of A. The weak form of the causal influence argument is hence satisfied by construction. In an HRT-like prescription (right), A can be deformed within its domain of dependence ♦ A as shown in figure 8. The holographic entangling surface then has to be in the causal shadow of ♦ A . Even for a partially static spacetime where RT and HRT are expected to be equivalent, this strong form of the causal influence argument is a severe restriction, as the part of the equal time slice inside of A would also be excluded by this argument. The region A in which signals can be both send to and received from ♦ A is called the causal wedge, see [69]. The complementĀ of A and its domain of dependence ♦Ā cause a similar causal shadow that extends into the bulk.
Similarly, the boundary region A on the conformal boundary does not need to lie on an equal time slice of the boundary time. As long as the boundary ∂A (to which E A will be anchored) is held fixed, the entanglement entropy of A, and E A , are supposed to be independent of deformations of A within its domain of dependence ♦ A , see figure 8. The domain of dependence ♦ A is defined as the set of all points on the boundary where every causal curve going through one of these points necessarily intersects A. Due to the possibility of deforming A in the HRT prescription, the causal influence argument can be formulated in what we refer to as its strong form: See also the right of figure 7 for an illustration.
It is very interesting to note that although RT-and HRT-like prescriptions are expected to agree on static spacetimes, the strong form of the causal influence argument is nontrivial in a static spacetime: It excludes a part of the spacelike slice on which E A would be located by construction in the RT-like prescription, and which in the RT-like prescription would not be excluded by any simple and obvious conditions. This means that the strong form of the causal influence argument points out a far from trivial difference between RT-like prescriptions and HRT-like prescriptions amended with a causality argument 12 in general gravitational theories of the form (2.3). For Einstein-Hilbert gravity it has been proven in [26,33,69] that holographic entangling surfaces anchored at the boundary can never enter the causal wedge of the corresponding boundary region. 13 So this impies that if the strong form of the causality condition is satisfied in a HRT-like prescription, and if the HRT-like prescription (supplemented with the causality condition) is to agree with the RT-like prescription on static spacetimes, then the latter needs to obey conditions that can be derived in the Euclidean setting and act as a precursor to the causality argument in a Lorentzian setting. Figure 8: In an HRT-like approach to entanglement entropy, the entangling region is not necessarily fixed to an equal time (t =const.) slice such as A. In fact, as long as the region stays spacelike and the boundary ∂A stays fixed (and hence stays inside of the domain of dependence ♦ A ), one can deform A to take shapes such as A ′ and A ′′ in the above picture.
In the previous sections 4.4 and 5 we showed that functionals of the form (2.4) may in many cases allow for additional closed extremal surfaces in global AdS and AdS black hole spacetimes, see e.g. figure 5. Due to standard AdS/CFT results, these surfaces can be expected to be unphysical. In the HRT prescription, the strong form of the causal influence argument presented above can be used to rule out these additional extremal surfaces: It is clear that at least when the additional extremal surfaces are outside of the black hole event horizon (case b) in figure 5), for a given time slice of the bulk spacetime there will always be an additional extremal surface on this slice homologous to the full boundary. In an RT-like prescription, the conditions on the extremal surfaces 14 would lead one to deduce that the additional extremal surface outside the black hole bifurcation surface determines the total entropies of CFT 1 and CFT 2 , since it gives a lower entropy. This would be a serious problem, as it would imply a mismatch between CFT and black hole entropy. Now let's look at this 12 Recall that for Einstein-Hilbert gravity, in [25] these causality arguments were argued to be necessary to make the RT and HRT prescriptions agree in certain cases. 13 Furthermore, in [27] a proof of the strong form of the causal influence argument in Einstein-Hilbert gravity is presented, assuming the null energy condition and some other technical details.
