# Euclidean scalar Green functions near the black hole and black brane horizons

###### Abstract

We discuss approximations of the Riemannian geometry near the horizon. If a dimensional manifold has a bifurcate Killing horizon then we approximate by a product of the two dimensional Rindler space and a dimensional Riemannian manifold . We obtain approximate formulas for scalar Green functions. We study the behaviour of the Green functions near the horizon and their dimensional reduction. We show that if is compact then the Green function near the horizon can be approximated by the Green function of the two-dimensional quantum field theory. The correction term is exponentially small away from the horizon. We extend the results to black brane solutions of supergravity in ten and eleven dimensions. The near horizon geometry can be approximated by . We discuss the Euclidean Green functions on and their behaviour near the horizon.

## 1 Introduction

The Hawking radiation shows that quantum phenomena accompanying the motion of a quantum particle in a neighborhood of a black hole require a description in the framework of relativistic quantum theory of many particle systems. It seems that quantum field theory supplies a proper method for a treatment of a varying number of particles. Quantum field theory can be defined by means of Green functions. In the Minkowski space the locality and Poincare invariance determine the Green functions and allow a construction of free quantum fields. In the curved space the Green function is not unique. The non-uniqueness can be interpreted as a non-uniqueness of the physical vacuum [1][2]. There is less ambiguity in the definition of the Green function on the Riemannian manifolds (instead of the physical pseudo-Riemannian ones). The Euclidean approach appeared successful when applied to the construction of quantum fields on the Minkowski space-time [3]. We hope that such an approach will be fruitful in application to a curved background as well. In contradistinction to the Minkowski space-time an analytic continuation of Euclidean fields to quantum fields from the Riemannian metric to the pseudoRiemannian one can be achieved only if the manifold has an additional reflection symmetry [4](for a possible physical relevance of the reflection symmetry see [5]).

The black hole can be defined in a coordinate independent way by the event horizon. The event horizon is a global property of the pseudoRiemannian manifold. It is not easy to see what is its counterpart after an analytic continuation to the Riemannian manifold. Nevertheless, there is a proper substitute:the bifurcate Killing horizon. As proved in [6] a manifold with the Killing horizon can be extended to the manifold with a bifurcate Killing horizon. Moreover, there always exists an extension with the wedge reflection symmetry [7] which seems crucial for an analytic continuation between pseudoRiemannian and Riemannian manifolds. The bifurcate Killing horizon is a local property which can be treated in local coordinates [8]. In local static coordinates close to the bifurcate Killing horizon the metric tensor tends to zero at the horizon [9]. This property is preserved after a continuation to the Riemannian metric. In sec.2 we approximate the Riemannian manifold with the bifurcate Killing horizon as , where is the two-dimensional Rindler space and enters the definition of the bifurcate Killing horizon as an intersection of past and future horizons. There is the well-known example of the approximation in the form of a product: the four-dimensional Schwarzschild solution can be approximated near the horizon by a product of the Rindler space and the two-dimensional sphere . However, we do not restrict ourselves to metrics which are solutions of Einstein gravity .

The Euclidean quantum fields are defined by Green functions. For an approximate metric near the bifurcate Killing horizon we consider an equation for the scalar Green functions. We expand the solution into eigenfunctions of the Laplace-Beltrami operator on . If is compact without a boundary then the Laplace -Beltrami operator has a discrete spectrum starting from (the zero mode). We show in sec.3 that the higher modes are damped by a tunneling mechanism. As a consequence, the position of the point on the manifold becomes irrelevant. The Green function near the bifurcate Killing horizon can be well approximated by the Green function of the two-dimensional free field. The splitting of the Green function near the horizon into a product of the two dimensional function and a function on has been predicted by Padmanabhan [10]. However, we obtain its exact form. In sec.4 we discuss an application of our earlier results [11] on the dimensional reduction of the Green functions. In sec.5 we apply the method to black brane solutions of string theory [12][13] in 10 and 11 dimensional supergravity which at the horizon have the geometry of . We show that Euclidean quantum field theory on the brane can be approximated by that on the hyperbolic space ( the Euclidean version of AdS). The propagator on the background manifold of the black brane can be applied for a construction of supergravity with the black brane as the vacuum state. Anti-de-Sitter space is the homogeneous space of the conformal group. Hence, at the level of the two-point functions we derive the relation between supergravity and conformal field theory on the boundary of of (which is )[14].

