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# Quantum-Corrected Black Hole Solutions from Loop Quantum Gravity and Their Canonical Ensemble Analysis

Wen-Xiang Chen

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Wen-Xiang Chen. Quantum-Corrected Black Hole Solutions from Loop Quantum Gravity and Their Canonical Ensemble Analysis. 2025. hal-04972898

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# Quantum-Corrected Black Hole Solutions from Loop Quantum Gravity and Their Canonical Ensemble Analysis

Wen-Xiang Chen

Department of Astronomy

School of Physics and Materials Science

GuangZhou University

wxchen4277@qq.com

March 2, 2025

## Abstract

This study investigates quantum-corrected black hole solutions derived from Loop Quantum Gravity (LQG) and explores their thermodynamic properties using the canonical ensemble framework. By incorporating higher-order quantum correction terms into classical black hole metrics, we construct regular black hole solutions that eliminate classical singularities. Advanced canonical ensemble techniques, including path integral formulations and stability analyses, are employed to examine the thermodynamic stability, phase transitions, and critical phenomena of these quantum-corrected black holes. The results indicate that quantum corrections significantly alter the thermodynamic landscape, introducing novel phase structures and stability conditions. Additionally, numerical simulations are conducted to visualize the behavior of thermodynamic quantities under varying quantum correction parameters. This work provides deeper insights into the interplay between quantum gravity effects and black hole thermodynamics, contributing to the broader understanding of quantum gravitational phenomena in strong gravitational fields.

**Keywords:** Loop Quantum Gravity; Black Hole Solutions; Quantum Corrections; Canonical Ensemble; Thermodynamics; Phase Transitions; Regular Black Holes

## 1 Introduction

Black holes are among the most intriguing predictions of General Relativity (GR), characterized by regions of spacetime from which nothing, not even light, can escape. [1, 2]. The study of black hole physics has profound implications for our understanding of gravity, quantum mechanics, and the fundamental nature of spacetime. Despite their theoretical elegance, classical black hole solutions in GR suffer from singularities where physical quantities become infinite, signaling the breakdown of the theory [3, 4].

To address the singularity problem, various approaches to quantum gravity have been proposed. Loop Quantum Gravity (LQG) stands out as a non-perturbative and background-independent framework aiming to quantize spacetime itself [1,5]. Recent advancements in LQG have shown
promise in resolving singularities by introducing discrete spacetime structures at the Planck scale [6,7].

Black hole thermodynamics bridges the gap between gravity and quantum theory by at-
tributing thermodynamic quantities such as temperature and entropy to black holes [3, 4, 8–10,
10, 11]. Understanding the thermodynamic behavior of black holes, especially in the context of
quantum gravity corrections, is essential for elucidating the microscopic structure of spacetime
and the nature of quantum gravitational interactions.

This document investigates the Klein-Gordon equation in curved spacetime and its appli-
cation to the thermodynamics of a modified Schwarzschild black hole. It also explores the
construction of functions that satisfy the functional equation through the use of Laurent series
and Fourier transforms. These mathematical constructs are subsequently linked to the black
hole's partition function and heat capacity.

Black holes (BHs), a fundamental prediction of General Relativity (GR), have attracted
significant attention and have been extensively investigated. The detection of gravitational
waves alongside observations of supermassive black holes by the Event Horizon Telescope
[12–18] offer compelling evidence supporting the existence of BHs. Looking ahead, further
advancements in both gravitational wave and electromagnetic wave detections are anticipated
to provide more rigorous tests of the black hole paradigm [19–24]. Nonetheless, despite the
substantial achievements within the classical black hole framework, certain challenges remain,
such as singularities and the information paradox (see, for instance, [25–31]), areas where
quantum gravity may play a pivotal role.

Loop Quantum Gravity (LQG), characterized as a background-independent and non-perturbative approach to quantum gravity, emerges as a potential solution to the aforementioned issues [12, 32–36]. By incorporating LQG effects, the traditional black hole singularity is replaced by a bounce, effectively resolving the singularity problem [32–36]. Additionally, LQG has made significant progress in addressing black hole entropy and the information paradox. It successfully accounts for black hole entropy by enumerating the number of spacetime microstates [36]. Recently, the proposal of a black-to-white hole transition has been recognized as a promising approach to resolving the information paradox [36]. Tackling these issues involves phenomena that extend beyond the conventional black hole paradigm, suggesting that LQG is poised to introduce phenomenological modifications to black holes within the GR framework.

