Renormalized vacuum polarization on rotating warped AdS3 black holes

We compute the renormalized vacuum polarization of a massive scalar field in the Hartle-Hawking state on (2+1)-dimensional rotating, spacelike stretched black hole solutions to Topologically Massive Gravity, surrounded by a Dirichlet mirror that makes the state well defined. The Feynman propagator is written as a mode sum on the complex Riemannian section of the spacetime, and a Hadamard renormalization procedure is implemented by matching to a mode sum on the complex Riemannian section of a rotating Minkowski spacetime. No analytic continuation in the angular momentum parameter is invoked. Selected numerical results are given, demonstrating the numerical efficacy of the method. We anticipate that this method can be extended to wider classes of rotating black hole spacetimes, in particular to the Kerr spacetime in four dimensions.


I. INTRODUCTION
The study of quantum field theory on black hole spacetimes has mostly been restricted to static, spherically symmetric spacetimes. Nevertheless, there have been attempts at considering stationary black hole spacetimes, with main focus on the Kerr spacetime [1][2][3][4][5][6]. One important task is the computation of expectation values of the renormalized stressenergy tensor for a matter field in a given quantum state [7,8]. This has proven to be very challenging and, so far, almost all calculations have only addressed the differences between expectation values for different quantum states [6] and the large field mass limit [9]. In [10], the stress-energy tensor for the rotating BTZ black hole [11,12] was renormalized with respect to AdS 3 , by using the fact that the black hole corresponds to AdS 3 with discrete identifications, but this method cannot be used for more general classes of rotating black hole solutions. We could summarize the main difficulties in three points: (i) the technical complexity of the computations required for the Kerr spacetime, due to the lack of spherical symmetry, (ii) the non-existence of generalizations of the (globally defined, regular and isometry-invariant) Hartle-Hawking state defined in static spacetimes, and (iii) the unavailability of Euclidean methods which simplify the task in static spacetimes.
To tackle point (i), we focus on a rotating black hole spacetime in 2+1 dimensions, the spacelike stretched black hole [13]. This is a vacuum solution of topologically massive gravity (TMG) [14,15], a deformation of (2+1)-dimensional Einstein gravity, and it can be thought as a "warped" version of the BTZ black hole. In contrast to the BTZ solution, the causal structure of the spacelike stretched black hole is similar to that of the Kerr spacetime [16]. In this setting, the matter field equations can be solved in terms of hypergeometric functions, which considerably simplify the technical issues in comparison with the Kerr spacetime. These black hole solutions are known to be classically stable to massive scalar field perturbations and, in particular, classical superradiance does not give rise to superradiant instabilities [17]. In this paper, we study a quantum scalar field on this black hole spacetime.
Concerning point (ii), it is well known that the Hartle-Hawking vacuum state in the Schwarzschild spacetime does not generalize to the Kerr spacetime [18]. This is linked to the speed-of-light surface, outside of which no observers can corotate with the horizon of the Kerr black hole. However, a quantum state with the same properties (regular at the horizon and invariant under the isometries of the spacetime) can be defined if appropriate mirror-like boundaries are introduced such that the region outside the speed-of-light surface is taken out of consideration. Here, we adopt this strategy and surround the spacelike stretched black hole with a mirror at a fixed radial coordinate.
Finally, regarding point (iii), we use a 'quasi-Euclidean' method which allows us to obtain the complex Riemannian section of the spacelike stretched black hole through a Wick rotation [19][20][21], in the same way a Euclidean (or Riemannian) section can be obtained for a static black hole spacetime. On this section, the matter field equation has a unique Green's function, related by analytical continuation to the Feynman propagator in the Hartle-Hawking state on the original Lorentzian section, and the renormalization procedure can be carried out using this Green's function.
In this paper, we compute the vacuum polarization Φ 2 (x) of a massive scalar field Φ in the Hartle-Hawking state on a spacelike stretched black hole surrounded by a mirror at which Dirichlet boundary conditions are imposed. We believe the method we use is general enough to be applicable to a wider class of rotating black hole solutions, in particular the Kerr spacetime in four dimensions. This calculation can be taken as a warm-up for the computation of the expectation values of the renormalized stress-energy tensor.
The contents of the paper are as follows. We begin in section II with the quantization of a massive scalar field on the spacelike stretched black hole bounded by a mirror, including a short description of the Hadamard renormalization. In section III, we outline the quasi-Euclidean method we use to obtain the complex Riemannian section of the black hole spacetime and renormalize the vacuum polarization. This is followed in section IV with the numerical evaluation of the renormalized vacuum polarization. Finally, our conclusions are presented in section V. Technical steps in the analysis are deferred to five appendices. Throughout this paper we use the (−, +, +) signature and units in which = c = G = k B = 1.

