Prospecting Black Hole Thermodynamics with Fractional Quantum Mechanics

Black hole (BH) physics constitutes an enthusiastic research domain for this century. On the one hand, gravitational waves resonating from colliding BHs have been measured Abbott and are now a regular astronomical probing tool. Outstandingly, astrophysicists have also captured the first-ever image of a BH at the center of galaxy M87 Aki . Furthermore, there is a collection of additional observational data Event hinting at the presence of an event horizon for BHs. In general, BH candidates may be stellar-mass objects in a X-ray binary system or supermassive at the center of typical galaxies Ritter ; Rees ; Stellar . All these achievements have been recently conveyed and recognized within the 2020 Nobel prize in Physics Nobel .

On the other hand, innovative work on BH physics in the past forty years or so have also brought us to the forefront of unanticipated research directions. In the early 70’s of the last century, Bekenstein proposed that a BH entropy is proportional to its horizon area Bec . Then, the evaporation of BHs (through the seminal use of quantum fields in curved spaces) was formulated by Hawking Hawking1 . Testing this property observationally has been recently appraised, bearing promising results GW-BH-S . Thereafter, physicists realized an intimate connection between geometrical horizons, thermodynamic temperature and quantum mechanics Fulling . Moreover, it was advanced Bekenstein that the BH horizon area may be quantized, and the corresponding eigenvalues would be given by

$\displaystyle A_{n}=\gamma L_{\text{P}}^{2}n,~{}~{}~{}~{}n=1,2,3,...~{},$

(1)

where $\gamma$ is a dimensionless constant of order one and $L_{\text{P}}=\sqrt{G}$ is the Planck length. The literature has since then been enriched with contributions strengthening in favor of the area spectrum (1), including information-theoretic considerations Area1 , ranging from string theory arguments Area2 to the periodicity of time Area3 , plus e.g., a Hamiltonian quantization of a dust collapse Area4 .

Within the viewpoint outlined in the previous paragraph, we suggest a complementary direction of exploration. Let us be more specific. Recently, a set of pertinent arguments and results have been put forward to apply fractional calculus par1 ; par2 ; par3 ; par4 in quantum physics. Such framework is known as Fractional Quantum Mechanics (FQM), see e.g. Arxiv . Fractional calculus has been embraced mainly within the last century. In essence, it follows from extending the meaning of derivatives to the case where the order is any number, i.e., irrational, fractional or complex. Its peculiar interest notwithstanding, the obstacles have been fruitfully addressed. In particular, it is currently known that non-integer order systems can describe the dynamical behavior of specific classes of materials and processes over different time and frequency scales. Fractional calculus has assisted in scattering theory, diffusion, probability, potential theory and elasticity. Therefore, it was only sensible to embrace and explore it within quantum physics.

FQM was built upon Feynman and Hibbs Bar1 assertion about quantum mechanics and path integrals. Moreover, Nelson Bar2 clarified about classical Brownian motion and quantum mechanical features, whereas Abbott and Wise Bar3 determined that in $1$-dimensional settings, the fractal (or Hausdorff) dimension of those paths is 2. The essential motivation for FQM emerges in that if we restrict the path integral (Feynman) description of quantum mechanics to Brownian paths only, it will be challenging to explain a few other pertinent quantum phenomena Laskin1 . Such difficulties have led to consider a generalization of the Feynman path integral, specifically by replacing the Gaussian probability distribution by Lévy’s Levy ; The Hausdorff dimension of the Lévy path is then equal to the fractional parameter $\alpha$.

Extended versions of the Schrödinger equation (SE) are then retrieved, namely from such broader path integral. More concretely, by including non-Brownian trajectories in the path integral formalism of quantum physics, either the space-fractional space , time-fractional time , and space-time-fractional space-time versions of the standard SE can be elaborated. Let us be precise and clarify that it is in space-fractional quantum mechanics, where the Feynman path integral method is modified such that the Gaussian probability distribution is replaced by Lévy’s Levy , that we obtain the following and broadly used modified SE. More particularly, starting from the Hamiltonian

$\displaystyle H=\frac{\mathbf{p}^{2}}{2m}+V(\mathbf{r},t),$

(2)

a generalization Arxiv ; Laskin1 ; space is produced by means of

$\displaystyle H_{\alpha}(\mathbf{p},\mathbf{r}):=D_{\alpha}|\mathbf{p}|^{\alpha}+V(\mathbf{r}),\hskip 28.45274pt1<\alpha\leq 2,$

(3)

in which $D_{\alpha}$ is a coefficient carrying dimension $[D_{\alpha}]={\rm erg}^{1-\alpha}{\rm cm}^{\alpha}{\rm sec}^{-\alpha}$. The parameter $\alpha$ is known as Lévy’s fractional parameter and is associated to the concept of Lévy path and its induced fractal dimension. In the Feynman path integral, the measure is generated by the process of the Brownian motion, and the path’s corresponding (fractal) dimension is $d^{\text{(Feynman)}}_{\text{fractal}}=2$, as we have remarked. The Lévy path integral leading to a SE would lead to a fractal dimension as $d_{\text{fractal}}^{\text{(L\'{e}vy)}}=\alpha$. For more details, please browse through, e.g., Rie ; par1 ; par2 ; par3 ; Arxiv ; Laskin1 ; space and other references therein.

Therefore, choosing a space representation with ${\hat{\mathbf{p}}}\rightarrow-i\hbar\nabla$ and ${\hat{\mathbf{r}}}\rightarrow{\mathbf{r}}$, we obtain the space-fractional SE

$$i\hbar\frac{\partial\psi(\mathbf{r},t)}{\partial t}=D_{\alpha}(-\hbar^{2}\Delta)^{\alpha/2}\psi(\mathbf{r},t)+V(\mathbf{r},t)\psi(\mathbf{r},t).$$

(4)

In the above fractional SE, the fractional Riesz derivative Rie ; Arxiv ; Kilbas , $(-\hbar^{2}\Delta)^{\alpha/2}$, is defined in terms of the Fourier transformation $\mathcal{F}$

$$\begin{array}[]{cc}(-\hbar^{2}\Delta)^{\alpha/2}\psi(\mathbf{r},t)=\mathcal{F}^{-1}|\mathbf{p}|^{\alpha}\mathcal{F}\psi(\mathbf{r},t)\\ \\ =\displaystyle\frac{1}{(2\pi\hbar)^{3}}\int d^{3}pe^{i\frac{\mathbf{p}\cdot\mathbf{r}}{\hbar}}|\mathbf{p}|^{\alpha}\int e^{-i\frac{\mathbf{p}\cdot\mathbf{r}^{\prime}}{\hbar}}\psi(\mathbf{r}^{\prime},t)d^{3}r^{\prime}.\end{array}$$

(5)

The infinite-well example was one of the first solutions of space-fractional SE, which Laskin solved Laskin1 . Despite its simplicity, this problem is critical since it is the prototype of a quantum detector with internal degrees of freedom.

