Statistical Entropy Based on the Generalized-Uncertainty-Principle-Induced Effective Metric

Abstract

We investigate the statistical entropy of black holes within the framework of the generalized uncertainty principle (GUP) by employing effective metrics that incorporate leading-order and all-order quantum gravitational corrections. We construct three distinct effective metrics induced by the GUP, which are derived from the GUP-corrected temperature, entropy, and all-order GUP corrections, and analyze their impact on black hole entropy using ’t Hooft’s brick wall method. Our results show that, despite the differences in the effective metrics and the corresponding ultraviolet cutoffs, the statistical entropy consistently satisfies the Bekenstein–Hawking area law when expressed in terms of an invariant (coordinate-independent) distance near the horizon. Furthermore, we demonstrate that the GUP naturally regularizes the ultraviolet divergence in the density of states, eliminating the need for artificial cutoffs and yielding finite entropy even when counting quantum states only in the vicinity of the event horizon. These findings highlight the universality and robustness of the area law under GUP modifications and provide new insights into the interplay between quantum gravity effects and black hole thermodynamics.

1. Introduction

The generalized uncertainty principle (GUP) emerges from several quantum gravity theories, including string theory [1,2,3,4] and loop quantum gravity [5,6,7,8]. It suggests that the classic Heisenberg uncertainty principle (HUP) needs to be modified at high energy scales or at extremely small distances. A central tenet of many GUP formulations is the existence of a minimal length [9,10,11,12,13], which is interpreted as a fundamental characteristic of quantum gravity. This minimal length introduces inherent limits to precision in position and momentum measurements [14,15,16].

In the context of black hole thermodynamics, the GUP plays a crucial role in averting the complete evaporation of black holes, thereby allowing for Planck-scale remnants that prevent singularities [17,18,19,20]. Without this intervention, small black holes would ultimately evaporate completely due to Hawking radiation [21]. Research utilizing frameworks like the brick-wall model [22] has also suggested potential modifications to black hole entropy. Furthermore, employing a modified state density inspired by the GUP can effectively eliminate divergences observed in the brick wall model, eliminating the necessity for arbitrary cutoffs [23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38].

Various forms of GUP have been proposed, including those with linear and quadratic terms in momentum or even higher-order terms, some of which also predict a maximal observable momentum [11,29,39,40,41,42,43,44,45,46]. To analyze the effects of GUP in curved spacetime, particularly around black holes, researchers often employ the concept of an effective metric [47,48,49,50,51,52], which encodes quantum gravitational corrections into a modified spacetime geometry. This approach is highly effective because the thermodynamic properties of black holes, such as entropy and temperature, are intrinsically tied to the spacetime metric (e.g., via surface gravity and horizon area). By constructing GUP-modified effective metrics, one can systematically study how quantum gravity alters black hole thermodynamics while retaining the mathematical framework of classical general relativity. For instance, the brick wall method for entropy calculation relies on the near-horizon metric structure, and GUP-induced corrections to the metric naturally regularize ultraviolet divergences without ad hoc cutoffs. This synergy between effective metrics and thermodynamic principles underscores how spacetime geometry serves as a bridge between quantum gravity phenomenology and observable black hole properties.

Despite its utility as a tool, the construction and application of GUP-based effective metrics face significant challenges. There is no universally accepted method for deriving the effective metric from a given GUP. Different assumptions or approaches can lead to different metric forms, raising questions about their validity and uniqueness. Concerns have also been raised that using series truncations in derivations rather than full GUP-corrected expressions can lead to incorrect results or artificial singularities [50]. This challenges the necessity of using full GUP expressions in quantum gravity phenomenology.

In this paper, we present a systematic and unified analysis of black hole statistical entropy in the framework of the GUP by constructing and comparing three distinct GUP-induced effective metrics: those derived from the leading-order GUP-corrected temperature, the leading-order GUP-corrected entropy, and the all-order GUP-corrected temperature. One of the main findings in this paper is that, despite differences in the effective metrics and corresponding ultraviolet cutoffs, all cases yield the same invariant distance near the horizon and lead to a universal recovery of the Bekenstein–Hawking area law for black hole entropy. Furthermore, we demonstrate that the GUP itself provides a natural regularization of the ultraviolet divergence in the density of states, eliminating the need for arbitrary cutoffs. This clarifies the universality of the area law under quantum gravity corrections and highlights the robustness and physical significance of the effective metric approach in understanding black hole thermodynamics.

The remainder of the paper is organized as follows: in Section 2,we introduce the GUP, addressing both the leading order and all orders in Planck length. In Section 3, we derive effective metrics based on the leading-order GUP-corrected temperature and entropy, as well as from the GUP-corrected temperature considering all orders in Planck length. In Section 4, we compute the free energy and entropy using the brick wall method based on the effective metrics identified in the previous section. This analysis demonstrates that the area laws are satisfied and that all three effective metrics remarkably yield the same invariant distance. In Section 5, we again calculate the free energy and entropy by carefully counting the number of quantum states near the event horizon without any artificial cutoff. Then, we compare the results with those from Section 4. Finally, in Section 6, we provide our conclusions.

2. GUP to Leading and All Orders in the Planck Length

The most common form of the GUP is expressed as

$$\Delta x\Delta p\ge {\displaystyle \frac{ħ}{2}}\left(1+\alpha L_p^2{\displaystyle \frac{\Delta p^2}{{ħ}^2}}\right).$$

(1)

Here, $\Delta x$ and $\Delta p$ represent the uncertainties in position and momentum, respectively. The reduced Planck constant is denoted by ħ, and the Planck length is denoted by $L_p$, which represents an extremely short distance at which quantum gravitational effects become significant. $\alpha$ is a dimensionless GUP parameter, typically of order unity. If the GUP parameter $\alpha$ vanishes (i.e., $\alpha =0$), this equation reverts to the standard HUP.

Solving the above GUP inequality for the momentum uncertainty $\Delta p$ yields the following range:

$${\displaystyle \frac{ħ}{\alpha L_p^2}}\Delta x\left(1-\sqrt{1-{\displaystyle \frac{\alpha L_p^2}{\Delta x^2}}}\right)\le \Delta p\le {\displaystyle \frac{ħ}{\alpha L_p^2}}\Delta x\left(1+\sqrt{1-{\displaystyle \frac{\alpha L_p^2}{\Delta x^2}}}\right).$$

(2)

A crucial consequence of this equation is that the term inside the square root must be positive. This condition imposes a minimum value on the position uncertainty $\Delta x$. This minimum measurable length is given by

$${(\Delta x)}_{\min }=\sqrt{\alpha }{L}_p.$$

(3)

This implies that GUP suggests the existence of a fundamental minimum length in space, a significant departure from standard quantum mechanics.

