Entanglement entropy of black holes
The entanglement entropy is a fundamental quantity which characterizes the correlations between sub-systems in a larger quantum-mechanical system. For two sub-systems separated by a surface the entanglement entropy is proportional to the area of the surface and depends on the UV cutoff which regulates the short-distance correlations. The geometrical nature of the entanglement entropy calculation is particularly intriguing when applied to black holes when the entangling surface is the black hole horizon. I review a variety of aspects of this calculation: the useful mathematical tools such as the geometry of spaces with conical singularities and the heat kernel method, the UV divergences in the entropy and their renormalization, the logarithmic terms in the entanglement entropy in 4 and 6 dimensions and their relation to the conformal anomalies. The focus in the review is on the systematic use of the conical singularity method. The relations to other known approaches such as ’t Hooft’s brick wall model and the Euclidean path integral in the optical metric are discussed in detail. The puzzling behavior of the entanglement entropy due to fields which non-minimally couple to gravity is emphasized. The holographic description of the entanglement entropy of the black hole horizon is illustrated on the two- and four-dimensional examples. Finally, I examine the possibility to interpret the Bekenstein-Hawking entropy entirely as the entanglement entropy.
- 1 Introduction
2 Entanglement entropy in Minkowski space-time
- 2.1 Definition
- 2.2 Short-distance correlations
- 2.3 Thermal entropy
- 2.4 Entropy of a system of finite size at finite temperature
- 2.5 Entropy in (1+1)-dimensional space-time
- 2.6 The Euclidean path integral representation and the replica method
- 2.7 Uniqueness of analytic continuation
- 2.8 Heat kernel and the Sommerfeld formula
- 2.9 An explicit calculation
- 2.10 Entropy of massive fields
- 2.11 An expression in terms of the determinant of the Laplacian on the surface
- 2.12 Entropy in theories with a modified propagator
- 2.13 Entanglement entropy in non-Lorentz invariant theories
- 2.14 Arbitrary surface in curved space-time: general structure of UV divergences
3 Entanglement entropy of non-degenerate Killing horizons
- 3.1 The geometric setting of black hole spacetimes
- 3.2 Extrinsic curvature of horizon, horizon as a minimal surface
- 3.3 The wave function of a black hole
- 3.4 Reduced density matrix and entropy
- 3.5 The role of the rotational symmetry
- 3.6 Thermality of the reduced density matrix of a Killing horizon
- 3.7 Useful mathematical tools
- 3.8 General formula for entropy in the replica method, relation to the Wald entropy
- 3.9 UV divergences of entanglement entropy for a scalar field
- 3.10 Entanglement Entropy of the Kerr-Newman black hole
- 3.11 Entanglement entropy as one-loop quantum correction
- 3.12 The statement on the renormalization of the entropy
- 3.13 Renormalization in theories with a modified propagator
- 3.14 Area law: generalization to higher spin fields
- 3.15 Renormalization of entropy due to fields of different spin
- 3.16 The puzzle of non-minimal coupling
- 3.17 Comments on the entropy of interacting fields
- 4 Other related methods
5 Some particular cases
- 5.1 Entropy of 2d black hole
- 5.2 Entropy of 3d BTZ black hole
- 5.3 Entropy of d-dimensional extreme black holes
- 6 Logarithmic term in the entropy of generic conformal field theory
- 7 Holographic description of entanglement entropy of black hole
- 8 Can entanglement entropy explain the Bekenstein-Hawking entropy of black holes?
- 9 Other directions of research
- 10 Concluding remarks
- 11 Acknowledgements
One of the mysteries in modern physics is why black holes have an entropy. This entropy, known as the Bekenstein-Hawking entropy, was first introduced by Bekenstein , ,  rather as a useful analogy. Soon after that, this idea was put on a firm ground by Hawking  who showed that black holes thermally radiate and calculated the black hole temperature. The main feature of the Bekenstein-Hawking entropy is its proportionality to the area of the black hole horizon. This property makes it rather different from the usual entropy, for example the entropy of a thermal gas in a box, which is proportional to the volume.
In 1986 Bombelli, Koul, Lee and Sorkin  published a paper where they considered the so-called reduced density matrix, obtained by tracing over the degrees of freedom of a quantum field that are inside the horizon. This procedure appears to be very natural for black holes, since the black hole horizon plays the role of a causal boundary, which does not allow anyone outside the black hole to have access to the events which take place inside the horizon. Another attempt to understand the entropy of black holes was made by ’t Hooft in 1985 . His idea was to calculate the entropy of the thermal gas of Hawking particles which propagate just outside the horizon. This calculation has uncovered two remarkable features: the entropy does turn out to be proportional to the horizon area, however, in order to regularize the density of states very close to the horizon, it was necessary to introduce the so-called “brick wall”, a boundary which is placed at a small distance from the actual horizon. This small distance plays the role of a regulator in the ’t Hooft’s calculation. Thus, the first indications that entropy may grow as area were found.
