Possibilities of Emerging Quantum-Conditioned Curvatures and Attenuating Space Singularity in Spacetime Surrounding Kerr Black Hole

1. Introduction

Over the last century, considerable research efforts have been made to establish a consensus between the principles of General Relativity (GR) and Quantum Mechanics (QM) [1,2,3]. Despite the attempts of various scientists to formulate alternative theories or to develop novel mathematical approaches that could facilitate the attainment of this long-standing scientific, cognitive, and philosophical aim, we are, unfortunately, still far from proposing an agreed-upon theory of quantum gravity [4,5,6]. A thorough examination conducted by one of the authors (AT) revealed that the reason both fundamental theories are orthogonal could be attributed to their mathematical formulations, which have prevented their unification over the last century [7,8,9,10,11,12,13,14,15,16]. Therefore, it was concluded that by implementing essential generalizations aimed at relaxing the various approximations that were imposed on both theories, it would be feasible to unify their fundamental principles. The notion that GR is predominant at large scales, in contrast to QM at minimal scales, seems to stem from the approximations that have been enforced. As the Einstein field equations (EFE) in GR can be fully constructed using the metric tensor, it was proposed that the metric tensor alone can be subject to geometric quantization. To accomplish this aim, it was crucial to adopt an ansatz that generalizes both theoretical frameworks. This is the main motivation for applying the geometric quantization approach [7,8,9,10,11,12,13,14,15,16]. In QM, it is presumed that finite gravitational fields are ought to be integrated, leading to the imposition of appropriate modifications. In this context, the Heisenberg uncertainty principle arises as the essential quantum theory that requires generalization and reformulation in a relativistic framework [17]. On the other hand, confining GR to Riemannian geometry is a presumption that results in approximations in the context of four-dimensional spacetime [18]. Riemannian geometry, in which quadratic restriction is assumed, is therefore a particular case of Finslerian geometry, in which such restriction is genuinely relaxed [19,20]. As a result, the proposal to replace Riemann with Finsler was naturally put forward [7,8,9,10,11,12,13,14,15,16]. This ansatz is associated with replacing manifold and tensor (Riemann) with manifold and structure (Finaler) [21,22]. By preserving all the mathematical properties of the tangent (Finsler) and the cotangent (Hamilton) structures, most notably the homogeneity of degree one in tangent quantities, momentum could be integrated on cotangent manifold. After verifying the validity of such transformations from Finsler to Hamilton geometry, it becomes possible to deduce the metric where quantum-mechanical ingredients are integrated via the Relativistic Generalized Uncertainty Principle (RGUP) [17]. Thereby, principles of QM are integrated into the metric obtained from Hamilton geometry. Its translation into Riemann geometry is achieved through further approximations, including the assumption that the line elements in both geometries are equivalent.

By deriving the fundamental metric tensor that integrates quantum components, one can straightforwardly reconstruct the entire theory of GR, thereby applying it to various phenomena, especially in cases where conventional GR faces substantial challenges. A thorough compilation of such implications is presented in recent publications [8,9,10,11,12,13,14,15,16]. To adhere to the scope of this manuscript, we highlight the implications related to black holes and the resulting effects on the spacetime curvatures and the nature of space singularities [23,24,25]. Such considerations are vital for a thorough exploration of the topic on intrinsic quantum curvatures and attenuating if not entirely regulating space singularity. This examination is carried out for the timelike geodesic congruence in the Kerr black hole, which is rotating, massive, non-charged, and axially symmetric. A similar analysis was performed in a homogeneous, isotropic, and spherically symmetric Schwarzschild, de Sitter–Schwarzschild and Friedmann–Lemaitre–Robertson–Walker (FLRW) metrics, as well as in a non-homogeneous Einstein–Gilbert–Straus (EGS) metric [8,9,10]. The intrinsic curvatures and space singularity in the vicinity of the Reissner-Nordström black hole were also examined as well [11]. Therefore, this investigation emphasizes the Kerr black hole and thus draws comparisons with other black holes.

The script is structured as follows. The formalism is detailed in Section 2. The geodesic equations, utilizing a quantum-mechanically revised metric tensor, are presented in Section 2.1. The timelike geodesic congruence with the conventional metric tensor $g_{\mu \nu }$ is discussed in Section 2.2, whereas the one with the quantized metric tensor ${\tilde{g}}_{\mu \nu }$ is covered in Section 2.3. The numerical results and discussions are thoroughly examined in Section 3. Finally, Section 4 is dedicated to the concluding remarks.

2. Formalism

As elaborated in the previous section, the notion of geometric quantization ansatz is based on the understanding that the fundamental metric tensor in GR conveys all essential information about spacetime and thereby the underlying geometry. For further details on the geometric quantization ansatz and its implications for various phenomena, one can refer to recent literature [7,8,9,10,11,12,13,14,15,16]. The derivation of the quantum-mechanically revisited metric tensor was introduced in ref. [7]. It was presumed that the quantized metric tensor could be expressed conformally in relation to the conventional one

$$\begin{array}{ccc}\hfill {\tilde{g}}_{\mu \nu }& \simeq & C(x,p)g_{\mu \nu }.\hfill \end{array}$$

(1)