With growing interest in quantum phenomenology involving GUP with higher orders in the Planck length, Nouicer [29] generalized the GUP in Equation (1) to include all orders in Planck length. This all-order GUP correction in the Planck length is expressed as

$$\Delta x\Delta p\ge {\displaystyle \frac{ħ}{2}}{e}^{\alpha L_p^2{\displaystyle \frac{\Delta p^2}{{ħ}^2}}}.$$

(4)

This expression can be reduced to the original GUP in Equation (1) if we only consider the leading order in the Planck length. By squaring the GUP in Equation (4),we arrive at the following inequality:

$$-2\alpha L_p^2{\displaystyle \frac{\Delta p^2}{{ħ}^2}}{e}^{-2\alpha L_p^2{\displaystyle \frac{\Delta p^2}{{ħ}^2}}}\le -{\displaystyle \frac{\alpha L_p^2}{2\Delta x^2}}.$$

(5)

Defining $W\left(\xi \right)$ and $\xi$ as

$$W\left(\xi \right)\equiv -2\alpha L_p^2{\displaystyle \frac{\Delta p^2}{{ħ}^2}},\xi \equiv -{\displaystyle \frac{\alpha L_p^2}{2\Delta x^2}},$$

(6)

it can be shown that the GUP for all orders in the Planck length satisfies

$$W\left(\xi \right)e^{W\left(\xi \right)}\le \xi ,$$

(7)

where $W\left(\xi \right)$ is a multi-valued Lambert function [53]. For the range $-1/e\le \xi \le 0$, it has two real values, $W_0\left(\xi \right)$ and $W_{-1}\left(\xi \right)$. For $\xi \ge 0$, it has one real value, $W_0\left(\xi \right)$. Here, $W_0\left(\xi \right)$ denotes the principal branch satisfying $W\left(\xi \right)\ge -1$, and $W_{-1}\left(\xi \right)$ denotes the branch satisfying $W\left(\xi \right)\le -1$. The branch point occurs at $\xi =-1/e$ and provides a minimum length for the GUP to all orders in the Planck length:

$$\Delta x\ge \sqrt{{\displaystyle \frac{e}{2}}}·\sqrt{\alpha }{L}_p\equiv {(\Delta x)}_{\min }.$$

(8)

The uncertainty in momentum from Equation (4) can be expressed using the Lambert W function as

$$\Delta p\ge {\displaystyle \frac{ħ}{2\Delta x}}{e}^{-{\displaystyle \frac{1}{2}}W(-{\displaystyle \frac{\alpha L_p^2}{2\Delta x^2}})}.$$

(9)

Throughout this paper, unless otherwise stated, we adopt natural units in which the Planck length ($L_p$) and the reduced Planck constant (ħ) are set to unity ($L_P=ħ=1$).

3. GUP-Induced Effective Metric

Following a proposal by Ong [50], we will briefly review the methods for finding effective metrics from the leading order GUP in the Planck length and then extend these methods to the all-order GUP in the Planck length. An effective metric represents a modified metric tensor that describes how GUP influences the geometry of spacetime.

3.1. Effective Metric from the Leading-Order GUP-Corrected Temperature

According to Adler, Chen, and Santiago (ACS) [17], by assuming that photons escape from a Schwarzschild black hole at its event horizon (radius $r_H=2M$, where M is the black hole’s mass) and that the spectrum of these escaping photons is thermal, one can derive the Hawking temperature for an asymptotically flat Schwarzschild black hole. The generalized momentum $\Delta p$ is identified with the characteristic energy of the Hawking particles, $E=k_{B}T=pc$ (where $k_B$ is the Boltzmann constant, T is temperature, and c is the speed of light), and the generalized position $\Delta x$ is identified with the horizon size $r_H$. Then, from the lower bound of the generalized momentum uncertainty in Equation (2), the Hawking temperature incorporating the GUP effect can be obtained as

$$T_{\mathrm{GUP}}={\displaystyle \frac{M}{\pi \alpha }}\left(1-\sqrt{1-{\displaystyle \frac{\alpha }{4M^2}}}\right).$$

(10)

A factor of $1/2\pi$ has been introduced here so that as $\alpha \to 0$, we recover the standard Hawking temperature of a Schwarzschild black hole, $T_{\mathrm{Sch}}=1/8\pi M$.

To incorporate GUP into an effective metric, one can consider a metric ansatz, without loss of generality, of the form

$$d{s}^2=-f\left(r\right)d{t}^2+f{\left(r\right)}^{-1}d{r}^2+r^{2}d{\Omega }^2.$$

(11)

Here, $f\left(r\right)$ is a function of the radial coordinate r, and $d{\Omega }^2$ is the line element of a unit 2-sphere. Assuming the areal radius remains intact (see discussion in [50]), $f\left(r\right)$ can be written as

$$f\left(r\right)=f_S\left(r\right)g\left(r\right),\mathrm{with}{f}_S\left(r\right)\equiv \left(1-{\displaystyle \frac{2M}{r}}\right).$$

(12)

This form of ansatz has the advantage that it modifies the Hawking temperature simply by a proportionality to the function $g\left(r\right)$

$$T={\displaystyle \frac{1}{4\pi }}{\left.f^{\prime }\left(r\right)\right|}_{r=r_H}={\displaystyle \frac{g\left(r_H\right)}{8\pi M}}.$$

(13)

Equating this temperature with the GUP-induced temperature (10), one obtains

$$g\left(r_H\right)={\displaystyle \frac{2r_H^2}{\alpha }}\left(1-\sqrt{1-{\displaystyle \frac{\alpha }{r_H^2}}}\right).$$

(14)

Thus, the explicit form of $g\left(r\right)$ can be inferred as

$$g\left(r\right)={\displaystyle \frac{2r^2}{\alpha }}\left(1-\sqrt{1-{\displaystyle \frac{\alpha }{r^2}}}\right).$$

(15)

This leads to the following candidate for the effective metric function $f_A\left(r\right)$ corresponding to the leading-order GUP corrected temperature as

$$f_A\left(r\right)=\left(1-{\displaystyle \frac{2M}{r}}\right){\displaystyle \frac{2r^2}{\alpha }}\left(1-\sqrt{1-{\displaystyle \frac{\alpha }{r^2}}}\right).$$

(16)

3.2. Effective Metric from the Leading-Order GUP-Corrected Entropy

The entropy from the GUP temperature (10) can be found by integrating $1/T_{GUP}$ with respect to M

$$\begin{array}{ccc}\hfill S_{\mathrm{GUP}}& =& \int {\displaystyle \frac{dM}{T_{\mathrm{GUP}}}}\hfill \\ & =& 2\pi M^2\left(1+\sqrt{1-{\displaystyle \frac{\alpha }{4M^2}}}\right)-{\displaystyle \frac{\pi \alpha }{2}}ln\left(M+M\sqrt{1-{\displaystyle \frac{\alpha }{4M^2}}}\right),\hfill \end{array}$$