An important step in the development of these ideas was made in 1993 when a paper of Srednicki  appeared. In this very inspiring paper Srednicki calculated the reduced density and the corresponding entropy directly in flat spacetime by tracing over the degrees of freedom residing inside an imaginary surface. The entropy defined in this calculation has became known as the entanglement entropy. Sometimes the term geometric entropy is used as well. The entanglement entropy, as was shown by Srednicki, is proportional to the area of the entangling surface. This fact is naturally explained by observing that the entanglement entropy is non-vanishing due to the short-distance correlations present in the system. Thus only modes which are located in a small region close to the surface contribute to the entropy. By virtue of this fact one finds that the size of this region plays the role of the UV regulator so that the entanglement entropy is a UV sensitive quantity. A surprising feature of Srednicki’s calculation is that no black hole is actually needed: the entanglement entropy of a quantum field in flat spacetime already establishes the area law. In an independent paper Frolov and Novikov  applied a similar approach directly to a black hole. These results have sparked the interest in the entanglement entropy. In particular, it was realized that the “ brick wall” model of t’Hooft studies a similar entropy and that the two entropies are in fact related. On the technical side of the problem, a very efficient method was developed to calculate the entanglement entropy. This method, first considered by Susskind , is based on a simple replica trick in which one first introduces a small conical singularity at the entangling surface, evaluates the effective action of a quantum field on the background of metric with a conical singularity and then differentiates the action with respect to the deficit angle. By means of this method one has developed a systematic calculation of the UV divergent terms in the geometric entropy of black holes, revealing the covariant structure of the divergences , , . In particular, the logarithmic UV divergent terms in the entropy were found . The other aspect, which was widely discussed in the literature, is whether the UV divergence in the entanglement entropy could be properly renormalized. It was suggested by Susskind and Uglum  that the standard renormalization of Newton’s constant makes the entropy finite provided one considers the entanglement entropy as a quantum contribution to the Bekenstein-Hawking entropy. This proposal however did not answer the question whether the Bekenstein-Hawking entropy itself can be considered as an entropy of entanglement. It was proposed by Jacobson  that, in models in which Newton’s constant is induced in the spirit of Sakharov’s ideas, the Bekenstein-Hawking entropy would also be properly induced. A concrete model to test this idea was considered in .
Unfortunately, in the 90-s the study of the entanglement entropy could not compete with the booming success of the string theory (based on D-branes) calculations of the black hole entropy . The second wave of interest in the entanglement entropy has started in 2003 with works studying the entropy in condensed matter systems and in lattice models. These studies revealed the universality of the approach based on the replica trick and the efficiency of the conformal symmetry to compute the entropy in two dimensions. The black holes again became in the focus of study in 2006 after work of Ryu and Takayanagi  where a holographic interpretation of the entanglement entropy was proposed. In this proposal, in the frame of the AdS/CFT correspondence, the entanglement entropy, defined on a boundary of anti-de Sitter, is related to the area of a certain minimal surface in the bulk of the anti-de Sitter spacetime. This proposal opened interesting possibilities for computing, in a purely geometrical way, the entropy and for addressing in a new setting the question of the statistical interpretation of the Bekenstein-Hawking entropy.
The progress made in the recent years and the intensity of the on-going research indicate that the entanglement entropy is a very promising direction which in the coming years may lead to a breakthrough in our understanding of the black holes and Quantum Gravity. A number of very nice reviews appeared in the recent years that address the role of the entanglement entropy for black holes , , , ; review the calculation of the entanglement entropy in quantum field theory in flat spacetime ,  and the role of the conformal symmetry ; focus on the holographic aspects of the entanglement entropy , . In the present review I build on these works and focus on the study of the entanglement entropy as applied to black holes. The goal of this review is to collect a complete variety of results and present them in a systematic and self-consistent way without neglecting neither technical nor principal aspects of the problem.