14 The homology condition and minimisation of the entropy.
problem in a HRT-like (i.e. manifestly Lorentzian) framework, where causality arguments can be used. Consider the setup depicted in figure 9. If A is a complete equal time slice of the right boundary (and henceĀ of the left), then ♦ A is the complete right boundary in its full extent in space and time (and similarly ♦Ā is the complete left boundary). Requiring that the entangling surface(s) corresponding to this division of the total system into subsystems A andĀ are not connected to any point on the boundaries via timelike curves leaves the black hole bifurcation surface as the only possible extremal surface, see figure 9. Therefore, the strong form of the causal influence argument leads (in the cases studied in this paper) to an agreement between black hole entropy and holographic CFT entropy. Figure 9: When the region A in the CFT is a full equal time slice of the right boundary, the causal wedges A and Ā together cover the entire region outside the event horizons of the black and white holes. As illustrated by the dashed green line, all closed extremal surfaces outside the black hole event horizon lie in this region and are thus ruled out by the strong form of the causal influence argument. The dotted red line representing an additional closed extremal surface inside the event horizon is also forbidden, as it is timelike connected to ♦ A and ♦Ā. The intersection of causal shadows of both ♦ A and ♦Ā is therefore the bifurcation surface B of the black hole (which is called the causal shadow of the spacetime), leaving it as the only permissible closed extremal surface.
Let us now come to the additional extremal curves anchored at the boundary that we found in section 4.3, see especially figure 4. The calculations of that section were carried out on AdS and BTZ background spacetimes. For such background spacetimes, it is known that (on an equal time slice) the boundary of the causal wedge of a certain boundary region A will be given by a spacelike geodesic of the form (4.10) and (4.11) respectively, see [33]. We can now easily see that the additional (non-geodesic) extremal curves found in section 4.3 are ruled out by the strong form of the causal influence argument by examining figure 4. There, we see two geodesics anchored at the boundary, drawn as thick dashed (red and blue) lines. The first one (drawn in red) is similar to the curve labeled by C in figure 2 and marks the boundary of the causal wedge A corresponding to the boundary region A. The second one (drawn in blue) is the equivalent of curve D in figure 2 and marks the boundary of Ā, the causal wedge ofĀ.
In figure 4 it is now easy to see that all of the additional, non-geodesic extremal curves found in section 4.3 are excluded by strong form of the the causal influence argument: They either enter A or Ā, and hence leave the causal shadow of either A orĀ. Only the geodesics which are located exactly at the boundary of the causal wedges are not excluded by the strong form of the causal influence argument.
In summary we find that, at least for (partially) static spacetimes in NMG and Gauss-Bonnet gravity, only the HRT-like prescription amended with the strong form of the causal influence argument can ensure physical results for entanglement entropy. This is in contrast to the Einstein-Hilbert case, where the RT and the HRT prescription amended with the causal influence argument agree on such spacetimes. For RT-like prescriptions, due to their manifestly Euclidean nature, it is not possible to make arguments based on causality. Nevertheless, if RTlike prescriptions and HRT-like prescriptions with causality conditions are supposed to agree on (partially) static spacetimes, the surfaces found in the RT setting need to obey the causality conditions which only make sense in a covariant framework. This implies the fascinating possibility that there exists a condition that has to be imposed in the RT-like approach in addition to the homology constraint, and which can be derived by arguments completely independent of causality. This condition would act as a precursor to the (strong form of the) causal influence argument that comes into effect in the Lorentzian setting. Perhaps a better understanding of holographic entanglement entropy would hopefully give conditions from first principles that exclude such pathological surfaces.
In the next section, we will show for Gauss-Bonnet gravity that indeed also in the RT-like setting one can derive conditions that rule out the additional closed extremal surfaces and fix the holographically computed CFT entropy to be equal to the black hole entropy calculated on the bifurcation surface. We will speculate on a possible connection between these conditions and the causality constraint in section 8.2.

Conical boundary condition method in Gauss-Bonnet gravity Method
In [12], Lewkowycz and Maldacena proposed a method of calculating entanglement entropy holographically for CFT duals to Einstein-Hilbert gravity by introducing the concept of generalised gravitational entropy. This is an extension of the usual Euclidean methods for calculating black hole entropy to solutions without U(1) symmetry in Euclidean time. In the present context, it is important to note that this method is applicable only to spatial regions in static spacetimes. To calculate entanglement entropy, the basic idea is to translate the replica trick into the bulk, where a co-dimension two hypersurface with conical defect is introduced. By expanding the bulk equations of motion about the conical singularity, and demanding finiteness of the energy-momentum tensor, the location of the holographic entangling surface can be determined. Thus entanglement entropy can be calculated. The details of this procedure for Einstein-Hilbert gravity were given in [12]. It was also applied to Gauss-Bonnet gravity in [48,64,65], and to general higher curvature theories of the form (2.3) in [16,17].
Here, i = 1, 2, ..., d − 1 and ε is a small parameter that is later taken to zero in the replica trick. k (α)ij (α ∈ {τ, q}) is the extrinsic curvature of the surface, and y i are its induced coordinates. The conical singularity is localised along the surface at q = 0.