II. SPACELIKE STRETCHED BLACK HOLES AND SCALAR FIELDS
In this section, we first give a short description of topologically massive gravity and review the basic features of the spacelike stretched black hole solutions, including their causal structure. We then proceed to quantize the massive scalar field and outline the Hadamard renormalization procedure.

A. Spacelike stretched lack holes
The (2+1)-dimensional rotating black holes we focus in this paper are vacuum solutions of Topologically Massive Gravity, whose action is obtained by adding a gravitational Chern-Simons term to the Einstein-Hilbert action with a negative cosmological constant [14,15] G is Newton's gravitational constant, ν is a dimensionless coupling, g is the determinant of the metric, R is the Ricci scalar, > 0 is the cosmological length (which will be set to ≡ 1 from now on), Γ ρ λσ are the Christoffel symbols, and λµν is the Levi-Civita tensor in three dimensions.
The spacelike stretched black hole is one of the several types of warped AdS 3 black hole solutions [13]. Its metric, in coordinates (t, r, θ), is given by with t ∈ (−∞, ∞), r ∈ (0, ∞), (t, r, θ) ∼ (t, r, θ + 2π) and There are outer and inner horizons at r = r + and r = r − , respectively, where the coordinates (t, r, θ) become singular, and a singularity at r = r 0 . The dimensionless coupling ν ∈ (1, ∞) is the warp factor and in the limit ν → 1 the above metric reduces to the metric of the BTZ black hole in a rotating frame. More details about this black hole solution can be found in [13,17,[22][23][24][25][26][27]. Here, we just describe a few relevant features.
The Carter-Penrose diagram for this spacetime when r 0 < r − < r + is shown in Fig. 1, which is essentially of the same form of those of asymptotically flat spacetimes in 3+1 dimensions.
Consider the exterior region r > r + . ∂ t and ∂ θ are Killing vector fields. However, ∂ t is spacelike everywhere, even though surfaces of constant t are still spacelike. Consequently, there is no stationary limit surface and no observers following orbits of ∂ t (the usual "static observers" in other spacetimes) anywhere. In fact, it is easy to show that there is not any timelike Killing vector field in the exterior region of the spacetime.
Nonetheless, there are observers at a given radius r following orbits of the vector field ξ(r) = ∂ t + Ω(r) ∂ θ , which are timelike as long as with Ω(r) is negative for all r > r + , approaches zero as r → +∞, and tends to as r → r + . In view of these observations, we can regard Ω H as the angular velocity of the outer horizon with respect to stationary observers close to infinity. One particular important class of observers are the 'locally non-rotating observers' (LNRO), whose worldlines are everywhere normal to constant-t surfaces. Because of this, they are sometimes also known as 'zero angular momentum observers' (ZAMO). In this case, Ω(r) = −N θ (r), which satisfies (2.6). They are the closest to the concept of 'static observers' in this spacetime.
Even though there is no stationary limit surface, there is still a speed-of-light surface, beyond which an observer cannot corotate with the outer horizon. Given the information above it is easy to check that the vector field χ = ∂ t + Ω H ∂ θ is the Killing vector field which generates the horizon. χ is null at the horizon and at which is the location of the speed-of-light surface.
In the context of quantum field theory, as it is detailed below, the non-existence of an everywhere timelike Killing vector field in the exterior region of the spacetime is directly related to the non-existence of a well-defined quantum vacuum state which is regular at the horizon and is invariant under the isometries of the spacetime. For the Kerr spacetime, this has been proven in [18]. A vacuum state with these properties can however be defined if we restrict the spacetime by inserting an appropriate mirror-like boundary which respects the Killing isometries of the spacetime. The simplest example is a boundary M at constant radius r = r M , in which the scalar field satisfies Dirichlet boundary conditions, Φ(t, r M , θ) = 0. If we choose the radius such that r M ∈ (r + , r C ), then χ is a timelike Killing vector field up to the boundary, and a vacuum state with the above properties is well-defined. Moreover, the introduction of a mirror with reflective boundary conditions also serves to remove superradiant modes and, thus, any ambiguities they might cause when defining positive frenquency mode solutions [5,6].
For convenience, we change coordinates such that χ is given by χ = ∂t. We shall denote these 'corotating coordinates' (t = t, r,θ = θ − Ω H t) and the metric is then given by (2.10) From now on, we consider as the spacetime manifold M the one constructed in the following way. In region I we insert a boundary M at constant radius r = r M , with r M ∈ (r + , r C ), in which Dirichlet boundary conditions are imposed. We denote by I the portion of the region I from the horizon up to the mirror. In region IV, a similar boundary M is inserted, which can be obtained by the action of a discrete isometry J which takes points in region I to points in region IV by a reflection about the bifurcation surface. In a similar way, a region IV is defined. We take as the new manifold M of interest the union of regions I, II, III and IV (see Fig. 1).
Even though the resulting manifold is not globally hyperbolic, the Dirichlet boundary conditions imposed on the boundaries are enough to make the time evolution of the Cauchy data in any spacelike surface unique [28][29][30]. This allows us to analyse quantum field theory in this bounded spacetime.