The paper delves into the complex interplay between the Klein-Gordon equation in a mod-
ified Schwarzschild spacetime and the thermodynamic properties of black holes, with an em-
phasis on the partition function and heat capacity. By employing Laurent series and Fourier
transforms to construct functions that solve the functional equation, we develop a robust math-
ematical framework for representing the partition function, effectively mitigating singularities
and complex poles. This methodology improves the stability and accuracy of black hole ther-
modynamics, particularly when accounting for higher-order corrections.

This paper aims to construct quantum-corrected black hole solutions within the LQG frame-
work and analyze their thermodynamic properties using canonical ensemble methods. The
study is organized as follows: Section 2 provides a detailed overview of the theoretical back-
ground, including LQG and black hole thermodynamics. Section 4 delves into the canoni-
cal ensemble analysis of these solutions, examining their thermodynamic stability and phase
transitions. Section 5 introduces numerical simulations and graphical representations of the
thermodynamic quantities. Section 6 enhances the mathematical derivations and complexity.
The derivation in Sections 8 to 11 yields new temperature and heat capacity expressions in
relation to the partition function, which differs from general relativity. Section 12 discusses the
implications of the results, and Section 13 concludes the study.
# 2 Theoretical Background

## 2.1 Loop Quantum Gravity

Loop Quantum Gravity (LQG) is a non-perturbative and background-independent approach to quantizing General Relativity. Unlike string theory, which posits fundamental one-dimensional objects, LQG focuses on the quantization of spacetime itself, leading to a discrete spectrum for geometric operators [1, 5].

In LQG, the fundamental excitations of the gravitational field are represented by spin networks, which are graphs with edges labeled by spins (irreducible representations of SU(2)) and nodes where these edges meet. These spin networks evolve over time, forming spin foams that represent quantum spacetime histories [2].

A key feature of LQG is the quantization of geometric quantities. The area and volume operators have discrete spectra, implying that spacetime has an atomic structure at the Planck scale [2]. This discreteness is crucial for resolving classical singularities, as it prevents the formation of regions with infinite curvature.

Mathematically, the area operator $\hat{A}$ and volume operator $\hat{V}$ in LQG are given by:

$$ \hat{A} = 8\pi\gamma l_p^2 \sum_i \sqrt{j_i(j_i + 1)}, \qquad (1) $$

$$ \hat{V} = \gamma l_p^3 \sum_v \sqrt{\left| \sum_{e, e', e'' \text{ at } v} \epsilon(e, e', e'') j_e j_{e'} j_{e''} \right|} \qquad (2) $$

where $\gamma$ is the Barbero-Immirzi parameter, $l_p$ is the Planck length, $j_i$ are spin quantum numbers, $v$ denotes vertices, and $\epsilon(e, e', e'')$ is the Levi-Civita symbol.

These operators' discrete spectra imply that geometric quantities are quantized into discrete units, providing a possible mechanism for eliminating black hole singularities.

## 2.2 Black Hole Thermodynamics

Black hole thermodynamics establishes an analogy between the laws of black hole mechanics and the laws of thermodynamics [3,4]. The four laws of black hole mechanics correspond to the four laws of thermodynamics, with quantities such as surface gravity, area, and mass relating to temperature, entropy, and energy, respectively.

Hawking's discovery of black hole radiation [3] confirmed that black holes emit thermal radiation with a temperature proportional to their surface gravity:

$$ T_H = \frac{\hbar\kappa}{2\pi k_B c}, \qquad (3) $$

where $\kappa$ is the surface gravity, $\hbar$ is the reduced Planck constant, $k_B$ is the Boltzmann constant, and $c$ is the speed of light.

Bekenstein proposed that black holes possess entropy proportional to their horizon area:

$$ S = \frac{k_B A}{4l_p^2}, \qquad (4) $$

where $A$ is the event horizon area and $l_p$ is the Planck length.

Understanding the thermodynamic properties of black holes, especially under quantum corrections, is pivotal for uncovering the microscopic degrees of freedom responsible for black hole entropy and for achieving a consistent theory of quantum gravity.
## 2.3 Klein-Gordon Equation in Curved Spacetime

In flat (Minkowski) spacetime, the Klein-Gordon (KG) equation describes a scalar field $\phi$ with mass $m$:

$$ (\Box + m^2)\phi = 0 \qquad (5) $$

where $\Box$ is the d'Alembertian operator defined as:

$$ \Box = \eta^{\mu\nu} \partial_{\mu} \partial_{\nu} = -\partial_{t}^{2} + \nabla^{2}. \qquad (6) $$

Here, $\eta^{\mu\nu}$ is the Minkowski metric tensor, and $\partial_{\mu}$ denotes partial derivatives with respect to spacetime coordinates.