In what concerns the time-fractional SE, the time evolution is given by the Caputo fractional derivative Caputo . In this case, the associated Hamiltonian is non-Hermitian and not local in time. The space-time fractional SE has been introduced by Wang and Xu space-time , where they employed a combination of space and time-fractional models to establish their equation. The space-time fractional SE is a generalized version of the equation (4):

$$\hbar_{\beta}i^{\beta}\partial_{t}^{\beta}\psi(\mathbf{r},t)=\left[D_{\alpha,\beta}(-\hbar_{\beta}^{2}\Delta)^{\alpha/2}+V(\mathbf{r},t)\right]\psi(\mathbf{r},t),$$

(6)

where $1<\alpha\leq 2,~{}0<\beta\leq 1$, $\hbar_{\beta}$ and $D_{\alpha,\beta}$ are two scale coefficients with physical dimensions $[\hbar_{\beta}]={\rm erg}.{\rm sec}^{\beta}$ and $[D_{\alpha,\beta}]={\rm erg}^{1-\alpha}.{\rm cm}^{\alpha}.{\rm sec}^{-\alpha\beta}$. In addition, $\partial_{t}^{\beta}$ denotes the left Caputo fractional derivative C67 of order $\beta$:

$$\partial_{t}^{\beta}f(t)=\frac{1}{\Gamma(1-\beta)}\int_{0}^{t}d\tau\frac{\dot{f}(\tau)}{(t-\tau)^{\beta}},$$

(7)

where $\dot{f}(\tau)=\frac{df(\tau)}{d\tau}$.

Proceeding towards a broader context, we recall that Wheeler Bar4 seminally suggested a foam structure for spacetime on the Planck scale. Hence, allowing that a fractal quality could therein be natural. Thus, contemplating a fractional WDW equation could unveil interesting aspects regarding the gravitational domain FQ2 ; FQ3 . In fact, in the last few years, FQM has indeed been proposed and employed as a tool to explore features within quantum cosmology and quantum gravity FQ1 . It has pointed to exciting opportunities and sidelong connections between unexpected mathematics and physics domains. See FQ2 ; FQ3 , for a recent survey about engaging FQM within a canonical route towards quantum gravity and cosmology. Additionally, a few more other references can be found in all-new .

Thus, in our paper we propose to apply space-FQM to investigate thermodynamic properties of the Schwarzschild BH. The Bekenstein-Hawking entropy expression for a Schwarzchild BH with mass $M$ is given by

$\displaystyle S_{\text{B-H}}=4\pi GM^{2}=\frac{A}{4G},$

(8)

where $A=16\pi G^{2}M^{2}$ is the BH horizon area. A suitably extended fractional version of the Wheeler-DeWitt (WDW) equation for the Schwarzschild BH will enable us to retrieve altered expressions for the entropy and other specific BH observables.

Moreover, Tsallis and Cirto recently proposed a new formulation for the Schwarzschild BH’s horizon entropy and the entropy-area relation Tsallis . They asserted that since the Bekenstein-Hawking entropy is proportional to the horizon area (instead of its volume), it would hint that Boltzmann-Gibbs entropy would be unsuitable to describe BHs. These authors have proposed a modified form upon the BH entropy given by Eq. (8), such that an extended expression, using a generalized non-additive entropy, would be

$\displaystyle S_{\delta}=\gamma A^{\delta},$

(9)

where $\gamma$ is an unspecified constant and $\delta$ denotes the non-additivity parameter. The above equation suggests that the entropy is a power-law function of its area.

In addition, by introducing a fractal structure for the horizon surface of the BH, Barrow Barrow conceived a 3-dimensional spherical analog of a Koch Snowflake–a sphereflake– using a hierarchy of touching spheres around the event horizon. This fractal structure of the BH surface changes its actual area, which in turn leads us to a new entropy relation, namely,

$\displaystyle S=\left(\frac{A}{A_{P}}\right)^{1+\frac{\triangle}{2}},$

(10)

where $A_{P}$ is the Planck area and $\triangle$ would suggestively denote an induced deformation of the horizon, with $\triangle=0$ reproducing the conventional Bekenstein-Hawking entropy (simplest horizon structure) and with $\triangle=1$ corresponding to the most intricate structure. Note that the Barrow modified entropy resembles Tsallis and Cirto’s non-additive entropy (9); nevertheless, the involved foundations and physical principles are entirely different. As an additional application of BH prospecting by means of FQM, we will retrieve these two extended entropy expressions but without using generalized non-additive features.

In essence, we propose and employ herewith a space-fractional formulation of the WDW equation for the Schwarzschild BH. Hence, let us then mention that the remaining of this paper is organized as follows. In section 3, we summarize the canonical quantization of the Schwarzschild BH, having briefly reviewed the corresponding Hamiltonian setting in section 2. In Section 4, we hypothesize a fractional calculus setting for the corresponding WDW equation as applied to the Schwarzschild BH. Specifically, we consider the observables reviewed in section 3 and present them (e.g., the Schwarzschild BH entropy) from within the framework of FQM. Section 5 contains our conclusions and a thorough discussion. Throughout this paper we shall work in natural units, $\hbar=c=k_{B}=1$.

We recall in this section results concerning the canonical quantization of the Schwarzschild BH that will be of relevance for our analysis in subsequent sections. The spherical symmetric ADM line element is

$$ds^{2}=-N(r,t)^{2}dt^{2}+\\ \Lambda(r,t)^{2}\Big{(}dr+N^{r}(r,t)dt\Big{)}^{2}+R(r,t)^{2}d\Omega^{2},$$

(11)

where $d\Omega^{2}$ is the line element for the unit two sphere $S^{2}$. We follow Kuchař’s fall-off conditions Kuchar1 . Firstly, it confirms that the coordinates $r$ and $t$ are extended to the Kruskal manifold, $-\infty<r,t<\infty$, and secondly, that the spacetime is asymptotically flat as well. Likewise, the fall-off conditions guarantee that the 4-momentum at the infinities, $r\rightarrow\pm\infty$, has no spatial component, which indicates that the BH is at rest concerning the left and right asymptotic Minkowski spacetimes. By fixing the asymptotic values of the lapse function, $N$, at infinities, $r\rightarrow\pm\infty$ to be $t$-dependent quantities (denoted by $N_{\pm}(t)$, respectively), the Hamiltonian form of the Einstein-Hilbert action functional, with appropriate boundary terms, reads then

$$\mathcal{S}=\displaystyle\int dt\int_{-\infty}^{\infty}\Big{\{}\Pi_{\Lambda}\dot{\Lambda}+\Pi_{R}\dot{R}-NH-N^{r}H_{r}\Big{\}}dr\\ -\displaystyle\int\Big{\{}N_{+}M_{+}+N_{-}M_{-}\Big{\}}dt,$$

