Interior of the Horizon of BTZ Black Hole

A quantum scalar field inside the horizon of a non-rotating BTZ black hole is studied. Not only near-horizon modes but also the normal modes deep inside the horizon are obtained. It is shown that the matching condition for normal modes of a scalar field between outside and inside of the horizon does not uniquely determine the normal-mode expansion of a scalar field inside the horizon.


Introduction
The interior of the black hole is not well understood. Understanding of its structure is necessary to resolve the information paradox [1] and the firewall problems. [2][3] [4][5] [6] [7] [8]. Recently there has been progress in the study of Hawking radiation. [9] [10] In this paper a scalar field in the BTZ black hole [11] is studied and the problem of duality between the region behind the horizon and the 2d conformal field theory (CFT) on the infinite boundary is revisited. This problem was studied before in [12] and it was concluded that the region inside the horizon can be described in terms of the boundary CFT. It was shown that BTZ solution is obtained by identifying points in AdS 3 described by a hyperboloid embedded in a flat space with signature (1, 1, −1, −1): x 2 0 +x 2 1 −x 2 2 −x 2 3 = 1. AdS 3 is classified into three types of regions and each region is covered by four separate coordinate patches. Each region has coordinates (r, t, φ) and coordinates of every pair of regions are related by imaginary number shift of t variable. It was argued that bulk to boundary scalar propagator for a point inside the horizon can be obtained via a shift of t by iβ/4, where β is the inverse of the temperature of the boundary CFT, in the bulk to boundary propagator for a point outside the horizon, when the Schwarzschild radial variable r is used. In [13] a bulk local state of a scalar field was constructed behind the horizon and it was also argued that two-point functions between points behind and outside the horizon of BTZ space are obtained from those between points outside horizons by the shift of t by iβ/4 . It is known that the metric tensor of BTZ space can be obtained from that for AdS 3 space by a coordinate transformation. [14] By using these coordinate transformations and the shift of t, it is formally possible to construct would-be two-point functions between points inside and outside horizons from those two-point functions for points outside horizons. The results can be expressed in terms of geodesic distance between the two points.
It is, however, not clear whether these observations are valid, until actual quantization of matter fields inside horizon is carried out explicitly. The roles of the time variable t and the radial variable r outside the horizon are interchanged inside the horizon. Hence it is not guaranteed that this simple shift of t would yield correct two-point functions between points inside and outside horizons. To check whether this is a correct procedure, it is necessary to carry out quantization of matter fields inside the horizon.
A general prescription for quantization of matter fields inside the horizon of an eternal black hole was proposed by Papadodimas and Raju [4][5] [6]; we must find out normal modes of matter fields inside the horizon, which have distinct power behaviors near the horizon, and impose suitable matching conditions on the fields on both sides of the horizons. Then we use the same set of creation and annihilation operators on both sides of the horizon. It was shown [4] that this prescription works in the case of Rindler space. Because there are not so many examples, where concrete calculations are possible, it is desirable to carry out studies of quantization of matter fields inside horizon in cases of black hole spacetimes, and show that the prescription works, or whether there arise any problems in the case of real black holes. Recently, a condition for quantum state to be smooth near the horizon under scalar field perturbations was also studied in Reissner-Nordstróm black holes and BTZ black holes. [15] Purpose of this paper is to study these matters, and clarify whether the interior of BTZ black hole has a description in terms of CFT's on AdS boundaries according to the principle of holography [16] [17]. The normal modes which we will study are those present throughout behind the horizon and are distinct from the near-horizon modes.
It will be shown that the matching condition for the scalar field at the horizon makes the near-horizon modes on both sides of the horizon smoothly connected, but are not sufficient to uniquely determine the normal mode expansion of scalar fields deep inside the horizon. There exist some undetermined functions of frequency and momentum in the normal modes for the scalar field inside the horizon. The two point functions of a scalar field between points inside and outside the horizon are not simply related to the two-point functions with both points outside by a shift of t by iβ/4. Furthermore unless appropriate additional conditions are imposed on the scalar theory inside the horizon, interior of the horizon will not be holographically dual to the boundary CFT's.
In this paper we will use unconventional approach for quantization of scalar field in the spacetime of BTZ black hole. The eternal AdS-Schwartzschild balck hole is described by a tensor product state called Thermo-Field Double (TFD) introduced by Israel [18].
This thermal state is a Hartle-Hawking-like state. [20] The spacetime is represented by Penrose diagram and it consists of four regions I, II, III and IV. For quantization of a matter (scalar) field it is decomposed into a complete orthonormal set of positive-frequency modes f ωℓm on a Cauchy surface Σ and regard the coefficient of the expansion as the annihilation operator a ωℓm , which satisfies together with the creation operator the usual commutator algebra. [19] Usually, this is carried out in regions I and III separately, which are outside the horizon. In this case a real scalar field is represented in terms of a single set of creation and annihilation operators in region I, and in terms of another independent set in region III. Then in order to make the scalar field smoothly connected at the horizon the state (1.1) is constructed in the tensor-product Hilbert space. [18] [20] In this paper the normal modes in black hole background are obtained from those of AdS 3 spacetime by suitable coordinate transformations. [14] These normal modes turn out not eigenstates of energy and momentum. We quantize scalar field on the constant-t slice, which is obtained by combining those slices in both regions I and III, by requiring that the normal modes form a complete orthonormal set on the combined constant-t slice.
This prescription ensures the smoothness of the scalar field at the horizon automatically.
We take time t in region III to flow upwards. By changing basis of the normal modes to that of eigenstates of energy and momentum it is found that the creation and annihilation operators in each regions I and III, which are diagonalized into the eigenstates of energy and momentum, are identified as single-trace operators of left and right boundary CFTs.
Then the vacuum state is shown to be the TFD (1.1). Furthermore, in this study each normal modes are represented in terms of integral representations, and this makes analysis of explicit asymptotic forms of these normal modes easy.
Then the region II behind the horizon will be studied. In region II integral representations of normal modes are technically more helpful. In region II some set of normal modes are obtained from those in AdS 3 by a coordinate transformation. It is also necessary to introduce yet a new set of them. Because in curved spacetimes without time-translation symmetry positive-and negative-frequency solutions cannot be defined, the number of independent basis functions must be doubled and then Klein-Gordon (K-G) inner products of normal modes in region II are not positive definite. It is also shown that K-G inner products of the normal modes in region II are discontinuous at the horizon. Using these normal modes solutions for matching condition at the horizon will be obtained.
This paper is organized as follows. In secs. 2 to 4 a scalar theory outside the horizon in BTZ black hole is quantized by using unconventional method. In sec. 5 a set of normal modes of a scalar field in region II is found, and in sec.6 it is found that there exists discontinuity in the inner products of these normal modes at the horizon. In sec.7 matching condition for the normal modes in region II and those in regions I and III are solved. In sec. 8 this solution is analyzed. Summary and conclusion are given in sec.9.
There are appendices A to E for details of calculations.