2 Entanglement entropy in Minkowski space-time
Consider a pure vacuum state of a quantum system defined inside a space-like region and suppose that the degrees of freedom in the system can be considered as located inside certain sub-regions of . A simple example of this sort is a system of coupled oscillators placed in the sites of a space-like lattice. Then for an arbitrary imaginary surface which separates the region in two complementary sub-regions and , the system in question can be represented as a union of two sub-systems. The wave function of the global system is given by a linear combination of the product of quantum states of each sub-system, . The states are formed by the degrees of freedom localized in the region while the states by those which are defined in the region . The density matrix that corresponds to a pure quantum state
has zero entropy. By tracing over the degrees of freedom in region we obtain a density matrix
with elements . The statistical entropy, defined for this density matrix by the standard formula
is by definition the entanglement entropy associated with the surface . We could have traced over the degrees of freedom located in region and formed the density matrix . It is clear that111For finite matrices this property indicates that the two density matrices have the same eigenvalues.
for any integer . Thus we conclude that the entropy (3) is the same for both density matrices and ,
This property indicates that the entanglement entropy for a system in a pure quantum state is not an extensive quantity. In particular, it does not depend on the size of each region or and thus is only determined by the geometry of .
2.2 Short-distance correlations
On the other hand, if the entropy (3) is non-vanishing, this shows that in the global system there exist correlations across the surface between modes which reside on different sides of the surface. In this review we shall consider the case when the system in question is a quantum field. The short-distance correlations that exist in this system have two important consequences:
the entanglement entropy becomes dependent on the UV cut-off which regularizes the short-distance (or the large-momentum) behavior of the field system
to leading order in the entanglement entropy is proportional to the area of the surface
For a free massless scalar field the 2-point correlation function in space-time dimensions has the standard form
where . Correspondingly, the typical behavior of the entanglement entropy in dimensions is
where the exact pre-factor depends on the regularization scheme. Although the similarity between (5) and (6) illustrates well the field-theoretical origin of the entanglement entropy, the exact relation between the short-distance behavior of 2-point correlation functions in the field theory and the UV divergence of the entropy is more subtle, as we shall discuss later in the paper.
2.3 Thermal entropy
Instead of a pure state one could have started with a mixed thermal state at temperature with density matrix , where is the Hamiltonian of the global system. In this case the relation (4) is no more valid and the entropy depends on the size of the total system as well as on the size of each sub-system. By rather general arguments, in the limit of large volume the reduced density matrix approaches the thermal density matrix. So that in this limit the entanglement entropy (3) reproduces the thermal entropy. For further references we give here the expression
for the thermal entropy of a massless field residing inside a spatial -volume at temperature .
2.4 Entropy of a system of finite size at finite temperature
In more general situation one starts with a system of finite size in a mixed thermal state at temperature . This system is divided by the entangling surface in two sub-systems of characteristic size . Then the entanglement entropy is a function of several parameters (if the field in question is massive then mass should be added to the parameters on which the entropy should depend on)
where is a UV cut-off. Clearly, the entanglement entropy in this general case is due to a combination of different factors: the entanglement between two sub-systems and the thermal nature of the initial mixed state. In dimensions even for simple geometries this function of 4 variables is not known explicitly. However, in two space-time dimensions, in some particular cases, the explicit form of this function is known.
2.5 Entropy in (1+1)-dimensional space-time
The state of a quantum field in two dimensions is defined on a union of intersecting intervals . The 2-point correlation functions behave logarithmically in the limit of coincident points. Correspondingly, the leading UV divergence of the entanglement entropy in two dimensions is logarithmic. For example, for a 2d massless conformal field theory, characterized by a central charge , the entropy is , , 
where is the number of intersections of intervals and where the sub-systems are defined, () is the length of the interval (). The second term in (9) is a UV finite term. In some cases the conformal symmetry in two dimensions can be used to calculate not only the UV divergent term in the entanglement entropy but also the UV finite term, thus obtaining the complete answer for the entropy, as was shown by Holzhey, Larsen and Wilczek  (see ,  for more recent developments). There are two different limiting cases when the conformal symmetry is helpful. In the first case one considers a pure state of the conformal field theory on a circle of circumference , the subsystem is defined on a segment of size of the circle. In the second situation the system is defined on an infinite line, the subsystem lives on interval of length of the line and the global system is in a thermal mixed state with temperature . In Euclidean signature both geometries represent a cylinder. For a thermal state the compact direction on the cylinder corresponds to Euclidean time compactified to form a circle of circumference . In both cases the cylinder can be further conformally mapped to a plane. The invariance of the entanglement entropy under conformal transformation can be used to obtain
in the case of a pure state on a circle and
for a thermal mixed state on an infinite line. In the limit of large the entropy (11) approaches
where the first term represents the entropy of the thermal gas (7) in a cavity of size while the second term represent the purely entanglement contribution (note that the intersection of and contains two points in this case so that ). The third term is an intermediate term due to the interaction of both factors, thermality and entanglement. This example clearly shows that for a generic thermal state the entanglement entropy is due to the combination of two factors: the entanglement between two subsystems and the thermal nature of the mixed state of the global system.