(12)

in which the conjugate momenta of $\Lambda$ and $R$ are

$$\begin{split}\Pi_{\Lambda}&=-\frac{M^{2}_{P}}{N}R\Big{(}\dot{R}-R^{\prime}N^{r}\Big{)},\\ \Pi_{R}&=-\frac{M_{P}^{2}}{N}\Big{[}\Lambda\left(\dot{R}-R^{\prime}N^{r}\right)+R\left(\dot{\Lambda}-(\Lambda N^{r})^{\prime}\right)\Big{]},\end{split}$$

(13)

$M_{P}=1/\sqrt{G}$ is the Planck mass, $\dot{f}=\partial_{t}f$, $f^{\prime}=\partial_{r}f$ and the quantities $M_{\pm}(t)$ are defined by the asymptotic fall-off of the configuration variables. Note that on a classical solution, $M_{\pm}$ are equal to the Schwarzschild mass of the BH. Furthermore, $H$ and $H_{r}$ are the super-Hamiltonian and the radial super-momentum constraints, respectively, given by

$$\begin{split}H=&-\frac{1}{RM_{P}^{2}}\Pi_{R}\Pi_{\Lambda}+\frac{1}{2R^{2}M_{P}^{2}}\Pi_{\Lambda}^{2}+\frac{RR^{\prime\prime}}{\Lambda}\\ &-\frac{RR^{\prime}\Lambda^{\prime}}{\Lambda^{2}}+\frac{R^{\prime 2}}{2\Lambda}-\frac{\Lambda}{2},\\ H_{r}&=\frac{1}{M_{P}^{2}}(\Pi_{R}R^{\prime}-\Lambda\Pi^{\prime}_{\Lambda}).\end{split}$$

(14)

Following Kuchař Kuchar1 introducing the following two $(M,\Pi_{M})$ and $(\mathcal{R},\Pi_{\mathcal{R}})$ pairs of canonical transformations

$$\begin{split}M&=\frac{\Pi_{\Lambda}^{2}}{2M_{P}^{4}R}-\frac{RR^{\prime 2}}{2\Lambda^{2}}+\frac{R}{2},\\ \Pi_{M}&=\frac{\Lambda\Pi_{\Lambda}}{M_{P}^{2}}\left[\left(\frac{R^{\prime}}{\Lambda}\right)^{2}-\frac{1}{M_{P}^{4}}\left(\frac{\Pi_{\Lambda}}{R}\right)^{2}\right]^{-1},\\ \mathcal{R}&=R,\\ \Pi_{\mathcal{R}}&=\left(\frac{\Pi_{\Lambda}H}{M^{2}_{P}R}+\frac{R^{\prime}H_{r}}{\Lambda^{2}}\right)\left[\left(\frac{R^{\prime}}{\Lambda}\right)^{2}-\frac{1}{M_{P}^{4}}\left(\frac{\Pi_{\Lambda}}{R}\right)^{2}\right]^{-1},\end{split}$$

(15)

the action (12) becomes

$$\mathcal{S}=\displaystyle\int dt\int_{-\infty}^{\infty}\left\{\dot{M}\Pi_{M}+\dot{\mathcal{R}}\Pi_{\mathcal{R}}-N^{r}H_{r}-NH\right\}dr\\ -\displaystyle\int\left\{M_{+}N_{+}-M_{-}N_{-}\right\}dt,$$

(16)

where the new super-Hamiltonian, $H$, and the super-momentum, $H_{r}$, are

$$H=\\ -\frac{\left(1-\frac{2M}{M_{P}^{2}R}\right)^{-1}M^{\prime}{\mathcal{R}}^{\prime}+M_{P}^{-4}\left(1-\frac{2M}{M_{P}^{2}R}\right)\Pi_{M}\Pi_{\mathcal{R}}}{\left[\left(1-\frac{2M}{M_{P}^{2}R}\right)^{-1}{\mathcal{R}}^{\prime 2}-M_{P}^{-4}\left(1-\frac{2M}{M_{P}^{2}R}\right)\Pi_{M}^{2}\right]^{\frac{1}{2}}},\\ H_{r}=\frac{1}{M_{P}^{2}}\left(\Pi_{M}M^{\prime}+\Pi_{\mathcal{R}}{\mathcal{R}}^{\prime}\right).$$

(17)

The variation of ADM action (17) with respect to the lapse function $N$ and $N^{r}$ imposes the Hamiltonian and momentum constraints

$\displaystyle H\approx 0,~{}~{}~{}~{}H_{r}\approx 0,$

(18)

or equivalently

$\displaystyle M^{\prime}\approx 0,~{}~{}~{}~{}\Pi_{\mathcal{R}}\approx 0.$

(19)

The constraint $M^{\prime}=0$ means that $M$ is homogeneous $M=M(t)$. Now, by substituting $\Pi_{\mathcal{R}}\approx 0$ and $M=M(t)$ back into the action (17) and in addition the new conjugate momenta of $M$ defined by

$$P=\displaystyle\int_{-\infty}^{\infty}\Pi_{M}dr=\\ -\displaystyle\int_{-\infty}^{\infty}\frac{\sqrt{\left(\frac{dR}{dr}\right)^{2}-\Lambda\left(1-\frac{2M}{M^{2}_{P}R}\right)}}{1-\frac{2M}{M^{2}_{P}R}}dr,~{}~{}~{}-\infty<P<\infty,$$

(20)

we obtain

$\displaystyle\mathcal{S}=\int\Big{\{}P\dot{M}-(N_{+}+N_{-})M\Big{\}}dt.$

(21)

Note that the new conjugate variables $(M,P)$ obey the Poisson bracket $\{M,P\}=1$. Following Louko and Mäkelä Louko , if we picked the right-hand side asymptotic Minkowski time as the observer time parameter, we should restrict $N_{+}=1$ and $N_{-}=0$. Next, the reduced action (21) reduces to the following simple form