Normal Modes and Klein-Gordon Inner Product Outside Horizon
The normal mode functions of a real scalar field in BTZ black hole background were obtained in [22], [23]. For simplicity only the BTZ black hole without angular momentum will be considered in this paper. The metric field is given by Here a = 8GM ℓ 2 is the mass of the black hole. The horizon is located at r = r + = √ a.
In the following the AdS length ℓ will be set to unity. The classical equation of motion for a scalar field φ(t, r, ϕ) with mass m is given by After separation of variables the normal mode functions of the scalar field are written as It was found that those solutions, f ωn (r) ∼ r 1+ν , which satisfy the normalizable boundary condition [24] at the infinite boundary (r ∼ ∞), are given by F (a, b; c; z) is a hypergeometric function. Here ν = √ 1 + m 2 and ∆ = 1 + ν is a scaling dimension of the single-trace operator in the dual conformal field theory (CFT). Only the case of ν = integer will be considered in this paper. The variable u is defined by u = r 2 a , and the parameters are In what follows we will use the following form of metric for the outer region of nonrotating BTZ black hole.  hole is divided into four regions, I, II, III, IV. The outer regions are I (right, 0 < y ≤ 2 √ a ) and III (left, y ≥ 2 √ a ), the inner regions are II (future) and IV (past). Normal modes of a scalar field in BTZ background are obtained from those in pure AdS 3 by a certain coordinate transformation. See Appendix A.
Normal modes for a scalar field in region I (0 < y ≤ 2 √ a ) are given by 1 Here ω and k are parameters which take values in ω ≥ 0 and |k| ≤ ω. The superscript I on Φ shows that these are modes in region I. As for dependence on y, Φ I ω,k (t, y, x) behaves as y ∆ in the limit y → 0. These modes are regular at the horizon y = 2 √ a . A mode Π = √ −g(−g tt )∂ t Φ I ω,k for the momentum conjugate to Φ I ω,k is also regular at the horizon. Actually, both the parameters ω and k are not energy and momentum. Later, the solution in the form (2.3) will be obtained by carrying out Fourier transformations over ζ and µ in (2.8), where ω and k are related to new ones ζ and µ by These new parameters take values in −∞ < ζ < ∞ and 0 ≤ µ < ∞. In this section quantization of a scalar field is studied by using these modes. Here the spatial variable x corresponds to a line −∞ < x < ∞ of planar BTZ, while ϕ in (2.3) corresponds to a circle 0 ≤ ϕ ≤ 2π for spherical BTZ. In this section quantization of a scalar field is studied by using these modes. The result for the BTZ case, where the spatial direction is a circle (0 ≤ ϕ ≤ 2π), will be obtained by using the method of images.
First, the Klein-Gordon (K-G) inner product for these modes in region I will be computed. The K-G inner product in regions I is defined for functions f (t, y, x) and g(t, y, x) as follows.
The integration region for y is 0 ≤ y ≤ 2 √ a . The expression in the integrand which depends on y is √ −g(−g tt ). The subscript I for the inner product on the left hand side means that the integration is to be carried out in region I. This inner product does not depend on t.
Hence this can be computed at t = 0. 1 Here ω is different from the one in (2.3).
By setting t = 0 and using (2.8) the inner product (Φ I ω,k , Φ I ω ′ ,k ′ ) I is given by After rescaling y → 2 √ a y, we set ρ = exp √ ax. Then we have Here we defined This integral appears to be imaginary. To check this is the case, let us change variables from x to ρ = exp( √ ax) and define new integration variables in place of ρ and y.
These take values in z, w ≥ 0 and then the above integral is evaluated as Here P stands for a principal-value prescription and here the following Fourier-Bessel formula is used.
So this inner product is a projection matrix.
We also need normal modes in region III (2/ √ a < y) and choose the following. We flip the direction of time t in region III compared to that in Penrose diagram and t is assumed to flow upwards. As for dependence on y, Φ III ω,k (t, y, x) behaves as y −∆ in the limit y → ∞. It can be shown that the following two sets of normal modes in the full outer region, (0 < y < ∞), form a complete set of orthonormal functions, although the two regions are causally disconnected.
At the horizon and t = 0 these modes are smoothly connected. In region III we have an inner product, Here the sign of the y-dependent factor is changed compared to (2.11), because integration region for y is 2 √ a ≤ y. The inner product of Φ III ω,k after rescaling y → 2 √ a y and change of variable ρ = exp √ ax is given by This coincides with a complex conjugate of the inner product (2.13). The sum of these two is a delta function.
As for the other inner products, it can be shown that and (Φ * ω,k , Φ ω ′ ,k ′ ) = 0. The scalar field Φ(t, y, x) is expanded into modes (2.8) and (2.18). In region I, we while in region III we have, The scalar field is represented in terms of the single set of operators, a(ω, k), a † (ω, k) in both regions I and III. By using the K-G inner products, the creation and annihilation operators are expressed as The commutation relations of these operators are obtained by imposing the canonical commutation relations (CCR's): when (t, y, x) and (t, y ′ , x ′ ) are both in region I or III, and when the two points are separated by the horizon. Furthermore, must hold for any separation of the two points. Here Π = √ −g(−g tt )∂ t Φ is a canonical momentum field. We also checked that these CCR's (2.29)-(2.31) are satisfied for 0 < y, y ′ < ∞ by using the mode expansions (2.24)-(2.25) and commutation relations (2.26)-(2.27).
The above annihilation operator defines a vacuum |0 β .
a(ω, k)|0 β = 0 (ω ≥ 0, |k| ≤ ω) (2.32) Here β = 2π/ √ a is an inverse temperature. As will be shown in sec.4, this is a thermal 'vacuum'. This vacuum is invariant under t and x translations. Under these transformations the normal mode transforms as k ′′ = k e √ aǫ . Same transformations are also valid for Φ III ω,k . Therefore, a(ω, k) and a † (ω, k) must be linear representations of the two translations, and the vacuum in (2.32) respects translation invariances of the vacuum.