The conformal coefficient $C(x,p)$ comprises all quantum-mechanical operators and quantities that are imposed on the revisited metric tensor. Although the approximations in deriving ${\tilde{g}}_{\mu \nu }$, Equation 1, a precise formulation of $C(x,p)$ still constitutes a mathematical challenge

$$\begin{array}{ccc}\hfill {\tilde{g}}_{\mu \nu }& =& \left({\varphi }^2\left(p_0\right)+2{\displaystyle \frac{\kappa }{{\left(p_0^0\right)}^2}}{F}^2\right)\left[1+{\displaystyle \frac{{\dot{p}}_0^{\alpha }{\dot{p}}_0^{\beta }}{{\mathcal{F}}^2}}\left(1+2\beta p_0^{\rho }{p}_{0\rho }\right)\right]g_{\mu \nu }\hfill \\ & +& \left[{\displaystyle \frac{d{x}_0^{\alpha }}{d{\zeta }^{\alpha }}}{\displaystyle \frac{d{x}_0^{\beta }}{d{\zeta }^{\beta }}}+\left(1+2\beta p_0^{\rho }{p}_{0\rho }\right){\displaystyle \frac{d{p}_0^{\alpha }}{d{\zeta }^{\alpha }}}{\displaystyle \frac{d{p}_0^{\beta }}{d{\zeta }^{\beta }}}\right]d_{\mu \nu },\hfill \end{array}$$

(2)

where the subscript 0 refers to auxiliary four-vector quantities that satisfy non-commutative relationships and $\mathcal{F}$ represents the maximum proper gravitational force that allows the quantum particles to achieve their maximum gravitational acceleration during their motion in the quantum-conditioned curvatures; sources of quantum gravity. The discovery of $\mathcal{F}$ as a new physical constant was reported in ref. [26], for example. It is apparent that $\mathcal{F}$ is linearly correlated with the maximum proper acceleration discovered by Caianiello [27,28,29], which was established on the basis of quantum corrections imposed on the spacetime metric through geometric phase space quantization [30]. The function $\varphi \left(p_0\right)$ is introduced through the RGUP ansatz [17], leading to a modification of the momentum operator as follows: $p_0^{\mu }=(1+\beta p_0^{\rho }{p}_{0\rho })p_0^{\mu }=\varphi \left(p_0\right)p_0^{\mu }$, where $\rho$ is a dummy index and $\beta$ is the RGUP parameter that can be determined either through observational or empirical methods [17,31]. Both $x_0$ and $p_0$ are auxiliary four-vectors for space and momentum, respectively, and they are canonically conjugate variables that under the general static metric $g^{\mu \nu }$ fulfill the relation $[x_0^{\mu },p_0^{\nu }]=i\hslash g^{\mu \nu }$. The quantity $\kappa$ is defined as $\kappa =\beta /{\left(p_0^0\right)}^2$. Furthermore, ${\dot{p}}_0^{\mu }=\partial p_0^{\mu }/\partial \xi$, where $\xi$ is a proper paramtrization that is imposed to ensure that the coordinates in Hamilton, Finsler, and Riemann manifolds are identical. It is obvious that ${\dot{p}}_0^{\alpha }{\dot{p}}_0^{\beta }$ represents the squared gravitational force acting on quantum particles with mass m.

The enduring mathematical challenge consists of reformulating the second line in Equation (2) in terms of $g_{\mu \nu }$, as is done with the first line. This issue has been explored in the literature, and for the sake of simplicity, it was presumed that Equation (2) would be truncated. Arguments for and against have been noted. As a result, we conclude that

$$\begin{array}{ccc}\hfill C(x,p)& =& \left({\varphi }^2\left(p_0\right)+2{\displaystyle \frac{\kappa }{{\left(p_0^0\right)}^2}}{F}^2\right)\left[1+{\displaystyle \frac{{\dot{p}}_0^{\mu }{\dot{p}}_0^{\nu }}{{\mathcal{F}}^2}}\left(1+2\beta p_0^{\rho }{p}_{0\rho }\right)\right],\hfill \end{array}$$

(3)

As stated in Equation (3), for instance, multiple quantum operators are included. The estimation, which is the measure of $C(x,p)$, is entirely contingent upon the evaluation of the various operators and quantities.

It is obvious that the RGUP ansatz [17,31] assumes a multi-dimensional generalization of the uncertainty principle, namely, auxiliary four-vector space and momentum, for example. Furthermore, a quantum operator representation was also discussed in literature [32]. Accordingly, the uncertainty in non-commutative quantities is related to quantum states and the operators are evaluated at the expectation value on the quantum state [33]. In curved spacetime, let us assume that,

$$\begin{array}{ccc}\hfill {\langle \widehat{g}\left(\widehat{x}\right)\rangle }_{\mu \nu }\equiv g_{\mu \nu }\left(\langle \widehat{x}\rangle \right),& & {\langle \widehat{g}\left(\widehat{p}\right)\rangle }_{\mu \nu }\equiv g_{\mu \nu }\left(\langle \widehat{p}\rangle \right).\hfill \end{array}$$

(4)