$\displaystyle\mathcal{S}=\int\left\{P\dot{M}-H(M)\right\}dt,$

(22)

where $H(M)=M$ is the reduced Hamiltonian of the BH. One can easily show that the field equations’ solution is $M=const.$ and $P=-t$, as expected. The BH’s mass constancy follows Birkhoff’s theorem, which declares that the mass is the only time-independent and coordinate invariant solution. Moreover, the conjugate momenta, $P$, describes the asymptotic time coordinate at the spacelike slice. Once again, by using the canonical transformations $(M,P)\rightarrow(x,p)$, introduced by Louko and Mäkelä Louko

$$\begin{split}|P|&=\displaystyle\int_{x}^{2MG}\frac{dy}{\sqrt{\frac{2MG}{y}-1}},\\ M&=\frac{1}{2G}\left(\frac{G^{2}p^{2}}{x}+x\right),\end{split}$$

(23)

the reduced action (22) takes the form

$\displaystyle\mathcal{S}=\int\left\{p\dot{x}-H\right\}dt,$

(24)

where $H=M$ is the Hamiltonian and is given (utilizing the second transformation of (23)), as

$\displaystyle H=\frac{M_{P}^{2}}{2}\Big{(}\frac{p^{2}}{M_{P}^{4}x}+x\Big{)}.$

(25)

It is of interest to acknowledge the following at this point. Transformations (23) map the BH solution into a wormhole solution, in which $x$ represents the wormhole throat Louko . The time evolution of this corresponding wormhole throat is given by Hamilton’s equations

$\displaystyle\dot{x}=\frac{p}{M_{P}^{2}x},~{}~{}~{}~{}\dot{p}=\frac{p^{2}}{2M_{P}^{2}x^{2}}-\frac{1}{2}M_{P}^{2}$

(26)

with a solution for the wormhole throat as

$$\begin{split}x&=\frac{M}{M_{P}^{2}}\Big{(}1+\cos(M_{P}\eta)\Big{)},\hskip 14.22636ptx\geq 0\\ t&=\frac{M}{M_{P}}\Big{(}\eta+\frac{1}{M_{P}}\sin(M_{P}\eta)\Big{)},\hskip 14.22636pt-\frac{M\pi}{M_{P}^{2}}\leq t\leq\frac{M\pi}{M_{P}^{2}}.\end{split}$$

(27)

As we can see from the above solutions, it is necessary to define transformations (23) so that the time parameter $t$ (or equivalently the conjugate momentum $P$) is restricted into the finite interval

$\displaystyle-\frac{1}{8T_{H}}\leq t\leq\frac{1}{8T_{H}},\hskip 14.22636ptT_{H}=\frac{M^{2}_{P}}{8\pi M},$

(28)

where $T_{H}$ is, remarkably, the Hawking temperature of the BH.

In the coordinate representation $\hat{p}\rightarrow-id/dx$, $\hat{x}\rightarrow x$, the canonical quantization procedure upon the previous section gives us a suitable time-independent WDW equation, for the simple one-dimensional minisuperspace of the Schwarzschild BH

$\displaystyle H\left(-i\frac{d}{dx},x\right)\psi(x)=M\psi(x).$

(29)

As usual, there is an operator-ordering problem in addressing the above WDW equation, but as we are interested in BH states whose mass $M\gg M_{P}$, its particular resolution will not surpass our semiclassical considerations. Hence, we adopt the following wide enough factor-ordering Jalalzadeh

$\displaystyle\frac{p^{2}}{x}=\frac{1}{3}\left(x^{i}px^{j}px^{k}+x^{k}px^{i}px^{j}+x^{j}px^{k}px^{i}\right),$

(30)

where $i+j+k=-1$. Note that because in the left hand side of (30) we have $p^{2}/x$, so, in its right side the power of $x$ is also $-1$. We thus write the WDW equation (29)

$$-\frac{1}{2M_{P}^{2}x}\frac{d^{2}}{d^{2}}\psi(x)+\frac{1}{2M_{P}^{2}x^{2}}\frac{d}{dx}\psi(x)+\\ \left(\frac{1}{2}M_{P}^{2}x+\frac{q}{2M_{P}^{2}x^{3}}\right)\psi(x)=M\psi(x),$$

(31)

where $q=(ij+ik+jk-2)/3$. If we redefine the wave function $\psi(x)\rightarrow\sqrt{x}\psi(x)$, and choose ordering in which $q=-3/4$, then Eq. (31) will reduce to

$$-\frac{1}{2M_{P}}\frac{d^{2}}{dx^{2}}\psi(x)+\frac{1}{2}M_{P}\omega_{P}^{2}\left(x-\frac{M}{M_{P}^{2}}\right)^{2}\psi(x)=\\ \frac{M^{2}}{2M_{P}}\psi(x),$$

(32)

which is a SE for the harmonic oscillator of the Planck’s mass $M_{P}$, the Planck’s angular frequency $\omega_{P}=1/t_{P}$ defined in terms of Planck’s time $t_{P}=1/M_{P}$. The domain of definition for $x$ is $x\geq 0$, and consequently, the Hamiltonian operator of the harmonic oscillator in (32) is defined on a dense domain $C^{\infty}(0,+\infty)$. Hence, $H$ is not an essentially self-adjoint operator. It may constitute an Hermitian operator if

$\displaystyle\langle\psi_{1}|H\psi_{2}\rangle=\langle H\psi_{1}|\psi_{2}\rangle,\hskip 14.22636pt\psi_{1},\psi_{2}\in{\mathcal{D}}(H).$

(33)

A necessary and sufficient condition for the validity of this condition is

$\displaystyle\left.\frac{\psi(x)}{\psi^{\prime}(x)}\right|_{x=0}=\gamma,\,\,\,\,\gamma\in\mathbb{R},$

(34)

where a prime symbol, $\prime$, denotes the derivative of $\psi(x)$ with respect to $x$. As pointed out by Tipler Tipler , the constant $\gamma$ (with dimension of length) would be a new fundamental constant of theory. To avoid such, we set it to be zero. Hence, we assume

$\displaystyle\psi(x)\Big{|}_{x=0}=0,$

(35)

which constitutes the DeWitt boundary condition DeWitt . In addition, we are interested in square-integrable wave functions in the interval $0\leq x<+\infty$, which implies the second boundary condition $\psi(x\rightarrow+\infty)=0$. The general solution of the WDW equation (32) is

$$\psi(z)=e^{-\frac{z^{2}}{4}}\Big{\{}A\,_{1}F_{1}\left(-\frac{\nu}{2};\frac{1}{2};\frac{z^{2}}{2}\right)+\\ Bz\,_{1}F_{1}\left(\frac{1-\nu}{2};\frac{3}{2};\frac{z^{2}}{2}\right)\Big{\}},$$