In Einstein-Hilbert gravity, by demanding that the equations of motion near the singularity are satisfied and regular, one arrives at the condition [12] k (α) = 0, (7.3) i.e. the co-dimension two surface is a minimal area surface, thus proving the RT prescription. Note that this is only valid in static scenarios, and is therefore not a proof of the HRT prescription.
For 4+1-dimensional Gauss-Bonnet gravity, upon inserting the metric (7.1) the following components of the equations of motion become singular [48]: i, j ∈ {1, 2, 3} are the indices of the induced metric on the co-dimension two surface, with intrinsic Ricci tensor R ij and k = Tr k ij . In the above equation, there is only one extrinsic curvature k ij as we are assuming a static background spacetime. In such a background, the two extrinsic curvatures of a surface on an equal time slice are always linear combinations of only one tensor k ij , independently of the choice of normal coordinates. By demanding that the singular components vanish, constraints on the entangling surfaces in 4+1-dimensional Gauss-Bonnet gravity can be derived. We utilise these constraints in the following to investigate whether the closed extremal surfaces of the Gauss-Bonnet entropy functional (5.3) are valid in the holographic calculation of entanglement entropy.

Conditions on closed bulk surfaces
As shown in section 5, assuming spherical symmetry the hypersphere of radius r = √ −2λL on a constant time slice t = const. is a closed extremal surface of the Gauss-Bonnet entropy functional (5.3) regardless of the topology of the bulk spacetime. That is, it is an extremal surface for a static spherically symmetric background of the form with the line element of the hypersphere independently of functions f (r) and h(r) in (7.7). For a static black hole spacetime, the black hole bifurcation surface is also an extremal surface of the functional.
To subject these extremal surfaces to the constraints arising from the equations of motion (7.4)-(7.6), the corresponding extrinsic curvatures need to be calculated. For a given co-dimension two surface characterised by embedding functions x µ = X µ (y), the extrinsic curvature can be calculated via [65] 15 Here, µ, ρ, σ ∈ {0, ..., 4} are indices of the full spacetime with Christoffel symbols Γ µ ρσ , i, j ∈ {1, 2, 3} are indices of the induced metric on the co-dimension two entangling surface and α ∈ {1, 2} labels the two orthonormal vectors (n (α) ) µ to this surface, see also appendix A.
For the extremal hyperspheres, we will use coodinates θ, φ, ψ, in which terms of the form ∂ i X µ become Kronecker-deltas. By symmetry, the extrinsic curvature k (1) ij with respect to the normal vector n (1) = ∂ t will vanish identically, so we are left with where by slight abuse of notation we introduced "shifted indices" m, n ∈ {θ, φ, ψ} 16 . The singular equations of motion (7.4)-(7.6) simplify greatly upon insertion of (7.10): The qicomponent vanishes trivially as it only contains covariant derivatives ∇ with respect to the induced metric γ ij and k ij = const.(r) · γ ij . Using that for the hyperspheres R ij = γ ij R/3 and R = 6/r 2 , the qq-component simplifies to Similarly, the ij-component reads (7.12) 15 In contrast to the quantity k (α) µν presented in appendix A where µ, ν were indices of the full spacetime, this quantity is the extrinsic curvature projected to the internal space of the hypersurface. 16 We refer to these as shifted because in the latin indices the θ-component corresponds to i = 1, while in the greek indices the same component corresponds to µ = 2. So in the usual notation the above equation would imply k 11 ∼ Γ 1 22 etc.

Implementing the conditions
It is clear that a bifurcation surface, which necessarily is a geometrical extremal surface with k = 0, 17 makes both (7.11) and (7.12) vanish identically. Additionally, we see that the leading divergence in (7.11) vanishes when the functional (5.4) is extremised, i.e. for any extremal curve of the functional (5.3). However for the additional extremal hypersphere of the Gauss-Bonnet entropy functional of radius r = √ −2λL, and extrinsic curvature (7.10), the remaining term in (7.11), and (7.12) do not vanish. It was exactly the vanishing of this leading divergence in the qq-component that was shown in [16] to be equivalent to the equations of motion derived from extremising the proposed entropy functional. Nevertheless, it is apparent from the above computations that the other components (and perhaps the less divergent terms) may contain vital information that is needed to rule out the unphysical extremal surfaces that one finds by varying the entropy functional. As (7.5) and (7.6) are proportional to λ, it is immediately apparent that these components do not play any role in Einstein-Hilbert gravity (λ = 0).