## 2 An approximation at the bifurcate Killing horizon

We consider a dimensional Riemannian manifold with a metric and a bifurcate Killing horizon. This notion assumes a symmetry generated by the Killing vector . Then, it is assumed that the Killing vector is orthogonal to a (past oriented) dimensional hypersurface and a (future oriented) hypersurface [8]. The Killing vector is vanishing (i.e.,) on an intersection of and defining a dimensional surface (which can be described as the level surface ) . The bifurcate Killing horizon implies that the space-time has locally a structure of an accelerated frame, i.e., the structure of the Rindler space [15]. In [6] it is proved that the space-time with a Killing horizon can be extended to a space-time with the bifurcate Killing horizon. Padmanabhan [10] describes such a bifurcate Killing horizon as a transformation from a local Lorentz frame to the local accelerated (Rindler) frame. In [6] it is shown that the extension can be chosen in such a way that the ”wedge reflection symmetry” [7] is satisfied.In the local Rindler coordinates the reflection symmetry is . The symmetry means that the metric splits into a block form (we denote coordinates on and its indices by capital letters)

The bifurcate Killing horizon distinguishes a two-dimensional subspace of the tangent space. At the bifurcate Killing horizon the two-dimensional metric tensor is degenerate. In the adapted coordinates such that we have and the metric does not depend on . Then, at the horizon means that or . We assume . As is non-negative its Taylor expansion must start with . Hence, if we neglect the dependence of the two-dimensional metric on and on then we can write it in the form

(1) |

If we neglect the dependence of on near the horizon then the metric ( denoted ) can be considered as a metric on the dimensional surface being the common part of and . Hence, in eq.(1) where is the two-dimensional Rindler space. As an example of the approximation of the geometry of we could consider the four dimensional Schwarzschild black hole when ( quantum theory with such an approximation is discussed in [16]) .

We shall work with Euclidean version of the metric (1)

(2) |

We consider the Laplace-Beltrami operator

on ().

In the approximation (2) we have (if is independent of )

(3) |

where is the Laplace-Beltrami operator on the two-dimensional Rindler space and is the Laplace-Beltrami operator for the metric

We are interested in the calculation of the Green functions

(4) |

Then, eq.(4) for reads

(5) |

After an exponential change of coordinates

(6) |

eq.(5) takes the form

(7) |

If is approximated by then the metric (1) is conformally related to the hyperbolic metric. This relation has its impact on the form of the Green functions (5) as will be seen in secs.3 and 4.

## 3 Green functions near the bifurcate Killing horizon

We investigate in this section the Green function (5) in dimensions under the assumption that is dimensional compact manifold without a boundary. We introduce the complete basis of eigenfunctions in the space of the remaining two coordinates

(8) |

where

(9) |

In eq.(8) denotes the set of all the parameters the solution depends on. The solutions satisfy the completeness relation

(10) |

with a certain measure and the orthogonality relation

(11) |

where again the function concerns all parameters characterizing the solution.

If is a compact manifold without a boundary then has a complete discrete set of orthonormal eigenfunctions [17]

(12) |

satisfying the completeness relation

Solutions of Eq.(12) always have a zero mode (we normalize the Riemannian volume element of to ) corresponding to . We expand (we drop the index in because there is only one Green function in this section) distinguishing the contribution of the zero mode

(13) |

where

(14) |

is the kernel of the inverse of the operator

i.e.,

(15) |

We have moved the zero mode in eq.(13) from the rhs to the lhs. is the solution of the equation

(16) |

Taking the Fourier transform in we express eq.(16) in the form

(17) |

where

(18) |

In order to obtain an approximate estimate on the behaviour of we write in the WKB form

(19) |

Then, from eq.(15) for large

(20) |

Hence, if then for large showing that is decaying exponentially fast away from the horizon (if then if ). We study this phenomenon in more detail now. First, we write the solution of eq.(8) in the form

(21) |

where

(22) |

Now, and in eqs.(10)-(11). When then the solution of eq.(22) is the plane wave

The normalized solution of eq.(22, which behaves like a plane wave with momentum for and decays exponentially for , reads

(23) |

where is the modified Bessel function of the third kind of order [18].