(36)

where $z=\sqrt{2M_{P}\omega_{P}}(x-M/M_{P}^{2})$ is the new dimensionless coordinate in the minisuperspace, ${}_{1}F_{1}(a;b;\zeta)$ is confluent hypergeometric function, $\nu=\frac{1}{2}\left(\frac{M^{2}}{M_{P}^{2}}-1\right)$, $A$ and $B$ are two integration constants. Moreover, from the asymptotic behavior of the confluent hypergeometric functions

${}_{1}F_{1}(a;b;\zeta\rightarrow\infty)\simeq\Gamma(b)e^{\zeta}\zeta^{a-b}/\Gamma(a),$

(37)

we find that the bracket $\{...\}$ in Eq. (36) brings the factor $e^{z^{2}/2}$, which would dominate over the Gaussian exponential factor explicit in (36). In other words, the wave function is square-integrable if the remaining factor provided from the bracket$\{...\}$ in Eq. (36), vanishes. This fixes the relative values of $A$ and $B$, leading to

$$\psi(z)=N2^{\frac{\nu}{2}}e^{-\frac{z^{2}}{4}}\Big{\{}\frac{{}_{1}F_{1}\left(-\frac{\nu}{2};\frac{1}{2};\frac{z^{2}}{2}\right)}{\Gamma(\frac{1-\nu}{2})}-\\ z\frac{\sqrt{2}\,_{1}F_{1}\left(\frac{1-\nu}{2};\frac{3}{2};\frac{z^{2}}{2}\right)}{\Gamma(-\frac{\nu}{2})}\Big{\}},$$

(38)

where $N$ is a normalization constant. Applying the boundary condition (34) on the obtained wave function (38) gives us

$$\frac{{}_{1}F_{1}\left(-\frac{\nu}{2};\frac{1}{2};(\frac{M}{M_{P}})^{2}\right)}{\Gamma(\frac{1-\nu}{2})}+\frac{2M\,_{1}F_{1}\left(\frac{1-\nu}{2};\frac{3}{2};(\frac{M}{M_{P}})^{2}\right)}{M_{P}\Gamma(-\frac{\nu}{2})}=0.$$

(39)

The solution of the above equation for $M$, gives us the mass spectrum of the BH.

We can see that the spectrum of $M^{2}$ is not equally spaced for small values of $\nu$. On the other hand, for large values of $\nu$ (which is defined by $\nu=\frac{1}{2}\left(\frac{M^{2}}{M_{P}^{2}}-1\right)$) if we use the asymptotic relation (37) in (39) we find

$$\frac{e^{2\nu+1}}{(2\nu+1)^{\frac{\nu+1}{2}}2^{\nu}\Gamma(-\nu)}=0\Rightarrow\Gamma(-\nu)=\pm\infty\Rightarrow\nu=n,$$

(40)

which provides us the mass spectrum

$\displaystyle M=M_{P}\sqrt{2n+1},$

(41)

where $n$ is an integer and $n\gg 1$. Note that the above result assumes that $M\gg M_{P}$, i.e., $\nu\gg 1$ and is therefore essentially semiclassical. To see this, let us apply the Bohr-Sommerfeld quantization rule to the Hamiltonian (25). The classical turning points of (25) are $x=0$ and $x=2M/M_{P}^{2}$. Hence, the conventional treatment of the Bohr-Sommerfeld quantization rule yields

$$2\pi\left(n+\frac{1}{2}\right)=2\displaystyle\int_{0}^{\frac{2M}{M_{P}^{2}}}pdx\\ =2\displaystyle\int_{0}^{\frac{2M}{M_{P}^{2}}}\sqrt{M^{2}-M_{P}^{4}\left(x-\frac{M}{M_{P}^{2}}\right)^{2}}dx,$$

(42)

which allows to extract the mass spectrum (41). Bekenstein Bekenstein firstly found a similar mass spectrum. Generally, the proportionality constant for the square root of $n$ in (41) is model dependent. Since then, several authors Bekenstein ; Kastrup have used different arguments for a quantum BH spectrum of the type (41). Before proceeding, let us keep in mind equation (41), as this and others in this section will bear alterations that will be brought from the intrinsic features of FQM, as applied to the Schwarzshild BH (see next section).

S. Hawking Hawking1 ; Hawking showed in 1974 that due to quantum fluctuations, BHs emit black-body radiation, consistently with the corresponding entropy being one fourth of the event horizon area, namely $A=16\pi G^{2}M^{2}$. Following Refs. Mukhanov and Xiang , let us assume that the Hawking radiation of a massive BH where $M\gg M_{P}$ and $n\gg 1$, is emitted when the BH system spontaneously jumps from the state $n+1$ towards the closest lower state level, i.e., $n$, as described by (41). Let us now denote the frequency of the emitted thermal radiation as $\omega_{0}$. Then

$$\omega_{0}=M(n+1)-M(n)\simeq\frac{M_{P}}{\sqrt{2n}}\\ \simeq\frac{M_{P}^{2}}{M}\left[1+\frac{1}{2}\left(\frac{M_{P}}{M}\right)^{2}\right],$$

(43)

which agrees with the classical BH oscillation frequencies which scales as $1/M$. We thus find a BH to radiate with a characteristic temperature $T\propto M_{P}^{2}/M$, matching the Hawking temperature.

The characteristic BH time (the lifetime of the BH at the state $M(n+1)$ before decaying into the lower state $M(n)$) can be defined Xiang as

$\displaystyle\tau_{n}^{-1}=\frac{\dot{M}}{\omega_{0}}\simeq\frac{M\dot{M}}{M_{P}^{2}}\left[1-\frac{1}{2}\left(\frac{M_{P}}{M}\right)^{2}\right],$

(44)

where $\dot{M}=dM/dt$ is the mass loss of the BH because of its evaporation; in the second equality we used the definition of $\omega_{0}$ expressed in (43). As discussed in Mukhanov ; Mukhanov2 , because of the interaction of the BH with the vacuum of the quantum fields, the width of the states, $W_{n}$, is not zero. The width of state $n$ can be estimated Mukhanov ; Mukhanov2 as

$\displaystyle W_{n}=\beta[M(n+1)-M(n)]=\beta\omega_{0},$

(45)

where $\beta\ll 1$ is a numerical dimensionless factor. Then, by inserting (43) into the uncertainty relation $W_{n}\tau_{n}\simeq 1$ and eliminating $\tau_{n}$ in resulting equation, with the assistance of (44) we can find

$\displaystyle\dot{M}=\frac{\beta M_{P}^{4}}{M^{2}}\left[1+\left(\frac{M_{P}}{M}\right)^{2}\right].$

(46)