Change of Basis of Normal Modes
In this section the basis of the normal modes will be changed to eigenstates of energy and momentum. We will denote ω and k as , it is easy to show that ζ in (2.8) and (2.18) plays the same roles as √ at. So we carry out the following Fourier transformations.
as (2.8) is. Now let us carry out the following shift of integration variables, ζ and ln µ.
We have Now energy E and momentum p are diagonalized. This normal mode will coincide with one of (2.3). In this representation, the imaginary periodicity of t is lost. This means that the move of the contour of ζ in the imaginary direction is not allowed. At the horizon y = 2 √ a (3.6) is proportional to δ(E) and vanishes for E = 0. In the AdS 3 /CFT 2 correspondence and in the leading order of 1/N expansion, the eigenvalues E and p for a scalar field are given by E = ∆ + 2n + |m| and p = m, such that n = 0, 1, . . . and m = 0, ±1, ±2, . . ..
However, in the large N limit, the energy eigenvalue becomes continuous. [4] In this paper continuous spectrum of energy and momentum eigenvalue will be adopted.
The other normal mode Φ III E,p is similarly given by Time t in region III is defined to flow upwards as opposed to the usual choice for the Penrose diagram. The coefficient of iEt in the first exponent is flipped w.r.t. that in (3.6).
Next the K-G inner product will be worked out. This is done by using (2.22), (3.2) and (3.3). We have The scalar field is expanded into the above modes (3.6) and (3.7) as follows. In region Here b ± (E, p) are an annihilation operator for positive (negative) frequency normal modes.
A notable point is that in the second line the negative-frequency mode Φ I −E,−p is associated with the annihilation operator b − (E, p). b † − creates a 'hole', while b † + creates a 'particle'. [25] [21] In sec.4 it will be shown that Φ I E,p and Φ I * −E,−p are linearly dependent. Similarly in region III, we have Therefore in region III operator b + is associated with the negative-frequency modes, and b − the positive-frequency modes. Therefore the scalar field contains negative-frequency operators as well as positive-frequency ones. Φ III E,p and Φ III * −E,−p will be also found to be linearly dependent.
b +,− annihilate the vacuum |0 β . The relation between a(ω, k) and b ± (E, p) is given by Hermitian conjugate of this equation gives a relation for a † (ω, k). The commutation relations of b ± and b † ± are given by Other commutators vanish.