(17)

up to some constant terms that depend only on $\alpha$. For large M, this entropy approximates to

$$S_{\mathrm{GUP}}\simeq 4\pi M^2-{\displaystyle \frac{\pi \alpha }{4}}ln\left(4\pi M^2\right)+{\displaystyle \frac{{\pi }^2{\alpha }^2}{16}}{\left(4\pi M^2\right)}^{-1}+{\displaystyle \frac{{\pi }^3{\alpha }^3}{64}}{\left(4\pi M^2\right)}^{-2}+\cdots .$$

(18)

Another way to obtain an effective metric is to start with the series-expanded entropy (18), noting that $d{S}_{\mathrm{GUP}}/dM=1/T$, where T is given by Equation (13). Then, in the lowest order in $\alpha$, one has

$$8\pi M\left(1-{\displaystyle \frac{\alpha }{16M^2}}\right)={\displaystyle \frac{8\pi M}{g\left(r_H\right)}}.$$

(19)

Thus, one can infer that

$$g\left(r\right)={\left(1-{\displaystyle \frac{\alpha }{4r^2}}\right)}^{-1}.$$

(20)

And we finally arrive at the effective metric function $f_B\left(r\right)$

$$f_B\left(r\right)=\left(1-{\displaystyle \frac{2M}{r}}\right){\left(1-{\displaystyle \frac{\alpha }{4r^2}}\right)}^{-1}.$$

(21)

Note that compared with Equation (16), this effective metric has a curvature singularity at $r=\sqrt{\alpha }/2$ [48] but one in Equation (16) at $r=\sqrt{\alpha }$ [50,51].

3.3. Effective Metric from the All-Order GUP-Corrected Temperature

As in the Section 3.1, according to ACS, the spectrum of escaping photons that satisfy the all-order GUP correction gives us the corresponding GUP temperature as

$$T_{\mathrm{GUP}}={\displaystyle \frac{1}{4\pi r_H}}{e}^{-{\displaystyle \frac{1}{2}}W(-{\displaystyle \frac{\alpha }{2r_H^2}})}.$$

(22)

Comparing this with the temperature in Equation (13) from the metric ansatz (12), one can infer

$$g\left(r\right)=e^{-{\displaystyle \frac{1}{2}}W(-{\displaystyle \frac{\alpha }{2r^2}})}.$$

(23)

As a result, one can arrive at an effective metric function $f_C\left(r\right)$ as

$$f_C\left(r\right)=\left(1-{\displaystyle \frac{2M}{r}}\right)e^{-{\displaystyle \frac{1}{2}}W(-{\displaystyle \frac{\alpha }{2r^2}})}.$$

(24)

This is the effective metric corrected by the GUP to all orders in the Planck length.

Finally, for later use, we summarize the surface gravities ${\kappa }_H(={\displaystyle \frac{1}{2}}{\displaystyle \frac{df}{dr}}{\mid }_{r=r_H})$ corresponding to the effective metrics as follows

$$\begin{array}{ccc}\hfill {\kappa }_S& =& {\left.{\displaystyle \frac{1}{2}}{\displaystyle \frac{d{f}_S}{dr}}\right|}_{r=r_H}={\displaystyle \frac{1}{4M}},\hfill \end{array}$$

(25)

$$\begin{array}{ccc}\hfill {\kappa }_A& =& {\left.{\displaystyle \frac{1}{2}}{\displaystyle \frac{d{f}_A}{dr}}\right|}_{r=r_H}={\displaystyle \frac{2M}{\alpha }}\left(1-\sqrt{1-{\displaystyle \frac{\alpha }{4M^2}}}\right),\hfill \end{array}$$

(26)

$$\begin{array}{ccc}\hfill {\kappa }_B& =& {\left.{\displaystyle \frac{1}{2}}{\displaystyle \frac{d{f}_B}{dr}}\right|}_{r=r_H}={\displaystyle \frac{1}{4M\left(1-\frac{\alpha }{16M^2}\right)}},\hfill \end{array}$$

(27)

$$\begin{array}{ccc}\hfill {\kappa }_C& =& {\left.{\displaystyle \frac{1}{2}}{\displaystyle \frac{d{f}_C}{dr}}\right|}_{r=r_H}={\displaystyle \frac{e^{-\frac{1}{2}W(-\frac{\alpha }{2r^2})}}{4M}}.\hfill \end{array}$$

(28)

Here, $f_S$, $f_A$, $f_B$, and $f_C$ are the original Schwarzschild metric (12) and the effective metrics (16), (21) and (24) for the leading-order GUP-corrected temperature, the leading-order GUP-corrected entropy and the all-order GUP-corrected temperature, respectively. Note also that when the GUP parameter $\alpha$ is very small, they become approximately

$$\begin{array}{ccc}\hfill {\kappa }_A& \simeq & {\displaystyle \frac{1}{4M}}+{\displaystyle \frac{\alpha }{64M^3}}+{\displaystyle \frac{{\alpha }^2}{512M^5}}+𝒪\left({\alpha }^3\right),\hfill \end{array}$$

(29)

$$\begin{array}{ccc}\hfill {\kappa }_B& \simeq & {\displaystyle \frac{1}{4M}}+{\displaystyle \frac{\alpha }{64M^3}}+{\displaystyle \frac{{\alpha }^2}{1024M^5}}+𝒪\left({\alpha }^3\right),\hfill \end{array}$$

(30)

$$\begin{array}{ccc}\hfill {\kappa }_C& \simeq & {\displaystyle \frac{1}{4M}}+{\displaystyle \frac{\alpha }{64M^3}}+{\displaystyle \frac{5{\alpha }^2}{2048M^5}}+𝒪\left({\alpha }^3\right),\hfill \end{array}$$

(31)

respectively, except for the pure Schwarzschild metric. As a result, one can see that

$${\kappa }_S<{\kappa }_B<{\kappa }_A<{\kappa }_C.$$

(32)

4. Revisiting the Brick Wall Model Using Effective Metrics

The statistical entropy of black holes, as derived using ’t Hooft’s brick wall method, arises from the analysis of quantum fields in curved spacetime. Starting with the Klein–Gordon equation for a scalar field $\Phi$ as

$$({\nabla }^2-m^2)\Phi =0,$$

(33)

the radial modes are decomposed as $\Phi =e^{-i\omega t}\phi \left(r\right)Y(\theta ,\varphi )$, where $Y(\theta ,\varphi )$ are spherical harmonics. Quantizing these modes yields the free energy F for bosonic fields

$$F=-{\displaystyle \frac{1}{\pi }}{\int }_0^{\infty }{\displaystyle \frac{g\left(\omega \right)}{e^{\beta \omega }-1}}d\omega ,$$

(34)

where $g\left(\omega \right)$ represents the density of states derived from the metric by

$$g\left(\omega \right)={\displaystyle \frac{2}{3}}{\int }_{r_0}^{r_1}{\displaystyle \frac{r^2}{f^{1/2}\left(r\right)}}{\left({\displaystyle \frac{{\omega }^2}{f\left(r\right)}}-m^2\right)}^{3/2}dr.$$

(35)

The statistical entropy is then given by

$$S={\beta }^2{\left.{\displaystyle \frac{\partial F}{\partial \beta }}\right|}_{\beta ={\beta }_H},$$

(36)

where ${\beta }_H=2\pi /{\kappa }_H$.