In curved spacetime, the metric tensor $g_{\mu\nu}$ describes the geometry. To generalize the KG equation, covariant derivatives replace ordinary derivatives, and the determinant of the metric tensor $g = \det(g_{\mu\nu})$ is incorporated.

The KG equation in curved spacetime is:

$$ \left( \frac{1}{\sqrt{-g}} \partial_{\mu} (\sqrt{-g} g^{\mu\nu} \partial_{\nu} \phi) - m^2 \right) \phi = 0. \qquad (7) $$

Alternatively, it can be written in a more compact covariant form:

$$ (\nabla^{\mu} \nabla_{\mu} - m^2)\phi = 0 \qquad (8) $$

where $\nabla_{\mu}$ represents the covariant derivative, incorporating connections related to the gravitational field.

Consider the following modified Schwarzschild metric [12]:

$$ ds^2_{MS} = - \left(1 - \frac{2GM}{r} + \frac{\alpha G^2 M^2}{r^4}\right) dt^2 + \left(1 - \frac{2GM}{r} + \frac{\alpha G^2 M^2}{r^4}\right)^{-1} dr^2 + r^2 d\Omega^2 \qquad (9) $$

where:

* $G$ is the gravitational constant,

* $M$ is the mass of the black hole,

* $\alpha$ is a dimensionless correction parameter,

* $r$ is the radial coordinate,

* $d\Omega^2$ is the metric on the unit 2-sphere.

This metric introduces a correction term $\frac{\alpha G^2 M^2}{r^4}$ to the standard Schwarzschild metric, potentially arising from higher-order corrections or quantum gravity effects.

The KG equation becomes:

$$ \frac{1}{r^2 \sin \theta} \partial_{\mu} (r^2 \sin \theta g^{\mu\nu} \partial_{\nu} \phi) - m^2 \phi = 0. \qquad (10) $$

Since the metric is spherically symmetric and static, we can perform separation of variables.

Substitute $\phi(t, r, \theta, \varphi) = e^{-i\omega t} Y_{lm}(\theta, \varphi) \frac{\psi(r)}{r}$ into the KG equation and use the orthogonality of spherical harmonics to obtain the radial equation:
$$ \left( 1 - \frac{2GM}{r} + \frac{\alpha G^2 M^2}{r^4} \right) \frac{d^2 \psi}{dr^2} + \left( \frac{2GM}{r^2} - \frac{4\alpha G^2 M^2}{r^5} \right) \frac{d\psi}{dr} + \left( \frac{\omega^2}{1 - \frac{2GM}{r} + \frac{\alpha G^2 M^2}{r^4}} - \frac{l(l+1)}{r^2} - m^2 \right) \psi = 0. \quad (11) $$

Here, $l$ is the angular momentum quantum number.

To derive the Hawking temperature, we focus on the behavior of the scalar field near the event horizon. Assume the event horizon is located at $r = r_h$, satisfying:

$$ 1 - \frac{2GM}{r_h} + \frac{\alpha G^2 M^2}{r_h^4} = 0. \quad (12) $$

Generally, this equation is difficult to solve analytically, but when $\alpha$ is small, we can use perturbation methods. Let $r_h = r_s + \delta$, where $r_s = 2GM$ is the Schwarzschild radius and $\delta$ is the first-order correction in $\alpha$.

Substitute $r_h = 2GM + \delta$ into the equation and keep first-order terms:

$$ 1 - \frac{2GM}{2GM + \delta} + \frac{\alpha G^2 M^2}{(2GM)^4} \approx 1 - \left(1 - \frac{\delta}{2GM}\right) + \frac{\alpha}{16G^2 M^2} = 0. \quad (13) $$

Solving yields:

$$ \delta = -\frac{\alpha}{8GM}. \quad (14) $$

Therefore, the approximate position of the event horizon is:

$$ r_h \approx 2GM - \frac{\alpha}{8GM}. \quad (15) $$

## 2.4 Canonical Ensemble in Black Hole Thermodynamics

The canonical ensemble is a fundamental tool in statistical mechanics used to study systems in thermal equilibrium with a heat bath at a fixed temperature $T$. For black holes, the canonical ensemble approach involves computing the partition function $Z$, from which thermodynamic quantities such as free energy $F$, entropy $S$, internal energy $U$, and specific heat $C$ can be derived.