Boundary limit of the Scalar Field and Bulk Reconstruction
In this section the boundary limit of Φ I,III will be considered, and the CFT operators on the two boundaries will be identified. Some detailed equations are summarized in Appendix B First, the normal modes Φ I E,p (3.6) has the following y → 0 limit.
where g(E, p) is a function defined by Then the scalar field in region I has the following y → 0 limit.
where O R (t, x) is a single trace operator on the boundary and the operator c R (E, p) is defined by c † R is its hermitian conjugate. These operators are boundary CFT operators on the right boundary, and satisfy the commutation relations.
Similarly, the boundary limit y → ∞ of the normal modes (3.6) in region III is given whereg(E, p) is a function defined bỹ Then the scalar field in region III has the following y → ∞ limit.
where the operator c L (E, p) is defined by Moreover, because regions I and III are causally disconnected, the scalar fields in both regions commute. Therefore c R and c † R commute with c L , and c † L . c L and c † L are CFT operators on the left boundary.
When (4.4) and (4.9) are simplified by using the results in Appendix B, it is found that these relations are Bogoliubov transformations.
The thermal expectation values of the number operators are the Bose-Einstein distribution.
Let us define a ground state of CFT's, |0 = |0 L ⊗ |0 R as the state annihilated by c R,L .
Then by solving (4.11), (4.12) in favor of b +,− it can be shown that the state |0 β annihilated by b +,− is a Hartle-Hawking state, or Thermo Field Double [18].
This is an entangled state. The Thermo-Field Hamiltonian is given by So by carrying out quantization of a scalar field in regions I and III in a single Hilbert space we obtained a Hartle-Hawking-like state as a 'vacuum' which is annihilated by the annihilation operators b +,− (E, p) for the scalar field.
By using the asymptotics (4.1) and (B.2) it can be shown that as y → 0 the following equation holds.
Because Φ I E,p is a solution to the K-G equation and the allowed asymptotic leading powers of y are 1 ± ν, the left hand side vanishes identically: Hence after some algebra the following eq is obtained.
This can be also expressed in terms of O R by using (4.3). Here it will not be attempted to rewrite (4.20) in terms of an integral as in [17]. Time evolution of Φ I is generated by Similarly in region III we obtain