B. Scalar field equation and basis modes
We consider a real massive scalar field Φ on the exterior region I. The field obeys the Klein-Gordon equation where m 0 is the mass of the field, R is the Ricci scalar and ξ is the curvature coupling parameter. The Ricci scalar is given by R = −6, which is a constant, so we can rewrite (2.11) as where m 2 ≡ m 2 0 + ξR is the "effective squared mass" of the scalar field. Since ∂t and ∂θ are Killing vector fields, we consider mode solutions of (2.12) of the form whereω ∈ R and k ∈ Z.
In [17], closed form solutions to (2.13) were obtained and bases of mode solutions were constructed for the unbounded spacetime. In particular, a set of 'up' basis modes was introduced for the exterior region, corresponding to flux coming from the black hole which is partially reflected back to the black hole and partially reflected to infinity. With a boundary in place, we define a new set of modes in I, Φ Iω k , withω > 0, which are the unique linearly independent solutions that satisfy the Dirichlet boundary conditions at the mirror. We take these solutions to be normalized, in the Klein-Gordon inner product on hypersurfaces of constantt in I. With the purpose of later defining the Hartle-Hawking state, we need to construct a new mode basis. First, we define modes in the region IV, Φ IṼ ωk , by the action of the discrete isometry J defined previously (which takes points in region I to points in region IV by a reflection about the bifurcate surface) We then understand Φ Ĩ ωk to vanish outside region I and Φ IṼ ωk to vanish outside region IV, and we define in the union of I and IV the new mode solutions Φ L ωk and Φ R ωk , by These L and R modes can now be analytically continued to all of M by crossing the horizon at r = r + in the lower half-plane in the affine parameters of the generators of the two branches at the horizon. Φ L ωk and Φ R ωk are hence of positive frequency in the affine parameters on the horizon. They are further orthonormal in the Klein-Gordon inner product on spacelike hypersurfaces from mirror to mirror (for more details of the construction, see e.g. [6] or appendix H of [31]).

C. Quantized field and Hartle-Hawking vacuum state
So far, only classical theory has been discussed. We now proceed to canonically quantize the scalar field using the standard Hilbert space approach. This is possible since, as seen above, there is a natural positive and negative frequency decomposition of the mode solutions for this spacetime.
Define H to be the one-particle Hilbert space of the positive frequency L and R solutions, and let F s (H ) be the corresponding Fock space, defined in the usual way. Denote the vacuum state by |H ∈ F s (H ). Since the L and R solutions are positive frequency with respect to the affine parameters of the past and future horizons, this vacuum state is regular at the horizons. Furthermore, it is invariant under the spacetime isometries. Therefore, we call |H the 'Hartle-Hawking vacuum state'.
The quantized scalar field Φ(x) is given by: The Feynman propagator is defined as where T is the time-ordering operator. The Feynman propagator is a bidistribution, G F ∈ D (M ×M ), and it is one of the Green's functions associated with the Klein-Gordon equation.