The original brick wall method imposes boundary conditions

$$\phi (r_H+h)=\phi \left(L\right)=0.$$

(37)

Here, $\phi \left(r\right)$ is the radial wave function, h is a UV cutoff near the horizon, and L is an IR cutoff confining the system. These cutoffs are necessary to regulate divergences in the density of states.

The free energy simplifies under a small mass approximation to the term

$$F\simeq -{\displaystyle \frac{2{\pi }^3}{45{\beta }^4}}{\int }_{r_H+h}^L{\displaystyle \frac{r^2}{f^2\left(r\right)}}dr,$$

(38)

to be integrated. Then, for the Schwarzschild black hole, this leads to

$$F_S\simeq -{\displaystyle \frac{2{\pi }^3}{45{\beta }^4}}\left({\displaystyle \frac{16M^4}{h_S}}+{\displaystyle \frac{L^3}{3}}+32M^{3}log{\displaystyle \frac{L}{h_S}}\right),$$

(39)

where the last two terms are the contribution from the vacuum surrounding the system at large distance. The $1/h_S$ term dominates because $h_S\ll L$, making vacuum contributions negligible. Therefore, we have

$$F_S\simeq -{\displaystyle \frac{32{\pi }^3{M}^4}{45{\beta }^4{h}_S}},$$

(40)

and can find the entropy as

$$S_S\simeq {\displaystyle \frac{1}{720\pi M{h}_S}}\left({\displaystyle \frac{A}{4}}\right).$$

(41)

Here, $A=16\pi M^2$ is the area of the event horizon. By choosing the cutoff $h_S$ as

$$h_S={\displaystyle \frac{1}{720\pi M}},$$

(42)

the statistical entropy matches the Bekenstein–Hawking entropy

$$S={\displaystyle \frac{A}{4}}.$$

(43)

It is important to note that the brick wall cutoff h is coordinate-dependent, which is considered a coordinate artifact. To address this, we introduce the invariant distance given by

$$l_{\mathrm{inv}}={\int }_{r_H}^{r_H+h_S}{\displaystyle \frac{dr}{\sqrt{f\left(r\right)}}}\simeq 2\sqrt{2M{h}_S}.$$

(44)

Then, the entropy can be rewritten in terms of this invariant distance, allowing for a mass-independent, constant cutoff while preserving the area law

$$S={\displaystyle \frac{1}{90\pi l_{\mathrm{inv}}^2}}\left({\displaystyle \frac{A}{4}}\right).$$

(45)

Therefore, if we choose the invariant distance as

$$l_{\mathrm{inv}}={\displaystyle \frac{1}{\sqrt{90\pi }}},$$

(46)

we have the mass-independent constant cutoff while keeping the area law intact.

As we have seen in the previous sections, modifications from the GUP alter the effective metric, introducing $\alpha$-dependent terms. For the effective metric derived from the leading-order GUP-corrected temperature, one can find the free energy as

$$\begin{array}{ccc}\hfill F_A\simeq & -& {\displaystyle \frac{2{\pi }^3}{45{\beta }^4}}\left({\displaystyle \frac{8M^4(1+\sqrt{1-\frac{\alpha }{4M^2}})-M^2\alpha }{h_A}}+{\displaystyle \frac{L^3}{6}}\left(1-\sqrt{1-{\displaystyle \frac{\alpha }{L^2}}}\right)\right.\hfill \\ & +& \left.16M\left(M^2-{\displaystyle \frac{\alpha }{16}}+{\displaystyle \frac{2M^3-\frac{3}{8}M\alpha }{\sqrt{4M^2-\alpha }}}\right)log{\displaystyle \frac{L}{h_A}}\right).\hfill \end{array}$$

(47)

Then, from the first term, after ignoring the vacuum contributing the last two terms, the entropy can be obtained as

$$S_A\simeq {\displaystyle \frac{1}{360\pi M{h}_A(1+\sqrt{1-\frac{\alpha }{4M^2}})}}\left({\displaystyle \frac{A}{4}}\right).$$

(48)

Therefore, if we choose the cutoff $h_A$ as

$$h_A={\displaystyle \frac{1}{360\pi M(1+\sqrt{1-\frac{\alpha }{4M^2}})}},$$

(49)

the statistical entropy recovers the area law in black hole thermodynamics. Moreover, the coordinate artifact, the brick wall cutoff $h_A$ can be replaced by the invariant distance

$$l_{A,\mathrm{inv}}={\displaystyle \frac{1}{\sqrt{90\pi }}},$$

(50)

by following the same procedure as before. This is the same invariant distance as the one of the original Schwarzschild black hole. Note that the entropy $S_A$ and the cutoff $h_A$ recover the correct limits of Equations (41) and (42) as $\alpha \to 0$, respectively. And the invariant distance $l_{A,\mathrm{inv}}^2$ remains the same with Equation (46).

For the effective metric from the leading-order GUP-corrected entropy, the free energy is

$$F_B\simeq -{\displaystyle \frac{2{\pi }^3}{45{\beta }^4}}\left({\displaystyle \frac{16{(M^2-\frac{\alpha }{16})}^2}{h_B}}+{\displaystyle \frac{L^3}{3}}+32M\left(M^2-{\displaystyle \frac{\alpha }{16}}\right)log{\displaystyle \frac{L}{h_B}}\right).$$

(51)

Ignoring the vacuum contributing the last two terms, as before, the entropy from the first term is given by

$$S_B\simeq {\displaystyle \frac{1}{720\pi M{h}_B(1-\frac{\alpha }{16M^2})}}\left({\displaystyle \frac{A}{4}}\right).$$

(52)

Therefore, if we choose the cutoff $h_B$ as

$$h_B={\displaystyle \frac{1}{720\pi M(1-\frac{\alpha }{16M^2})}},$$

(53)

the statistical entropy recovers the area law in black hole thermodynamics. Moreover, the brick wall cutoff $h_B$ can be replaced by the invariant distance

$$l_{B,\mathrm{inv}}={\displaystyle \frac{1}{\sqrt{90\pi }}},$$

(54)

by following the same procedure as before. Note that the entropy $S_B$ and the cutoff $h_B$ recover the correct limits of Equations (41) and (42), respectively, as $\alpha \to 0$.

