Starting from the WheelerDeWitt equation for the Schwarzschild black hole interior which is derived from a Hamiltonian formulated in terms of canonical phase space coordinates we show that by applying a simple reparametrization this equation can be expressed as the eigenvalue equation of a quantum linear harmonic oscillator. Within the standard quantization framework we find that the resulting wave function diverges in the region of the classical singularity and the expectation value of the Kretschmann scalar is undefined for all states within the black hole. However when we apply the minimal uncertainty approach to the quantization process we obtain a wave function that is both welldefined and squareintegrable. Additionally the expectation value of the Kretschmann scalar for these states remains finite throughout the black holes interior suggesting that the classical singularity is resolved in this approach replaced it by a minimum radius.
['General Relativity and Quantum Cosmology', 'High Energy Physics - Theory']
Quantum Black Hole as a Harmonic Oscillator from the Perspective of the Minimum Uncertainty Approach
{'Octavio Obregón, a Wilfredo Yupanqui. a': '- a Departamento de Física, División de Ciencias e Ingenierías, Universidad de Guanajuato Loma del Bosque 103, León 37150, Guanajuato, México.  \nE-mail: octavio@fisica.ugto.mx , w.yupanquicarpio@ugto.mx', 'Abstract:': "Starting from the Wheeler-DeWitt equation for the Schwarzschild black hole interior, which is derived from a Hamiltonian formulated in terms of canonical phase space coordinates, we show that by applying a simple reparametrization, this equation can be expressed as the eigenvalue equation of a quantum linear harmonic oscillator. Within the standard quantization framework, we find that the resulting wave function diverges in the region of the classical singularity, and the expectation value of the Kretschmann scalar is undefined for all states within the black hole. However, when we apply the minimal uncertainty approach to the quantization process, we obtain a wave function that is both well-defined and square-integrable. Additionally, the expectation value of the Kretschmann scalar for these states remains finite throughout the black hole's interior, suggesting that the classical singularity is resolved in this approach, replaced it by a minimum radius.", '1 Introduction': "A quantum black hole is a theoretical concept that arises from the formulation of a quantized Hamiltonian, which is defined within the framework of general relativity. This concept is central to the ongoing efforts to develop a theory of quantum gravity, which would reconcile the two fundamental theories of modern physics: quantum mechanics and general relativity.  \nSeveral approaches have been explored to investigate the interior of a black hole within the framework of quantum gravity. For instance, in Loop Quantum Gravity (LQG) [1], which is one of the leading non-perturbative approaches to the quantization of gravity, numerous studies have been conducted on both the interior and the entire spacetime of black holes. In these studies, the Hamiltonian describing the black hole's interior, expressed in terms of Ashtekar-Barbero variables, is quantized using the socalled polymer quantization [2-7]. This quantization method effectively introduces a parameter that establishes a minimal scale in the model, which allows for the avoidance of the singularity and also introduces a bounce from a black hole to a white hole in the vacuum case [8-14]. Polymer quantization also induces certain modifications in the algebra of the theory at the quantum level, which can be viewed as an effective modification of the classical algebra. Other methods for quantizing the interior of black holes can be found in the works [15-24].  \nWhen gravity is incorporated into quantum measurement processes, led to the generalization of Heisenberg's uncertainty relation (HUR). This modification to the HUR is known as the Generalized Uncertainty Principle (GUP), which implements a minimal uncertainty in the position by modifying the ordinary uncertainty relation of quantum mechanics to accommodate deformations at high energies, typically at the Planck scale [25]. On the other hand, GUP can be understood as an alternative quantization procedure that imposes a modified commutator between the position and a generalized momentum (or between generalized position and momentum [25], depending on the chosen representation) [26, 27], resulting in a minimum uncertainty in position or momentum [28]. In [29], it is demonstrated that the GUP arises from the consideration of non-extensive entropies that depend only on the probabilities. GUP is also derived from different proposals: in [30], the scattering of strings at ultra-high energies is considered to analyze the divergences of quantum gravity at the Planck scale; in [31], a gedanken experiment is proposed to measure the area of the apparent horizon of black holes in the context of quantum gravity; and [32] explores the idea that spacetime in the Planck region fluctuates, leading to the possibility of virtual micro-black holes affecting the measurement process.  \nAs mentioned, the effects of GUP are significant in systems with energies close to the Planck scale. Particularly relevant examples of such systems include the early universe and the interiors of black holes, where quantum gravity effects are expected to dominate [33-36]. Therefore, quantum cosmology, the branch of physics that studies these systems, is the appropriate field where this modified quantization rule is expected to have a considerable impact. In this context, taking advantage of the fact that the interior of a Schwarzschild black hole is isometric to the Kantowski-Sachs cosmological model, for the first time, the quantization based on the minimal uncertainty approach has been applied to the minisuperspace variables that describe the dynamics inside the black hole [37]. This implies a modification of the Wheeler-DeWitt equation, which governs the quantum cosmological model, thereby characterizing a modified dynamic of the solution.  \nFollowing this line of applying the minimal uncertainty approach to Quantum Cosmology, several works have been published. In [38], the classical Hamiltonian of the Schwarzschild black hole interior is considered within the Ashtekar-Barbero connection formalism. Inspired by models based on the Generalized Uncertainty Principle, the canonical algebra of the model is deformed, leading to the derivation of the effective dynamics. This deformation results in the resolution of the black hole singularity by introducing a minimum nonzero radius for the infalling two-spheres. Recently, in [39], starting from the proposal of a new reduced Hamiltonian, the classical black hole singularity is resolved by replacing it with an effective bounce that connects the interior  \nof a black hole with the interior of a white hole. This bounce occurs in the region near the Planck scale, where a new event horizon emerges. Crossing this horizon changes the nature of the interval from spatial to temporal outside the white hole. Finally, in [40], the interior of a Schwarzschild black hole was quantized using the minimal uncertainty approach suitable for the Ashtekar-Barbero connection variables. As a result, it was found that all interior states remain well-defined and square-integrable. Moreover, the expectation value of the Kretschmann scalar remains finite throughout the entire interior region of the black hole, particularly in the area where the classical singularity used to reside, indicating the resolution of the black hole singularity. Additionally, a minimum value for the radius of the 2-spheres was also identified.  \nIn this work, we focus on the Wheeler-DeWitt equation derived in [41] from a Hamiltonian that describes the spherically symmetric spacetime within the interior of the Reissner-Nordström black hole. The phase space characterizing this spacetime is parameterized by the charge Q , the mass M , and their respective conjugate momenta. Through a canonical transformation, configuration variables are obtained that naturally describe the dynamical properties of the black hole's interior. In [42], the simplest case of this model is considered, taking Q = 0 , and interestingly, by reparametrizing the black hole's radial coordinate, the eigenvalue equation of a linear harmonic oscillator is obtained. As a result, it is found that the area spectrum, and therefore the Schwarzschild radius, is discrete and proportional to the square of the Planck length [41, 43]. Upon solving the eigenvalue equation for the black hole, modeled as a linear harmonic oscillator, we find that the black hole singularity persists in the standard quantization model. To address this issue, we introduce quantization under the minimum uncertainty approach, which resolves the singularity by imposing a minimum radius on the 2-sphere. Furthermore, this approach modifies the area spectrum and the black hole's radius.  \nThe paper is organized as follows: In section 2, we provide a brief overview of the interior of the Schwarzschild black hole from the Hamiltonian perspective derived in [41, 43]. In section 3, we apply an appropriate transformation to express the eigenvalue equation for the black hole mass as a quantum harmonic oscillator-like eigenvalue equation. Then, following the standard quantization procedure, i.e., using the usual commutation relation between canonical variables, we quantize the interior of the black hole. Section 4 is dedicated to the quantization of the black hole interior using the minimal uncertainty approach, in which the usual commutation relation is modified. We demonstrate that the resulting wave function is finite within the black hole, as is the expectation value of the Kretschmann scalar. Finally in section 5 we summarize our results and conclude.", '2 Hamiltonian description of the black hole interior': 'The Reissner-Nordström metric is a static solution to the Einstein-Maxwell field equations, which corresponds to the gravitational field of a charged, non-rotating, spherically symmetric body of mass M . In spherical coordinates ( t, r, θ, φ ) , this metric is  \nds 2 = -( 1 -R s r + R 2 Q r 2 ) c 2 dt 2 + ( 1 -R s r + R 2 Q r 2 ) -1 dr 2 + r 2 d Ω 2 , (2.1)  \nwhere R s = 2 GM/c 2 is the Schwarzschild radius and R 2 Q = Q 2 G/ 4 πϵ 0 c 4 the characteristic