If we further assume, on the one hand, that the origin of the Hawking radiation emerges from the highly blueshifted modes just outside the horizon, and, on the other hand, take the BH as a black-body, then the radiated power is given by the Stefan–Boltzmann law Radiation ; Radiation2

$\displaystyle\dot{M}=\sigma_{S}AT^{4},$

(47)

where $\sigma_{S}=\pi^{2}/60$ is the Stefan–Boltzmann constant and $A=16\pi M^{2}/M_{P}^{4}$ is the horizon area. Eliminating the mass loss of the BH, using Eqs.(46) and (47), gives us the effective temperature

$\displaystyle T=\left(\frac{\beta}{16\pi\sigma_{S}}\right)^{\frac{1}{4}}\frac{M_{p}^{2}}{M}\left[1+\frac{1}{4}\left(\frac{M_{P}}{M}\right)^{2}\right].$

(48)

The BH entropy can then be expressed as

$\displaystyle S=\int\frac{dM}{T}.$

(49)

By choosing $\beta=1/15360\pi$ Correction , Eqs. (48) and (49) yields

$\displaystyle S=S_{\text{B-H}}-2\pi\ln{\left(S_{\text{B-H}}\right)}+const.,$

(50)

where $S_{\text{B-H}}=4\pi GM^{2}$ is the Bekenstein-Hawking entropy. The logarithmic correction to the Bekenstein-Hawking entropy is obtained using other methods Correction except that the overall factor $2\pi$ is model dependent.

In this section, we will establish and then investigate the alterations brought from employing FQM features towards a Schwarzschild BH. This will provide broader expressions for some thermodynamic observables, specifically in terms explicitly dependent on the fractional parameter, $1<\alpha\leq 2$, since the limit $\alpha=2$ will conveys us to the standard case (summarized in sections 2 and 3).

Concretely, a fractional WDW equation for a BH will be employed, in a corresponding minisuperspace. Our approach stems from Laskin’s seminal papers Laskin1 ; Leskin1 , leading to an Hamiltonian, which includes a fractional kinetic term in terms of the quantum Riesz fractional operator; this method has been extended to the WDW equation in Refs. FQ2 ; FQ3 .

The fractional extension of the WDW equation (32) is given by

$$\frac{1}{2}M_{P}^{1-\alpha}(-\Delta)^{\frac{\alpha}{2}}\psi(z)+\frac{1}{2}M_{P}^{\alpha-1}\omega_{P}^{2}z^{\alpha}\psi(z)=\\ \frac{M^{2}}{2M_{P}}\psi(z),$$

(51)

where $z=x-M/M_{P}^{2}$ is the new coordinate in the $1$-dimensional minisuperspace, $\Delta=d^{2}/dz^{2}$, $(-\Delta)^{\frac{\alpha}{2}}$ is the Riesz fractional derivative Rie ; par1 ; par2 ; par3 ; Butzer and $1<\alpha\leq 2$. Unfortunately, there is no known general solution, explicitly bearing a dependence on $\alpha$ for the above fractional WDW equation. Therefore, we may resort to employ the Bohr-Sommerfeld quantization rule. By replacing $|p|=(-\nabla)^{\frac{1}{2}}$, the fractional WDW equation leads to the following relation

$\displaystyle\frac{M_{P}^{1-\alpha}}{2}|p|^{\alpha}+\frac{M_{P}^{\alpha-1}}{2}\omega_{P}^{2}z^{\alpha}=\frac{M^{2}}{2M_{P}},$

(52)

which is, in fact, the fractional version of (25). Hence, the classical turning points (where $|p|=0$) are $z=\pm(M^{2}/M_{P}^{2-\alpha})^{\frac{1}{\alpha}}$. Moreover,

$$2\pi\left(n+\frac{1}{2}\right)=\displaystyle\oint pdz=2\left(\frac{M}{M_{P}}\right)^{\frac{4}{\alpha}}\displaystyle\int_{0}^{1}\left(1-y^{\alpha}\right)^{\frac{1}{\alpha}}dy=\\ \frac{2M^{\frac{4}{\alpha}}}{\alpha M_{P}^{\frac{4}{\alpha}}}B\left(\frac{1}{\alpha},1+\frac{1}{\alpha}\right),$$

(53)

where $y=(M_{P}^{\alpha+2}/M^{2})^{\frac{1}{\alpha}}z$ and $B(a,b)$ is the beta function. Thus, the application of the standard Bohr-Sommerfeld quantization rule in this setting gives us the following semi-classical mass spectrum

$\displaystyle M=\left(\frac{\alpha\pi\Gamma(\frac{2}{\alpha})}{\Gamma(\frac{1}{\alpha})^{2}}\right)^{\frac{\alpha}{4}}M_{P}\left(n+\frac{1}{2}\right)^{\frac{\alpha}{4}}.$

(54)

For $\alpha=2$ we recover spectrum (41) as written in the previous section. Fig. (1) shows the BH mass spectrum for three values of $\alpha$. As we show in this figure, the mass of the BH increases with $n$ but at a faster rate for $\alpha=2$, namely the standard case.

Similarly to the previous section 3, if we write the frequency of the radiation emitted by the BH, $\omega_{0}$ , we now obtain

$$\omega_{0}(\alpha)=M(n+1,\alpha)-M(n,\alpha)\simeq\frac{\alpha BM_{P}}{4}n^{\frac{\alpha}{4}-1}\\ \simeq\frac{\alpha B^{\frac{4}{\alpha}}M_{P}^{\frac{4}{\alpha}}}{4M^{\frac{4}{\alpha}-1}}\left[1+\frac{1}{2}(1-\frac{\alpha}{4})\left(\frac{BM_{P}}{M}\right)^{\frac{4}{\alpha}}\right],$$

(55)

where $B=\frac{\left(\alpha\pi\Gamma(\frac{2}{\alpha})\right)^{\frac{\alpha}{4}}}{{\Gamma(\frac{1}{\alpha})^{\frac{\alpha}{2}}}}$. Note that the energy of the emitted radiation is a function of the fractional-order $\alpha$. We then find that for a massive BH, where $M\gg M_{P}$, the energy of emitted radiation is minimal for $\alpha\rightarrow 1$, and it will increases up to a maximum at $\alpha=2$. Additional elements are conveyed within Fig. (2).