This solution is inserted into the formulas (13)-(14) for the Green function with the normalization (11)

(24) |

(25) |

Then, performing the integral over in eq.(14)

we obtain

(26) |

We expect that the rhs of eq.(26) is decaying fast to zero away from the horizon. Each term on the rhs of eq.(26) is decaying exponentially fast when because the Bessel function is decreasing exponentially. We wish to show that the sum is decreasing exponentially as well. This is not a simple problem because we need some estimates on eigenfunctions and eigenvalues of the Laplace-Beltrami operator on .

Let us consider the simplest example (the circle of radius which can be related to the three-dimensional BTZ black hole [21]). Then,

(27) |

We have

(28) |

Inserting (28) in eq.(26) and approximating the denominator in eq.(28) by 1 we obtain an asymptotic estimate for large positive and (large and ;this is eq.(26) neglecting )

(29) |

We can estimate the integral over if (if this condition is not satisfied then the estimates are much more difficult because we must estimate the behaviour of simultaneously for large and large ). In such a case inserting the asymptotic expansion of the Bessel function we obtain

(30) |

where (for )

is the solution of the equation

(31) |

It follows that close to the horizon tends exponentially fast to the Green function for the two-dimensional quantum field theory.

In a similar approach we treat the case , where is the two-dimensional sphere with radius . This case is interesting because it describes the near horizon geometry of the four-dimensional Schwarzschild black hole [22]. The approximate near horizon geometry of the four-dimensional Schwarzschild black hole is (in dimensions this is ) where is the two-dimensional Rindler space. Now, the formula (26) reads

(32) |

where , is the Legendre polynomial and is the geodesic distance on . In the spherical angles

(33) |

A finite number of terms on the rhs of eq.(32) is decaying exponentially in (because is decaying exponentially). Hence, it is sufficient to estimate the sum in eq.(32) for . For large we can use the approximation . Applying the representation of the Legendre polynomials

(34) |

and the representation (27) of the Bessel functions we obtain

(35) |

where

and

(36) |

If in we neglect (or expand it in a power series of ) and apply the asymptotic expansion for the Bessel functions then we can conclude that the sum starting from is decaying as

(37) |

where is the solution of eq.(16).

For general compact manifolds we must apply some approximations in order to estimate the infinite sums. We estimate the rhs of eq.(13) for large and by means of a simplified argument applicable when , and . Then,

(38) |

Let be the lowest non-zero eigenvalue. The finite sum on the rhs of eq.(38) is decreasing as . For this reason we can begin the sum starting from large eigenvalues. For large eigenvalues ( with sufficiently large) we can apply the Weyl approximation for the eigenvalues distribution [23] with the conclusion

(39) |

The finite sum as well as the integral on the rhs of eq.(39) are decaying exponentially in with the rate whereas for we can get the estimate

for non-negative . Hence, for any the rhs of eq.(13) is decreasing to zero faster than both and .

## 4 Green functions on a product manifold

In secs.2 and 3 we have approximated a manifold with a bifurcate Killing horizon by a product manifold and discussed the Green functions in such an approximation. In this section we consider Green functions on product manifolds in a formulation based on our earlier paper [11]. Here, we emphasize some methods which have applications to the black brane solutions to be discussed in the next section.

Let us consider a manifold in the form of a product where has dimensions and is a dimensional manifold. The metric on takes the form

(40) |

where the coordinates on are denoted by the capital , the ones on by and the coordinates on are denoted by . A solution of eq.(4) can be expressed by the fundamental solution of the diffusion equation

(41) |

with the initial condition . Then

(42) |

We may write eq.(4) in the form (here and )

(43) |

From eq.(43) we have a simple formula ( in the sense of a product of semigroups)

(44) |

where the upper index of the heat kernel denotes the manifold of its definition.

