1 Introduction

The Schwarzschild solution [1] in the framework of Einstein’s general relativity has the problem of a singularity [2, 3] in the interior spacetime of a black hole [4]. All known laws of physics must break down at this singularity where it is impossible to have a meaningful description of the spacetime itself [5, 6]. Thus existence of a singularity questions the validity of all laws of physics described in the background of the same spacetime. On the other hand, the success of a plethora of laws of physics suggests the possibility that the existence of singularity is due to the classical nature of general relativity. This viewpoint has been professed by many physicists in their research works [7,8,9,10,11,12,13,14].

The only hope to resolve this problem can be offered by a quantum theory of gravity. The quantum mechanical dynamics of spacetime near the black hole singularity is therefore an important arena to explore which may contain signatures of singularity resolution.

Attempts have been made to formulate a quantum theory of gravity following the lines of standard quantum field theory. However, such a theory, requiring a perturbative approach, suffers from the problem of non-renormalizability due to the fact that Newton’s constant has a mass dimension of \(-2\) [15].

A natural way to avoid this difficulty is to construct a theory which either does not require renormalization or it is non-perturbative in nature. String theory [16,17,18,19,20] is a theory of quantum gravity which is intrinsically ultraviolet regularized although it is perturbative in nature. On the other hand, Wheeler–DeWitt quantization [21, 22] and loop quantum gravity [23,24,25] are non-perturbative approaches to quantum gravity. For a review on these subjects, see Refs. [26,27,28]. These approaches, among others, have been employed to study the quantum nature of spacetime at the black hole singularity.

Since the singularity in a black hole is hidden behind its event horizon [29,30,31,32,33], it is impossible for an external observer to obtain any information regarding the singularity. Consequently, black hole singularity avoidance is an important requirement as the process of Hawking radiation [34, 35] would make the black hole shrink sufficiently so as to collide with the singularity towards the end of evaporation.

Greene et al. [36] argued that at the singularity black hole condensation can happen in the moduli space of type II Calabi–Yau string vacua. This condensate can make a smooth transition to a new Calabi–Yau vacuum towards the end of black hole evaporation, thus achieving the requirement of singularity avoidance.

Witten [37] constructed a regular conformal field theory for a black hole with a two dimensional target space from a gauged Wess-Zumino-Witten action. The metric of the target space asymptotically approaches a flat Euclidean geometry as \(r\rightarrow \infty \), whose generalization to higher dimensions can be interpreted as describing a Euclideanized black hole. Including a dilaton field by integrating out the gauge field, it was found that the string coupling vanishes as \(r\rightarrow \infty \). Wick rotating the angular coordinate, a Lorentzian spacetime corresponding to the Euclidean black hole was constructed in which the radial coordinate could be identified as the Liouville mode. He further considered the graviton-dilaton action whose field equations yield the flat spacetime Liouville solution with a dilaton field proportional to the space coordinate. The Liouville flat solution was perturbed and the ADM mass of the black hole was obtained from the conserved current of the perturbation fields at the flat end of the solution. This mass was found to be related to the dilaton field at the horizon. Witten further suggested that the spacetime towards the end point of black hole evaporation by Hawking radiation may be regarded to be analogous to an extremal Reissner-Nordström black hole.

Maldacena [38] showed the existence of a sector in the Hilbert space of large N conformal field theories that can describe supergravity on an AdS background including other manifolds. This conclusion was made by taking the low energy limit of the string theory whereby the field theory living on the brane decouples from the bulk giving the near horizon behavior in the large N limit.

Based on Maldacena’s finding, Horowitz and Ross [39] argued that the black hole singularity is naturally resolved because the background spacetime of string theory also describes a black hole in AdS space and the field theory in the interior region of the black hole is perfectly regular.

Gibbons et al. [40] took a dilatonic black hole coupled to an abelian gauge field in 4 dimensions and extended the system to \(4+p\) dimensions by including a p-brane. For appropriate choices of the conformal scalings in the original 4 dimensional spacetime and the brane space, they showed that the \(4+p\) dynamical action can be rendered as a system without any dilaton field. For odd values of p, they showed that the exterior spacetime is isometric with the interior spacetime and the only singularity in the system is the coordinate singularity at the horizon. Thus with extra dimensions the system is free from any dangerous singularity.

Recently, Codina et al. [41] studied black hole models by considering a general action with T-dual invariant derivative corrections to all orders of \(\alpha ^{\prime }\) in a two dimensional string background having timelike isometry. In the Gasperini and Veneziano scheme [42], they parametrized the duality invariant corrections in the exterior and interior regions of the black hole. They showed that a T-dual black hole with a regular horizon occurs with a curvature singularity. It is however possible to construct regular (no singularity) solutions with a horizon by suitable parameterization, which appears to exist in a subregion not containing a string theory. Black hole singularity in string theory is further discussed in Refs. [43, 44].

Another approach, loop quantum gravity [23,24,25], is non-perturbative and it resembles gauge theories without a reference background. It involves nonlocal operators such as area and volume which are finite without renormalization and can be regulated diffeomorphism invariantly. This leads to discreteness in the geometry so that the area and volume operators have discrete spectra, having the lowest eigenvalues \(\sim l^2_P\) and \(\sim l^3_P\), respectively, where \(l_P\) is the Planck length [23, 25, 45,46,47,48]. The discreteness in the geometry naturally resolves the black hole singularity, leading to a regular spacetime inside the black hole [49,50,51]. The interior spacetime of the black hole is further discussed in Refs. [52, 53].