Finally, for the effective metric from the all-order GUP-corrected temperature, the free energy is approximately

$$F_C\simeq -{\displaystyle \frac{2{\pi }^3}{45{\beta }^4}}\left(16M^3{e}^{W(-{\displaystyle \frac{\alpha }{8M^2}})}\left({\displaystyle \frac{M}{h_C}}+log{\displaystyle \frac{L}{h_C}}\right)\right).$$

(55)

After ignoring the vacuum contributing the last term, as before, the entropy from the first term is given by

$$S_C\simeq {\displaystyle \frac{e^{-\frac{1}{2}W(-\frac{\alpha }{8M^2})}}{720\pi M{h}_C}}\left({\displaystyle \frac{A}{4}}\right).$$

(56)

Therefore, if we choose the cutoff $h_C$ as

$$h_C={\displaystyle \frac{e^{-\frac{1}{2}W(-\frac{\alpha }{8M^2})}}{720\pi M}},$$

(57)

the statistical entropy recovers the area law in the black hole thermodynamics. Moreover, the brick wall cutoff $h_C$ can be replaced by the invariant distance

$$l_{C,\mathrm{inv}}={\displaystyle \frac{1}{\sqrt{90\pi }}},$$

(58)

by following the same procedure as before. Note that the entropy $S_C$ and the cutoff $h_C$ recover the correct limits of Equations (41) and (42), respectively, as $\alpha \to 0$.

Table 1 summarizes the cutoffs and invariant distances for the GUP-corrected effective metrics of the Schwarzschild black hole. The UV cutoffs near the event horizon are apparently different from each other; however, interestingly, the invariant distances are the same in all cases. One can compare their relative differences by expanding in $\alpha$ as

$$\begin{array}{ccc}{h}_A& \simeq & {\displaystyle \frac{1}{720\pi M}}+{\displaystyle \frac{\alpha }{11520\pi M^3}}+{\displaystyle \frac{{\alpha }^2}{92160\pi M^5}}+𝒪\left({\alpha }^3\right),\hfill \\ \hfill h_B& \simeq & {\displaystyle \frac{1}{720\pi M}}+{\displaystyle \frac{\alpha }{11520\pi M^3}}+{\displaystyle \frac{{\alpha }^2}{184320\pi M^5}}+𝒪\left({\alpha }^3\right),\hfill \\ \hfill h_C& \simeq & {\displaystyle \frac{1}{720\pi M}}+{\displaystyle \frac{\alpha }{11520\pi M^3}}+{\displaystyle \frac{{\alpha }^2}{73728\pi M^5}}+𝒪\left({\alpha }^3\right),\hfill \end{array}$$

(59)

showing that

$$h_S<h_B<h_A<h_C.$$

(60)

Table 1. Effective metrics, cutoff h formulas, and invariant distances $l_{\mathrm{inv}}$.

Metric	Cutoff h Formula	Invariant Distance ${\mathit{l}}_{\mathbf{inv}}$
Schwarzschild	$h_S={\displaystyle \frac{1}{720\pi M}}$	${\displaystyle \frac{1}{\sqrt{90\pi }}}$
$f_A\left(r\right)$	$h_A={\displaystyle \frac{1}{360\pi M(1+\sqrt{1-\frac{\alpha }{4M^2}})}}$	${\displaystyle \frac{1}{\sqrt{90\pi }}}$
$f_B\left(r\right)$	$h_B={\displaystyle \frac{1}{720\pi M(1-\frac{\alpha }{16M^2})}}$	${\displaystyle \frac{1}{\sqrt{90\pi }}}$
$f_C\left(r\right)$	$h_C={\displaystyle \frac{e^{-\frac{1}{2}W(-\frac{\alpha }{8M^2})}}{720\pi M}}$	${\displaystyle \frac{1}{\sqrt{90\pi }}}$

5. Statistical Entropy Based on the GUP Induced Effective Metrics

Let us calculate the statistical entropy of a free scalar field on the Schwarzschild black hole with the effective metrics considering the near-horizon contributions of quantum states. First of all, it is well-known that when the gravity is turned on, the number of quantum states in a volume element in phase space are changed from ${\left(2\pi \right)}^3$ into ${\left(2\pi \right)}^3{(1+\alpha p^2)}^3$ for the leading-order GUP correction, and ${\left(2\pi \right)}^3{e}^{\alpha p^2}$ for the all-order GUP correction, respectively [23,26,27,28]. Specifically, in (3 + 1) dimensions, they are

$$dn={\displaystyle \frac{d^{3}x{d}^{3}p}{{\left(2\pi \right)}^3{(1+\alpha p^2)}^3}},$$

(61)

for the leading-order GUP corrections, and

$$dn={\displaystyle \frac{d^{3}x{d}^{3}p}{{\left(2\pi \right)}^3}}{e}^{-\alpha p^2},$$

(62)

for the all-order GUP correction. Note that both the leading-order GUP-corrected temperature and entropy approaches yield the same phase space modification, as given in Equation (61). The square module of momentum $p^2$ is given by

$$p^2\equiv g^{rr}{p_r}^2+g^{\theta \theta }{p_{\theta }}^2+g^{\varphi \varphi }{p_{\varphi }}^2={\displaystyle \frac{{\omega }^2}{f}}-{\mu }^2.$$

(63)

When $\alpha \to 0$, they are reduced to the usual number of quantum states in HUP [22].

Now, substituting the ansatz of the wave function $\Phi (t,r,\theta ,\varphi )=e^{-i\omega t}\psi (r,\theta ,\varphi )$ in the Klein–Gordon Equation (33), we have

$${\partial }_r^2\psi +\left({\displaystyle \frac{f^{\prime }}{f}}+{\displaystyle \frac{2}{r}}\right){\partial }_r\psi +{\displaystyle \frac{1}{f}}\left({\displaystyle \frac{1}{r^2}}\left[{\partial }_{\theta }^2+\cot \theta {\partial }_{\theta }+{\displaystyle \frac{1}{{\sin }^2\theta }}{\partial }_{\varphi }^2\right]+{\displaystyle \frac{{\omega }^2}{f}}-{\mu }^2\right)\psi =0,$$

(64)

where $f^{\prime }$ denotes the differentiation with respect to r. By using the Wenzel–Kramers–Brillouin approximation [22] with $\psi \sim e^{iS(r,\theta ,\varphi )}$ and keeping the real parts, we have the following modified dispersion relation

$$p_{\mu }{p}^{\mu }=-{\displaystyle \frac{{\omega }^2}{f}}+f{p_r}^2+{\displaystyle \frac{p_{\theta }^2}{r^2}}+{\displaystyle \frac{p_{\varphi }^2}{r^2{\sin }^2\theta }}=-{\mu }^2,$$

(65)

where $p_r={\displaystyle \frac{\partial S}{\partial r}}$, $p_{\theta }={\displaystyle \frac{\partial S}{\partial \theta }}$ and $p_{\varphi }={\displaystyle \frac{\partial S}{\partial \varphi }}$. Then, one can easily calculate the volume of the momentum phase space as

$$\begin{array}{c}\hfill V_p(r,\theta )=\int d{p}_{r}d{p}_{\theta }d{p}_{\varphi }={\displaystyle \frac{4\pi }{3}}{\displaystyle \frac{r^2\sin \theta }{\sqrt{f}}}{\left({\displaystyle \frac{{\omega }^2}{f}}-{\mu }^2\right)}^{{\displaystyle \frac{3}{2}}},\end{array}$$

(66)

which satisfy $\omega \ge \mu \sqrt{f}$.