length scale, in which Q is the electric charge. In the limit that the charge Q (or equivalently, the length scale R Q ) goes to zero, one recovers the Schwarzschild metric  \nds 2 = -( 1 -R s r ) c 2 dt 2 + ( 1 -R s r ) -1 dr 2 + r 2 d Ω 2 , (2.2)  \nwith r ∈ (0 , ∞ ) being the radial coordinate. It is well-known that upon crossing the event horizon of the Schwarzschild black hole which is located at R s , the timelike and spacelike curves switch their causal nature. This is because when an object crosses the event horizon of a Schwarzschild black hole, its timelike trajectory inside the event horizon becomes spacelike, and it can no longer escape the gravitational attraction of the black hole. So, one can obtain the interior metric by switching t ↔ r in (2.2) [44]  \nds 2 = -( R s t -1 ) -1 dt 2 + ( R s t -1 ) dr 2 + t 2 d Ω 2 . (2.3)  \nHere t is the Schwarzschild time coordinate which has a range t ∈ (0 , R s ) in the interior. Consider the interior solution as measured by an observer at rest relatively to the space coordinates, i.e., dr = d Ω = 0 . In this case, from the metric (2.3), we have [44]  \n( dt dτ ) 2 = R s t -1 , (2.4)  \nwhere τ is the proper time.  \nIn [41, 45], a classical Hamiltonian corresponding to the metric in equation (2.1) is derived from the Einstein-Maxwell action. The constraint equations derived in this model allow us to define the Hamiltonian of the Reissner-Nordström black hole in terms of the variables m and q , which can be identified as the mass M and the charge Q of the hole. These variables, along with their conjugate canonical momenta p m and p q , form a finite phase space. If the charge Q is fixed as an external parameter (in our case, we set Q = 0 ), it is possible to perform a canonical transformation from the phase space  \nvariables ( m,p m ) to the new phase space variables ( a, p a ) , which naturally describe the dynamic properties inside the Reissner-Nordström black hole. Consequently, the classical Hamiltonian, in terms of the variables a and p a , takes the form [42]  \nH = p 2 a 2 a + 1 2 a, (2.5)  \nwhere the new variable a and its conjugate momentum p a satisfy the classical algebra  \n{ a, p a } = 1 . (2.6)  \nAs noted in [41], the geometric interpretation of a is related to the radial coordinate of the black hole. In our case, where an uncharged hole ( R Q = 0 ) is considered, we conclude that 0 < a < R s . Therefore, we interpret the variable a as describing the dynamics exclusively within the interior of the Schwarzschild black hole.  \nSince the Hamiltonian, (2.5), describing the dynamic properties of the interior of the Schwarzschild spacetime is known, we can proceed with the canonical quantization of the spacetime in question. For this process, we choose a Hilbert space of the type L 2 ( R n , d x ) , in which the variable a acts as a multiplicative operator and its conjugate momentum p a is represented as a differential operator, p a = -i∂/∂a . In [42], a simplified form of the Hamiltonian is considered by setting Q = 0 , and the full-dimensional form of its differential equation is derived  \nℏ 2 G 2 c 6 a -s -1 d da ( a s d da ) Ψ( a ) = ( a -R s ) Ψ( a ) , (2.7)  \nwhere s is an arbitrary factor ordering parameter. It is important to note that (2.7) is a Wheeler-DeWitt-type equation. Therefore, the wave function in this model represents a quantum black hole. Additionally, the phase space coordinates represented by the pair ( a, p a ) are not affected at the quantum level by their classical dependence on time t or radius r . Thus, they can be interpreted as the variables of a minisuperspace.  \nThe equation of motion for a , in the case where Q = 0 , is given by [41]  \n˙ a 2 = R s a -1 , (2.8)  \nand by comparing this expression with (2.4), it becomes evident that the Schwarzschild time, t , and the dynamical variable, a , exhibit similar behavior. Therefore, we can express the interval (2.3) in terms of a  \nds 2 = -( R s a -1 ) -1 da 2 + ( R s a -1 ) dr 2 + a 2 d Ω 2 . (2.9)  \nThis justification is based on the fact that both variables t and a are confined to the interior of the black hole, where t ∈ (0 , R s ) and a ∈ (0 , R s ) . Furthermore, they describe the interior dynamics of the black hole in the same manner, as can be verified by comparing equations (2.4) and (2.8).  \nA way of detecting singularities is to find where the energy density or the spacetime curvature become infinite and the usual description of the spacetime breaks down. However, to be sure that there is an essential singularity which cannot be transformed away by a coordinate transformation, invariants are constructed from the curvature tensor. These quantities, particularly the Kretschmann scalar K = R µναβ R µναβ , are coordinate invariant measures of whether a singularity exists in some region, given that the said region is not an infinite affine parameter away. For the black hole region, given by the metric (2.9), the scalar Kretschmann polynomial, K , is given by  \nK = 48 G 2 M 2 a 6 . (2.10)  \nOn the horizon a → R s = 2 GM and K = 3 4 G 4 M 4 , which indicates that in this region there is no real singularity. In a → 0 we have that K →∞ indicating the presence of a physical singularity there.', '3 Black hole as harmonic oscillator': "It is interesting and somewhat surprising to note that the Wheeler-DeWitt equation for a Schwarzschild black hole, as expressed in (2.7), can be reparametrized as follows [42]  \nψ ( a ) = U ( a ) a , x = a -R s 2 , (3.1)  \nwhich leads to an eigenvalue equation resembling that of a linear harmonic oscillator  \n-ℏ 2 G 2 c 6 ( d 2 dx 2 + s -2 x -R s / 2 d dx -s -2 ( x -R s / 2) 2 ) U ( x ) + x 2 U ( x ) = R 2 s 4 U ( x ) , (3.2)  \nin particular, by selecting the factor ordering parameter as s = 2 , the expression in (3.2) transforms into a quantum linear oscillator  \n( -l 2 Pl d 2 dx 2 + x 2 l 2 Pl ) U ( x ) = R 2 s 4 l 2 Pl U ( x ) , (3.3)  \nwhere l 2 Pl = ℏ G/c 3 denotes the Planck length.  \nFor our purposes, we can define the momentum operator, conjugate to x , as a differential operator  \nˆ p x = -il Pl d dx , (3.4)  \nwhich clearly corresponds to the differential operator on the left-hand side of equation (3.3). Therefore, we can rewrite (3.3) as follows  \n( ˆ p 2 x + ˆ x 2 l 2 Pl ) U ( x ) = R 2 s 4 l 2 Pl U ( x ) . (3.5)  \nOur goal is to solve this eigenvalue equation in the momentum representation, where ˆ p x acts as a multiplicative operator and ˆ x as a differential operator, meaning that  \nˆ p x = p x , ˆ x = il Pl d dp x . (3.6)  \nOn the other hand, it can be verified that ˆ x and ˆ p x satisfy a Heisenberg-type algebra, taking the form  \n[ˆ x, ˆ p x ] = il Pl . (3.7)  \nWith this in mind, we express the differential equation (3.5) in momentum representation  \nd 2 ¯ U ( p x ) dp 2 x +( κ 2 -p 2 x ) ¯ U ( p x ) = 0 , (3.8)  \nwhere κ 2 = R 2 s / 4 l 2 Pl . Equation (3.8) is known as Weber's differential equation, and its solution is given by  \n¯ U n ( p x ) = N n e -p 2 x / 2 H n ( p x ) , (3.9)  \nwhere H n ( z ) is the Hermite polynomials and N n a normalization constant  \nN n = √ 1 2 n √ πn ! . (3.10)  \nAccording to this solution, the possible eigenvalues of the black hole horizon area are of the form  \nA s ( n ) = 32 π ( n + 1 2 ) l 2 Pl . (3.11)  \nThe equation (3.11) implies that the area of the quantum black hole is quantized and proportional to the Planck length, where n is a quantum number with values n = 0 , 1 , 2 , · · · . This result coincides with those obtained in [41, 42, 46]. On the other hand, given the quantization of the area (3.11), the Schwarzschild radius must also be  \nquantized due to the relation A s ( n ) = 4 πR 2 s ( n ) , and consequently the mass M ( n ) of the black hole. Thus, from (3.11) we obtain  \nR s ( n ) = 2 √ 2 ( n + 1 2 ) l Pl , (3.12)  \nwhich implies that the radial coordinate inside the black hole will be constrained between 0 and R s ( n ) .  \nAlthough we have found the wave function of the interior of a Schwarzschild black hole in standard quantum mechanics as a function of the momentum variable p x , more direct physical information about the black hole can be obtained from configuration variable x , as the variable x is related to the radius of the black hole by (3.1). It is therefore convenient to change the representation by performing a standard Fourier transform  \nϕ ( x ) = 1 √ 2 πl Pl ∫ e ip x x/l Pl ¯ ϕ ( p x ) dp x . (3.13)  \nThe Fourier transformation of (3.9) into position space can be achieved using the exponential generating function of the Hermite polynomials. Then, from (3.13) one have  \nU n ( x ) = ( i ) n N n √ l Pl e -x 2 / 2 l Pl H n ( x/l Pl ) . (3.14)  \nIt is our interest to derive the wave function in terms of the variable a , arising from the modification in (3.1), given that the metric components (2.9) are expressed in terms of this variable. So, we obtain  \nΨ n ( x ) = ( i ) n N n √ l Pl e -x 2 / 2 l Pl x + R s ( n ) 2 H n ( x/l Pl ) , (3.15)  \nThe domain of x , according to (3.1), is x ∈ ( -R s ( n ) / 2 , R s ( n ) / 2) . It is evident that the integral of the square of the wave function (3.15) within the integration domain of x does not converge, indicating it is not square-integrable in the interior of the black hole. This behavior can be attributed to the presence of the physical singularity at a = 0 (or in x = -R s ( n ) / 2 ); in this region, the wave function (3.15) diverges. This suggests that the classical singularity remains unresolved within the context of the standard quantization approach employed above.  \nIn Figure 1, we depict the square of the wave function (3.15) plotted against the variable x , which is associated with the radius of the 2-sphere. As the quantum number n escalates, the number of probable states also rises. Notably, the probability density | Ψ n ( a ) | 2 is more pronounced in the vicinity of x = -R s ( n ) / 2 (or a = 0 ) and diminishes  \nFigure 1 . Graph showing the square of the wave function (3.15) plotted against different values of the quantum number n .  \n<!-- image -->  \n(a) Square wave function for  \nn  \n= 0  \n<!-- image -->  \n- (c) Square wave function for n = 5  \n<!-- image -->  \n(b) Square wave function for  \nn  \n= 2  \n<!-- image -->  \n(d) Square wave function for  \nn  \n= 10  \nas we approach the event horizon. In the region where a equals zero, the probability density becomes infinite due to the classical singularity present in that region.  \nTo investigate the fate of the singularity in the quantum regime, we calculate the expectation value of K , as defined in (2.10), with respect to the quantum state given in (3.15). Upon quantization, K becomes an operator due to its dependence on the operator ˆ a . For simplicity, we evaluate its expectation value in the position representation, where ˆ a acts multiplicatively. Therefore, we utilize the wave function given in (3.15) to compute the expectation value of the Kretschmann scalar  \n⟨ ˆ K ⟩ n = ∫ Ψ ∗ n ( x ) ˆ K Ψ n ( x ) dx ∫ | Ψ n ( x ) | 2 dx . (3.16)  \nSince the wave function in (3.15) is not square-integrable, the expectation value of the Kretschmann scalar is indeterminate in the standard quantization process. This outcome was anticipated due to the unresolved physical singularity inside the black hole in the approach taken.", '4 Quantization employing the minimum uncertainty approach': "As observed in the previous section, standard quantization of the interior of the Schwarzschild black hole does not resolve the physical singularity present in the region a = 0 . In this section, we will implement quantization based on the minimum uncertainty approach, which introduces a minimal length description, typically at high energies.  \nTo introduce the quantum effects of gravity in the measurement process, the uncertainty relation for position and momentum is extended to [25]  \n∆ q ∆ p ≥ ℏ 2 ( 1 + β (∆ p ) 2 ) , (4.1)  \nwhere β is know as the deformed parameter. In ordinary quantum mechanics, ∆ q can be made arbitrarily small by letting ∆ p grow correspondingly, this is no longer the case when (4.1) is considered. If for decreasing ∆ q , ∆ p increases, the new term β (∆ p ) 2 on the right-hand side of (4.1) will eventually grow faster than the left-hand side. Hence ∆ q can no longer be made arbitrarily small, searching a minimal uncertainty of the order ℏ √ β . It allows one to express the idea that a minimal length l min should quantum theoretically be described as a minimal uncertainty in position measurements. The modified commutation relation for the ˆ q and ˆ p operators associated with (4.1) is expressed as [25]  \n[ˆ q, ˆ p ] = i ℏ ( 1 + β ˆ p 2 ) . (4.2)  \nDue to the deformed commutator (4.2) the operators ˆ q and ˆ p are not conjugates anymore. Now, these fundamental variables are to be high energy operators valid, in particular, at or near the Planck scale. They have non-linear representations, ˆ q = q (ˆ q 0 ) , ˆ p = p (ˆ p 0 ) in terms of the variables ˆ q 0 , ˆ p 0 which are position and momentum operators at low energies, obeying the standard Heisenberg algebra [ˆ q 0 , ˆ p 0 ] = i ℏ .  \nThis minimal uncertainty approach has yielded intriguing results when considering the quantum effects of gravity. For instance, we can cite some works [37, 38]. Motivated by these successes, we will apply this quantization approach in this section to resolve the singularity of the black hole under consideration. To implement the minimal uncertainty approach in the quantization procedure, the algebra (3.7) will be modified according to (4.2) in order to achieve  \n[ˆ x, ˆ p x ] = il Pl ( 1 + β ˆ p 2 x ) , (4.3)  \nfrom which one can find the generalized uncertainty relation  \n∆ x ∆ p x ≥ l Pl 2 [ 1 + β (∆ p x ) 2 ] , (4.4)  \nwhich correspond to minimal uncertainty in x of the order l Pl √ β . Therefore, β effectively defines the magnitude of the minimal uncertainty effects.  \nTo simplify our calculations and proceed with a standard quantization procedure, it is convenient to introduce a new variable, p x 0 , conjugate to x , that satisfies the usual commutation relation [26, 27]  \n[ˆ x, ˆ p x 0 ] = il Pl . (4.5)  \nFor simplicity in the calculations, we will work in the representation where ˆ p x 0 acts as a multiplicative operator and the position operator is represented as a differential operator ˆ x = il Pl ∂/∂p x 0 . The reason for choosing this representation is that it is not possible to clearly define the momentum operator ˆ p x as a differential operator that satisfies the commutation relation (4.3) [25]. Therefore, from (4.3), we can find the relation between the physical variable p x and the auxiliary p x 0 by  \np x = 1 √ β tan ( √ βp x 0 ) , (4.6)  \nwhich satisfies the modified commutation relation (4.3). Also, the domain of p x 0 is restricted to -π/ 2 √ β < p x 0 < π/ 2 √ β [26].  \nIn this approach, the differential equation (3.8) is expressed in terms of the new canonically conjugate variables x and p x 0 . Thus, in the p x 0 space, it reads  \nd 2 ¯ U ( p x 0 ) dp 2 x 0 + ( κ 2 -tan 2 ( √ βp x 0 ) β ) ¯ U ( p x 0 ) = 0 . (4.7)  \nUsing the new variable ξ by the change ξ = √ βp x 0 , the above equation can be written as  \nd 2 ¯ U ( ξ ) dξ 2 + ( ϵ -tan 2 ξ β 2 ) ¯ U ( ξ ) = 0 , (4.8)  \nhere we denoted ϵ = κ 2 /β = R 2 s / 4 βl 2 Pl . This differential equation reduces to the hypergeometric one through the transformation z = sin 2 ξ and y = ¯ U cos m ξ [47]  \nz ( z -1) y '' ( z ) + [ (1 -m ) z -1 2 ] y ' ( z ) -1 4 ( m + ϵ ) y ( z ) = 0 , (4.9)  \nwhere m is a root of the quadratic equation m 2 + m -1 β 2 = 0 , from which one have for m  \nm = -β ± √ 4 + β 2 2 β . (4.10)  \nThen, the general solution of (4.9) is given by  \ny ( z ) = C 1 2 F 1 ( β + √ β 2 +4+2 √ ϵβ 2 +1 4 β , β + √ β 2 +4 -2 √ ϵβ 2 +1 4 β ; 1 2 ; z ) + C 2 z 1 / 2 2 F 1 ( β + √ β 2 +4+2 √ ϵβ 2 +1 4 β + 1 2 , β + √ β 2 +4 -2 √ ϵβ 2 +1 4 β + 1 2 ; 3 2 ; z ) . (4.11)  \nHere, 2 F 1 ( A, B ; C ; z ) represents the hypergeometric function, which converges if one of the first two arguments, A or B , is a non-positive integer. For example, if we consider the hypergeometric function appearing in the first line of equation (4.11), this condition for the first argument results in  \nβ + √ β 2 +4+2 √ ϵβ 2 +1 4 β = -n, with n = 0 , 1 , 2 , · · · . (4.12)  \nFrom this, one can derive the eigenvalues of the black hole's area, considering that ϵ = R 2 s / 4 βl 2 Pl and A s = 4 πR 2 s , within the framework of the minimum uncertainty approach. That is  \nA GUP s (2 n ) = 32 π ( 2 n + 1 2 ) l 2 Pl ( √ 1 + β 2 4 + β 2 ) +16(2 n ) 2 πβl 2 Pl . (4.13)  \nIf, on the other hand, we apply the convergence condition to the second argument in the hypergeometric function 2 F 1 ( A, -n ; C ; z ) , we obtain exactly (4.13), thus ensuring the convergence of the first term in the wave function (4.11).  \nWhen the same convergence criterion is applied to the hypergeometric function appearing in the second line of equation (4.11), we find that the spectrum of the area, in this case, is  \nA GUP s (2 n +1) = 32 π ( (2 n +1) + 1 2 ) l 2 Pl ( √ 1 + β 2 4 + β 2 ) +16(2 n +1) 2 πβl 2 Pl , (4.14)  \nand combining both expressions, (4.13) and (4.14), we obtain the discrete spectrum of the black hole's area  \nA GUP s ( n ) = 32 π ( n + 1 2 ) l 2 Pl ( √ 1 + β 2 4 + β 2 ) +16 n 2 πβl 2 Pl . (4.15)  \nIt can be readily observed from this expression that in the limit where β → 0 , the usual area spectrum in (3.11) is recovered. Conversely, if β remains finite, the area  \nlevels depend on the square of the quantum number n , and for large n , they grow like n 2 . Just as the area is modified, the Schwarzschild radius will also be, by  \nR GUP s ( n ) = l Pl √ √ √ √ 8 ( n + 1 2 ) ( √ β 2 4 +1+ β 2 ) +4 βn 2 . (4.16)  \nAccording to these convergence conditions, (4.12), the wave function (4.11) is written, in terms of the original variable ξ , as  \ny n ( ξ ) = C 1 2 F 1 ( -n, n + α ; 1 2 ; sin 2 ξ ) + C 2 sin ξ 2 F 1 ( -n, n + α +1; 3 2 ; sin 2 ξ ) , (4.17)  \nwhere we have denoted  \nα = β + √ 4 + β 2 2 β . (4.18)  \nHere, if the negative sign is chosen instead of the positive in (4.10), we find that -m = α . Additionally, considering the special doublen formulas of the Gegenbauer polynomials [48]  \nC ( α ) 2 n (sin ξ ) =( -1) n ( n + α -1 n ) 2 F 1 ( -n, n + α ; 1 2 ; sin 2 ξ ) , (4.19)  \nC ( α ) 2 n +1 (sin ξ ) =2( -1) n α ( n + α n ) sin ξ 2 F 1 ( -n, n + α +1; 3 2 ; sin 2 ξ ) , (4.20)  \nwe can write the solution (4.17) in a compact form  \n¯ U n ( p 0 x ) = N n C ( α ) n (sin ( √ βp 0 x )) cos α ( √ βp 0 x ) , (4.21)  \nwhere the substitution ξ = √ βp x 0 has been used, and N n is a constant that can be determined from the normalization condition of the wave function (4.21). That is  \nN 2 n ∫ π 2 √ β -π 2 √ β ( 1 -sin 2 ( √ βp 0 x ) ) α [ C ( α ) n (sin ( √ βp 0 x )) ] 2 dp 0 x = 1 , (4.22)  \nand considering that the Gegenbauer polynomials are normalized by [49]  \n∫ 1 -1 (1 -h 2 ) α -1 / 2 [ C ( α ) n ( h )] 2 dh = 2 1 -2 α π Γ( n +2 α ) ( n + α )Γ 2 ( α )Γ( n +1) , (4.23)  \nwe obtain  \nN n = [ √ β ( n + α )Γ 2 ( α )Γ( n +1) 2 1 -2 α π Γ( n +2 α ) ] 1 / 2 . (4.24)  \nUnlike the constant in (3.10), this integration constant is expressed not only in terms of the quantum number n but also in terms of the deformation parameter β , as shown in (4.18).", '4.1 Limit β → 0': 'In the standard limit, where β → 0 , the usual solution (3.9) should be recovered. To verify this, consider from (4.18) that for small values of the deformation parameter β , we have α = 1 /β , which implies that when β → 0 , α →∞ . On the other hand, from the relationship between the Gegenbauer polynomial C ( α ) n and the relativistic Hermite polynomial H ( α ) n [50], given by  \nH ( α ) n ( √ αu ) = n ! α n/ 2 (1 + u 2 ) n/ 2 C ( α ) n ( u √ 1 + u 2 ) , (4.25)  \nwe can express (4.21) as follows  \n¯ U n ( p 0 x ) = [ √ β ( n + α )Γ 2 ( α ) α n 2 1 -2 α π Γ( n +2 α ) n ! ] 1 / 2 cos n + α ( √ βp 0 x ) H ( α ) n ( √ α tan ( √ βp 0 x )) , (4.26)  \nwhere u = tan ( √ βp 0 x ) was consider. Using the Stirling and asymptotic formulas [49]  \nΓ( α ) ∼ √ 2 πe -α α α -1 / 2 , (4.27)  \nΓ( n +2 α ) ∼ √ 2 πe -2 α (2 α ) 2 α + n -1 / 2 , (4.28)  \nis easy to verify that the term in brackets in (4.26) reduces to  \nlim α →∞ [ √ β ( n + α )Γ 2 ( α ) α n 2 1 -2 α π Γ( n +2 α ) n ! ] 1 / 2 = 1 √ π 1 / 2 2 n n ! . (4.29)  \nOn the other hand, using the expansion ln cos x = -x 2 / 2 -x 4 / 12 -··· , we can make the following approximation  \nlim β → 0 cos n + α ( √ βp 0 x ) = lim β → 0 e n ln cos ( √ βp 0 x ) e α ln cos ( √ βp 0 x ) ∼ e -p 2 0 x / 2 . (4.30)  \nIn the limit α → ∞ (non-relativistic limit) the relativistic Hermite polynomial H ( α ) n turns into the Hermite polynomial H n ( ξ ) [50], and equivalently, its argument reduces to lim β → 0 tan ( √ βp 0 x ) / √ β = p x 0 . So, we have that  \nlim α → 0 H ( α ) n ( √ α tan ( √ βp 0 x )) = H n ( p 0 x ) . (4.31)  \nCombining this limit with those found in (4.29) and (4.30), we find that the wave function (4.21), derived using the minimal uncertainty approach, reduces to, as expected, the wave function (3.9) in the limit β → 0 .', '4.2 Fourier transform': 'The wave function in (4.21) is defined in momentum space, in terms of the auxiliary variable p 0 x . To extract physical information from our results, we need to express the wave function inside the black hole in terms of the radial coordinate, as the components of the metric (2.9) depend on this variable. This can be accomplished using the Fourier transform (3.13), which in this case would be  \nU n ( x ) = N n √ 2 πl Pl ∫ π 2 √ β -π 2 √ β e ip 0 x x/l Pl cos α ( √ βp 0 x ) C ( α ) n (sin ( √ βp 0 x )) dp 0 x . (4.32)  \nTo perform this integration, we will use the explicit expression of the Gegenbauer polynomials [49]  \nC ( α ) n (sin ( √ βp 0 x )) = [ n 2 ] ∑ k =0 ( -1) k Γ( n -k + α ) Γ( α ) k !( n -2 k )! (2 sin ( √ βp 0 x )) n -2 k , (4.33)  \nso, (4.32) becomes in  \nU n ( x ) = [ n 2 ] ∑ k =0 W n,k ∫ π 2 √ β -π 2 √ β e ip 0 x x/l Pl ( 1 -sin 2 ( √ βp 0 x ) ) α 2 sin n -2 k ( √ βp 0 x ) dp 0 x , (4.34)  \nwhere W n,k is a constant denoted as  \nW n,k = ( -1) k N n 2 n -2 k Γ( n -k + α ) √ 2 πl Pl Γ( α ) k !( n -2 k )! . (4.35)  \nThe Fourier transform in (4.34) yields the wave function in x space, which can be expressed as  \nΨ  \nGUP n ( x ) = -i √ β ( x + R s 2 ) [ n 2 ] ∑ k =0 W n,k e -1 2 iπ (2 α +10 k -5 n ) × { Γ( α +1) [ e 1 2 iπ ( α +6 k -x √ βl Pl -3 n ) Γ ( -l Pl ( α -2 k + n ) + x √ β 2 l Pl ) × 2 ˜ F 1 ( α +1 , 1 2 ( α -2 k + n -x l Pl √ β +2 ) ; 1 2 ( α +2 k -n -x l Pl √ β +2 ) ; -1 ) + e 1 2 iπ ( α +10 k + x √ βl Pl -5 n ) Γ ( 1 2 ( -α +2 k -n + x l Pl √ β )) × 2 ˜ F 1 ( α +1 , 1 2 ( α -2 k + x √ βl Pl + n +2 ) ; 1 2 ( α +2 k + x √ βl Pl -n +2 ) ; -1 ) ] -  \nΓ( -2 k + n +1) [ e iπ ( α +2 k -n ) Γ ( -l Pl ( α -2 k + n ) + x √ β 2 l Pl ) × 2 ˜ F 1 ( -2 k + n +1 , 1 2 ( α -2 k -x √ βl Pl + n +2 ) ; 1 2 ( -α -2 k -x √ βl Pl + n +2 ) ; -1 ) + e iπ ( α +4 k -2 n ) Γ ( 1 2 ( -α +2 k -n + x l Pl √ β )) × 2 ˜ F 1 ( -2 k + n +1 , 1 2 ( α -2 k + x √ βl Pl + n +2 ) ; 1 2 ( -α -2 k + x √ βl Pl + n +2 ) ; -1 ) ]} , (4.36)  \n̸  \nwith the conditions β > 0 and α > -1 . Here, 2 ˜ F 1 ( A, B ; C ; z ) = 2 F 1 ( A,B ; C ; z ) Γ( C ) is the regularized hypergeometric function. To ensure that the wave function is convergent, both the gamma functions and the hypergeometric functions appearing in (4.36) must be finite. Based on the definition of Γ( z ) , where z cannot be a negative integer, we find from Γ ( -l Pl ( α -2 k + n )+ x √ β 2 l Pl ) that x = l Pl √ β (2 l -( α -2 k + n )) , where l = 0 , 1 , 2 , . . . . Similarly, from Γ ( 1 2 ( -α +2 k -n + x l Pl √ β )) , we find that x = -l Pl √ β (2 l -( α -2 k + n )) . These convergence conditions for x can be summarized as follows  \n̸  \n̸  \nx = ± l Pl √ β (2 l -( α -2 k + n )) , (4.37)  \nand, as shown in (3.1), the variable x is related to the radius of the 2-sphere, with the condition a > 0 . This leads us to conclude, from (4.37), that the physically acceptable condition for x is  \nx > ∣ ∣ ∣ l Pl √ β (2 l -( α -2 k + n )) ∣ ∣ ∣ . (4.38)  \nFor the ground state n = 0 , the floor function ⌊ n 2 ⌋ restricts the summation in (4.36) to k = 0 , which simplifies condition (4.38) to x > l Pl √ β α . Similarly, for n = 1 , the condition becomes x > l Pl √ β ( α +1) , with l = 0 in both cases. This shows that the minimal uncertainty approach introduces a minimum radius on the 2-sphere, which is related to the quantum parameter β . Notice that this is a universal bound, meaning that it is independent of the mass of the black hole and purely a quantum effect, which vanishes for β → 0 . This can also be seen as a weaker argument for the resolution of the black hole singularity in this approach. On the other hand, due to condition (4.38), the domain of x is restricted between  \n∣ ∣ ∣ l Pl √ β (2 l -( α -2 k + n )) ∣ ∣ ∣ < x < R GUP s ( n ) , (4.39)  \nwhich clearly differs from the domain in the standard case.  \nMoreover, it is noteworthy that condition (4.38) ensures the convergence of the hypergeometric functions appearing in (4.36). This guarantees that the wave function, derived from the quantization based on the minimal uncertainty approach, is wellbehaved within the domain of interest (4.39). Figure 2 shows the plots of the wave function squared for various values of n .  \nBy comparing the wave function obtained in the standard quantization scheme (3.15) with the one derived from the minimal uncertainty approach (4.36), we observe that they behave differently both at the horizon and in the region where the classical singularity of the black hole is located. For instance, in the standard case, the classical singularity is found at x = -R s ( n ) / 2 , where the wave function Ψ n ( x ) diverges. In contrast, due to condition (4.38), which imposes a minimum radius for the black hole, x cannot reach the region where the classical singularity resides, and therefore the wave function Ψ GUP n remains finite in that region and is square-integrable (see Table 1). This result suggests that the singularity is resolved in the quantization approach implemented in this section. A strong indication that the singularity has been resolved is demonstrating that the expectation value of the Kretschmann scalar is finite within the interior of the black hole.', '4.3 Expectation value of Kretschmann scalar': 'As in the standard case, to determine whether the singularity is truly resolved in the minimal uncertainty approach, we must calculate the expectation value of the Kretschmann scalar for the states characterizing the interior of the modified black hole. Given the complex form of the wave function (4.36) for each state characterized by the quantum number n , we perform the calculation of ⟨ K ⟩ n in (3.16) numerically. Table 1 shows the expected values for various n .  \nTable 1 . Expected values of the Kretschmann scalar for different quantum numbers n .  \nClearly, the expected value of the Kretschmann scalar is finite inside the black hole for all the states that characterize it, which is not the case in the standard approach, where the expected value of the Kretschmann scalar is undefined due to the presence of the singularity. This result allows us to argue that the classical singularity that once  \nFigure 2 . Graphical representation of the wave function (4.36), obtained using the minimal uncertainty quantization scheme, for different values of n .  \n<!-- image -->  \n(a) Square wave function for  \nn  \n= 0  \n0.04  \n0.03  \n0.02  \n0.01  \n0.00  \n2  \nP  \nU  \nG  \n2  \nΨ  \n0  \n1  \n2  \n3  \n4  \nx  \n- (c) Square wave function for n = 2  \nexisted inside the black hole has been resolved as a direct consequence of implementing quantization based on the minimal uncertainty approach.  \nIn a previous work [40], the interior of a Schwarzschild black hole was quantized using the minimal uncertainty approach appropriate for the Ashtekar-Barbero connection variables. Some of these results align with those found in this work, derived from a different formulation, yet physically consistent with each other.', '5 Discussion and conclusion': "In this work, we explore the interior of a black hole through the perspective of two quantization approaches. We begin with the Wheeler-DeWitt equation (2.7), which describes the dynamics within the Schwarzschild black hole's interior. By rescaling the wave function and the radial coordinate, as shown in (3.1), we reformulate this equation as the eigenvalue equation of a quantum linear harmonic oscillator (3.3) [42].  \nn  \n=  \n2,  \nl  \nPl  \n2  \nP  \nU  \nG  \n1  \nΨ  \n=  \n1,  \nR  \ns  \n0.05  \n0.04  \n0.03  \n0.02  \n0.01  \n0.00  \n0.0  \n0.5  \n1.0  \n1.5  \n2.0  \n2.5  \n3.0  \n3.5  \nx  \n(b) Square wave function for  \nn  \n= 1  \nPl  \n, G  \n=  \n1  \n=  \n4.7  \nl  \nn  \n=  \n1,  \nl  \nPl  \n=  \n1,  \nR  \ns  \n=  \n3.6  \nl  \nPl  \n, G  \n=  \n1  \nThe first quantization approach we apply is based on standard quantization, in which the usual commutation relation for a pair of canonical variables is satisfied. In this approach, we find that the wave function (3.15) is not square-integrable within the black hole, due to the fact that the wave function diverges in the region where the classical singularity is located. This result can be interpreted as a sign that the singularity still persists after the standard quantization process. Additionally, it is concluded that the expectation value of the Kretschmann scalar is undefined within the black hole's interior.  \nThe second quantization method we implement is based on the minimal uncertainty approach (4.3). Using this method, we find that the wave function (4.36), and therefore its squared modulus, remain finite throughout the entire interior of the black hole, implying that all states within it are square-integrable. To ensure the convergence of the wave function (4.36), we discover that the black hole's radius acquires a minimum value (4.38) that depends only on the quantum number n , which determines the possible states within the black hole, and the deformation parameter β , which defines the magnitude of minimal uncertainty effects. This minimum radius is universal as it does not depend on the black hole's mass.  \nIn order to investigate the fate of the singularity within the minimal uncertainty approach, we calculate the expectation value of the Kretschmann scalar (3.16). In this case, unlike in the standard quantization scheme, we find that the expectation value is well-defined and finite throughout the interior of the black hole (see Table 1), bounded by the domain (4.39). Ultimately, based on these results, we can conclude that both the finiteness of the wave function and the expectation value of the Kretschmann scalar indicate that the classical singularity is resolved and replaced by a minimum radius on the 2-sphere in the minimal uncertainty quantization scheme.  \nWe want to emphasize that the methods applied in this work, as well as the results obtained by treating the black hole as a harmonic oscillator, both in standard quantization and in the minimal uncertainty approach, are novel in the literature. This opens up a potential area of study that will be explored in future projects.", 'Acknowledgments': 'O. O and W. 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