2.6 The Euclidean path integral representation and the replica method
A technical method very useful for the calculation of the entanglement entropy in a field theory is the so-called the replica trick, see ref.. Here we illustrate this method for a field theory described by a second order Laplace type operator. One considers a quantum field in a -dimensional spacetime and chooses the Cartesian coordinates , where is Euclidean time, such that the surface is defined by the condition and are the coordinates on . In the subspace it will be convenient to choose the polar coordinate system and , where the angular coordinate varies between and . We note that if the field theory in question is relativistic then the field operator is invariant under the shifts , where is an arbitrary constant.
One first defines the vacuum state of the quantum field in question by the path integral over a half of the total Euclidean spacetime defined as such that the quantum field satisfies the fixed boundary condition on the boundary of the half-space,
where is the action of the field. The surface in our case is a plane and the Cartesian coordinate is orthogonal to . The co-dimension 2 surface defined by the conditions and naturally separates the hypersurface in two parts: and . These are the two sub-regions and discussed in section 2.1.
The boundary data is also separated into and . By tracing over one defines a reduced density matrix
where the path integral goes over fields defined on the whole Euclidean spacetime except a cut . In the path integral the field takes the boundary value above the cut and below the cut. The trace of the -th power of the density matrix (14) is then given by the Euclidean path integral over fields defined on an -sheeted covering of the cut spacetime. In the polar coordinates the cut corresponds to values . When one passes across the cut from one sheet to another, the fields are glued analytically. Geometrically this -fold space is a flat cone with angle deficit at the surface . Thus we have
where is the Euclidean path integral over the -fold cover of the Euclidean space, i.e. over the cone . Assuming that in (15) one can analytically continue to non-integer values of , one observes that
where is the renormalized matrix density. Introduce the effective action , where is the partition function of the field system in question on a Euclidean space with conical singularity at the surface . In the polar coordinates the conical space is defined by making the coordinate periodic with period , where is very small. The invariance under the abelian isometry helps to construct without any problem the correlation functions with the required periodicity starting from the -periodic correlation functions. The analytic continuation of to different from 1 in the relativistic case is naturally provided by the path integral over the conical space . The entropy is then calculated by the replica trick
One of the advantages of this method is that we do not need to care about the normalization of the reduced density matrix and can deal with a matrix which is not properly normalized.
2.7 Uniqueness of analytic continuation
The uniqueness of the analytic continuation of to non-integer may not seem obvious, especially if the field system in question is not relativistic so that there is no isometry in the polar angle which would allow us without any trouble to glue together pieces of the Euclidean space to form a path integral over a conical space . However, some arguments can be given that the analytic continuation to non-integer is in fact unique.
Consider a renormalized density matrix . The eigenvalues of lie in the interval . If this matrix were a finite matrix we could use the triangle inequality to show that
For infinite size matrices the trace is usually infinite so that a regularization is needed. Suppose that is the regularization parameter and is the regularized trace. Then
Thus is a bounded function in the complex half-plane, . Now suppose that we know that for integer values of . Then, in the region , we can represent in the form
where the function is analytic (for ). Since by condition (17) the function is bounded, we obtain that, in order to compensate for the growth of the sine in (18) for complex values of , the function should satisfy the condition
By Carlson’s theorem  an analytic function, which is bounded in the region and which satisfies condition (19), vanishes identically. Thus we conclude that and there is only one analytic continuation to non-integer , namely the one given by function .
2.8 Heat kernel and the Sommerfeld formula
Consider for concreteness a quantum bosonic field described by a field operator so that the partition function is . Then the effective action defined as
where parameter is a UV cutoff, is expressed in terms of the trace of the heat kernel . The latter is defined as a solution to the heat equation
In order to calculate the effective action we use the heat kernel method. In the context of manifolds with conical singularities this method was developed in great detail in , . In the Lorentz invariant case the invariance under the abelian symmetry plays an important role. The heat kernel (where we omit the coordinates other than the angle ) on regular flat space then depends on the difference . This function is periodic with respect to . The heat kernel on a space with a conical singularity is supposed to be periodic. It is constructed from the periodic quantity by applying the Sommerfeld formula 
That this quantity still satisfies the heat kernel equation is a consequence of the invariance under the abelian isometry . The contour consists of two vertical lines, going from to and from to and intersecting the real axis between the poles of the : , and , respectively. For the integrand in (22) is a -periodic function and the contributions of these two vertical lines cancel each other. Thus, for a small angle deficit the contribution of the integral in (22) is proportional to .