Thus, at least for the closed extremal curves in Gauss-Bonnet gravity, the conditions derived from the conical boundary condition method yield a restriction on the RT-like prescription that ensures physicality of the resulting entropy. As explained in section 6, such conditions are needed in order to play the role in the RT-like prescription that the causal influence argument plays in the HRT-like prescriptions. We would however like to add that although the conditions (7.11) and (7.12) are successful in ruling out the additional closed extremal curves in the RT-like prescription, there are cases when these constraints prove to be too restrictive. For example, it was shown in [64] that the extremal surfaces of the entropy functional corresponding to cylindrical boundary regions fail to satisfy the conditions coming from the subleading divergent component (7.4) near their turning point in the bulk. 18 The conditions we have discussed in this section are therefore not yet complete for the RT-like prescription. Perhaps the additional conditions coming from subleading terms and the other components (such as the ij-component in (7.6)) can only be imposed, if at all, when there is not a unique curve extremising the functional for given boundary conditions.
It is interesting to note that similar conditions on the entangling surfaces in static settings can be derived independent of the conical boundary condition method (i.e. the replica trick), by considering the Brown-York stress tensor T computed on a static co-dimension one hypersurface extending into the bulk [64,71]. The profile of this hypersurface that defines the extension of it into the bulk is then a spacelike co-dimension two hypersurface located on an equal time slice, just as in the calculation of entanglement entropy. Indeed, for Einstein-Hilbert gravity, demanding T tt = 0 results in a minimal surface condition on the profile of the hypersurface, in agreement with the RT prescription [71]. This seems to imply a connection between entanglement entropy and the Brown-York stress tensor. In [64] the same approach was investigated for Gauss-Bonnet gravity, and it was shown that demanding T tt = 0 imposes the following equation on the profile of the co-dimension one hypersurface: Although not equal, this equation bears a remarkable similarity with the divergent part of the qq-component (7.4) derived above from the conical boundary condition approach. While the first two terms in (7.13) are exactly the equation of motion derived from the Jacobson-Myers functional, the last term corresponds to the subleading term in (7.4). This means that from the discussion above, the additional closed extremal surfaces fail to satisfy (7.13) while only a black hole bifurcation surface would do so. For a spherical region on the boundary, the holographic entangling surfaces make the first two terms and the third term in (7.13) vanish separately. Hence they are extremal curves of the Jacobson-Myers functional and additionally satisfy the condition of third order in extrinsic curvature that we needed to rule out the additional closed extremal curves in the bulk. However, as mentioned in the above, the second term remains too restrictive for other boundary regions, such as a cylinder.
For NMG, in principle it is possible to perform analyses similar to those presented in this section, yet this is a very tedious task. Nevertheless, at least for the closed extremal curves encircling the BTZ black hole it would likely yield the same results. It was already mentioned in [12] that in cases with a full U(1) symmetry in Euclidean time, the conical boundary conditions method reproduces Wald's entropy for Einstein-Hilbert gravity as well as for higher derivative theories.

Summary
Let us begin this final section with a summary of this paper. We have investigated functional prescriptions of calculating entanglement entropy holographically in higher derivative gravity theories, using New Massive gravity (short NMG) and Gauss Bonnet gravity as concrete examples. We emphasised that in using entropy functionals to calculate entanglement entropy of a given region in a dual CFT, a non-trivial step is to find the particular surface upon which the corresponding functional is to be evaluated. The location of these surfaces can in principle be determined by solving the equations of motion with conical boundary conditions [12], however this is very involved and it is hoped that they could be alternatively determined by extremising the entropy functional [16]. In this work we considered the latter approach in (partially) static spacetimes (i.e. AdS and black hole backgrounds), introducing the RT-like and HRTlike prescriptions which are based on extremising the entropy functional (2.4) proposed for higher derivative theories of square order in the curvature. The RT-like prescription translates the Ryu and Takayanagi prescription for (partially) static spacetimes restricted to a constant time-slice in Einstein Hilbert gravity to higher curvature theories. The HRT-like prescription on the other hand is based on the Hubeny-Rangamani-Takayanagi prescription, which applies also to dynamical set ups. In both the RT-and HRT-like prescriptions, the extremal surfaces used to calculate CFT entropy holographically, for physicality, must be homologous to the full boundary. However in the HRT-like prescription, additional constraints can be imposed which arise from the Lorentzian nature of the spacetimes.