There is another important approach to quantum gravity based on canonical quantization. This approach consists of an ADM \(3+1\) decomposition [54], so that Hamiltonian and momentum constraints naturally emerge on three dimensional space-like hypersurfaces at constant times. Applying the standard rules of canonical quantization, the Hamiltonian constraint gives rise to the Wheeler–DeWitt equation [21, 22] with a wave function depending on the metric variables of the three dimensional hypersurface. For singularity resolution, it requires to satisfy the DeWitt criterion, implied by vanishing of the wave function at the singularity. This approach to quantum gravity is remarkably appealing because it is similar to standard quantum mechanics which has been successful in many areas of physics.

In the context of quantum black holes, Chien et al. [55] obtained a solution of the Wheeler–DeWitt equation and showed that the wave function vanishes much before arriving at the singularity. Bouhmadi-López et al. [56] also analyzed this problem and showed that the choice of wave function is intimately related with the arrow of time. The Wheeler–DeWitt equation has also been explored in the context of holographic duality by Chowdhury et al. [57] and Hartnoll [58].

It is known from quantum field theory that the cosmological constant (or dark energy) turns out to be enormously high, \(10^{120}\) times the observed value [59]. To explain this discrepancy, Carlip [60] argued that Wheeler-type foamy geometry at the Planckian scales can have expanding and contracting regions with high values of cosmological constant, averaging out to a nearly zero value for the effective cosmological constant at large scales. Furthermore, he [61] constructed a quantum gravity model by Wheeler–DeWitt quantization of a midisuperspace metric. The solution gives rise to a self-reproducing foamy geometry expanding on an average with a very small cosmological constant.

The Wheeler–DeWitt equation has been employed by Barvinsky and Kiefer [62] to obtain semiclassical gravitational corrections to the matter Hamiltonian. This was achieved by splitting the full wave function into a product of gravitational and matter wave functions, with WKB ansatz for the gravitational part. This yielded a Schrodinger-like equation governing the matter wave function with Hamilton-Jacobi terms appearing as corrections to the matter Hamiltonian. The corrections responsible for back-reaction could then be treated as perturbation which naturally leads to a perturbation expansion. In the language of Feynman diagrams, this perturbation series contains graviton and graviton-matter loops.

A discrete version of the Wheeler–DeWitt equation [63] has also been explored in the spirit of Regge calculus [64, 65]. In such lattice approach, the wave function suggested the existence of a finite correlation length and a critical point separating weak and strong coupling regimes [66, 67].

We see that a lot of efforts have been expended in order to formulate and explain the quantum nature of spacetime in various formulations. However, it still remains unclear how the spacetime behaves near the black hole singularity. An understanding can be achieved in this direction if we know the nature of the black hole wave function near the singularity.

In this paper, we make an effort to address this question. We represent the interior spacetime of the black hole with the Kantowki–Sachs metric [68]. As we know from quantum field theory [69, 70], a Klein–Gordon matter field, upon quantization, gives rise to zero-point vacuum fluctuations of the field. This property should be preserved when the Klein–Gordon field is minimally coupled to gravity. This is because, as we switch off gravity slowly, we should recover quantum field theory in the zero-gravity limit, preserving consistency with quantum field theory. Moreover, in quantum field theory on a curved classical background [71, 72], the vacuum fluctuations of matter fields play a crucial role in modifying the Einstein field equation.

Consequently, we include a Klein–Gordon matter field minimally coupled to the spacetime geometry in order to incorporate the vacuum fluctuations. Following an ADM decomposition of this system, we obtain the Wheeler–DeWitt equation in the standard canonical quantization scheme. This equation, a partial differential equation in three variables, is solved by the standard method of separation of variables. We analyze this general solution and obtain admissible wave functions near the singularity of the black hole. We find that the Hilbert space splits into three nonoverlapping sectors and the DeWitt admissibility criterion holds only in two of those sectors, where regular quantum black holes can exist. In the third sector, this possibility does not exist.

The remainder of the paper has the following layout. In Sect. 2.1 we introduce the Kantowki–Sachs metric and obtain the classical Hamiltonian employing ADM decomposition. We further introduce a Klein–Gordon field in Sect. 2.2 and carry out a canonical quantization of the full system to obtain a Wheeler–DeWitt equation in Sect. 3.1. This equation is exactly solved in Sect. 3.2 giving a general solution. This solution is further studied in Sect. 4 to obtain admissible wave functions and their nature near the singularity, leading to three nonoverlapping sectors in the Hilbert space, as described earlier. We finally give a detailed discussion and conclude the paper in Sect. 5.

2 Geometry and matter

To find the quantum nature of spacetime near the black hole singularity, we need to carry out canonical quantization of the black hole interior geometry. This requires the classical Hamiltonian of the interior geometry.

Moreover, in order to describe the black hole interior quantum mechanically, spontaneous vacuum fluctuations of matter fields cannot be neglected. This requires the classical Hamiltonian of some matter field.

In this section, we lay out mathematically tractable models for these Hamiltonians that we shall further consider for quantization in the subsequent sections.