2.9 An explicit calculation
Consider an infinite -plane in -dimensional space-time. The calculation of the entanglement entropy for this plane can be done explicitly by means of the heat kernel method. In flat space-time, if the operator is the Laplace operator,
one can use the Fourier transform in order to solve the heat equation. In spacetime dimensions one has
Putting and choosing in the polar coordinate system , that we have that , where and is the angle between the -vectors and . The radial momentum and angle , together with the other angles form a spherical coordinate system in the space of momenta . Thus one has for the integration measure , where is the area of a unit radius sphere in dimensions. Performing the integration in (23) in this coordinate system we find
For the trace one finds
Collecting everything together one finds that in flat Minkowski space-time
where is the volume of space-time and is the area of the surface . Substituting (27) into equation (20) we obtain that the effective action contains two terms. The one proportional to the volume reproduces the vacuum energy in the effective action. The second term proportional to the area is responsible for the entropy. Applying the formula (16) we obtain the entanglement entropy
of an infinite plane in space-time dimensions. Since any surface, locally, looks like a plane and a curved spacetime, locally, is approximated by Minkowski space, this result gives the leading contribution to the entanglement entropy of any surface in flat or curved space-time.
2.10 Entropy of massive fields
The heat kernel of a massive field described by the wave operator is expressed in terms of the heat kernel of massless field,
Thus one finds
where the trace of the heat kernel for vanishing mass is given by (27). Therefore the entanglement entropy of a massive field is
In particular, if , one finds that
The logarithmic term in the entropy that is due to the mass of the field appears in any even dimension . The presence of a UV finite term proportional to the -th power of mass is the other general feature of (30), (31).
2.11 An expression in terms of the determinant of the Laplacian on the surface
Even though the entanglement entropy is determined by the geometry of the surface , in general this can be not only its intrinsic geometry but also how the surface is embedded in the larger space-time. The embedding is determined by the extrinsic curvature. The curvature of the larger spacetime enters through the Gauss-Cadazzi relations. But in some particularly simple cases the entropy can be given a purely intrinsic interpretation. To see this for the case when is a plane we note that the entropy (28) or (30) originates from the surface term in the trace of the heat kernel (27) (or (31)). To leading order in , the surface term in the case of a massive scalar field is
can be interpreted as the trace of the heat kernel of operator , where is the intrinsic Laplace operator defined on -plane . The determinant of the operator is determined by
Thus we obtain an interesting expression for the entanglement entropy
in terms of geometric objects defined intrinsically on the surface . A similar expression in the case of an ultra-extreme black hole was obtained in  and for a generic black hole with horizon approximated by a plane was obtained in .
2.12 Entropy in theories with a modified propagator
In certain physically interesting situations the propagator of a quantum field is different from the standard and is described by some function as . The quantum field in question then satisfies a modified Lorentz invariant field equation
Theories of this type naturally arise in models with extra dimensions. The deviations from the standard form of propagator may be both in the UV regime (large values of ) or in the IR regime (small values of ). If the function for large values of grows faster than this theory is characterized by improved UV behavior.
The calculation of the entanglement entropy performed in section 2.9 can be generalized to include theories with operator (33). This example is instructive since, in particular, it illuminates the exact relation between the structure of 2-point function (the Green’s function in the case of free fields) and the entanglement entropy .
In spacetime dimensions one has
Note that we consider Euclidean theory so that . The Green’s function
is a solution to the field equation with a delta-like source
and can be expressed in terms of the heat kernel as follows
Obviously, the Green’s function can be represented in terms of the Fourier transform in a manner similar to (34),
where the functions are defined as
The entanglement entropy takes the form (we remind the reader that for simplicity we take the surface to be a -dimensional plane) 
(i) the area law in the entanglement entropy is universal and is valid for any function ;
(ii) the entanglement entropy is UV divergent independently of the function , with the degree of divergence depending on the particular function ;
(iii) in the coincidence limit, , the Green’s function (38)
may take a finite value if is decaying faster than . However, even for this function the entanglement entropy is UV divergent.
As an example, consider a function which grows for large values of as . The 2-point correlation function in this theory behaves as
and for it is regular in the coincidence limit. On the other hand, the entanglement entropy scales as
and remains divergent for any positive value of . Comparison of (43) and (44) shows that only for (the standard form of the wave operator and the propagator) the short-distance behavior of the 2-point function is similar to the UV divergence of the entanglement entropy.