D. Hadamard renormalization
The Feynman propagator, evaluated for certain quantum states, as defined in (2.19), is a bidistribution of Hadamard type, i.e. it has an Hadamard expansion of the form Here, we assume that x and x belong to a geodesically convex neighborhood N ⊂ M , that is, they are linked by a unique geodesic which lies entirely in N . Additionally, σ ∈ C ∞ (N ×N ) is the Synge's world function, defined such that σ(x, x ) is the half of the square of the geodesic distance between x and x ; and U ∈ C ∞ (N × N ) and W ∈ C ∞ (N × N ) are symmetric and regular biscalar functions. A quantum state for which the short-distance singularity structure of G F is given by It can be shown (see e.g. [33]) that U (x, x ) only depends on the geometry along the geodesics joining x to x , whereas W (x, x ) contains the quantum state dependence of the Feynman propagator. Therefore, the singular, state-independent part of the Feynman propagator is This is known as the 'Hadamard singular part' and it is singular at x → x.
The biscalar U (x, x ) can be expanded as For the computation of the vacuum polarization, it is sufficient to know the zeroth term, Given G Had , we may obtain the renormalized vacuum polarization Φ 2 (x) in any Hadamard state as To Φ 2 (x) , as defined in (2.24), one can add terms proportional to m, as can be verified by dimensional analysis. This is a usual feature of any renormalization procedure. For instance, for a scalar field of mass m on Minkowski spacetime in the Minkowski vacuum |0 , the renormalized vacuum polarization computed as in (2.24) is We are free to set this quantity to any desired value by adding a term proportional to m.
In the case of the expectation value of the stress-energy tensor, it is conventional to set 0|T µν (x)|0 ≡ 0 for the Minkowski vacuum. In this paper we shall not attempt to introduce a criterion for fixing this ambiguity and shall just define Φ 2 (x) as in (2.24).

III. COMPLEX RIEMANNIAN SECTION OF THE SPACELIKE STRETCHED BLACK HOLE
In this section, we first consider the complex Riemannian section of the spacelike stretched black hole and obtain the unique Green's function associated with the Klein-Gordon equation as a mode sum. This is followed by a detailed account of the Hadamard renormalization procedure, in which we subtract the divergences in the mode sum by a sum over Minkowski modes with the same singularity structure.

A. Complex Riemannian section
Euclidean methods are a powerful tool to study quantum field theory on static spacetimes. A static spacetime is a real Lorentzian section of a complex manifold, for which it is always possible to find a real Riemannian (or 'Euclidean') section by performing an appropriate analytical continuation (usually a Wick rotation t → −it, where t is a global timelike coordinate). In many cases, it is much easier to perform calculations in the Riemannian section (e.g. computing the unique Green's function associated with the scalar field equation) and then analytically continue the results back to the Lorentzian section.
These methods are not easily generalized to stationary, but non-static, spacetimes and, in particular, to rotating black hole spacetimes, which not only do not possess a global timelike Killing vector field, but in many cases do not have a timelike Killing vector field in their exterior regions (as in the Kerr and spacelike stretched cases). To obtain a real Riemannian metric, apart from Wick rotating t → −it, a further analytical continuation is necessary, usually by allowing the real angular momentum of the spacetime (when defined) to be imaginary [32]. It is not clear at all this is a real section of a complex manifold which as the original spacetime as a real Lorentz section. Here, we take the view of [20] that the resulting metric is not in general related to the original Lorentzian metric.
However, for our purposes, we only need to generalize the procedure to the region I of our spacetime. In this region there is an everywhere timelike Killing vector field, χ = ∂t. If we now perform a Wick rotationt = −iτ , with τ ∈ R,the metric (2.10) becomes This is the complex-valued metric g C of the 'complex Riemannian' (or 'quasi-Euclidean') section I C of a complex manifold, in which region I is a real Lorentzian section. This metric is regular at the horizon if τ is periodic with period 2π/κ + , where κ + is the surface gravity, The resulting manifold has then two periodic (and thus compact) directions and a third direction that is compact by virtue of the mirror at r = r M . The complex Riemannian section of certain rotating spacetimes has been briefly discussed in [19] and [34] in the context of the Kerr-Newman black hole. In [21], a more general concept of 'local Wick rotation' is discussed for any Lorentzian manifold, even without a timelike Killing vector field, as long as its metric is a locally analytic function of the coordinates.