2.1 Hamiltonian of geometry

The black hole interior geometry can be obtained by analytic continuation of the exterior metric. We shall consider the simplest geometry given by the Schwarzschild metric, with \(c=1\),

$$\begin{aligned} ds_{{\textrm{ext}}}^2=-\left( 1-\frac{2GM}{r}\right) dt^2+\frac{dr^2}{1-\frac{2GM}{r}}+r^2 d\Omega ^2, \end{aligned}$$
(1)

in the exterior region, \(r>2GM\). Analytical continuation to the interior region, \(r<2GM\), gives the interior metric,

$$\begin{aligned} ds_{{\textrm{int}}}^2=\left( \frac{2GM}{r}-1\right) dt^2-\frac{dr^2}{\frac{2GM}{r}-1}+r^2 d\Omega ^2. \end{aligned}$$
(2)

It is clear that t and r interchange their roles in the interior, so that the general form of the interior metric can be written as

$$\begin{aligned} ds_{\textrm{int}}^2=-\frac{dt^2}{\frac{2GM}{t}-1}+\left( \frac{2GM}{t}-1\right) dr^2+t^2 d\Omega ^2. \end{aligned}$$
(3)

Thus a simple representation of the interior geometry can be taken to be the Kantowski–Sachs metric [68], given by

$$\begin{aligned} ds^2=-\alpha ^2(t) \,dt^2+\xi ^2(t) \,dr^2+\zeta ^2(t) \, (d\theta ^2+\sin ^2\theta \,d\varphi ^2), \nonumber \\ \end{aligned}$$
(4)

where \(\alpha (t)\) is the lapse function.

To obtain the classical Hamiltonian, we have to employ ADM decomposition [54] of the metric

$$\begin{aligned} g_{\mu \nu }=\left( \begin{array}{cc} -\alpha ^2(t) &{}\quad 0 \\ 0 &{}\quad h_{ij}(t) \end{array} \right) . \end{aligned}$$
(5)

which is already in the required form, and

$$\begin{aligned} h_{ij}(t)=\left( \begin{array}{ccc} \xi ^2(t) &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad \zeta ^2(t) &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad \zeta ^2(t) \sin ^2\theta \ \end{array} \right) , \end{aligned}$$
(6)

represents metric of constant time space-like hypersurface.

The Ricci curvature of this hypersurfaces is given by

$$\begin{aligned} {\,}^{(3)}R=\frac{2}{\zeta ^{2}(t)}, \end{aligned}$$
(7)

and the extrinsic curvature tensor (or the second fundamental form) of this hypersurface, \(K_{ij}=-\frac{1}{2\alpha }\frac{d h_{ij}}{dt}\), has its elements

$$\begin{aligned} K_{ij}=-\frac{1}{\alpha }\left( \begin{array}{ccc} \xi \dot{\xi }&{}\quad 0 &{}\quad 0 \\ 0 &{}\quad \zeta \dot{\zeta }&{}\quad 0 \\ 0 &{}\quad 0 &{}\quad \zeta \dot{\zeta }\sin ^2\theta \ \end{array} \right) , \end{aligned}$$
(8)

where an overdot represents differentiation with respect to time. The trace and the contracted form therefore assume the forms

$$\begin{aligned} K=h^{ij}K_{ij}=-\frac{1}{\alpha }\, \frac{\dot{\xi } }{\xi }-\frac{2}{\alpha }\, \frac{\dot{\zeta }}{\zeta } \end{aligned}$$
(9)

and

$$\begin{aligned} K_{ij}K^{ij}=\frac{1}{\alpha ^2}\frac{\dot{\xi }^2}{\xi ^2}+\frac{2}{\alpha ^2}\frac{\dot{\zeta }^{2}}{\zeta ^{2}}. \end{aligned}$$
(10)

The gravitational Lagrangian density, occurring in the Einstein-Hilbert action \(S_g=\int \sqrt{-g}\,\,\,\, ^{(4)}R \,\,d^4 x\), when expressed in terms of the intrinsic and extrinsic curvature, becomes

$$\begin{aligned} {\mathscr {L}}_g=\frac{1}{16\pi G}\sqrt{-g}\,(K_{ij}K^{ij}-K^2+ {\,}^{(3)}R), \end{aligned}$$
(11)

Employing Eqs. (7), (9) and (10), we thus have

$$\begin{aligned} {\mathscr {L}}_g=\frac{\sin \theta }{16\pi G}\left( -\frac{2}{ \alpha }\xi \dot{\zeta }^{2}-\frac{4}{\alpha }\zeta \dot{\xi }\dot{\zeta }+ 2\alpha \xi \right) . \end{aligned}$$
(12)

The Lagrangian \(L_g= \int {\mathscr {L}}_g \,d^3 x\) then reads

$$\begin{aligned} L_g= \frac{r_0}{4 G} \left( -\frac{2}{ \alpha } \xi \dot{\zeta }^2 -\frac{4}{\alpha }\zeta \dot{\xi }\dot{\zeta }+ 2 \alpha \xi \right) , \end{aligned}$$
(13)

where \(r_0=\int dr\) is a nondynamical constant.