2.13 Entanglement entropy in non-Lorentz invariant theories
Non-Lorentz invariant theories are characterized by a modified dispersion relation, , between the energy and the 3-momentum . These theories can be described by a wave operator of the following type
where is the spatial Laplace operator. Clearly, the symmetry with respect to the Lorentz boosts is broken in operator (45) if .
As in the Lorentz invariant case to compute the entanglement entropy associated with a surface we choose spatial coordinates , where is the coordinate orthogonal to the surface and are the coordinates on the surface . Then, after going to Euclidean time , we switch to the polar coordinates, , . In the Lorentz invariant case the conical space which is needed for calculation of the entanglement entropy is obtained by making the angular coordinate periodic with period by applying the Sommerfeld formula (22) to the heat kernel. If Lorentz invariance is broken, as it is for the operator (45) there are certain difficulties in applying the method of the conical singularity when one computes the entanglement entropy. The difficulties come from the fact that the wave operator , if written in terms of the polar coordinates and , becomes an explicit function of the angular coordinate . As a result of this, the operator is not invariant under shifts of to arbitrary . Only shifts with , where is an integer are allowed. Thus, in this case one cannot apply the Sommerfeld formula since it explicitly uses the symmetry of the differential operator under shifts of angle . On the other hand, a conical space with angle deficit is exactly what we need to compute for the reduced density matrix. In ref., by using some scaling arguments it was shown that the trace of the heat kernel on a conical space with periodicity, is
where is the bulk contribution. By the arguments presented in Section 2.7 there is a unique analytic extension of this formula to non-integer . A simple comparison with the surface term in the heat kernel of the Lorentz invariant operator, which was obtained in the previous section, shows that the surface terms of the two kernels are identical. We thus conclude that the entanglement entropy is given by the same formula
where is defined in (40), as in the Lorentz invariant case (41). A similar property of the entanglement entropy was observed for a non-relativistic theory described by the Schrödinger operator  (see also  for a holographic derivation). For polynomial operators, , some scaling arguments can be used  to get the form of the entropy that follows from (47).
In the rest of the review we shall mostly focus on the study of Lorentz invariant theories with field operator quadratic in derivatives, of the Laplace type, .
2.14 Arbitrary surface in curved space-time: general structure of UV divergences
The definition of the entanglement entropy and the procedure for its calculation generalize to curved spacetime. The surface can then be any smooth closed co-dimension two surface222If the boundary of is not empty there could be extra terms in the entropy proportional to the “area” of the boundary as was shown in . We do not consider this case here. which divides the space in two sub-regions. In the next section we will consider in detail the case where this surface is a black hole horizon. Before proceeding to the black hole case we would like to specify the general structure of UV divergent terms in the entanglement entropy. In -dimensional curved spacetime entanglement entropy is presented in the form of a Laurent series with respect to the UV cutoff (for see )
where is proportional to the area of the surface . All other terms in the expansion (48) can be presented as integrals over of local quantities constructed in terms of the Riemann curvature of the spacetime and the extrinsic curvature of the surface . The intrinsic curvature of the surface of course can be expressed in terms of and using the Gauss-Codazzi equations. Since nothing should depend on the direction of vectors normal to , the integrands in expansion (48) should be even powers of extrinsic curvature. The general form of the term can be symbolically presented in the form
where stands for components of the Riemann tensor and their projections onto the sub-space orthogonal to and labels the components of the extrinsic curvature. Thus, since the integrands are even in derivatives, only terms , appear in (48). If is even then there also may appear a logarithmic term . The term in (48) is a UV finite term, which also may depend on the geometry of the surface , as well as on the geometry of the space-time itself.
3 Entanglement entropy of non-degenerate Killing horizons
3.1 The geometric setting of black hole spacetimes
The notion of entanglement entropy is naturally applicable to a black hole. In fact, probably the only way to separate a system in two sub-systems is to place one of them inside a black hole horizon. The important feature that, in fact, defines the black hole is the existence of a horizon. Many useful definitions of a horizon are known. In the present paper we shall consider only the case of the so-called eternal black holes for which different definitions of the horizon coincide. The corresponding spacetime then admits a maximal analytic extension which we shall use in our construction. The simplest example is the Schwarzschild black hole, the maximal extension of which is demonstrated on the well known Penrose diagram. The horizon of the Schwarzschild black hole is an example of a so-called Killing horizon. The spacetime in this case possesses a global Killing vector, , which generates time translations. The Killing horizon is defined as a null hypersurface on which the Killing vector is null, . The null surface in the maximal extension of an eternal black consists of two parts: the future horizon and the past horizon. The two intersect on a co-dimension two compact surface called the bifurcation surface. In the maximally extended spacetime a hypersurface of constant time is a Cauchy surface. The bifurcation surface naturally splits the Cauchy surface in two parts, and , respectively inside and outside the black hole. For asymptotically flat spacetime, a such as the Schwarzschild metric, the hypersurface has the topology of a wormhole. (In the case of the Schwarzschild metric it is called the Einstein-Rosen bridge.) The surface is the surface of minimal area in . In fact the bifurcation surface is a minimal surface not only in the -dimensional Euclidean space but also in the -dimensional spacetime. As a consequence, as we show below, the components of the extrinsic curvature defined for two vectors normal to , vanish on .