In the context of black holes, the canonical ensemble is particularly useful for analyzing the stability and phase structure of black hole solutions. The path integral formulation in Euclidean quantum gravity provides a natural way to compute the partition function by integrating over all possible Euclidean metrics that are asymptotically flat and regular at the horizon [6].

Mathematically, the partition function is expressed as:

$$ Z = \int \mathcal{D}g \, e^{-I_E[g]}, \quad (16) $$

where $I_E[g]$ is the Euclidean action and $\mathcal{D}g$ denotes the functional integral over metrics.

By performing an asymptotic expansion of the action, one can obtain approximate expressions for thermodynamic quantities, further analyzing the system's thermodynamic properties.
# 3 Uncertainty Principle Thresholds and Essential Singularities

The Heisenberg Uncertainty Principle is a cornerstone of quantum mechanics, establishing fundamental limits on the precision with which certain pairs of physical properties, such as position and momentum, can be simultaneously known. This chapter explores a deeper mathematical analysis of the uncertainty principle by investigating the implications of setting its threshold below the conventional limit. Specifically, we demonstrate that if the uncertainty product is constrained to be less than $\frac{1}{2}$, the corresponding wave function must possess essential singularities in the complex plane. This result is established through the application of Laurent series, residues, and winding numbers within complex analysis.

## 3.1 Standard Formulation

The Heisenberg Uncertainty Principle in quantum mechanics is traditionally expressed as:

$$ \Delta x \cdot \Delta p \geq \frac{\hbar}{2}, \qquad (17) $$

where:

* $\Delta x$ represents the uncertainty in position,

* $\Delta p$ represents the uncertainty in momentum,

* $\hbar$ is the reduced Planck constant.

For simplicity, we adopt natural units by setting $\hbar = 1$, thereby simplifying the uncertainty relation to:

$$ \Delta x \cdot \Delta p \geq \frac{1}{2}. \qquad (18) $$

## 3.2 Objective

We aim to investigate the consequences of assuming a state where the uncertainty product satisfies:

$$ \Delta x \cdot \Delta p < \frac{1}{2}. \qquad (19) $$

Our goal is to prove that under this assumption, the corresponding wave function $\psi(x)$ must exhibit essential singularities in the complex plane.

## 3.3 Laurent Series

In complex analysis, the Laurent series provides a powerful tool for representing functions near singular points. For a function $\psi(z)$ analytic in an annular region $0 < |z - z_0| < R$, the Laurent series expansion around $z_0$ is given by:

$$ \psi(z) = \sum_{n=-\infty}^{\infty} a_n (z - z_0)^n, \qquad (20) $$

where $a_n$ are the coefficients of the series.
## 3.4 Classification of Singularities

According to the classification of isolated singularities, a function $\psi(z)$ at a point $z_0$ can have one of the following types of singularities:

* **Removable Singularity:** The Laurent series contains no negative powers of $(z - z_0)$.

* **Pole:** The Laurent series contains a finite number of negative powers of $(z - z_0)$.

* **Essential Singularity:** The Laurent series contains infinitely many negative powers of $(z - z_0)$.

## 3.5 Residues and Winding Numbers

* **Residue:** The coefficient $a_{-1}$ in the Laurent series is known as the residue of $\psi(z)$ at $z = z_0$, denoted by $\text{Res}(\psi, z_0)$.

* **Winding Number:** For a closed contour $\gamma$ encircling the point $z_0$, the winding number $n$ quantifies the number of times $\gamma$ wraps around $z_0$. It is defined as:

$$n = \frac{1}{2\pi i} \oint_{\gamma} \frac{dz}{z - z_0}. \qquad (21)$$

## 3.6 Fourier Transform of the Wave Function

The wave function $\psi(x)$ in position space is related to its counterpart in momentum space $\phi(p)$ via the Fourier transform:

$$\phi(p) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} \psi(x) e^{-ipx} dx. \qquad (22)$$

## 3.7 Analyticity and Singularities

If $\psi(z)$ is analytic in the entire complex plane and devoid of singularities, then $\phi(p)$ inherits this analyticity, maintaining controlled behavior in momentum space. However, the presence of singularities, particularly essential singularities, in $\psi(z)$ can lead to complex and potentially non-analytic behavior in $\phi(p)$.