Corresponding to the degrees of freedom \(\xi \) and \(\zeta \) of the black hole interior geometry, the canonical momenta are defined in the usual way, yielding

$$\begin{aligned} P_\xi =\frac{\partial L_g}{\partial \dot{\xi }}=-\frac{r_0}{\alpha G} \zeta \dot{\zeta } \end{aligned}$$
(14)

and

$$\begin{aligned} P_\zeta =\frac{\partial L_g}{\partial \dot{\zeta }}=-\frac{r_0}{\alpha G}\left( \xi \dot{\zeta }+\zeta \dot{\xi }\right) . \end{aligned}$$
(15)

Thus the gravitational Hamiltonian \(H_g= P_\xi \dot{\xi }+ P_\zeta \dot{\zeta }-L_g\) of the black hole interior geometry is expressed as

$$\begin{aligned} H_g=\frac{\alpha G}{r_0 \xi \zeta ^2}\left( -\xi \zeta P_\xi P_\zeta +\frac{1}{2} (\xi P_\xi )^2-\frac{r_0^2}{2 G^2}\xi ^2 \zeta ^2\right) . \end{aligned}$$
(16)

This Hamiltonian describes the classical dynamics of the black hole interior geometry in the minisuperspace phase variables (\(\xi ,\zeta ,P_\xi , P_\zeta \)).

2.2 Hamiltonian of matter

For simplicity, we shall assume the matter field to be described by a Klein–Gordon field having the Lagrangian density

$$\begin{aligned} {\mathscr {L}}_{\phi }=-\frac{1}{2}\sqrt{-g}\,g^{\mu \nu }\partial _{\mu } \phi \,\partial _{\nu } \phi . \end{aligned}$$
(17)

For simplicity of description, we assumed the black hole interior geometry to be described by the Kantowski–Sachs metric (4), with the metric coefficients depending only on time. Likewise, we shall assume the Klein–Gordon field \(\phi \) to be dependent only on time in the black hole interior. Thus the Klein–Gordon Lagrangian assumes the simple form

$$\begin{aligned} L_{\phi }=\int {\mathscr {L}}_{\phi } \,d^3 x =\frac{2\pi r_0}{\alpha } \xi \zeta ^2\,\dot{\phi }^2. \end{aligned}$$
(18)

The canonical momentum conjugate to the Klein–Gordon degree of freedom \(\phi \) is immediately obtained as

$$\begin{aligned} P_{\phi }=\frac{\partial L_{\phi }}{\partial \dot{\phi }}= \frac{4\pi r_0}{\alpha } \xi \zeta ^2\,\dot{\phi }, \end{aligned}$$
(19)

so that the Klein–Gordon Hamiltonian reads

$$\begin{aligned} H_{\phi }=P_{\phi } \dot{\phi }-L_{\phi }=\frac{\alpha }{ 8\pi r_0}\frac{P_{\phi }^2}{ \xi \zeta ^2}. \end{aligned}$$
(20)

The Hamiltonian expressions given by Eqs. (16) and (20) describe the gravity-matter system only classically. We shall quantize this system in the next section in obtain a quantum mechanical description of the black hole interior.

3 Wheeler–DeWitt equation

In this section, we shall formulate the Wheeler–DeWitt equation by means of canonical quantization employing the Hamiltonian constraint. We shall also obtain an exact solution of this partial differential equation.

3.1 Canonical quantization

The total classical Hamiltonian \(H=H_g+H_{\phi }\) of the gravity-matter system can be obtained from Eqs. (16) and (20),

$$\begin{aligned} H= & {} \frac{\alpha G}{r_0\xi \zeta ^2}\Bigg (-\xi \zeta P_\xi P_\zeta +\frac{1}{2} (\xi P_\xi )^2\nonumber \\ {}{} & {} -\frac{r_0^2}{2 G^2}\xi ^2\zeta ^2+\frac{P_{\phi }^2}{8\pi G}\Bigg ). \end{aligned}$$
(21)

Canonical quantization of this gravity-matter system is achieved by promoting the momenta into operators: \(P_\xi \rightarrow \hat{P_\xi }=-i\hbar \frac{\partial }{\partial \xi }\), \(P_\zeta \rightarrow \hat{P_\zeta }=-i\hbar \frac{\partial }{\partial \zeta }\) and \(P_{\phi }\rightarrow \hat{P_{\phi }}=-i\hbar \frac{\partial }{\partial \phi }\), in the Schrodinger picture.

Consequently, the Hamiltonian (21) is promoted to an operator \({\hat{H}}\), and the Wheeler–DeWitt equation is given by

$$\begin{aligned} \hat{H}\,\Psi =0, \end{aligned}$$
(22)

where \(\Psi (\xi ,\zeta ,\phi )\) is the wave function of the gravity-matter system.

This Eq. (22) describes the dynamics of the black hole interior and it can be expressed in differential form in the Kantowski–Sachs and Klein–Gordon minisuperspace coordinates (\(\xi , \zeta , \phi \)) as

$$\begin{aligned}{} & {} \Bigg ( \xi \zeta \frac{\partial ^2}{\partial \xi \partial \zeta }-\frac{\xi ^2}{2} \frac{\partial ^2}{\partial \xi ^2}-\frac{\xi }{2} \frac{\partial }{\partial \xi }-\frac{r_0^2}{2 \hbar ^2 G^2}\xi ^2 \zeta ^2\nonumber \\ {}{} & {} \quad -\frac{1}{8\pi G} \frac{\partial ^2}{\partial \phi ^2} \Bigg )\Psi =0, \end{aligned}$$
(23)

This is the Wheeler–DeWitt equation for the black hole interior in the minisuperspace approximation. Evidently, it is a partial differential equation in the three minisuperspace variables (\(\xi ,\zeta ,\phi \)). We shall solve this equation in the next section with appropriate boundary conditions.