B. Green's function associated to the Klein-Gordon equation
In the real Lorentzian section, we defined the Feynman propagator G F ∈ D ( I × I) as one of the Green's functions associated with the Klein-Gordon equation satisfied by the scalar field. Here, we find the Green's function G ∈ D (I C × I C ) associated with the Klein-Gordon equation in the complex Riemannian section. It satisfies the distributional equation where g(x) := | det(g C µν )| and ∇ 2 := (g C ) µν ∇ µ ∇ ν is the covariant d'Alembertian operator. In contrast to the real Lorentzian section, in the complex Riemannian section there is a unique solution to this equation which satisfies the following boundary conditions: (i) G(x, x ) is regular at r = r + , and (ii) G(x, x ) satisfies the Dirichlet boundary conditions at r = r M . This is due to the fact that two of the directions of the spacetime are periodic, while the third direction is compact. Compare this to the situation on static spacetimes without any boundary (and suitable asymptotic properties at infinity), whose Euclidean section has a unique Euclidean Green's function, due to the ellipticity of the Klein-Gordon operator.
Given the periodicity conditions of τ andθ, one has understood as distributional identities. If we expand G(x, x ) as and use (3.4) and (3.5) one obtains a differential equation for G nk The solutions of this equation can be given in terms of solutions of the corresponding homogeneous equation. Two independent solutions of the homogeneous equation are where we introduce a new radial coordinate and where the parameters of the hypergeometric functions are given by Our convention for the the branch of the square roots in (3.12) is the one with non-negative real part. Considering again the equation (3.7) for G nk , the regular solution near the event horizon at z = 0 is whereas the Dirichlet solution near the mirror at z = z M is given by (3.14) The radial part of the Green's function is then where z < := min{z, z }, z > := max{z, z } and C nk is the normalization constant. If we rewrite (3.6) as then C nk is given by . (3.17)

C. Hadamard renormalization
As we did before with the Feynman propagator in the real Lorentz section, we want to investigate the short-distance singularity structure of the Green's function G obtained in the complex Riemannian section. That is done in some detail in appendix C, which follows [21].
The main idea is the notion of a geodesically linearly convex neighborhood of p ∈ I C , which is essentially a neighborhood of p, N p ⊂ I C , such that, for any q, q ∈ N p , there is only one real-parameter geodesic segment which links q and q and which lies completely in N p (see appendix C for more details). It was shown in [21] that, given a complex Riemannian manifold such as the one considered in this paper, for any given point, there is always a geodesically linearly convex neighborhood N . Therefore, we can define the complex Synge's world function σ ∈ C ω (N × N ), which reduces to the usual definition for Riemannian and Lorentzian manifolds. In particular, suppose we choose x and x in a way such that two of their coordinates in a given coordinate system are the same and the induced metric on the submanifold defined by this condition is either Riemannian or Lorentzian. Then, we can use the previous definition as half of the square of the geodesic distance between x and x .
Having checked that the Synge's world function can be defined in the complex Riemannian section, we can now write the Hadamard singular part of the Green's function G as In an analogous way to the Lorentzian case, we now subtract the Hadamard singular part from the Green's function G from which one obtains the vacuum polarization at x ∈ I (In a slight abuse of notation, on the RHS of the equation x, x ∈ I C , such that x ∈ I C is the result of a Wick rotation of x ∈ I.) By construction, the Green's function G is regular at r = r + , satisfies the Dirichlet boundary conditions at r = r M and is invariant under the spacetime isometries. Therefore, after analytically continuing back to the Lorentz section, Φ 2 (x) as given by (3.20) is the vacuum polarization for a scalar field in the Hartle-Hawking state.