Interestingly, the mass spectrum (54) (which includes the standard case (41)) may induce observable signatures that gravitational waves GW may inform about. Eq. (43) shows that the frequency of the emitted Hawking radiation scales as $1/M$. Hence, if we put $M\simeq 10M_{\odot}-50M_{\odot}$ as typical of the BHs registered by LIGO-Virgo, then we obtain from Eq. (43) that $f=\frac{\omega_{0}}{2\pi}=\mathcal{O}(10^{2}-10^{4})$ Hz. On the other hand, the fractional frequency of radiation $f(\alpha)=\frac{\omega_{0}(\alpha)}{2\pi}$, whereas defined instead with (55), gives us a much wider possibility for the frequencies. For example, with $M=10M_{\odot}$ the range $1<\alpha\leq 2$ yields $\mathcal{O}(10^{-74})\text{Hz}<f(\alpha)\leq\mathcal{O}(10^{4})\text{Hz}$. Figure (2) shows how $\omega_{0}(\alpha)$ depends on $\alpha$.

To be more clear, let us contemplate the mass spectrum as in (54) and further discuss it within a wider analysis as follows. Take the interaction of a gravitational wave with a BH. Suppose the BH is initially at state $n_{1}$. Regarding the mass spectrum in (54), we can also consider process whereby the BH could alternatively absorb a gravitational wave whose frequency, $f_{\text{GW}}$ satisfies

$$f_{\text{GW}}=\frac{\Delta M}{2\pi}=\frac{\alpha\pi\Gamma(\frac{2}{\alpha})\Delta n}{8\Gamma(\frac{1}{\alpha})^{2}M^{\frac{4}{\alpha}-1}},$$

(56)

where $\Delta n=n_{1}-n_{2}$ and $n_{2}$ denotes the final state of the BH after absorption of the gravitational wave. As known in classical GR, the effective potential that describes the motion of a test particle has a maximum at the $3/2$ Schwarzschild radius, called photon sphere. Essentially, this (potential barrier) screens the near-horizon region from external observers Potential . The gravitational radiation proceeding towards the horizon will be scattered at the horizon if its frequency does not match, e.g., (56). Otherwise, radiation with the frequencies such as (56) in our example, would be absorbed. The previous assertions notwithstanding, if then we consider back-reaction quantum effects, BHs are not perfect absorbers of gravitational radiation and part of the radiation will be reflected from the horizon area. The reflected part of the radiation then interacts with the potential barrier at the photon sphere. Then, it will partially be transmitted, and the other portion will be reflected again back towards the horizon. This outcome generates a series of so-called gravitational-wave echoes Potential1 . This feature could eventually be detected either in the inspiral stage of BH binaries or during the last stages of BH relaxation following a merger. For further details and detection methods please see GWnew . Therefore, BHs may act as “magnifying lenses” in the sense that they could bring new features of the BH horizon-area within the realm of gravitational waves observations.

Furthermore, using the formula of the characteristic time of an evaporating BH, $\tau^{-1}=\dot{M}/\omega_{0}$, the uncertainty relation for width of states, $W_{n}\tau_{n}\simeq 1$ and the relation $W_{n}=\beta\omega_{0}$, we compute

$\displaystyle\dot{M}=\frac{\alpha^{2}\beta B^{\frac{8}{\alpha}}M_{P}^{\frac{8}{\alpha}}}{16M^{\frac{8}{\alpha}-2}}\left[1+\left(1-\frac{\alpha}{4}\right)\left(\frac{BM_{P}}{M}\right)^{\frac{4}{\alpha}}\right],$

(57)

which will increase as mass is lost. Inserting the above relation into the Stefan–Boltzmann law (47), it gives the corresponding temperature of the BH¹¹1In our paper we assume that fractional calculus only induces alterations at quantum geometrodynamics level, and all the thermodynamic laws are essentially unchanged. We might extend the discussion to include fractional thermodynamics, though. In this case, we should consider a fractional Stefan-Boltzmann law; see, e.g., Korichi .

$$T=\\ \left(\frac{\beta\alpha^{2}B^{\frac{8}{\alpha}}}{16^{2}\pi\sigma_{S}}\right)^{\frac{1}{4}}\frac{M_{P}^{\frac{2}{\alpha}+1}}{M^{\frac{2}{\alpha}}}\left[1+\frac{1}{4}\left(1-\frac{\alpha}{4}\right)\left(\frac{BM_{P}}{M}\right)^{\frac{4}{\alpha}}\right],$$

(58)

$\displaystyle\beta=\frac{\Gamma(\frac{1}{\alpha})^{4}}{15(\alpha+2)^{4}4^{\frac{4}{\alpha}-1}\Gamma(\frac{2}{\alpha})^{2}\pi^{\frac{4}{\alpha}+1}}.$

(59)

Note that for $\alpha=2$ we will recover equations of the previous section. Moreover, using (49), we find the fractional entropy of the BH to be

$$S=\\ S_{\text{B-H}}^{\frac{2+\alpha}{2\alpha}}+\frac{\alpha(4-\alpha)(2+\alpha)4^{\frac{\alpha+2}{2\alpha}}\pi^{\frac{3\alpha+2}{2\alpha}}\Gamma(\frac{2}{\alpha})}{(2-\alpha)\Gamma(\frac{1}{\alpha})^{2}}\frac{1}{S_{\text{B-H}}^{\frac{2-\alpha}{2\alpha}}},\,\,\,\,\alpha\neq 2.$$

(60)

To further contrast a BH with FQM features with a standard Schwarzschild BH, let us examine Eqs. (55), (57), (58) and (60) and for a stellar BH with mass approximately $10M_{\odot}$. For the limiting $\alpha=2$ and $\alpha\rightarrow 1$ values of the Lévy index, the values of $S(\alpha)$, $T(\alpha)$, $\omega_{0}(\alpha)$ and $\dot{M}(\alpha)$ are given in Table (1).

Its first row, together with Fig. (2), shows that the entropy of fractional BH can be much larger than in the standard Schwarzschild case. The second row shows that a stellar BH temperature (with mass $10M_{\odot}$) in fractional formalism is always less than $10^{-9}K$. On the other hand, in the extreme case $\alpha\rightarrow 1$, the temperature of a fractional BH is approximately zero: $T(\alpha\rightarrow 1)\simeq 10^{-48}$ K. Moreover, the manner the mass decay $\dot{M}(\alpha)$ proceeds is indicated by by the third row, in agreement with the temperature range: it is quite negligible for the extreme case, $\alpha\rightarrow 1$. Since the average temperature of the Universe at the present epoch is about 2.7 K, all stellar BHs with $M=10M_{\odot}$ and $1<\alpha\leq 2$ are absorbing more matter and radiation than they emit Hawking radiation and will not begin to evaporate until the Universe has expanded and cooled below their corresponding temperature. Therefore, in our setting, the BHs with fractional features could be almost eternal within $\alpha\rightarrow 1$.