We see that the classical time t has disappeared from the Wheeler–DeWitt equation. This problem is quite general in quantum gravity and the matter field \(\phi \) is often regarded as playing the role of time. Different ways of incorporating the time in quantum gravity are given in Refs. [73,74,75,76].

We however note that the issue of time is immaterial for our purpose of discussion in this paper. Since the Hamiltonian does not depend on \(\phi \) explicitly, the Klein–Gordon variable \(\phi \) is separable in the wave function \(\Psi (\xi ,\zeta ,\phi )\), as we shall see while solving the Wheeler–DeWitt equation.

3.2 Exact solution

The Wheeler–DeWitt equation (23) can be separated in the Klein–Gordon variable \(\phi \) by writing \(\Psi (\xi ,\zeta ,\phi )=\psi (\xi ,\zeta )e^{-i \kappa \phi }\), leading to

$$\begin{aligned}{} & {} \Bigg ( \xi \zeta \frac{\partial ^2}{\partial \xi \partial \zeta }-\frac{\xi ^2}{2} \frac{\partial ^2}{\partial \xi ^2}-\frac{\xi }{2} \frac{\partial }{\partial \xi }-\frac{r_0^2}{2 \hbar ^2 G^2}\xi ^2 \zeta ^2\nonumber \\ {}{} & {} \quad +\frac{\kappa ^2}{8\pi G} \Bigg )\psi =0, \end{aligned}$$
(24)

where \(-\kappa ^2\) is the separation constant. Equation (24) is a partial differential equation in two independent variables, (\(\xi ,\zeta \)), and it requires special methods [77, 78] for its solution.

To deal with Eq. (24), we transform (\(\xi ,\zeta \)) into a new set of independent variables (x, z), with \(x=f_1(\xi ,\zeta )\) and \(z=f_2(\xi ,\zeta )\), \(f_1\) and \(f_2\) having different functional forms. Choosing \(f_1(\xi ,\zeta )=\xi \) and \(f_2(\xi ,\zeta )=\xi \zeta \), we transform the variables as

$$\begin{aligned} z=\xi \zeta \hspace{14.22636pt}\textrm{and} \hspace{14.22636pt}x=\xi , \end{aligned}$$
(25)

reducing Eq. (24) to

$$\begin{aligned}{} & {} \Bigg ( z^2\frac{\partial ^2}{\partial z^2}+ z \frac{\partial }{\partial z}- x^2 \frac{\partial ^2}{\partial x^2}- x \frac{\partial }{\partial x}-\frac{r_0^2}{ \hbar ^2 G^2} z^2\nonumber \\ {}{} & {} \quad +\frac{\kappa ^2}{4\pi G} \Bigg )\psi =0. \end{aligned}$$
(26)

This partial differential equation can be separated in the two variables z and x by writing \(\psi (z,x)=Z( z)X(x)\), yielding two ordinary differential equations,

$$\begin{aligned} z^2 \frac{\partial ^2 Z}{\partial z^2}+ z \frac{\partial Z}{\partial z}-\frac{r_0^2}{\hbar ^2 G^2} z^2 Z-\lambda Z=0 \end{aligned}$$
(27)

and

$$\begin{aligned} x^2 \frac{\partial ^2 X}{\partial x^2}+ x \frac{\partial X}{\partial x}-\frac{\kappa ^2}{4\pi G}X-\lambda X=0, \end{aligned}$$
(28)

where \(\lambda \) is the separation constant.

Equation (27) is the Bessel equation with general solution,

$$\begin{aligned} Z( z)=c_1 \, J_{\nu }(i{\tilde{z}})+c_2 \, Y_{\nu }(i{\tilde{z}}), \end{aligned}$$
(29)

where \(J_{\nu }(i{\tilde{z}})\) and \(Y_{\nu }(i{\tilde{z}})\) are respectively the Bessel functions of first and second kind of order \(\nu ={\sqrt{\lambda }}\) and \({\tilde{z}}=\frac{r_0}{\hbar G} z\).

Equation (28) is much simpler and its general solution is given by

$$\begin{aligned} X(x)=c_3 \,x^p+c_4 \,x^{-p}, \end{aligned}$$
(30)

where \(p={\sqrt{\lambda +\frac{\kappa ^2}{4\pi G}}}\). (In the above solutions, \(c_1\), \(c_2\), \(c_3\) and \(c_4\) are arbitrary constants.)

Thus Eq. (26) has the general solution

$$\begin{aligned} \psi (z,x)=\left[ c_1 \, J_{\nu }(i{\tilde{z}})+c_2 \, Y_{\nu }(i{\tilde{z}})\right] \,\left[ c_3 x^p+c_4 x^{-p}\right] . \end{aligned}$$
(31)

and the Wheeler–DeWitt equation (23) yield the solution

$$\begin{aligned} \Psi (\xi ,\zeta ,\phi )&=\left[ c_1 \, J_{\sqrt{\lambda }} \left( ia_0 \xi \zeta \right) +c_2 \, Y_{\sqrt{\lambda }} \left( ia_0 \xi \zeta \right) \right] \nonumber \\ {}&\quad \times \left[ c_3\, \xi ^{\sqrt{\lambda +k^2}}+c_4\,\xi ^{-\sqrt{\lambda +k^2}}\right] \,e^{-i \kappa \phi }. \end{aligned}$$
(32)

where \(a_0=\frac{r_0}{\hbar G}\) and \(k^2=\frac{\kappa ^2}{4\pi G}\).