As noted in ref. [33], the metric ${\widehat{g}}_{\mu \nu }$, which is a tensor whose generic element corresponds to $g_{\mu \nu }$ evaluated at the expectation value of the quantum state, is viewed as being made up of operators that act on the state of the underlying quantum system. An analogous evaluation of metric tensors constructed from operators was also reported in ref. [34]. For the Kerr black hole, Section 2.2, the corresponding metric elements can be given as

$$\begin{array}{ccc}\hfill g_{\mu \nu }& =& \mathtt{diag}\left(-{\displaystyle \frac{\Sigma [r,\theta ]-r_{s}r}{\Sigma [r,\theta ]}},{\displaystyle \frac{\Sigma [r,\theta ]}{\Delta \left[r\right]}},\Sigma [r,\theta ],{\displaystyle \frac{\Sigma [r,\theta ](a^2+r^2)+r_{s}r{a}^2{sin}^2\left(\theta \right)}{\Sigma [r,\theta ]}}{sin}^2\left(\theta \right)\right),\hfill \end{array}$$

where $\Delta \left[r\right]=r^2+a^2-r_{s}r$, $\Sigma [r,\theta ]=r^2+a^2{cos}^2\left(\theta \right)$. Here, the spherical coordinates $(t,r,\theta ,\varphi )$ are utilized. The RGUP ansatz [17,31] appears to regard all components of space and momentum on an equal footing. This might be acceptable in terms of space, at least orthogonal coordinates. However, the equivalence of $p_t\equiv E$, $p_r$, $p_{\theta }$, and $p_{\varphi }$, and whether the function $\varphi (x,p)$ is uniformly applied to each, needs to be investigated elsewhere. In view of this discussion, one can conclude that the considerable challenge involves estimating the quantum operators and improving the RGUP ansatz. Until this issue is resolved, the approximate methods are inevitably applied.

The truncation imposed on Equation (2), regardless of the mathematical and physical limitations, appears to be unavoidable and, on the other hand, seems to facilitate the retention of the quantum revisions. As the proposed geometric quantization approach is exclusively based on the fundamental metric tensor, it is imperative to first confirm its accuracy and effectiveness, even if only approximately. Whatever the numerical intentions may be, the quantum operators likely undergo multiple approximations. In this context, at least, qualitative analyses can be undertaken by treating $C(x,p)$, the conformal coefficient, as a perturbative correction to $g_{\mu \nu }$, where $0<C(x,p)\le 1$ is evidently positive. Notice that the values assigned to $C(x,p)$ are exclusively applied by only in the contexts where the quantity $C(x,p)$ is found. The derivatives that contribute to various formulations, including affine corrections, Riemann and Ricci tensors, and scalars, are approached in an approximate quantum-mechanical manner. With this approximation, it is possible to gain valuable insights into the quantization of spacetime surrounding Kerr black hole. The conformal coefficient $C(x,p)$ plays the role of an independent parameter that signifies the impact of quantization on the emergence of quantum-conditioned curvatures, which are sources of additional gravity. This also allows for the evaluation of space singularity in Kerr black holes.

The geodesic equations derived from ${\tilde{g}}_{\mu \nu }$ are introduced in Section 2.1. A comparison with the conventional geodesic equations shows that the proposed quantization ansatz results in important contributions, causing substantial modifications in the related velocity fields and timelike geodesic congruence. The aim of this manuscript is to identify these modifications, particularly in the context of spacetime defined by the Kerr metric.

2.1. Geodesic Equations with Quantum-Mechanically Revisited Metric Tensor

For the fundamental metric tensor of conventional GR, $g_{\mu \nu }$, the proper time is formulated as $-c^{2}d{\tau }^2=d{s}^2=g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }$. Based on Equation (1), ${\tilde{g}}_{\mu \nu }$ is linearly related to $g_{\mu \nu }$, which results in the corresponding proper time being given by $-c^{2}d{\tilde{\tau }}^2=d{\tilde{s}}^2={\tilde{g}}_{\mu \nu }d{x}^{\mu }d{x}^{\nu }$. In geometrized units, we find that

$$\begin{array}{ccc}\hfill {\tilde{\tau }}_{ab}& =& {\int }_0^1\sqrt{-{\tilde{g}}_{\mu \nu }{\displaystyle \frac{d{x}^{\mu }}{d\zeta }}{\displaystyle \frac{d{x}^{\nu }}{d\zeta }}}={\int }_0^1\mathcal{L}\left({\displaystyle \frac{d{x}^{\mu }}{d\zeta }},x^{\mu }\right)d\zeta ,\hfill \end{array}$$

(5)

where $\mathcal{L}$ is the Lagrangian. For maximal timelike geodesics, one can directly apply the Euler–Lagrange equation

$$\begin{array}{ccc}\hfill -{\displaystyle \frac{d}{d\zeta }}{\displaystyle \frac{\partial \mathcal{L}}{\partial (d{x}^{\gamma }/d\zeta )}}+{\displaystyle \frac{\partial \mathcal{L}}{\partial x^{\gamma }}}& =& 0,\hfill \end{array}$$

(6)

resulting in a set of equations that are equivalent to the geodesic ones. On the other hand, this ansatz seems to avoid the calculations associated with Christoffel symbols. Accordingly, one derives the quantized geodesic equations as

$$\begin{array}{ccc}\hfill {\displaystyle \frac{d^2{x}^{\mu }}{d{\tau }^2}}+{\tilde{\Gamma }}_{\delta \nu }^{\mu }{\displaystyle \frac{d{x}^{\delta }}{d\tau }}{\displaystyle \frac{d{x}^{\nu }}{d\tau }}& =& -{\displaystyle \frac{g_{\delta \nu }}{2{\tilde{g}}_{\mu \gamma }}}{F}_{,\gamma }^{2}C(x,p){\displaystyle \frac{d{x}^{\delta }}{d\tau }}{\displaystyle \frac{d{x}^{\nu }}{d\tau }},\hfill \end{array}$$