D. Subtraction of the Hadamard singular part
We have obtained the Green's function G(x, x ) as the mode sum (3.16), whereas G Had (x, x ) is given in the closed form (3.18) by the Hadamard procedure. For computational purposes, it is convenient to consider a particular choice of point separation. Consider the complex Riemannian section in (τ, z,θ) coordinates and suppose that x and x are in the region I C and are angularly separated. In this case, x and x are in a geodesically linearly convex neighborhood and the complex Synge's world function can be obtained for small angular separation using the standard Riemannian relation. It is given by Thus, the Hadamard singular part of the Green's function is Without loss of generality, let x = (τ, r, 0) and x = (τ, r,θ), withθ > 0, such that As G(x, x ) is known only as the mode sum (3.16), the evaluation of Φ 2 (x) as the limit (3.20) requires G BH Had to be rewritten as a mode sum that can be combined with (3.16) so that the divergences in the coincidence limit get subtracted under the sum term by term. We shall acomplish this by comparing G BH Had to the Hadamard singular part for a scalar field in rotating Minkowski spacetime in the complex Riemannian section, which is computed in appendix B. The advantage of using the Minkowski spacetime is that its symmetries allow us to compute the Green's function in both closed form and as a mode sum.
The Hadamard singular part of the Green's function for a scalar field in the Minkowski vacuum for angularly separated points can be written as where the notation is described in appendices A and B. Suppose one identifies the leading terms of the Hadamard singular parts of both spacetimes by 1 4π where γ(r) > 0 is a function to be specified below. This provides a matching between the two radial coordinates: ρ(r) = γ −1 (r) R(r) . (3.26) Given this identification, we can now write: The parameters on the Minkowski Green's function (ρ, T , Ω and m 2 M ) can now be chosen such that the double sum is convergent whenθ → 0. After this matching is performed, the vacuum polarization is just given by which is a well-defined smooth function for x ∈ I.

E. Fixing of the Minkowski free parameters
At least some of the parameters of the Minkowski Green's function must be fixed such that the double sum in (3.27) is convergent in the coincidence limit. To motivate the choice of the parameters, we look at the large n and k behaviour of the summand by performing a WKB-like expansion, as explained in appendix D.
Using Proposition D.2, one can write the asymptotic expansions and with and This allows us to write The terms O(χ −1 ) in the two expressions match if the parameters T , Ω and γ(z) are chosen as This choice corresponds to have the temperature T of the scalar field in Minkowski to match the Hawking temperature of the black hole and to have the angular velocity Ω to be equal to the one measured by a locally non-rotating observer at radius z in the black hole spacetime. We now claim that, with this choice of parameters, the double sum in (3.27) is convergent in the coincidence limit.
Theorem III.1. If the parameters T , Ω and γ(z) are chosen as in (3.35), then is finite.
Proof. It is enough to consider where n,k stands for the double sum over k and n excluding the k = n = 0 term. The first terms in the WKB-like expansion cancel each other, thus With the choice (3.35), one has where W(z) does not depend on n and k. Note that: In appendix E, it is shown that the latter series is convergent. This proves the absolute convergence of the limit comparison test implies the absolute convergence of Therefore, we conclude that the ∆G(z) is finite.

IV. NUMERICAL EVALUATION OF THE VACUUM POLARIZATION
We numerically compute the vacuum polarization of the scalar field in the Hartle-Hawking state in region I using the expression (3.27) and (3.28) with the Minkowski parameters chosen as in (3.35): with → 0+ indicating the choice of branch of the square root (see details in appendix B). As described previously, the sums in (4.1) are convergent. For the numerical evaluation of the sums, cutoffs are imposed appropriately. Note that the parameter m 2 M is not fixed and it is chosen in such a way to improve the numerical convergence of the double sum over k and n.
The numerical results for selected values of the parameters are presented in Fig. 2. In the plot, Φ 2 (x) is shown as a function of the normalized radial coordinate z/z M . Note that Φ 2 (x) gets arbitrarily large and negative as the mirror is approached, as expected (see e.g. chapter 4.3 of [7]). Note also that the plot is very similar to the one found in Ref. [36] for a scalar field in the (3+1)-dimensional Minkowski spacetime surrounded by a mirror with Dirichlet boundary conditions.
We re-emphasize that the result shown in Fig. 2 is the full renormalized vacuum polarization in the Hartle-Hawking state. To find the renormalized vacuum polarization in other Hadamard states of interest, such as the Boulware vacuum state, it would suffice to use the Hartle-Hawking state as a reference and just to calculate the difference, which is finite without further renormalization. For comparison, we note that in Kerr with a mirror the difference of the vacuum polarization in the Boulware and Hartle-Hawking states was found in [6], while the renormalized vacuum polarization in the individual states appears to be still unknown.