Equation (32) gives the wave function stationary in \(\phi \) for the black hole interior in the minisuperspace approximation. Apart from the minisuperspace variables (\(\xi ,\zeta ,\phi \)), the wave function also depends on the separation constants (\(\lambda ,\kappa ^2\)). The separation constants have the natural interpretations of eigenvalues to which the wave function belongs.

4 Quantum spacetime near the singularity

As in ordinary quantum mechanics, the Wheeler–DeWitt wave function must be regular and well-behaved at and around the region of the classical singularity. For example, although the hydrogen atom Hamiltonian has a classical singularity at \(r=0\), the hydrogen atom wave function is regular and well-behaved at and around \(r=0\).

In this context it is important to note that the DeWitt criterion consists of the boundary condition that the Wheeler–DeWitt wave function must approach zero as the singularity is approached. Consequently, we must pick up only those solutions that vanish at the classical singularity.

4.1 Admissible wave functions

Noting that the classical metric coefficient \(\xi ^2(t)=\frac{2GM}{t}-1\) is singular at \(t=0\), and this singularity occurs at \(\zeta ^2(t)=t^2=0\), we have the asymptotic limits \(\zeta \rightarrow 0\), \(\xi \rightarrow \infty \), and \(\xi \zeta \rightarrow 0\) as the classical singularity, \(t\rightarrow 0\), is approached. Accordingly, we shall use these asymptotic limits in order to analyze the behaviour of the interior wave function (32) near the classical singularity.

In the above limits, the Bessel functions have the asymptotic forms [79, 80]

$$\begin{aligned} J_{\sqrt{\lambda }} \left( ia_0 \xi \zeta \right) \sim ( \xi \zeta )^{\sqrt{\lambda }} \end{aligned}$$
(33)

and

$$\begin{aligned} Y_{\sqrt{\lambda }} \left( ia_0 \xi \zeta \right) \sim ( \xi \zeta )^{-\sqrt{\lambda }}, \end{aligned}$$
(34)

So that the wave function (32) asymptotes to

$$\begin{aligned} \Psi _{\textrm{asy}}(\xi ,\zeta ,\phi )&\sim \left[ c_1 \, ( \xi \zeta )^{\sqrt{\lambda }}+c_2 \, ( \xi \zeta )^{-\sqrt{\lambda }}\right] \nonumber \\ {}&\quad \times \left( c_3\, \xi ^{\sqrt{\lambda +k^2}}+c_4\,\xi ^{-\sqrt{\lambda +k^2}}\right) \,e^{-i \kappa \phi }, \end{aligned}$$
(35)

near the singularity, \(\zeta \rightarrow 0\) and \(\xi \rightarrow \infty \). Writing \( \Psi _{\textrm{asy}}=\left[ C_1\psi _1+C_2\psi _2+C_3\psi _3+C_4\psi _4\right] \,e^{-i \kappa \phi }\), where

$$\begin{aligned}&\psi _1=( \xi \zeta )^{\sqrt{\lambda }}\, \xi ^{\sqrt{\lambda +k^2}},\end{aligned}$$
(36)
$$\begin{aligned}&\psi _2=( \xi \zeta )^{\sqrt{\lambda }}\, \xi ^{-\sqrt{\lambda +k^2}}, \end{aligned}$$
(37)
$$\begin{aligned}&\psi _3=( \xi \zeta )^{-\sqrt{\lambda }}\, \xi ^{\sqrt{\lambda +k^2}}, \end{aligned}$$
(38)
$$\begin{aligned}&\psi _4=( \xi \zeta )^{-\sqrt{\lambda }}\, \xi ^{-\sqrt{\lambda +k^2}}, \end{aligned}$$
(39)

and \(C_1=c_1c_3\), \(C_2=c_1c_4\), \(C_3=c_2c_3\) and \(C_4=c_2c_4\).

It is obvious that the asymptotic behaviours of the above components (\(\psi _1\), \(\psi _2\), \(\psi _3\) and \(\psi _4\)) are governed by the arithmetic nature of the eigenvalues \(\lambda \) and \(k^2\). This naturally leads to three different cases to consider, that we shall analyze in the following subsections.

Fig. 1
figure 1

Sketch representing nonoverlapping sectors of the Hilbert space belonging to different subregions of eigenvalues (\(\lambda , \kappa ^2\)). Regular quantum black holes (QBHs) can exist in Sectors I and II where the black hole wave function is well-behaved near the singularity and it vanishes at the singularity in consistency with the DeWitt boundary condition. In Sector III, the DeWitt criterion is violated and regular quantum black holes cannot exist

Full size image

4.2 Sectors in the Hilbert space

We define a Sector in the Hilbert space which belongs to a subregion in the two dimensional eigenvalue plane (\(\lambda , \kappa ^2\)). We may define three different nonoverlapping Sectors with common boundaries as shown in Fig. 1. We shall find admissible wave functions consistent with the DeWitt criterion in each of these Sectors.

For an admissible the wave function, we shall apply the DeWitt criterion, necessitating the boundary condition that the wave function must vanish as the singularity is approached and it must be well-behaved in the region around the singularity, where \(\xi \rightarrow \infty \), \(\zeta \rightarrow 0\), along with \(\xi \zeta \rightarrow 0\).