(7)

where $C(x,p)$ is the conformal coefficient defined in Equation (3), while $F^2$ represents the structure on either tangent (Finsler) or cotangent (Hamilton) manifolds. It is evident that the proposed geometric quantization introduces an extra term that integrates the quantum operators and quantities imposed. Additionally, the quantized affine connections, ${\tilde{\Gamma }}_{\delta \nu }^{\mu }$, add another quantum term,

$$\begin{array}{c}\hfill {\tilde{\Gamma }}_{\delta \nu }^{\mu }={\Gamma }_{\delta \nu }^{\mu }+{\displaystyle \frac{1}{2}}{\displaystyle \frac{F_{,\gamma }^2}{C(x,p)}}\left({\delta }_{\nu }^{\mu }+{\delta }_{\delta }^{\mu }-g^{\mu \gamma }{g}_{\delta \nu }\right).\end{array}$$

(8)

Regarding the structure, let us consider the most basic metric, namely the Klein metric, which is defined on the complement of a fixed quadric within a projective space, utilizing a cross-ratio [35]. In non-Euclidean geometry, it is fundamental and provides a framework for hyperbolic and elliptic geometry. Accordingly, the comma derivative reads

$$\begin{array}{ccc}\hfill F_{,\gamma }^2& =& 2{\displaystyle \frac{〈x_0^{\mu }·p_0^{\nu }〉}{{\left[{\left(x_0^{\mu }\right)}^2-1\right]}^2}}\left\{\left[1-{\left(x_0^{\mu }\right)}^2\right]p_0^{\nu }-x_0^{\mu }〈x_0^{\mu }·p_0^{\nu }〉\right\}.\hfill \end{array}$$

(9)

The present study, both analytical and numerical, examines the contributions of the proposed geometric quantization ansatz. It evaluates their effects on the space singularity of Kerr black hole and the emergence of intrinsic curvatures at quantum scales. In particular, we analyze the additional contributions to the timelike geodesic congruence in the spacetime surrounding the Kerr black hole. To achieve this, a comprehensive comparison between the conventional and quantized timelike geodesic congruence is provided. Section 2.2 introduces the timelike geodesic congruence utilizing the conventional metric tensor $g_{\mu \nu }$.

2.2. Timelike Geodesic Congruence with $g_{\mu \nu }$

The solution of EFE which describes the geometry of spacetime characterized by rotating, massive, non-charged, axially symmetric black hole, the Kerr metric, in Boyer–Lindquist coordinates $(t,r,\theta ,\varphi )$ is given as

$$\begin{array}{ccc}\hfill d{s}^2& =& -{\displaystyle \frac{\Sigma [r,\theta ]-r_{s}r}{\Sigma [r,\theta ]}}{c}^{2}d{t}^2+{\displaystyle \frac{\Sigma [r,\theta ]}{\Delta \left[r\right]}}d{r}^2+\Sigma [r,\theta ]d{\theta }^2\hfill \\ & & +{\displaystyle \frac{\Sigma [r,\theta ](a^2+r^2)+r_{s}r{a}^2{sin}^2\left(\theta \right)}{\Sigma [r,\theta ]}}{sin}^2\left(\theta \right)d{\varphi }^2-{\displaystyle \frac{2r_{s}ra{sin}^2\left(\theta \right)}{\Sigma [r,\theta ]}}cdtd\varphi ,\hfill \end{array}$$

(10)

where $a=J/M$ is length scale, with M is the mass and J is the angular momentum that characterizes the rotation. Given the natural units, that are $G=c=1$, for instance, all occurrences of $2M$ relate to the Schwarzschild radius, $r_s=2GM/c^2=2M$. The Kerr black hole is characterized by its angular momentum, which exhibits distinct features when compared to non-rotating black holes, including frame-dragging and a quasispherical event horizon. Furthermore, the existence of ergosphere, which is the region surrounding the black hole where spacetime is dragged along with rotation of the black hole, comes up with additional features. Objects within the ergosphere are compelled to rotate in the same direction as the black hole, allowing for the extraction of energy from the rotation of the black hole.

In co-moving coordinate, i.e., vanishing $u^{\theta }$ and $u^{\varphi }$, velocity components in $\theta$ and $\varphi$ directions, the first geodesic equation is derived as

$$\begin{array}{ccc}\hfill {\displaystyle \frac{d{u}^t}{d\tau }}& =& -{\displaystyle \frac{2Mr(a^2+r^2)}{-2M{r}^2\left(a^2+r^2\right)+r^2\left[r^3+a^3\left(2M+r\right)\right]}}{u}^t{u}^r,\hfill \end{array}$$

(11)

where $u^r$ is the r-component of velocity. Here, for the sake of simplicity, it was assumed that $\theta =\pi /2$. Then, Equation (11) can be rewritten as

$$\begin{array}{ccc}\hfill {\displaystyle \frac{d{u}^t}{u^t}}& =& -{\displaystyle \frac{2Mr(a^2+r^2)}{-2M{r}^2\left(a^2+r^2\right)+r^2\left[r^3+a^3\left(2M+r\right)\right]}}.\hfill \end{array}$$