Let us now remark the following. Tsallis and Cirto Tsallis investigated the entropy of a Schwarzschild BH, using appropriate non-additive generalizations for $d-$dimensional systems and suggested a generalized entropy. In their study Tsallis , a non-additive entropy is defined by

$\displaystyle S_{\delta}=\sum_{i}^{N}p_{i}\left(\ln(\frac{1}{p_{i}})\right)^{\delta},$

(61)

for a set of $N$ discrete states, where $\delta>0$ denotes the non-additivity parameter and $p_{i}$ is a probability distribution Tsallis2 . For $\delta=1$ we recover the standard Boltzmann-Gibbs entropy. Then, as Tsallis and Cirto demonstrated Tsallis , the generalized BH entropy can be written as

$\displaystyle S_{\text{Tsallis}}\propto S_{\text{B-H}}^{\delta}.$

(62)

If we identify the non-additivity parameter of Tsallis and Cirto as $\delta=\frac{2+\alpha}{2\alpha}$, the entropy (60) retrieved from fractional quantum mechanical methods can be rewritten as

$\displaystyle S=S_{\text{B-H}}^{\delta}+\frac{\kappa}{S_{\text{B-H}}^{\delta-1}},~{}~{}~{}~{}~{}~{}~{}1<\delta<\frac{3}{2},$

(63)

where

$\displaystyle\kappa=\frac{\delta(2\delta-3)4^{\delta+1}\pi^{\delta+\frac{2}{2\delta-1}}\Gamma(2\delta-1)}{(\delta-1)(2\delta-1)^{2}\Gamma(\delta-\frac{1}{2})^{2}}.$

(64)

This shows that the leading term of the FQM computed BH entropy can be expressed in terms of the Tsallis and Cirto result for a BH. Noteworthy, for $1<\alpha\leq 2$ the fractional entropy of a BH varies in the corresponding interval as

$$S_{\text{B-H}}^{\frac{3}{2}}+\frac{36\pi^{\frac{5}{2}}}{S^{\frac{1}{2}}_{\text{B-H}}}<S\leq S_{\text{B-H}}-2\pi\ln{\left(S_{\text{B-H}}\right)}.$$

(65)

A natural question that arises is the meaning of the entropy relation, ranging from Eqs. (60) to (65). To assist in addressing this question, let us first remind the fractal nature of the Lévy path Levy in FQM. As we know, in the Feynman path integral representation of quantum mechanics, the measure is generated by the process of the Brownian motion, and the fractal dimension of the Feynman’s path is $d^{\text{(Feynman)}}_{\text{fractal}}=2$. In addition,, FQM is based on the fractional path integral Laskin1 ; Levy . The Lévy path integral corresponding to the SE (51) is

$\displaystyle K_{L}(z_{f},t_{f}|z_{i},t_{i})=\displaystyle\int_{z_{i}}^{z_{f}}\mathcal{D}ze^{-i\int_{z_{i}}^{z_{f}}V(z(t))dt},$

(66)

where $V(z(t))$ is the potential as a functional of the Lévy path and $z_{i}$ ($z_{f}$) denotes the initial (final) point. In this case, the fractional path integral measure is

$$\mathcal{D}z(t)=\displaystyle\lim_{N\rightarrow\infty}dz_{1}...dz_{N-1}\times\\ \displaystyle(iD_{\alpha}\varepsilon)^{-\frac{N}{\alpha}}\prod_{j=1}^{N}L_{\alpha}\left\{\left(\frac{1}{iD_{\alpha}\varepsilon}\right)^{\frac{1}{\alpha}}|z_{j}-z_{j-1}|\right\},$$

(67)

where $D_{\alpha}=M_{P}^{1-\alpha}$, $\varepsilon=(t_{f}-t_{i})/N$, $z_{0}=z_{i}$, $z_{N}=z_{f}$ and the Lévy distribution function, $L_{\alpha}$, is indicated in terms of Fox’s $H$ function

$$\left(D_{\alpha}t\right)^{-\frac{1}{\alpha}}L_{\alpha}\left\{\left(\frac{1}{D_{\alpha}t}\right)^{\frac{1}{\alpha}}|z|\right\}=\\ \frac{1}{\alpha|z|}H_{2,2}^{1,1}\left[\left(\frac{1}{D_{\alpha}t}\right)^{\frac{1}{\alpha}}|z|\Big{|}^{(1,\frac{1}{\alpha}),(1,\frac{1}{2})}_{(1,1),(1,\frac{1}{2})}\right].$$

(68)

The measure (67) indicates that a length increment, $\Delta z=z_{j}-z_{j-1}$, and a time increment, $\Delta t$, satisfy the fractional scaling relation

$\displaystyle\Delta z\propto D_{\alpha}^{\frac{1}{\alpha}}(\Delta t)^{\frac{1}{\alpha}}.$

(69)

The above scaling relation implies that the fractal dimension of the Lévy path is $d_{\text{fractal}}^{\text{(L\'{e}vy)}}=\alpha$. Thus, fractional features within FQM could be interpreted as being generated by a Lévy stochastic process Levy and suggesting a fractal structure²²2According to Mandelbrot “A fractal is by definition a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension” Benoit . For example, the Hausdorff dimension of a regular Brownian surface is 2.79 and a triangular von Koch fractal Surface is $1+\log_{2}(3)=2.5849$ . . Hence, it may be possible to theorize within the FQM framework herewith proposed, that the horizon of the BH may unveil a fractal structure whose relevance will be $\alpha$-dependent. In this context, let us rewrite the first term on the right-hand side of (60) as follows

$$S_{\text{B-H}}^{\frac{2+\alpha}{2\alpha}}=\left(\frac{A}{4G}\right)^{\frac{2+\alpha}{2\alpha}}=\frac{A_{\text{fractal}}}{4G},$$

(70)

by means of which we hence define $A_{\text{fractal}}$. Being more concrete, the entropy is proportional to a surface area, but one explicitly given by

$$A_{\text{fractal}}=4L_{p}^{2}\left(\frac{A}{4G}\right)^{\frac{2+\alpha}{2\alpha}}.$$

(71)

Furthermore, the above expression additionally suggests a fractal dimension Benoit of the BHs surface, namely

$$D_{\text{fractal}}=\frac{2+\alpha}{\alpha},~{}~{}~{}~{}~{}2\leq D_{\text{fractal}}<3.$$

(72)

Moreover, equation (60) will take the following form

$$S=\\ \frac{A_{\text{fractal}}}{4G}+\kappa\left(\frac{A_{\text{fractal}}}{4G}\right)^{1-\frac{1}{2}D_{\text{fractal}}},~{}~{}~{}~{}2<D_{\text{fractal}}<3.$$

(73)