4.2.1 Sector I: \(\lambda >0\)

In this Sector, \({\sqrt{\lambda }}>0\) and \({\sqrt{\lambda +k^2}}>0\), since \(k^2=\frac{\kappa ^2}{4\pi G}>0\). Thus, as the singularity is approached, \(\psi _1\), \(\psi _3\) and \(\psi _4\) are not admissible solutions since they approach infinity. We must therefore set \(C_1=C_3=C_4=0\), and the only admissible asymptote comes from \(\psi _2\), given by \(\Psi _{\textrm{asy}}\sim C_2\psi _2\), which is regular and well-behaved around the singularity, and it vanishes at the singularity, respecting the DeWitt criterion. Thus, the wave function in this Sector is given by

$$\begin{aligned} \Psi _I(\xi ,\zeta ,\phi )=C_2\, J_{\sqrt{\lambda }}\left( ia_0 \xi \zeta \right) \xi ^{-\sqrt{\lambda +k^2}}e^{-i \kappa \phi }. \end{aligned}$$
(40)

4.2.2 Sector II: \(\lambda <0\) and \(k^2>|\lambda |\)

In this Sector, \({\sqrt{\lambda }}\) is imaginary and \({\sqrt{\lambda +k^2}}\) is real and positive. We therefore write \(\lambda =-\eta ^2\) with \(\eta \) real. As the singularity is approached, the first factors in \(\psi _1\), \(\psi _2\), \(\psi _3\) and \(\psi _4\), all have the asymptotic form \((\xi \zeta )^{\pm \sqrt{\lambda }}\sim (\xi \zeta )^{\pm i\eta }\), that oscillate and do not vanish at the singularity. Thus the singular nature of these components (\(\psi _j\)’s) is governed by the second factor \(\xi ^{\pm \sqrt{\lambda +k^2}}\).

Consequently, \(\psi _1\) and \(\psi _3\) are not admissible solutions since they approach infinity as the singularity is approached. We must therefore set \(C_1=C_3=0\), and the admissible asymptotes come from \(\psi _2\) and \(\psi _4\), given by \(\Psi _{\textrm{asy}}\sim C_2\psi _2+C_4\psi _4\), which are regular and well-behaved around the singularity, and they vanish at the singularity, respecting the DeWitt criterion. Thus, the wave function in this Sector is expressed as

$$\begin{aligned} \Psi _{II}(\xi ,\zeta ,\phi )&=\left[ C_2\, J_{i\eta } \left( ia_0 \xi \zeta \right) +C_4\, Y_{i\eta } \left( i a_0 \xi \zeta \right) \right] \nonumber \\ {}&\quad \times \xi ^{-\sqrt{k^2-\eta ^2}}e^{-i \kappa \phi }. \end{aligned}$$
(41)

4.2.3 Sector III: \(\lambda <0\) and \(k^2<|\lambda |\)

In this Sector, \({\sqrt{\lambda }}\) and \({\sqrt{\lambda +k^2}}\) are both imaginary. We therefore write \(\sqrt{\lambda }=i\eta \) and \({\sqrt{\lambda +k^2}}=i\gamma \) with both \(\eta \) and \(\gamma \) real. As the singularity is approached, the first and the second factors in \(\psi _1\), \(\psi _2\), \(\psi _3\) and \(\psi _4\), behave like \((\xi \zeta )^{\pm \sqrt{\lambda }}\sim (\xi \zeta )^{\pm i\eta }\) and \(\xi ^{{\pm \sqrt{\lambda +k^2}}}\sim \xi ^{\pm i \gamma }\), respectively. Both these factors oscillate and do not vanish as the singularity is approached. This implies that none of the asymptotes among \(\psi _1\), \(\psi _2\), \(\psi _3\) and \(\psi _4\) is admissible as they do not satisfy the DeWitt criterion.

Thus in this Sector regular quantum black holes cannot exist as there is no admissible wave function respecting the DeWitt criterion.

This may indicate two possibilities: either black holes do not exist in Sector III or black holes belonging to Sector III continue to have the singularity even in quantum gravity. The second possibility means that the singularity of a black hole belonging to Sector III is not resolved by quantum gravity. However, the first possibility seems to be the most likely one since we expect that quantum mechanics should be free from any singularity. In this case, the wave function of the black hole must vanish, implying non-existence of quantum black holes with a singularity.

We therefore see that it is only in Sectors I and II where regular quantum black holes can exist. The wave function of the black hole existing in Sector I or II vanishes at the singularity and it is well-behaved around the singularity, making a regular spacetime in the quantum black hole. Thus the singularity of a black hole in these Sectors are naturally resolved by quantum gravity.

Figure 1 displays the above three Sectors of the Hilbert space belonging to different subregions of eigenvalues (\(\lambda \), \(\kappa ^2\)). Sector I and II are marked “QBH” denoting existence of regular quantum black holes in these Sector. Sector III is marked “NO QBH” indicating that regular quantum black holes cannot exist in this Sector.

5 Discussion and conclusion

Since the classical black hole solution contains a non-removable singularity in the interior, it is intriguing to ask what would be the fate of this singularity when the spacetime is described by quantum mechanics. Our general experience with quantum mechanics suggests that the black hole singularity should be removed in quantum gravity. This intuition is further strengthened when we see that the hydrogen atom wave function is regular and well behaved at the origin although the classical Hamiltonian diverges at the origin.