(12)

Obviously, the integration wrt r results in

$$\begin{array}{ccc}\hfill u^t& =& exp\left\{2\left[{\displaystyle \frac{M}{r}}-{\displaystyle \frac{M^2}{a^2}}\left({\displaystyle \frac{2Marctan\left(\frac{r-M}{\sqrt{a^2-M^2}}\right)}{\sqrt{a^2-M^2}}}-2ln\left(r\right)+ln\left(\Delta \left[r\right]\right)\right)\right]\right\}.\hfill \end{array}$$

(13)

Now with the normalization condition $u^t{u}^r{g}_{tr}=-1$, the second velocity component, $u^r$, can be derived as well

$$\begin{array}{ccc}\hfill u^r& =& -{\displaystyle \frac{1}{r^2}}{\left[-r^2+e^{4{\displaystyle \frac{M}{r}}+{\displaystyle \frac{8M^{3}arctan\left(\frac{M-r}{g(a,M)}\right)}{a^{2}g(a,M)}}+r^{-8{\displaystyle \frac{M^2}{a^2}}}\Delta {\left[r\right]}^{4{\displaystyle \frac{M^2}{a^2}}+1}\left(r-2M\right)}\right]}^{1/2},\hfill \end{array}$$

(14)

where $g(a,M)=\sqrt{(a-M)(a+M)}$.

The covariant derivatives of the velocity fields express the geodesic congruence expansion, namely $\Theta \left[r\right]=u_{;\alpha }^{\alpha }=u_{,\alpha }^{\alpha }+u^{\sigma }{\Gamma }_{\sigma \alpha }^{\alpha }$, so that

$$\begin{array}{ccc}\hfill \Theta \left[r\right]& =& {\displaystyle \frac{1}{2\Delta \left[r\right]F[r,\theta ]}}\left\{r^{-3-8{\displaystyle \frac{M^2}{a^2}}}\left[-r^{2-8{\displaystyle \frac{M^2}{a^2}}}\Delta \left[r\right]\right]\left(2\Delta \left[r\right]+r{\Delta }^{\prime }\left[r\right]\right)-{\displaystyle \frac{\Delta \left[r\right]{\Delta }^{\prime }\left[r\right]}{{(2M-r)}^{-1}{r}^{-2}}}\right.\hfill \\ & & \left.+{\displaystyle \frac{G[r,\theta ]}{\Delta {\left[r\right]}^{-4\frac{M^2}{a^2}}}}\left[2\Delta \left[r\right]\left(r^2(6M^2-5Mr+r^2)+a^2(4M^2-3Mr+r^2)\right)\right]\right\}\hfill \\ & +& {\displaystyle \frac{1}{2r^2}}F[r,\theta ]\left[{\displaystyle \frac{1}{r}}+{\displaystyle \frac{(a^2+3r^2)(2Mr-r^2)}{2M{r}^2(a^2+r^2)-[r^3+a^2(M-r)-a^{2}Mcos\left(2\theta \right)]r^2}}-{\displaystyle \frac{{\Delta }^{\prime }\left[r\right]}{\Delta \left[r\right]}}+{\displaystyle \frac{{\Sigma }^{(1,0)}[r,\theta ]}{\Sigma [r,\theta ]}}\right.\hfill \\ & & \left.+{\displaystyle \frac{2Mr\left(a^2+r^2\right)\left[-\Sigma [r,\theta ]+r{\Sigma }^{(1,0)}[r,\theta ]\right]}{\Sigma [r,\theta ]\left[-2M{r}^2\left(a^2+r^2\right)+\left(a^{2}r+r^3+2a^{2}M{sin}^2\left(\theta \right)\right)\Sigma [r,\theta ]\right]}}\right]\hfill \\ & -& {\displaystyle \frac{2}{r^3}}F[r,\theta ],\hfill \end{array}$$

(15)

where

$$\begin{array}{ccc}\hfill G[r,\theta ]& =& exp\left[4{\displaystyle \frac{M}{r}}+{\displaystyle \frac{8M^{3}arctan\left(\frac{M-r}{g(a,M)}\right)}{a^{2}g(a,M)}}\right],\hfill \end{array}$$

(16)

$$\begin{array}{ccc}\hfill F[r,\theta ]& =& {\left\{r\Delta \left[r\right]\left[-r+G[r,\theta ]r^{-8{\displaystyle \frac{M^2}{a^2}}}(r-2M){[a^2+r(r-2M)]}^{4{\displaystyle \frac{M^2}{a^2}}}\right]\right\}}^{1/2}.\hfill \end{array}$$

(17)

Now, the evolution of the timelike geodesic congruence expansion, Equation (15), can be derived straightforwardly. The expression obtained from $d\Theta \left[r\right]/d\tau =(d\Theta \left[r\right]/dr)\left(u^r\right)$ is very long to be displayed in the main text. Appendix A is devoted to.

The timelike geodesic congruence and its evolution obtained from the proposed quantization approach, ${\tilde{g}}_{\mu \nu }$ are presented in Section 2.3.