5.1. Leading-Order GUP Correction

From Equations (61) and (63), the number of quantum states related to the radial mode with energy less than $\omega$ is given by

$$\begin{array}{c}\hfill n\left(\omega \right)=\int dn={\displaystyle \frac{2}{3\pi }}{\int }_{r_H}dr{\displaystyle \frac{r^2{\left(\frac{{\omega }^2}{f}-{\mu }^2\right)}^{\frac{3}{2}}}{\sqrt{f}{\left(1+\alpha (\frac{{\omega }^2}{f}-{\mu }^2)\right)}^3}}.\end{array}$$

(67)

For the bosonic case, the free energy of a thermal ensemble of scalar fields at inverse temperature $\beta$ is given by

$$\begin{array}{ccc}\hfill F& =& {\displaystyle \frac{1}{\beta }}\sum _{K}ln\left(1-e^{-\beta {\omega }_K}\right)\hfill \\ & =& -{\displaystyle \frac{2}{3\pi }}{\int }_{r_H}dr{\displaystyle \frac{r^2}{\sqrt{f}}}{\int }_{\mu \sqrt{f}}^{\infty }d\omega {\displaystyle \frac{{\left(\frac{{\omega }^2}{f}-{\mu }^2\right)}^{\frac{3}{2}}}{(e^{\beta \omega }-1){\left(1+\alpha (\frac{{\omega }^2}{f}-{\mu }^2)\right)}^3}}.\hfill \end{array}$$

(68)

Here, we have considered the continuum limit, integrated it by parts, and used the number of quantum states in Equation (67).

Now, we are only interested in the contribution from just the vicinity near the event horizon in the range of $(r_H,r_H+ϵ)$ where $ϵ$ is the brick wall cutoff used to remove ultraviolet divergences. Since $f\to 0$ near the event horizon, the ${\displaystyle \frac{{\omega }^2}{f}}$ term is dominant in ${\displaystyle \frac{{\omega }^2}{f}}-{\mu }^2$, and we do not need to require the small mass approximation. Then, the free energy can be rewritten as

$$F=-{\displaystyle \frac{2}{3\pi }}{\int }_{r_H}^{r_H+ϵ}dr{\displaystyle \frac{r^2}{f^2}}{\int }_0^{\infty }d\omega {\displaystyle \frac{{\omega }^3}{(e^{\beta \omega }-1){\left(1+\frac{\alpha {\omega }^2}{f}\right)}^3}}.$$

(69)

Then, from F in Equation (69), one can find the entropy as

$$\begin{array}{c}\hfill S={\displaystyle \frac{{\beta }_H^2}{6\pi }}{\int }_{r_H}^{r_H+ϵ}dr{\displaystyle \frac{r^2}{f^2}}{\int }_0^{\infty }d\omega {\displaystyle \frac{{\omega }^4}{{sinh}^2\left(\frac{{\beta }_H}{2}\omega \right){\left(1+\frac{\alpha {\omega }^2}{f}\right)}^3}}.\end{array}$$

(70)

Introducing $x=\beta \omega /2$, the entropy can be rewritten as

$$S={\displaystyle \frac{16}{3\pi {\beta }_H^3}}{\int }_{r_H}^{r_H+ϵ}dr{\displaystyle \frac{r^2}{f^2}}{\int }_0^{\infty }dx{\displaystyle \frac{x^4}{{sinh}^2\left(x\right){\left(1+\frac{4\alpha x^2}{{\beta }_H^{2}f}\right)}^3}}.$$

(71)

Making use of the inequality

$${sinh}^2\left(x\right)\ge x^2,$$

(72)

and after performing $\omega$ integration, one can obtain

$$\begin{array}{ccc}\hfill S& <& {\displaystyle \frac{16}{3\pi {\beta }_H^3}}{\int }_{r_H}^{r_H+ϵ}dr{\displaystyle \frac{r^2}{f^2}}{\int }_0^{\infty }dx{\displaystyle \frac{x^2}{{\left(1+\frac{4\alpha x^2}{{\beta }_H^{2}f}\right)}^3}}\hfill \\ & =& {\displaystyle \frac{1}{24{\alpha }^{3/2}}}{\int }_{r_H}^{r_H+ϵ}dr{\displaystyle \frac{r^2}{\sqrt{f}}}.\hfill \end{array}$$

(73)

Since we are only interested in the contribution from just the vicinity near the horizon in the range $(r_H,r_H+ϵ)$, this integration finally becomes

$$S<{\displaystyle \frac{1}{24\pi {\alpha }^{3/2}}}\sqrt{{\displaystyle \frac{2ϵ}{{\kappa }_H}}}\left({\displaystyle \frac{A}{4}}\right)+𝒪\left({ϵ}^{3/2}\right).$$

(74)

In the leading-order GUP correction, the minimum length is given by Equation (3), and by identifying the invariant length with this, we have

$$S<{\displaystyle \frac{1}{24\pi \alpha }}\left({\displaystyle \frac{A}{4}}\right).$$

(75)

When we choose the GUP parameter $\alpha$ as

$$\alpha ={\displaystyle \frac{1}{24\pi }},$$

(76)

we can finally obtain the entropy as

$$S={\displaystyle \frac{A}{4}}.$$

(77)

As a result, we have obtained the Bekenstein–Hawking entropy satisfying the area law exactly.

It is appropriate to comment that the GUP parameter $\alpha$ is $\alpha \approx 0.0133$. On the other hand, in ref. [23], the GUP parameter $\lambda$, which is the same as our $\alpha$, was $\lambda ={\displaystyle \frac{3}{4\pi }}\approx 0.2387$. Therefore, the correction in this paper is much better than the ones in [23] and stricter than in [31] where $\lambda ={\displaystyle \frac{\sqrt{e}}{6\sqrt{2}{\pi }^{3/2}}}\approx 0.0349$.

5.2. All-Order GUP Correction

In the case of all-order GUP correction in the Planck length, the number of quantum states associated with the radial mode is given by

$$\begin{array}{c}\hfill n\left(\omega \right)=\int dn={\displaystyle \frac{2}{3\pi }}{\int }_{r_H}dr{\displaystyle \frac{r^2}{\sqrt{f}}}{\left({\displaystyle \frac{{\omega }^2}{f}}-{\mu }^2\right)}^{{\displaystyle \frac{3}{2}}}{e}^{-\alpha ({\displaystyle \frac{{\omega }^2}{f}}-{\mu }^2)}.\end{array}$$

(78)

It is important to note that $n\left(\omega \right)$ remains finite at the horizon without the need for any artificial cutoff, due to the presence of the exponential suppression term $e^{-\alpha {\omega }^2/f}$ induced by the GUP.