The fundamental criterion to remove any spacetime singularity in general relativity is expressed by the DeWitt boundary condition, requiring that the wave function must vanish at the singularity. However, this criterion has not been meet so far in various quantum black hole solutions [81]. Among various schemes for quantum gravity, the wave function of a black hole is immediately formulated in the framework of Wheeler–DeWitt quantization. Since this advantage of having a wave function is usually not available in other formulations of quantum gravity, such as, string theory and loop quantum gravity, the question of DeWitt criterion has not been addressed in these formulations. Although there exist routs to singularity resolution without a black hole wave function in these formulations, they do not address the question of DeWitt criterion (a brief account appears in Sect. 1). It is nevertheless tempting to demand the DeWitt criterion which gives a straightforward and definite answer to singularity resolution.

Consequently, in this paper we took the Kantowski–Sachs metric to represent the interior geometry of a Schwarzschild black hole. With this metric, we carried out a \(3+1\) decomposition to obtain the classical Hamiltonian of the black hole interior geometry. Quite like the case of the hydrogen atom, this Hamiltonian is also plagued with singularity. Our expectation is that this singularity should be removed when we quantize the system, similar to the case of the hydrogen atom.

As we know from quantum field theory, quantum vacuum fluctuations of matter should spontaneously occur in the spacetime. Consequently, we included a Klein–Gordon matter field in the model.

Having constructed the gravity-matter model in the above scheme, we quantized this system following standard rules of quantum mechanics, such as, by promoting the momenta and hence the Hamiltonian into operators in the Schrodinger picture. The Hamiltonian constraint then naturally led to a Wheeler–DeWitt equation with the wave function of the interior geometry in the minisuperspace variables. Since the matter field does not occur explicitly in the Hamiltonian, it was immediately separable, leading to a partial differential equation in two variables. Following standard methods [77, 78], we made this equation separable, the emerging separation constants identified as eigenvalues of the system. The ensuing ordinary differential equations could be immediately solved, one of them giving the general solution in terms of the Bessel functions of the first and second kind with imaginary argument, while the other gave rise to power law solutions. As a product of these solutions, we thus obtained the general wave function in the minisuperspace variables, with the eigenvalues appearing as parameters in its expression.

This general solution could be written as a linear superposition of four wave functions (\(\psi _1, \psi _2, \psi _3\) and \(\psi _4 \)). We found that their behaviors near the singularity are intimately related with the relative magnitudes of the two eigenvalues. This gave rise to three different Sectors in the Hilbert space, each sector behaving quite differently from the other two.

In one of the Sectors, dubbed Sector I, only one of the terms (\(\psi _2\)) in the linear combination of the wave functions was admissible, whereas the rest three wave functions (\(\psi _1, \psi _3, \psi _4\)) were found to be unbounded near the singularity. Thus the three corresponding coefficients were set equal to zero, following the usual notion of picking up admissible solutions in quantum mechanics. This solution (\(\Psi _I\)), admissible near the singularity, turns out to be given by the Bessel function of first kind with imaginary argument, multiplied with a power-law term. This solution is regular and well-behaved near the singularity. Moreover, this wave function vanishes at the singularity, satisfying the DeWitt criterion. Thus regular quantum black hole can exist in this Sector.

In another Sector, called Sector II, two terms (\(\psi _2,\psi _4\)) in the linear combination of the wave function are admissible, and the rest two (\(\psi _1,\psi _3\)) were found to be unbounded near the singularity. Thus the two corresponding coefficients (of \(\psi _1,\psi _3\)) were set equal to zero, as before. This solution (\(\Psi _{II}\)), admissible near the singularity, turns out to be given by a linear combination of Bessel functions of the first and second kinds with imaginary argument, multiplied with a power-law term. This solution, unlike \(\Psi _I\), makes a damped oscillation as the singularity is approached, and it is regular and well-behaved near the singularity. Moreover, this solution vanishes at the singularity, satisfying the DeWitt criterion. Thus regular quantum black hole can exist in this Sector as well.

In the last Sector, named Sector III, none of the terms (\(\psi _1,\psi _2,\psi _3,\psi _4\)) in the linear combination of the wave function was found to be admissible, as all of them make sinusoidal oscillations without damping as the singularity is approached. In order to satisfy the DeWitt criterion, we must set all four coefficients (of \(\psi _1,\psi _2,\psi _3,\psi _4\)) equal to zero. This implies that the wave function of the black hole vanishes in this Sector, that is, \(\Psi _{III}=0\). This immediately implies that regular quantum black holes cannot exist in this sector.

If, however, one denies the DeWitt criterion, then the wave function \(\Psi _{III}\) will not vanish, and there would exist quantum black holes with undamped oscillating wave function near the singularity. In such black holes, the singularity will continue to persist as a consequence of violation of the DeWitt criterion.

On the other hand, the DeWitt criterion being a fundamental aspect for singularity resolution in quantum gravity, it ought to be respected for singularity resolution. Thus regular quantum black holes can exist only in Sectors I and II of the Hilbert space, and there exist no regular quantum black holes in Sector III.

We also note that if we simultaneously assume bounded wave functions everywhere and fulfillment of the DeWitt boundary condition at the classical singularity, we may speculate that the wave function should change its behavior as the horizon is approached. Since our present analysis is focused in the region near the classical singularity, the nature of change in behavior of the wave function cannot be determined from this analysis. The Klein–Gordon field may play a crucial role in this regard, which remains a matter of further study.