2.3. Timelike Geodesic Congruence with ${\tilde{g}}_{\mu \nu }$

As outlined in Section 2.2, for ${\tilde{g}}_{\mu \nu }$, the first and second velocities, respectively, read

$$\begin{array}{ccc}\hfill {\tilde{u}}^t[r,C(x,p)]& =& exp\left\{{\displaystyle \frac{M}{r}}-{\displaystyle \frac{F_{,\gamma }^2}{2C(x,p)}}+{\displaystyle \frac{M^2}{a^2}}\left[-2ln\left(r\right)+ln(\Delta \left[r\right])+2M{\displaystyle \frac{g(a,M)arctan\left(\frac{M-r}{g(a,M)}\right)}{g^2(a,M)}}\right]\right\},\hfill \\ \hfill {\tilde{u}}^r[r,C(x,p)]& =& {\displaystyle \frac{\sqrt{\Delta \left[r\right]}{e}^{-\frac{F_{,\gamma }^2}{2C(x,p)}}{r}^{-2\frac{M^2}{a^2}}}{\sqrt{C(x,p)}\Sigma [r,\theta ]}}\left\{-{\displaystyle \frac{r^{2+4\frac{M^2}{a^2}}}{e^{-\frac{F_{,\gamma }^2}{C(x,p)}}}}-2C(x,p)e^{2{\displaystyle \frac{M}{r}}+4M^3{\displaystyle \frac{\sqrt{a^2-M^2}arctan\left(\frac{M-r}{g(a,M)}\right)}{a^2{g}^2(a,M)}}}Mr\Delta {\left[r\right]}^{2{\displaystyle \frac{M^2}{a^2}}}\right.\hfill \\ & & +C(x,p)e^{2{\displaystyle \frac{M}{r}}+4M^3{\displaystyle \frac{\sqrt{a^2-M^2}arctan\left(\frac{M-r}{g(a,M)}\right)}{a^2{g}^2(a,M)}}}{r}^2\Delta {\left[r\right]}^{2{\displaystyle \frac{M^2}{a^2}}}-a^2{e}^{{\displaystyle \frac{F_{,\gamma }^2}{C(x,p)}}}{r}^{4{\displaystyle \frac{M}{r}}}{cos}^2\left(\theta \right)\hfill \end{array}$$

(18)

$$\begin{array}{ccc}& & {\left.+a^{2}C(x,p)e^{2{\displaystyle \frac{M}{r}}+4M^3{\displaystyle \frac{\sqrt{a^2-M^2}arctan\left(\frac{M-r}{g(a,M)}\right)}{a^2{g}^2(a,M)}}}\Delta {\left[r\right]}^{2{\displaystyle \frac{M^2}{a^2}}}{cos}^2\left(\theta \right)\right\}}^{1/2}.\hfill \end{array}$$

(19)

Because of the linearity expressed in Equation (1), the revisited timelike geodesic congruence expansion can be derived from

$$\begin{array}{ccc}\hfill \tilde{\Theta }[r,C(x,p)]& =& {\tilde{u}}^{\alpha }{[r,C(x,p)]}_{;\alpha }={\tilde{u}}^{\alpha }{[r,C(x,p)]}_{,\alpha }+u^{\sigma }[r,C(x,p)]{\tilde{\Gamma }}_{\sigma \alpha }^{\alpha }.\hfill \end{array}$$

(20)

This results in

$$\begin{array}{ccc}\hfill \tilde{\Theta }[r,C(x,p)]& =& {\displaystyle \frac{\sqrt{\Delta \left[r\right]}{e}^{-\frac{F_{,\gamma }^2}{2C(x,p)}}{r}^{-3-2\frac{M^2}{a^2}}}{2\sqrt{C(x,p)}\Delta \left[r\right]{\left\{-e^{\frac{F_{,\gamma }^2}{C(x,p)}}{r}^{2+4\frac{M^2}{a^2}}+C(x,p)e^{2\frac{M}{r}+\frac{4M^{3}arctan\left(\frac{M-r}{g(a,M)}\right)}{a^{2}g(a,M)}}r(r-2M)\Delta {\left[r\right]}^{2\frac{M^2}{a^2}}\right\}}^{1/2}}}\hfill \\ & & \times \left\{-{\displaystyle \frac{1}{C(x,p)}}\left[e^{{\displaystyle \frac{F_{,\gamma }^2}{C(x,p)}}}{r}^{2+4{\displaystyle \frac{M^2}{a^2}}}\left(2C(x,p)\left(2a^2+{\displaystyle \frac{r}{{\left(3r-5M\right)}^{-1}}}\right)+7F_{,\gamma }^2\Delta \left[r\right]\right)\right]\right.\hfill \\ & & +e^{2{\displaystyle \frac{M}{r}}+4{\displaystyle \frac{M^{3}arctan\left(\frac{M-r}{g(a,M)}\right)}{a^{2}g(a,M)}}}\Delta {\left[r\right]}^{2{\displaystyle \frac{M^2}{a^2}}}\left[{\displaystyle \frac{r^2}{{(r-2M)}^{-1}}}\left({\displaystyle \frac{C(x,p)}{{(6r-10M)}^{-1}}}+{\displaystyle \frac{7F_{,\gamma }^2}{{(r-2M)}^{-1}}}\right)\right.\hfill \\ & & \left.\left.+a^2\left({\displaystyle \frac{2C(x,p)}{{(M-r)}^{-2}}}+7F_{,\gamma }^2{r}^2(r-2M)\right)\right]\right\}\hfill \end{array}$$

(21)

Now, the evolution of the timelike geodesic congruence expansion can be straightforwardly derived from $d\tilde{\Theta }\left[r\right]/d\tau =(d\tilde{\Theta }\left[r\right]/dr)\left({\tilde{u}}^r\right)$. The resulting expression is also long to be given in the main text. All details are outlined in Appendix A.