The space-time in question admits a Euclidean version by analytic continuation . It is a feature of regular metrics with a Killing horizon that the direction of Euclidean time is compact with period which is determined by the condition of regularity, i.e. the absence of a conical singularity. In the vicinity of the bifurcation surface , the spacetime then is a product of a compact surface and a two dimensional disk, the time coordinate playing the role of the angular coordinate on the disk. The latter can be made more precise by introducing a new angular variable which varies from to . We consider the static space-time with Euclidean metric of the general type
The radial coordinate is such that the surface is defined by the condition . Near this point the functions and can be expended as
where is the metric on the bifurcation surface equipped with coordinates . This metric describes what is called a non-degenerate horizon. The Hawking temperature of the horizon is finite in this case and equal to .
It is important to note that the metric (50) does not have to satisfy any field equations. The entanglement entropy can be defined for any metric which possesses a Killing type horizon. In this sense the entanglement entropy is an off-shell quantity. It is useful to keep this in mind when one compares the entanglement entropy with some other approaches in which an entropy is assigned to a black hole horizon. Even though the metric (50) with (51) does not have to satisfy the Einstein equations we shall still call the complete space described by the Euclidean metric (50) the Euclidean black hole instanton and will denote it by .
3.2 Extrinsic curvature of horizon, horizon as a minimal surface
The horizon surface defined by the condition in the metric (50) is a co-dimension 2 surface. It has two normal vectors: a spacelike vector with the only non-vanishing component and a timelike vector with the non-vanishing component . With respect to each normal vector one defines an extrinsic curvature, , . The extrinsic curvature identically vanishes. It is a consequence of the fact that is a Killing vector which generates the time translations. Indeed, the extrinsic curvature can be also written as a Lie derivative, , so that it vanishes if is a Killing vector. The extrinsic curvature associated to the vector ,
is vanishing when restricted to the surface defined by the condition . It is due to the fact that the term linear in is absent in the -expansion for in the metric (50). This is required by the regularity of the metric (50): in the presence of such a term the Ricci scalar would be singular at the horizon, .
The vanishing of the extrinsic curvature of the horizon indicates that the horizon is necessarily a minimal surface. It has the minimal area considered as a surface in -dimensional spacetime. On the other hand, in the Lorentzian signature, the horizon has the minimal area if considered on the hypersurface of constant time , , the latter thus has the topology of a wormhole.
3.3 The wave function of a black hole
Although the entanglement entropy can be defined for any co-dimension two surface in spacetime when the surface is a horizon particular care is required. In order to apply the general prescription outlined in section 2.1, we first of all need to specify the corresponding wave function. Here we will follow the prescription proposed by Barvinsky, Frolov and Zelnikov . This prescription is a natural generalization of the one in flat spacetime discussed in section 2.6. On the other hand, it is similar to the “no-boundary” wave function of the Universe introduced in . We define the wave function of a black hole by the Euclidean path integral over field configurations on the half-period Euclidean instanton defined by the metric (50) with angular coordinate changing in the interval from to . This half-period instanton has Cauchy surface (on which we can choose coordinates ) as a boundary where we specify the boundary conditions in the path integral,
where is the action of the quantum field . The functions and are the boundary values defined on the part of the hypersurface which is respectively inside () and outside () the horizon . As was shown in  the wave function (53) corresponds to the Hartle-Hawking vacuum state .