The free energy of a thermal ensemble of scalar fields is then

$$\begin{array}{c}\hfill F=-{\displaystyle \frac{2}{3\pi }}{\int }_{r_H}dr{\displaystyle \frac{r^2}{\sqrt{f}}}{\int }_{\mu \sqrt{f}}^{\infty }d\omega {\displaystyle \frac{{\left(\frac{{\omega }^2}{f}-{\mu }^2\right)}^{\frac{3}{2}}}{e^{\beta \omega }-1}}{e}^{-\alpha ({\displaystyle \frac{{\omega }^2}{f}}-{\mu }^2)}.\end{array}$$

(79)

Near the event horizon, the dominant contribution to the free energy simplifies to

$$F=-{\displaystyle \frac{2}{3\pi }}{\int }_{r_H}^{r_H+ϵ}dr{\displaystyle \frac{r^2}{f^2}}{\int }_0^{\infty }d\omega {\displaystyle \frac{{\omega }^3}{e^{\beta \omega }-1}}{e}^{-{\displaystyle \frac{\alpha {\omega }^2}{f}}}.$$

(80)

After integrating over $\omega$, the entropy can be expressed as

$$\begin{array}{c}\hfill S={\displaystyle \frac{{\beta }_H^2}{6\pi }}{\int }_{r_H}^{r_H+ϵ}dr{\displaystyle \frac{r^2}{f^2}}{\int }_0^{\infty }d\omega {\displaystyle \frac{{\omega }^4}{{sinh}^2\left(\frac{{\beta }_H}{2}\omega \right)}}{e}^{-{\displaystyle \frac{\alpha {\omega }^2}{f}}}.\end{array}$$

(81)

By introducing the substitution $x=\sqrt{\alpha }\omega$, this becomes

$$\begin{array}{c}\hfill S={\displaystyle \frac{{\beta }_H^2}{6\pi {\alpha }^2\sqrt{\alpha }}}{\int }_0^{\infty }dx{\displaystyle \frac{x^4}{{sinh}^2\left(\frac{{\beta }_H}{2\sqrt{\alpha }}x\right)}}\Lambda (x,ϵ),\end{array}$$

(82)

where

$$\begin{array}{c}\hfill \Lambda (x,ϵ)\equiv {\int }_{r_H}^{r_H+ϵ}dr{\displaystyle \frac{r^2}{f^2}}{e}^{-{\displaystyle \frac{x^2}{f}}}.\end{array}$$

(83)

Focusing on the near-horizon region, Equation (83) reduces to

$$\Lambda (x,ϵ)\approx {\int }_{r_H}^{r_H+ϵ}dr{\displaystyle \frac{r^2}{{\left[2{\kappa }_H(r-r_H)\right]}^2}}{e}^{-{\displaystyle \frac{x^2}{2{\kappa }_H(r-r_H)}}}.$$

(84)

This integral can be evaluated exactly by substituting $t=x^2/2{\kappa }_H(r-r_H)$, yielding

$$\begin{array}{ccc}\hfill \Lambda (x,ϵ)& =& {\displaystyle \frac{1}{2{\kappa }_H{x}^2}}{\int }_{{\displaystyle \frac{x^2}{2{\kappa }_Hϵ}}}^{\infty }dt\left(r_H^2+{\displaystyle \frac{r_H{x}^2}{{\kappa }_{H}t}}+{\displaystyle \frac{x^4}{4{\kappa }_H^2{t}^2}}\right)e^{-t}\hfill \\ & =& {\displaystyle \frac{r_H^2}{2{\kappa }_H{x}^2}}\Gamma (1,{\displaystyle \frac{x^2}{2{\kappa }_Hϵ}})+{\displaystyle \frac{r_H}{2{\kappa }_H^2}}\Gamma (0,{\displaystyle \frac{x^2}{2{\kappa }_Hϵ}})+{\displaystyle \frac{x^2}{8{\kappa }_H^3}}\Gamma (-1,{\displaystyle \frac{x^2}{2{\kappa }_Hϵ}}),\hfill \end{array}$$

(85)

where the incomplete Gamma function is defined as

$$\Gamma (a,z)={\int }_z^{\infty }dt{t}^{a-1}{e}^{-t}.$$

(86)

The all-order GUP-corrected entropy can thus be written as

$$\begin{array}{c}\hfill S=S_1+S_2+S_3\end{array}$$

(87)

where

$$\begin{array}{ccc}\hfill S_1& =& {\displaystyle \frac{{\beta }_H^2{r}_H^2}{12\pi {\alpha }^2\sqrt{\alpha }{\kappa }_H}}{\int }_0^{\infty }dx{\displaystyle \frac{x^2}{{sinh}^2\left(\frac{{\beta }_{H}x}{2\sqrt{\alpha }}\right)}}\Gamma (1,{\displaystyle \frac{x^2}{2{\kappa }_Hϵ}}),\hfill \end{array}$$

(88)

$$\begin{array}{ccc}\hfill S_2& =& {\displaystyle \frac{{\beta }_H^2{r}_H}{12\pi {\alpha }^2\sqrt{\alpha }{\kappa }_H^2}}{\int }_0^{\infty }dx{\displaystyle \frac{x^4}{{sinh}^2\left(\frac{{\beta }_{H}x}{2\sqrt{\alpha }}\right)}}\Gamma (0,{\displaystyle \frac{x^2}{2{\kappa }_Hϵ}}),\hfill \end{array}$$

(89)

$$\begin{array}{ccc}\hfill S_3& =& {\displaystyle \frac{{\beta }_H^2}{48\pi {\alpha }^2\sqrt{\alpha }{\kappa }_H^3}}{\int }_0^{\infty }dx{\displaystyle \frac{x^6}{{sinh}^2\left(\frac{{\beta }_{H}x}{2\sqrt{\alpha }}\right)}}\Gamma (-1,{\displaystyle \frac{x^2}{2{\kappa }_Hϵ}}).\hfill \end{array}$$

(90)

Redefining $y={\displaystyle \frac{{\beta }_{H}x}{2\sqrt{\alpha }}}$ and using the minimum length (8) with ${\beta }_H{\kappa }_H=2\pi$, these terms become

$$\begin{array}{ccc}\hfill S_1& =& {\displaystyle \frac{r_H^2}{3{\pi }^2\alpha }}{\int }_0^{\infty }dy{\displaystyle \frac{y^2}{{sinh}^{2}y}}\Gamma (1,{\displaystyle \frac{2y^2}{{\pi }^{2}e}}),\hfill \end{array}$$

(91)

$$\begin{array}{ccc}\hfill S_2& =& {\displaystyle \frac{r_H{\kappa }_H}{3{\pi }^4}}{\int }_0^{\infty }dy{\displaystyle \frac{y^4}{{sinh}^{2}y}}\Gamma (0,{\displaystyle \frac{2y^2}{{\pi }^{2}e}}),\hfill \end{array}$$