Regarding the curvatures which emerge with the proposed geometric quantization ansatz, i.e., replacing $g_{\alpha \beta }$ by ${\tilde{g}}_{\alpha \beta }$, one needs to assess whether they are real, essential, and intrinsic or merely artifacts that appear in some coordinate systems while not in others? To this end, both Ricci and Kretschmann invariant scalars - among others - can be utilized. The polynomial scalars in GR are, as the name says, invariant whatever the coordinate system is being. Notice that the curvature of spacetime could be determined quantitatively despite the coordinate-system being taken into consideration. It is apparent that the Ricci scalar can be contracted from the Ricci curvature tensor, which in turn can be contracted from the Riemann curvature tensor [15,16,21,22]. Therefore, we find that

$$\begin{array}{ccc}\hfill R& =& g^{\beta \nu }\left(g_{\gamma }^{\mu }{R}_{\beta \mu \nu }^{\gamma }\right).\hfill \end{array}$$

(22)

The Kretschmann invariant scalar, on the other hand, is defined as the quadratic polynomial invariant composed of the sum of the squares of the Riemann curvature tensor components [36,37,38]

$$\begin{array}{ccc}\hfill K& =& R^{\gamma \beta \mu \nu }{R}_{\gamma \beta \mu \nu }\hfill \end{array}$$

(23)

where $R^{\alpha \beta \mu \nu }$ and $R_{\alpha \beta \mu \nu }$ are the Riemann curvature tensors.

It is straightforward to confirm that the last two expressions (23) and (22) are obviously valid for the quantized metric tensor, Equation (1), i.e., both Ricci and Kretschmann scalar is invariant under the proposed conformal transformation so that

$$\begin{array}{ccc}\hfill \tilde{R}& =& {\tilde{g}}^{\beta \nu }\left({\tilde{g}}_{\gamma }^{\mu }{\tilde{R}}_{\beta \mu \nu }^{\gamma }\right).\hfill \end{array}$$

(24)

$$\begin{array}{ccc}\hfill \tilde{K}& =& {\tilde{R}}^{\gamma \beta \mu \nu }{\tilde{R}}_{\gamma \beta \mu \nu }.\hfill \end{array}$$

(25)

The derivation of expressions for R, K, $\tilde{R}$, and $\tilde{K}$ in relation to the Kerr metric is algebraically straightforward, although the derivation for K and $\tilde{K}$ is computationally intensive.

The numerical analyses and discussion of the results are elaborated in Section 3.

3. Numerical Results and Discussions

This section discusses the results of the evolution of the timelike geodesic congruence expansion in spacetime defined by the Kerr metric. Figure 1 illustrates a systematic comparison of the results for $d\Theta [r,C(x,p\left)\right]/d\tau$ and $d\tilde{\Theta }[r,C(x,p)]/d\tau$, which were derived using the conventional (left panel) and quantized metric tensor (right panel), respectively. These calculations are performed using arbitrary values for M, a, and $F_{,\gamma }^2$. These values have been chosen at random, but to ensure real evolution. It is observed that $d\Theta [r,C(x,p\left)\right]/d\tau$ approaches near zero at large radial distances r. As r decreases, the values of $d\Theta [r,C(x,p\left)\right]/d\tau$ decline rapidly. The negative sign indicates the presence of a space singularity. The slight increase to positive values before the prevailing negative results suggests a spherically asymmetric structure.

The results of $d\tilde{\Theta }[r,C(x,p)]/d\tau$ (right panel) exhibit a rich structure. At larger values of r, the vanishing evolution becomes dominant. It is only at very small $C(x,p)$, which indicates an almost vanishing metric tensor, that negative values begin to appear rapidly. For small r, a nonmonotonic behavior is observed, where the results gradually alternate between negative and positive values. The sign of the results is exclusively dictated by $C(x,p)$. According to Equation (1), as $C(x,p)\to 1$, the geometric quantization nearly ceases, resulting in ${\tilde{g}}_{\mu \nu }\equiv g_{\mu \nu }$. This phenomenon is - as anticipated - linked to a significant negative $d\tilde{\Theta }[r,C(x,p)]/d\tau$, similar to that of $d\Theta [r,C(x,p\left)\right]/d\tau$ (left panel). Conversely, a different trend is observed as $C(x,p)$ moves away from 1. The values of $d\Theta [r,C(x,p\left)\right]/d\tau$ jump to positive, indicating a controllable attenuation in the space singularity [23,24,25].

As can be concluded from Equation (1), the proposed quantization ansatz retains all the assumptions of conventional GR. This suggests that $d\tilde{\Theta }[r,C(x,p)]/d\tau$ apparently combines results from $d\Theta [r,C(x,p\left)\right]/d\tau$ with an additional contribution that only appears at quantum scales. The latter can be clearly deduced when $d\Theta [r,C(x,p\left)\right]/d\tau$ is subtracted from $d\tilde{\Theta }[r,C(x,p)]/d\tau$. The relevant results are depicted in Figure 2. Expectedly, we find that almost all finite values of $C(x,p)$, with the exception of very small values that correspond to a vanishing metric tensor, change the sign of $d\tilde{\Theta }[r,C(x,p)]/d\tau$ to positive. Thus, we conclude that the proposed geometric quantization offers a mechanism to control the space singularity in Kerr black hole.