In a (partially) static scenario, the RT-and HRT-like prescriptions should agree, at least when a causality condition is imposed on the latter. To investigate this, we studied the nature of the extremal surfaces of the entropy functionals in NMG and Gauss-Bonnet gravity in sections 4 and 5 respectively. In particular we focused on closed extremal surfaces in black hole backgrounds, whose significance in relation to dual CFT entropy was explained in section 3. The black hole bifurcation surface will always be a closed extremal surface of the entropy functional, however we discovered that for certain parameter ranges in the two theories a closed extremal surface additional to the bifurcation surface also exists. For the non-rotating BTZ black hole background in NMG, this surface can encircle the black hole event horizon (see figures 5 and 6), and evaluates the entropy functional at a lower value than that given by the black hole bifurcation surface. As explained in sections 2 and 3, a naive implementation of the RTand HRT-like proposals which employs only the homology constraint would hence require us to identify the CFT-entropy with the value given by the additional closed extremal surface, instead of the expected black hole entropy.
Since NMG is plagued by several problems concerning unitarity in the bulk and on the boundary, in section 5 we investigated closed extremal surfaces for Gauss-Bonnet gravity, which is believed to be much better behaved and understood in a holographic context. We found that this theory also allows for additional closed bulk extremal surfaces, although for the rather unconventional choice of the Gauss-Bonnet coupling parameter λ < 0 (or λ > 0 in hyperbolic spacetimes).
To us, our findings suggest that any complicated enough functional of the type (2.4) can in principle, at least for certain choices of the parameters, allow for additional closed bulk extremal curves around black holes.
This would have the implication that, when naively employing the prescriptions for calculating holographic entanglement entropy, the phenomenon of a seeming mismatch between CFT entropy and black hole entropy could be quite common in higher curvature theories. We hence adopted the view that the additional bulk extremal surfaces described above are unphysical.
In section 6 we identified the strong form of the causal influence argument [27,28] as a possible way to rule out the additional extremal surfaces encountered in this work. Nevertheless, this argument is intrinsically only applicable to entanglement entropy in an HRT-like approach, i.e. crucially it is only applicable in a Lorentzian setting. Since the additional extremal surfaces found appear in both Lorentzian and Euclidean settings, it would certainly be desirable to find a way to rule them out in an RT-like approach too.
In search of a means to dismiss the additional extremal surfaces in the RT-like approach, in section 7 for the example of Gauss-Bonnet gravity we turned to the alternative conical boundary condition method of finding entangling surfaces, based on translating the replica trick into the bulk [12]. Although it was argued in [16] that the equations of motion derived by extremising the functional (2.4) are equivalent to the conditions arising from the conical boundary condition method, we showed that this method could in fact give rise to further conditions that rule out the additional closed extremal surfaces, leaving only the black hole bifurcation surface (if present) as the correct answer for the surface determining the full CFT entropy. 19

Outlook
Spacetime from entanglement We have given arguments throughout this work why the extremal surfaces additional to the black hole bifurcation surface should not be considered as defining the physical entanglement entropy of the CFT. Recently in [35] a number of results about entanglement entropy in the RT prescription for Einstein-Hilbert gravity with matter satisfying the weak energy condition were proven. Amongst other properties, it was shown that in this case, the extremal surface computing the full CFT entropy will always be the bifurcation surface of the bulk event horizon, if it exists. Hence the existence of additional extremal surfaces is excluded in this setting.
Although there is no straightforward generalisation of many of the proofs given in [35] to higher curvature theories and functionals of the form (2.4), it is interesting to note that the proof of the theorem mentioned above was the only one in that paper which made use of the equations of motion in Einstein-Hilbert gravity (and an energy condition on matter) 20 . In fact, our findings show that when varying functionals of the form (2.4) without imposing causality constraints (as proposed in [16]), this theorem does not generalise to higher curvature theories. We therefore think that this property is in fact the most non-trivial, and hence the physically most interesting of the results proven in [35]. In the past, ideas have been proposed that the holographic entangling surface E A and the homology surface F corresponding to a boundary region A (see figure 1) define the part of the spacetime that can be holographically reconstructed from knowledge of the density matrix ρ A , see [26,35,72]. In the framework of this conjecture, the additional extremal surfaces would be unphysical, as the corresponding homology surface F would not reach as deep into the bulk as for the bifurcation surface, meaning that even full knowledge of the CFT would not be enough to reconstruct the entire spacetime from the boundary down to the event horizon.