(92)

$$\begin{array}{ccc}\hfill S_3& =& {\displaystyle \frac{\alpha {\kappa }_H^2}{12{\pi }^6}}{\int }_0^{\infty }dy{\displaystyle \frac{y^6}{{sinh}^{2}y}}\Gamma (-1,{\displaystyle \frac{2y^2}{{\pi }^{2}e}}).\hfill \end{array}$$

(93)

Evaluating these integrals yields

$$\begin{array}{ccc}\hfill {\delta }_1& \equiv & {\int }_0^{\infty }dy{\displaystyle \frac{y^2\Gamma (1,\frac{2y^2}{{\pi }^{2}e})}{{sinh}^{2}y}}\approx 1.4509,\hfill \end{array}$$

(94)

$$\begin{array}{ccc}\hfill {\delta }_2& \equiv & {\int }_0^{\infty }dy{\displaystyle \frac{y^4\Gamma (0,\frac{2y^2}{{\pi }^{2}e})}{{sinh}^{2}y}}\approx 3.0709,\hfill \end{array}$$

(95)

$$\begin{array}{ccc}\hfill {\delta }_3& \equiv & {\int }_0^{\infty }dy{\displaystyle \frac{y^6\Gamma (-1,\frac{2y^2}{{\pi }^{2}e})}{{sinh}^{2}y}}\approx 18.4609.\hfill \end{array}$$

(96)

Thus, the entropy can be expressed as follows

$$\begin{array}{c}\hfill S={\displaystyle \frac{{\delta }_1}{3{\pi }^3\alpha }}\left({\displaystyle \frac{A}{4}}\right)+{\displaystyle \frac{{\delta }_2}{3{\pi }^4}}{r}_H{\kappa }_H+{\displaystyle \frac{{\delta }_3}{12{\pi }^6}}\alpha {\kappa }_H^2.\end{array}$$

(97)

Now, by using the surface gravity ${\kappa }_C$ from Equation (28) for the all-order GUP correction and considering the small $\alpha$ limit, we can simplify this to

$$S\simeq {\displaystyle \frac{4{\delta }_1{M}^2}{3{\pi }^2\alpha }}+{\displaystyle \frac{{\delta }_2}{6{\pi }^4}}+{\displaystyle \frac{(2{\pi }^2{\delta }_2+{\delta }_3)\alpha }{192{\pi }^6{M}^2}}+{\displaystyle \frac{(5{\pi }^2{\delta }_2+2{\delta }_3){\alpha }^2}{3072{\pi }^6{M}^4}}+𝒪\left({\alpha }^3\right).$$

(98)

In terms of the surface area $A=16\pi M^2$, this expression can be further rewritten as

$$S\simeq {\displaystyle \frac{{\delta }_1}{3{\pi }^3\alpha }}\left({\displaystyle \frac{A}{4}}\right)+{\displaystyle \frac{\alpha }{6{\pi }^3}}\left({\delta }_2+{\displaystyle \frac{{\delta }_3}{2{\pi }^2}}\right)A^{-1}+{\displaystyle \frac{5{\alpha }^2}{12{\pi }^2}}\left({\delta }_2+{\displaystyle \frac{2{\delta }_3}{5{\pi }^2}}\right)A^{-2}+𝒪\left({\alpha }^3{A}^{-3}\right).$$

(99)

This formulation includes a constant term and reveals that, as the order of $\alpha$ increases, the terms grow larger inversely with respect to the surface area A. This property, except for a logarithmic term, exhibits a characteristic of quantum-gravity-corrected black hole entropy [54,55].

When we choose the GUP parameter $\alpha$ as

$$\alpha ={\displaystyle \frac{{\delta }_1}{3{\pi }^3}}\approx 0.0156,$$

(100)

we arrive at the final entropy expression:

$$\begin{array}{c}\hfill S\simeq {\displaystyle \frac{A}{4}}+c_1{A}^{-1}+c_2{A}^{-2}+𝒪\left(A^{-3}\right),\end{array}$$

(101)

where $c_1=3.3589\times {10}^{-4}$ and $c_2=3.9223\times {10}^{-5}$. Thus, through the all-order GUP correction in the effective metric, we have finally obtained an entropy expression that upholds the area law, along with correction terms that are inversely proportional to the surface area.

6. Discussion

In this paper, we have studied effective metrics derived from leading-order GUP-corrected temperature and entropy and extended it to effective metric concerning all-order GUP correction in Planck length. Among these effective metrics, the approach based on the leading-order GUP-corrected temperature represents a specific limit of the all-order GUP-corrected temperature case, while the one based on the leading-order GUP-corrected entropy serves as an alternative construction within the GUP framework. Each approach is applicable in different regimes of quantum gravity corrections and is dependent on the initial assumptions made for the metric construction. Despite introducing three distinct GUP-corrected effective metrics, the statistical entropy calculated via t’ Hooft’s brick wall method consistently recovers the Bekenstein–Hawking area law, which shows the universality of the area law even in quantum-corrected spacetimes.

We have also shown that the divergent near-horizon contribution to entropy is regulated by a UV cutoff h, which differs for each effective metric, as shown in Equations (49), (53) and (57). These cutoffs ensure the entropy matches the area law, with all reducing to the Schwarzschild case as $\alpha \to 0$. On the other hand, to eliminate coordinate dependence, the UV cutoff h is replaced with the invariant distance $l_{\mathrm{inv}}^2$, and remarkably, all three metrics yield the same invariant distance

$$l_{\mathrm{inv}}={\displaystyle \frac{1}{\sqrt{90\pi }}},$$

(102)

confirming the regularization’s physical consistency across different effective metrics and the robustness of the area law under GUP modifications.

These results are further confirmed by explicitly counting quantum states in the vicinity of the event horizon without introducing any artificial cutoff. Both the leading-order and all-order GUP corrections naturally regularize the density of states, yielding finite entropy and reproducing the area law exactly when the GUP parameter is appropriately chosen. The analysis, particularly in all-order GUP-corrected temperature, shows that subleading corrections, including constant and inverse-area terms, arise in the entropy expansion, reflecting the quantum gravitational structure encoded by the GUP. These findings collectively demonstrate that the use of full, non-perturbative GUP corrections in effective metrics provides a consistent and physically meaningful framework for understanding black hole thermodynamics and that the area law remains universal even in the presence of quantum gravity effects.

For further study, it would be interesting to extend the analysis to the extended uncertainty principle and the generalized extended uncertainty principle. These alternative frameworks introduce large-distance corrections and new phenomenological features, particularly in cosmological or (A)dS backgrounds. They represent an important direction for a broader understanding of black hole thermodynamics. Additionally, it would be valuable to investigate rotating or charged black holes, as well as those in higher dimensions. Such studies will reinforce the connection between quantum gravity phenomenology and black hole thermodynamics, emphasizing the resilience of the area law under GUP modifications.