For a better comparison, Figure 3 merges the three varieties of results. It is noted that at large r, the curvature, whether it is conventional or quantized, disappears. Lowering r results in the emergence of finite curvatures. The results obtained from the proposed quantization approach demonstrate the interplay between pure quantum and pure classical phenomena.

Figure 3 graphically demonstrates the emergence of quantum-conditioned curvatures resulting from the proposed quantization. It is now legitimate to question their intrinsic nature. One might argue that these curvatures were simply artifacts found within certain coordinate systems. An ultimate assessment of this matter is to verify whether they are preserved after various coordinate transformations. On the other hand, such an evaluation would not be entirely thorough, due to the fact that, whatever we have discovered in coordinate systems and performed transformations across all of them, there is no mathematical or physical evidence indicating that there are no other coordinate systems yet to be discovered. Thus, the ultimate evaluation would involve the invariant scalars. We have discussed two of them in the preceding section. Let us exclude the Ricci scalar, as its value is perpetually vanishing in a vacuum background. Figure 3 illustrates the results from the Kretschmann invariant scalar.

As depicted in Figure 3, a large r is associated with vanishing curvatures. In this region, the Kretschmann invariant scalar, Figure 4, is correspondingly vanishing as well. With a reduction in r, the positive values of K increase substantially, indicating that the conventional curvatures are indeed real, essential, and intrinsic. The behavior of $\tilde{K}$ is notably structured. We observe that $\tilde{K}$ remains positive throughout. As $C(x,p)$ increases, there is a large positive increase even at smaller values of r.

As shown in Figure 2, the pure quantum contributions can be inferred by subtracting K from $\tilde{K}$. The results in this case are non-monotonic. Moreover, large r is marked by the absence of quantum curvatures. There is a noteworthy relationship between r and $C(x,p)$. When $C(x,p)$ is small, a reduction in r results in a large negative Kretschmann scalar, indicating the presence of negative quantum curvatures, while an increase in $C(x,p)$ changes this to positive, thus indicating positive curvatures.

Let us first discuss negative and positive curvatures. Negative curvatures found in black holes reveal the formation of a singularity. On the other hand, positive curvatures are directly related to spacelike singularities, which are defined by trapped surfaces in spacetime. These surfaces occur when gravitational forces modify the path of light, creating focal points that converge the light rays and stop them from escaping. Such a trapped surface signifies the existence of a singularity, where the gravitational field reaches infinite strength, resulting in the disruption of spacetime itself. The finite Kretschmann scalar holds significance not merely in indicating the types of curvature. Instead, it reflects the nature of curvature, particularly its intrinsic characteristics. Consequently, we conclude that the resulting quantum curvatures are indeed real, essential, and intrinsic. They do not represent artifacts that can be removed in a particular coordinate system.

4. Conclusions

The analysis of the expansion of the timelike geodesic congruence in the Kerr metric is conducted using both the conventional $g_{\mu \nu }$ and quantized metric tensor ${\tilde{g}}_{\mu \nu }$. A systematic numerical examination of its evolution is carried out for both versions of the metric tensor. It is important to emphasize that the proposed geometric quantization ansatz deliberately broadens the scope of conventional GR to encompass low (quantum) scales, thereby maintaining the fundamental postulates of Einsteinian GR. Our focus is directed towards the gravitational field associated with a rotating, massive, non-charged, and axially symmetric Kerr black hole.

We would like to first emphasize that the numerical results presented are likely reliant on the approximate values assigned to M (mass) and $a=J/M$ (length scale) of a Kerr black hole, where J represents its angular momentum, and $F_{,\gamma }^2$, the derivative of the tangent or cotangent structure, of the proposed quantization approach. It is found that as the radial distance r is reduced, an extremely large evolution of the geodesic congruence expansion and thus curvatures is achieved. The proposed geometric quantization ansatz, qualitatively estimated by the conformal coefficient that relates ${\tilde{g}}_{\mu \nu }$ to $g_{\mu \nu }$, contributes significantly to the quantum-conditioned curvatures that emerge at quantum (low) scales. For instance, the sign of the evolution of the quantum-conditioned curvatures abruptly changes from negative to positive as the conformal coefficient increases.

To evaluate the intrinsic nature of the quantum-conditioned curvatures, the Kretschmann invariant scalar is utilized, which yields non-monotonic results for pure quantum contributions. For vanishing quantization, or unity conformal coefficient, the Kretschmann invariant scalar indicates positive curvatures, which then reverses to a negative sign as the conformal coefficient decreases. This seemingly manifests the ability to regulate space singularity through the proposed quantization. Nevertheless, the crucial function of the Kretschmann invariant scalar is to evaluate whether the curvatures that arise at low (quantum) scales are artifacts of a certain coordinate system or are genuinely intrinsic eigencurvatures. The finite Kretschmann scalar obtained appears to signify the characteristics of these distinguished curvatures that have emerged. We conclude that the quantum-conditioned curvatures in the gravitational field of a rotating, massive, non-charged, and axially symmetric Kerr black hole are indeed real, essential, and intrinsic. They do not merely represent removable artifacts within any coordinate system, either known or to yet to be discovered.