3.4 Reduced density matrix and entropy
The density matrix defined by tracing over -modes is given by the Euclidean path integral over field configurations on the complete instanton with a cut along the axis where the field in the path integral takes the values and below and above the cut respectively. The trace is obtained by equating the fields across the cut and doing the unrestricted Euclidean path integral on the complete Euclidean instanton . Analogously, is given by the path integral over field configurations defined on the n-fold cover of the complete instanton. This space is described by the metric (50) where angular coordinate is periodic with period . It has a conical singularity on the surface so that in a small vicinity of the total space is a direct product of and a two-dimensional cone with angle deficit . Due to the abelian isometry generated by the Killing vector this construction can be analytically continued to arbitrary (non-integer) . So that one can define a partition function
by the path integral over field configurations over , the -fold cover of the instanton . For a bosonic field described by the field operator one has that . Defining the effective action as , the entanglement entropy is still given by (16), i.e. by differentiating the effective action with respect to the angle deficit. Clearly, only the term linear in contributes to the entropy. The problem thus reduces to the calculation of this term in the effective action.
3.5 The role of the rotational symmetry
We emphasize that the presence of the so called rotational symmetry with respect to the Killing vector , which generates rotations in the 2-plane orthogonal to the entangling surface , plays an important role in our construction. Indeed, without such a symmetry it would be impossible to interpret for an arbitrary as a partition function in some gravitational background. In general, two points are important for this interpretation:
i) that the spacetime possesses, at least locally near the entangling surface, a rotational symmetry so that, after the identification we get a well-defined spacetime with no more than just a conical singularity; this holds automatically if the surface in question is a Killing horizon;
ii) and that the field operator is invariant under the “rotations”, ; this is automatic if the field operator is a covariant operator.
In particular, the point ii) allows us to use the Sommerfeld formula (more precisely its generalization to a curved spacetime) in order to define the Green’s function or the heat kernel on the space . As is shown in  (see also discussion in section 2.13) in the case of the non-Lorentz invariant field operators in flat Minkowski spacetime the lack of the symmetry ii) makes the whole “conical space” approach rather obscure. On the other hand, in the absence of the rotational symmetry i) there may appear terms in the entropy that are “missing” in the naively applied conical space approach: the extrinsic curvature contributions  or even some curvature terms .
In what follows we consider the entanglement entropy of the Killing horizons and deal with the covariant operators so that we do not have to worry about i) or ii).
3.6 Thermality of the reduced density matrix of a Killing horizon
The quantum state defined by equation (53) is the Hartle-Hawking vacuum . The Green’s function in this state is defined by analytic continuation from the Euclidean Green’s function. The periodicity is thus inherent in this state. This periodicity indicates that the correlation functions computed in this state are in fact thermal correlation functions when continued to the Lorentzian section. This fact generalizes to an arbitrary interacting quantum field as shown in . On the other hand, being globally defined, the Hartle-Hawking state is a pure state which involves correlations between modes localized on different sides of the horizon. This state however is described by a thermal density matrix if reduced to modes defined on one side of the horizon as was shown by Israel . That the reduced density matrix obtained by tracing over modes inside the horizon is thermal can be formally seen by using angular quantization. Introducing the Euclidean Hamiltonian which is the generator of rotations with respect to the angular coordinate defined above, one finds that , i.e. the density matrix is thermal with respect to the Hamiltonian with inverse temperature . This formal proof in Minkowski space was outlined in . The appropriate Euclidean Hamiltonian is then the Rindler Hamiltonian which generates the Lorentz boosts in the direction orthogonal to the surface . In  the proof was generalized to the case of generic static spacetimes with bifurcate Killing horizons admitting a regular Euclidean section.
3.7 Useful mathematical tools
3.7.1 Curvature of space with a conical singularity
Consider a space which is an -fold covering of a smooth manifold along the Killing vector generating an Abelian isometry. Let surface be a stationary point of this isometry so that near the space looks like a direct product, , of the surface and a two-dimensional cone with angle deficit . Outside the singular surface the space has the same geometry as a smooth manifold . In particular, their curvature tensors coincide. However, the conical singularity at the surface produces a singular (delta-function like) contribution to the curvatures. This was first demonstrated by Sokolov and Starobinsky  in two-dimensions by using topological arguments. These arguments were generalized to higher dimensions in . One way to extract the singular contribution is to use some regularization procedure, replacing the singular space by a sequence of regular manifolds . This procedure was developed by Fursaev and Solodukhin in . In the limit one obtains the following results :
where is the delta-function, ; are two orthonormal vectors orthogonal to the surface , and the quantities , and are computed in the regular points by the standard method.
These formulas can be used to define the integral expressions333It should be noted that formulas (55) and (56), (57), (58), (59) are valid even if subleading terms (as in (51)) in the expansion of the metric near singular surface are functions of . Such more general metrics describe what might be called a “local Killing horizon”. 
where and . We use a shorthand notation for the surface integral .