Hence

(45) |

We expand the Green function (distinguishing the zero mode) in eigenfunctions (12) of the Laplace-Beltrami operator

(46) |

is a solution of the equation

(47) |

where is defined in eq.(9). The zero mode is the solution of the equation

(48) |

As in sec.3 we ask the question whether can be approximated by . As a first rough approximation for large distances we apply the WKB reresentation expressing the Green function (47) in the form

(49) |

Assuming that is growing uniformly in each direction we obtain in the leading order for large distances the equation

(50) |

for . Eq.(50) is an equation for the geodesic distance on the manifold with the metric [24]. Hence, the geodesic distance is the solution of eq.(50) which is symmetric under the exchange of the points and satisfies the boundary condition . We insert the approximate solutions (49) into the sum (46). Then, we can express the sum by the heat kernel of or the heat kernel of

(51) |

Eq.(51) gives a better approximation than eq.(49) (together with eq.(50)) of the Green function on because the sum over eigenvalues has been performed. If the heat kernel on is known (to be discussed in the next section) then from eq.(51) we can obtain an approximation for .

## 5 Green functions for approximate geometries of black holes and black branes

In sec.3 we have derived an approximation for the Riemannian metric near the bifurcate Killing horizon. We could see that at the horizon . In the case of the Schwarzschild black hole in four dimensions we have . We are interested in this section also in the black brane solutions in 10 dimensional supergravity which have the near horizon geometry and the black brane solutions in 11 dimensional supergravity with the near horizon geometry or [13]. These manifolds can be considered as solutions to the string theory compactification problem [22]. We can see that for a description of the black hole (black brane) Green functions the formulas for the heat kernel on the Rindler space and AdS space will be useful. We shall set the radius of the black hole (brane) in this section. The radius can be inserted in our formulas by a restoration of proper dimensionality of the numbers entering these formulas (as will be indicated further on). For the Rindler space the heat kernel equation reads

(52) |

If we take the Fourier transform in in eq.(52) then we obtain an equation for the Bessel function [18]. Hence, we can see that the solution of the heat kernel for with the initial condition can be expressed in the form

(53) |

Eq.(52) for the heat kernel on coincides with the heat equation on the plane but expressed in cylindrical coordinates,i.e., if

then

As a consequence the heat kernel with periodic boundary conditions imposed on ( with the period ) must coincide with the heat kernel on the plane (see also [25]). Hence,

(54) |

where

(55) |

First, we apply the methods of sec.4 to the case discussed already in another way in sec.3. The approximate near horizon geometry of the Schwarzschild black hole in four dimensions is (in dimensions this is ). We may apply eq.(45) in order to express the Green function by the heat kernels. For this purpose the eigenfunction expansion on is useful

(56) |

where is the Legendre polynomial and the geodesic distance is defined in eq.(33). Applying the representation (34) of the Legendre polynomials we can sum up the series (56) and express it by an integral

(57) |

where

(58) |

If we expand in then we obtain the expansion (56) of the heat kernel.

Applying eqs.(45),(54) and (57) we can represent the scalar Green function on the four-dimensional black hole of temperature ( this is the conventional Hawking temperature in dimensionless units; the coordinate is made an angular variable in order to make the singular conical manifold regular [26][22][27][28])

(59) |

where

We obtain an exponential decay of the rhs of eq.(59) if we make the approximation and estimate the correction to this approximation. In this way we reach the approximation (37) of sec.3 but now at the Hawking temperature. For zero temperature or a temperature different from the Hawking temperature it would be difficult to obtain a useful approximation for the Green function from eq.(45) because the formula (53) for the heat kernel on the Rindler space is rather implicit.

We can obtain simple formulas for the Schwarzschild black hole in an odd dimension . For an odd the formula for the heat kernel on reads [29]

(60) |

(the derivative over annihilates all with ). Performing the integral (45)(of the heat kernels (54) and (60)) we obtain the Green function of the black hole in dimensions at temperature