ER=EPR
Another nice (but far from rigorous) argument against the validity of these additional extremal surfaces is the "ER=EPR" conjecture (see [73] and the ever growing list of papers citing this). It postulates a connection between non-traversable wormholes (ER) and entanglement (EPR), and is best explained for the example of an eternal black hole such as in figure 3. Here, the two CFTs are proposed to be in an entangled thermodouble state [74], and in the bulk their entanglement is supposed to be described by the presence of the Einstein-Rosen bridge (left side of figure 2). Concretely, the entanglement entropy between the two CFTs that can be calculated from the thermodouble state is equal to the bulk black hole entropy, given by the area of the bifurcation surface B which is exactly the throat of the wormhole. If the additional closed extremal surfaces found in sections 4.4 and 5 could not be ruled out, it would mean that the entanglement entropy between the two CFTs would no longer be equal to the black hole entropy. One might hope that the ER=EPR conjecture could at least still hold qualitatively in such cases, but even this is not true: While the existence of the bifurcation surface B is intimately related to the presence of the wormhole and hence the topology of the spacetime, this is not the case for the additional closed extremal surfaces A. For example in the Gauss-Bonnet case we saw that any spherically symmetric spacetime will have an additional closed extremal surface at radius r a = √ −2λL 2 , i.e. where the functional (5.4) attains its minimum, independently of the spacetime topology. In contrast, when working in Schwarzschild coordinates this functional seems to be completely oblivious to the presence of a black hole bifurcation surface. Indeed, in section 3 we explained that one would have to either work in Kruskal coordinates or invoke arguments employing the global flows of the Killing vector field ∂ t to argue that the black hole bifurcation surface B indeed extremises this functional. From the perspective of ER=EPR, when the full spacetime contains a wormhole entanglement between the two asymptotic boundaries is connected to spacetime topology as well as to extremal surfaces. 21 As the additional extremal closed curves are not dependent on the spacetime topology, they are not consistent with ER=EPR. Yet, as concluded in this paper, these curves can be ruled out for example by the causal influence argument.

Causality and other additional conditions
As explained in section 6, in a Lorentzian (HRT-like) setting the strong form of the causal influence argument elegantly rules out the additional extremal surfaces. In that section we also pointed out that even in (partially) static spacetimes this argument implies that further restrictions should also be imposed in the RT-like approach if both are supposed to agree. It would hence be of great interest to better understand this argument, and whether it maybe is only the Lorentzian corollary of a more general restriction that has to be imposed both on HRTand RT-like prescriptions. In section 7, we showed for Gauss-Bonnet gravity in a Euclidean setting that such conditions can indeed be derived from an approach using conical boundary conditions. It would be interesting to find out whether these conditions for general surfaces and theories also have the effect of ruling out surfaces that would violate the causality condition in a Lorentzian setting. This would have the fascinating implication that the Euclidean computations already "know" in a sense about the causality in the Lorentzian setting.

f (R) gravity
Another possible way to further investigate the conditions that have to be placed on holographic entanglement entropy in higher curvature theories would be to look at f (R) theories, where the gravitational Lagrangian is an arbitrary function of the Ricci scalar R. As pointed out in [16], using a field redefinition these theories can be mapped to Einstein-Hilbert gravity minimally coupled to a scalar field, where holographic entanglement entropy can be studied using the area functional (2.1). It would certainly be interesting to study whether f (R) theories allow for additional extremal surfaces similarly to NMG and Gauss-Bonnet gravity, and how these surfaces are mapped into the Einstein frame. This might shed some light on the conditions that have to be imposed on extremal surfaces in higher curvature theories.

C Closed extremal curves of other black hole solutions in NMG
For the more general rotating BTZ black hole we find through calculations similar to those done in section 4 (best performed in Eddington-Finkelstein coordinates [75]) additional extremal curves at radial distances for σm 2 < 0, or when σm 2 > 0 and |J| < M 2 6σm 2 . As r 2 > 0, it depends on the values and signs of M, J, σ, m 2 which of the branches in (C.1) gives a valid solution.
One can also look at Lifshitz black holes [76]  which are solutions to NMG for ℓ 2 m 2 = − 1 2 and σ = +1. In this metric, additional extremal curves appear only for M < 0.