Normal Modes of a Real Scalar Field behind the Horizon
In this section solutions to the K-G equation behind the horizon will be considered.
The metric inside the horizon can be obtained from (2.1) by a coordinate transformation.
Then the metric tensor behind the horizon is written as In region II η is a time variable and takes values in η ≥ 2 √ a , and in region IV 0 ≤ η ≤ 2 √ a .
In this paper region IV is not considered. One set of the normal modes in region II obtained by a coordinate transformation from that for AdS 3 is given by (A.23), where ω = µ cosh ζ and k = µ sinh ζ. Then the eigenfunctions of momenta corresponding to t and x translations are given by Fourier transformation of (5.3) with respect to ζ and ln µ.
Here E and p are momentum eigenvalues conjugate to t and x, and take values in −∞ < E, p < ∞. This solution can also be represented as a suitable linear combination of solutions constructed in terms of a hypergeometric functions as in [22] [23]. This integration formula, however, allows explicit asymptotic formulas.
This set of normal modes, however, does not form a complete set of linearly independent functions. In (C.5) of Appendix C it is shown that the following relation holds for η > Let us note that E is a component of spatial momentum. Then we will introduce a new set of normal modes.
This is related to (5.4) by a relatioñ The Klein-Gordon (K-G) inner products for these modes in region II, η ≥ 2/ √ a, is defined for functions f (η, t, x) and g(η, t, x) as follows.
The factor in the integrand which depends on η is √ −g(−g ηη ). The subscript II for the inner product on the left hand side means that the integration is to be carried out in region II. This inner product does not depend on η owing to K-G equation (at least away from the horizon. See next section.) Hence the inner products of the normal modes will be computed in the limit η → +∞. They are evaluated in Appendix C. Some of the results These inner products are not positive definite: for example, (5.9) is negative for p > 0 and positive for p < 0. Furthermore, (5.4) behaves as Φ at large η. Similarly (5.6) behaves asΦ . By using these properties of mode functions it can be shown that In addition to Φ Although usually, only two of these functions are chosen to be a basis of linearly independent functions, we will consider a vector space V of functions spanned by this set of the normal modes, and expand the scalar field inside horizon into these modes. This is because in curved spacetime without time-translation symmetry positive-and negative-frequency solutions cannot be defined and it is not known beforehand which normal modes to assign to annihilation operators. So the number of independent basis functions must be doubled.
Another reason is that it will be shown in the next section that the inner products of these functions are not independent of η at the horizon, i.e., take distinct values at the horizon from those for η > 2/ √ a. Therefore it is necessary to keep all of the modes (5.13) as the basis in the analysis of matching conditions at the horizon. Furthermore, Φ II(ν) E,p and Φ II(ν) * −E,−p are not linearly independent as shown above. So the mode functions with p > 0 and those with p < 0 must be treated separately. The above four mode functions (5.13) are all proportional to e −iEt+ipx .
The scalar field must satisfy matching conditions at the horizon [4] as well as normalization condition of the inner products of the normal modes. For this prescription to work the inner products of the normal mode functions which are used to expand the scalar field must be constructed in such a way that they are continuous in the interior region of the black hole including the horizon.
Here the factor g tt √ −g in the first line is given by This has a singularity exactly at the location of the horizon η = 2/ √ a. Because the integrand is a total derivative, the right hand side will vanish for η > 2/ √ a. Then the inner products are independent of η as long as η > 2/ √ a. Just at the horizon, however, special care must be taken. This problem can be studied by computing the inner product of the mode functions as η → ∞ and at η = 2/ √ a, separately, and showing that the results of the two cases are same or distinct. 2 Inner products of (5.13) at η > 2/ √ a are given in (C.9)-(C.16). These are computed by using the asymptotic behaviors (C.4) for η → ∞. Inner products of the four modes in 2 In the case of Rindler space it can be shown that there is also a singularity at the horizon in an equation similar to (6.1). In this case, however, inner products of mode functions are not discontinuous at the horizon.
(5.13) can also be computed in a region near the horizon. In (D.6) a behavior of Φ II(ν) E,p near the horizon is presented. By using this result some inner products of (5.13) are computed in Appendix D. The results are completely different from those in (C.9)-(C.16). Hence the inner products of the mode functions have discontinuities at η = 2/ √ a.
Next, we will study linear dependence of the mode functions (5.13). In the η → ∞ limit, Φ II(ν) E,p behaves as (C.4). This shows that there exist following two relations in this limit.
Here D(E, p) is defined in (C.17). It can be shown that inner products of the left hand sides of (6.3) and (6.4) with the four functions in (5.13) all vanish, as long as η > 2/ √ a.
These relations are valid for η > 2/ √ a. There are only two linearly independent modes, as should be the case.
On the contrary by using the behavior (D.6) of Φ