V. CONCLUSIONS
In this paper, we have computed Φ 2 (x) for a massive scalar field Φ in the Hartle-Hawking state on a spacelike stretched black hole with a mirror. We have employed a 'quasi-Euclidean' method to obtain a complex Riemannian section of the original spacetime, in which we found the unique Green's function associated with the Klein-Gordon equation. This Green's function is given as a mode sum and its singular behaviour in the coincidence limit can be subtracted by a sum over Minkowski modes with the same singularity structure. This renormalization procedure renders a smooth function whose coincidence limit is precisely the renormalized value of Φ 2 (x) . In the future, we intend to extend this method to compute the expectation values of the stress-energy tensor.
A key ingredient in our implementation of the Hadamard renormalization was to match the mode sum for the Green's function on the complex Riemannian section of the black hole to a mode sum on the complex Riemannian section of a rotating Minkowski spacetime. We anticipate that this method can be extended to wider classes of rotating black hole spacetimes, and in particular in four dimensions to the Kerr spacetime. In Kerr, the relevant mode solutions to the Klein-Gordon equation on the complex Riemannian section would need to be constructed fully numerically, but the asymptotic properties of the solutions in the limit of large quantum numbers should be within analytic reach, and it is only these asymptotic properties that are required in the matching to mode solutions on a complex section of rotating Minkowski. Also, the freedom in the shape of the mirror in Kerr should not present complications for the matching since boundary terms in the rotating Minkowski mode functions do not enter the final subtraction terms. The implementation of our method in Kerr would hence seem feasible in principle, and it should prove interesting to attempt the implementation in practice.
If one now expands the Green's function G(x, x ) as then G nk (ρ, ρ ) satisfies (A12) Consider the homogeneous equation associated with (A12) and let p nk (ρ) be the regular solution near ρ = 0 and q nk (ρ) be the Dirichlet solution near ρ = ρ M . Then, the unique solution to the inhomogeneous equation is where C nk is a normalization constant which is determined from the Wronskian relation Comparing (A6) and (A12) one concludes that the solutions to the homogeneous equation corresponding to (A12) are Moreover, Eq. (A14) leads to C nk = 1, thus with ρ < := min{ρ, ρ } and ρ > := max{ρ, ρ }. In the appendix A, the Green's function G(x, x ) for a scalar field at temperature T in the complex Riemannian section of Minkowski spacetime was obtained as a mode sum over k and n. Its Hadamard singular part G Had is given in closed form by (3.23). For the purposes of this paper, we also want to express the Hadamard singular part of this Green's function as a mode sum.
We can write the Green's function G(x, x ) (A11) as where G reg (x, x ) is finite when x → x. As G Had has no mirror dependence, it is convenient to express it as andĜ reg (x, x ) finite when x → x. In this form, neither of the terms on the RHS of (B2) has any mirror dependence. We have written G Had as a mode sum (plus a regular term), which can be used to subtract the divergences in the black hole Green's function, as detailed in section III D. It remains to computeĜ reg (x, x ). Since this term is finite in the coincidence limit, we only need to determine the limit of this term when x → x.
In the case of angular separation, the Green's function becomes G(τ, ρ, 0; τ, ρ,θ) = 1 4π coordinate ξ such that the radial field equation can be written in the form and the Wronskian relation is given by where C is a constant. Here, χ 2 (ξ) contains all the n and k dependence and is large whenever λ 2 := k 2 + n 2 is large. We assume then that f (ξ; λ) := − (χ 2 (ξ) + η 2 (ξ)) has an asymptotic expansion of the form where {a j (λ)} is an asymptotic sequence such that a 0 (λ) = 1. In this case, standard WKB theory guarantees that there is an asymptotic expansion for the solutions φ i (ξ), i = 1, 2, when λ → +∞, given by the so-called WKB method (see e.g. [38]). We are interested in obtaining the large χ expansion of