II(ν)
E,p near the horizon, it can be shown that the following relation holds near the horizon.
Another relation obtained by a replacement ν → −ν also holds. In this way linear dependence of mode functions may be discontinuous at the horizon. It is then unavoidable to use all the four modes altogether behind the horizon. So we will work in the vector space of normal modes V.
Now a question arises as to how it is possible to match a scalar field in regions I and III outside the horizon to that in region II. Even if it could be possible to connect the mode functions of a scalar field on both sides of the horizon smoothly, the inner products of the mode functions might be discontinuous at the horizon. How the creation and annihilation operators must be assigned to the mode functions inside region II? If the inner products at η = 2/ √ a were used, the quantum theory obtained might not be appropriate in the whole region inside the horizon. If the inner products for η > 2/ √ a were used, then the quantum theory inside the horizon might be discontinuous from that outside. In the following analysis we will start by using the inner products for η > 2/ √ a, (C.9)-(C. 16) and solve the matching conditions. It will be shown in sec.8 that when these conditions are satisfied, then special linear combinations of normal modes, ψ E,p (7.2) defined below, have appropriate inner products which are continuous at the horizon.

Quantization Behind the Horizon
Behind the horizon the scalar field will be expanded into linear combinations of the normal modes (5.13) introduced above. For the horizon to be smooth, the operators which multiply the normal modes must be chosen to coincide with those in regions I and III, and the operators inside and outside the horizon must be fully entangled.

Normal Mode Expansion of a Scalar Field in Region II
Choice of normal modes is carried out in such a way that the exponentials e ±iEt±ipx in the normal modes in region II coincide with those in the corresponding normal modes in regions I and III, respectively. Normal mode expansion of a scalar field in region II is given by Here c R (E, p) and c L (E, p) with E ≥ 0 and −∞ < p < +∞ are two sets of annihilation operators which are the same as those in regions I and III. It is important for smoothness of the horizon to use the same operators inside the horizon as those outside. [4] In the above expansion of a scalar field, integration region for p is devided into two, p > 0 and p < 0, where E ≥ 0 and p ≥ 0. If only Φ    It can be shown that in order to ensure that (ψ E ′ ,−p ′ ) = 0 it is necessary and sufficient to require γ 4 . In the following the following constraints are imposed.
Although these constraints are a bit stronger than necessary, it is possible to show that there exist solutions to the matching conditions. Similarly, in order to ensure that E ′ ,p ′ ) = 0 the following constraints are imposed.
On the other hand orthogonality of ψ Although in general these norms are not positive definite, by imposing γ

Matching Conditions
In the maximally extended Penrose diagram the black hole spacetime is expressed in terms of the Kruskal-Szeckeres coordinates, U and V . In region I these coordinates are given by Here r * is the tortoise coordinate. In region II they are given by Similarly, in region III they are Here the time direction in region III is flipped compared to that in the usual maximally-  The matching conditions are given as follows.
• At the horizon between regions I and II • At the horizon between regions II and III The conditions for vanishing of terms in ψ (1,2) proportional to (−U ) iE/ √ a at the horizon between regions I and II are Exactly the same conditions are obtained for vanishing of terms in ψ (3,4) proportional to √ a at the horizon between regions III and II. Here C ±ν are defined by

Solutions
By solving (7.10)-(7.15) the following solution for γ (i) n 's are obtained. Details are given in Appendix F.  By using these results norms of ψ (i) E,∓p (i = 1, · · · , 4) are found to be (ψ E,p 's are phase factors, (7.5) hold. These norms are positive-definite and normalized to unity. Due to the pre-factor 1/(4π √ πa) in the scalar field (7.1), the operators c R,L (E, p) must satisfy and a similar relation for c L and c † L . These relations agree with (4.5) and (4.10) and the horizon is smooth for the scalar field. The prescription of [6] works and there are no firewalls for a scalar at the horizon.
The above solution, however, contains phase factors e iδ i (E,p) (i = 1, · · · , 4), N n can be parametrized as Then γ 1 and γ (1) 3 are defined to be solutions to the following equations   E,−p . By substitution of (7.21) and (7.22) it is found that At the horizon those factors in the first line which multiply γ 3 , respectively, vanish due to the relation (6.5). Actually, the latter relation can be rewritten as Φ E,−p and N (1) E,p at the horizon. When the inner product of the last term of (8.1) is computed at the horizon by using (D.9)-(D.13), we obtain at η = 2/ √ a (ψ Here H stands for horizon. It can be shown that the other inner products, (ψ This has the correct K-G norm (7.5) owing to (7.26), (C.9) and (C.12). Similarly, the other normal modes are found to be These also have correct K-G norm (7.5). This means that the matching condition cannot determine the quantum scalar field inside the horizon uniquely. The near-horizon normal mode, the last term of (8.1), which is unique, is connected to the deep-inside mode (8.3).
The latter is not unique and depends on arbitrary functions F (1) n (E, p) (n = 1, 2). Because there is no isometry for the time variable η inside the horizon and it is not possible to distinguish between positive-and negative-frequency solutions inside the horizon. So the number of independent normal modes is doubled and as a result the K-G inner product (7.6) contains a term with a negative sign. Therefore solutions which contain continuous parameters can exist.

Summary and Conclusion
In this paper quantization of a free scalar field in BTZ black hole including the region inside the horizon is studied. The normal modes of a scalar field in black hole background are obtained from those of AdS 3 spacetime by suitable coordinate transformations. These normal modes turn out not eigenstates of energy and momentum. We quantized scalar field on the same constant-t slice in both regions I and III: K-G inner products are computed on the constant t slice obtained by combining those in both regions. By changing basis of the normal modes to that of eigenstates of energy and momentum it is found that the creation and annihilation operators in each regions I and III can be identified as singletrace operators in boundary CFTs. The vacuum state is shown to be the TFD (1.1).
Then a scalar field behind a horizon of a two-sided BTZ black hole is quantized by using the matching condition of [4] for normal modes. It is shown that the scalar field can be smoothly connected across the horizon by using the matching condition and the scalar field also satisfies correct equal time commutation relations behind the horizon. It is, however, shown that there are still infinitely many ways to expand a scalar field into normal modes deep inside the horizon.
To conclude, the matching condition at the horizon of [4] is not sufficient to determine a scalar theory inside the horizon of BTZ black hole uniquely. Correlation functions behind the horizon depend on extra functions F E,p . These two are not obtained by a same coordinate transformation from the corresponding normal modes in AdS 3 . These results mean that a scalar theory behind the horizon is not reconstructed in terms of the CFT's on the boundaries, unless appropriate additional conditions are imposed on the matter theory inside the horizon. This constraint cannot be arranged by any further conditions at the horizon, because the near-horizon normal modes, the last term of (8.1), are uniquely determined by the matching condition. What is an appropriate additional condition to determine the quantum theory far inside the horizon? It is interesting to study whether there is similar arbitrariness of the structure of matter theories behind the horizon in higher-dimensional black holes. These questions will be left for future work.

A Connection of Normal Modes of a Scalar Field in BTZ
Black Hole to those in AdS 3 The metric for massless BTZ black hole is given by Here AdS length is set to unity. u is a radial coordinate and w + = χ+τ and w − = χ−τ and τ is a time, and χ the spatial one. We perform the following coordinate transformations (u, w + , w − ) → (y, z + , z − ) in the above equation. [26][14] [27] w Then we obtain a new metric.
Here S[f, z] is a Schwarzian derivative.
Here a = 4GM is a parameter related to the black hole mass M , and G is a 3d Newton constant. Then the metric is transformed to then the metric reads Because r ≥ √ a, (A.8) describes the space-time outside the black hole. (A.9) is solved for y as The horizon is located at r = r + = √ a. Each 0 < y ≤ 2 √ a and 2 √ a ≤ y < ∞ correspond to the exterior region r ≥ √ a.
By separation of variables mode functions of this scalar field will be obtained in the form φ ωk (τ, u, χ) = e −iωτ +ikχ f ωk (u), (A. 13) where ω and k are constants, and f ωk (u) is a solution to the following equation By BDHM dictionary [24] the scalar field dual to the 2d CFT must satisfy a boundary condition φ ∼ u 1+ν near the boundary u → 0, where ν = √ 1 + m 2 , and for non-integer ν, The mode functions are then given by Here J ν (z) is a Bessel function of the ν-th order.
In region (I) the map connecting the uniformization coordinates and those of black hole space-time is given by (A.7 In region (III) we need to make a shift t → t − iβ/2 in (A.7). However, y must be mapped to the region 2π √ a ≤ y by a transformation y → 4 ay . Then the relation (A.7) is unchanged. To obtain the normal modes in region III the direction of time must be flipped:t → −t in (A.7). Then the normal modes in region III are given by To describe the interior of the horizon, we introduce a new radial coordinate η(> 0) by y = 8η Note that this is a complex transformation. Then the metric is transformed to Now t is a space-like variable and x the time-like one. Note that η = 0 and η = ∞ are singularities. This metric describes the region behind the horizon of the spacetime (A.10), This can also be confirmed by The relation between η and r is also two-fold, To go to region (II) the transformation (A.18) is used. The range of η is 0 < η < 2/ √ a.
In addition the shift t → t − iβ/4 must be carried out in order to make coordinates realvalued. Here β is the inverse temperature. Hence we have

B Functions g(E, p) andg(E, p)
In this appendix some formulas related to the functions g(E, p) andg(E, p) are presented. g(E, p) is defined in (4.2). By using a representation of the modified Bessel function of the second kind, this function is evaluated as Furthermore by using a formula which is valid for Reµ > |Reν|, the following equation is finally obtained.
Then the following quantity related to the normalization factor is positive semi-definite.
In a similar fashion the following results can be derived for the functiong(E, p) (4.7).
C Inner Products of Φ Here ζ integral is carried out by using a formula Then µ integration is performed for Re(a ± ib) > 0, Re(ν − λ + 1) > |Reµ| by [28] ∞ 0 The following result is obtained.
This asymptotics also shows that Φ Because the inner product does not depend on η as far as η > 2/ √ a, it can be evaluated in the limit η → +∞ by using (C.4). The following results are obtained.
Similarly, formulas forΦ II(ν) E,p (5.6) can be derived. Asymptotic form for (5.6) in the limit η → ∞ is given bỹ Inner products of (5.6) are given by In the remainder of this appendix, inner products of Φ Here −∞ < E, E ′ , p, p ′ < ∞. On the other hand we have Here D(E, p) is defined by

D Near-Horizon Behavior of Normal Modes
The near horizon behavior of the normal mode Φ II(ν) E,p in region II, (5.4), are obtained as follows. For η ∼ 2 √ a , this mode asymptotes to Φ II(ν) (D.1) By using formula (C.2) the integration over ζ in (D.1) is carried out.
The near horizon behavior is then estimated by using The normal mode in region III, (3.7), is also obtained.
In what follows some inner products of the four mode functions near the horizon are presented. Inner products evaluated at the horizon will be denoted with subscript H.

E Solutions to the Matching Conditions
In this Appendix appropriate solutions to the matching conditions (7.10)-(7.15) will be obtained. First (7.10) and (7.14) are analyzed. When γ (1) 4 is eliminated from these equations, it is found that γ 3 also disappears, and we have where M 1 and M 2 are defined in (E.4) and (E.5) below. This is because the following relations hold Similarly, when γ 2 is eliminated from (7.10) and (7.12), then γ 1 also dropps out. b 1 is defined by removing e −ipx V −iE/ √ a from the righthand side of (7.10) In order to make the norm (7.6) with i = 1 take the form (7.23), it is assumed that 2 ) 1/2 e iδ 2 − J (1) (E.12) Here δ 1,2 are arbitrary real constants. These equations are solved for γ Also by comparing (7.10) and (7.12) it is found that where b 3 is defined by removing e ipx U −iE/ √ a from the righthand side of (7.12) Then (E.7) and (E.18) imply that N where δ N (E, p) is an arbitrary real number.