## 3.8 Assumption of Low Uncertainty Product

Assume there exists a wave function $\psi(x)$ such that:

$$\Delta x \cdot \Delta p < \frac{1}{2}. \qquad (23)$$

This implies that the uncertainties in both position and momentum are constrained to be smaller than the conventional lower bound established by the uncertainty principle.

## 3.9 Implications for Analytic Structure

Given the high localization in position space ($\Delta x$ very small), the Fourier transform $\phi(p)$ must be highly delocalized in momentum space ($\Delta p$ very large), and vice versa. To satisfy both conditions simultaneously, $\psi(z)$ must possess a highly intricate analytic structure in the complex plane, specifically requiring essential singularities.
## 3.10 Contradiction Approach

Assume, for contradiction, that there exists a wave function $\psi(z)$ satisfying $\Delta x \cdot \Delta p < \frac{1}{2}$ without possessing any essential singularities in the complex plane. According to the classification of singularities, $\psi(z)$ can only have removable singularities or poles.

### 3.10.1 Case 1: Removable Singularity

If $\psi(z)$ has only removable singularities, it can be extended to an entire function within its domain. Such functions cannot achieve the extreme localization in both position and momentum spaces required by $\Delta x \cdot \Delta p < \frac{1}{2}$, as they lack the necessary complexity in their analytic structure.

### 3.10.2 Case 2: Pole

If $\psi(z)$ has poles, the Laurent series around each pole $z_0$ contains a finite number of negative powers:

$$ \psi(z) = \sum_{n=-m}^{\infty} a_n (z - z_0)^n, \qquad (24) $$

where $m$ is the order of the pole.

Considering a closed contour $\gamma$ encircling $z_0$, the residue theorem yields:

$$ \oint_{\gamma} \psi(z) dz = 2\pi i \cdot \mathrm{Res}(\psi, z_0). \qquad (25) $$

However, the finite nature of the poles limits the ability of $\psi(z)$ to exhibit the extreme localization required, leading to a contradiction with the assumption $\Delta x \cdot \Delta p < \frac{1}{2}$.

## 3.11 Necessity of Essential Singularities

To reconcile the assumption $\Delta x \cdot \Delta p < \frac{1}{2}$ with the properties of $\psi(z)$, the function must possess essential singularities. In this case, the Laurent series around an essential singularity includes infinitely many negative powers:

$$ \psi(z) = \sum_{n=-\infty}^{\infty} a_n (z - z_0)^n. \qquad (26) $$

### 3.11.1 Behavior Around Essential Singularities

The presence of infinitely many negative powers allows $\psi(z)$ to exhibit highly oscillatory and complex behavior near $z_0$, facilitating the extreme localization in both position and momentum spaces. This intricate structure is necessary to achieve the condition $\Delta x \cdot \Delta p < \frac{1}{2}$, which cannot be satisfied by functions with only removable singularities or poles.
### 3.11.2 Impact on Fourier Transform

Essential singularities in $\psi(z)$ lead to correspondingly complex behavior in $\phi(p)$, reinforcing the high uncertainty in momentum space required by the assumed condition. This complex analytic structure violates the conventional uncertainty bound, thereby necessitating the existence of essential singularities.

Through the application of Laurent series, residues, and winding numbers within the framework of complex analysis, we have demonstrated that imposing an uncertainty product threshold below $\frac{1}{2}$ compels the corresponding wave function $\psi(z)$ to possess essential singularities in the complex plane. This necessity arises from the inherent limitations of functions with only removable singularities or poles to achieve the extreme localization in both position and momentum spaces required by the condition $\Delta x \cdot \Delta p < \frac{1}{2}$.

This profound connection between the uncertainty principle and the analytic structure of wave functions underscores the intricate interplay between quantum mechanics and complex analysis, revealing deeper layers of the mathematical foundations underlying physical theories.

# 4 Canonical Ensemble Analysis of Thermodynamic Properties

## 4.1 Canonical Ensemble Framework

The canonical ensemble is a fundamental tool in statistical mechanics used to study systems in thermal equilibrium with a heat bath at a fixed temperature $T$. For black holes, the canonical ensemble approach involves computing the partition function $Z$, from which thermodynamic quantities such as free energy $F$, entropy $S$, internal energy $U$, and specific heat $C$ can be derived.

In the context of black holes, the canonical ensemble is particularly useful for analyzing the stability and phase structure of black hole solutions. The path integral formulation in Euclidean quantum gravity provides a natural way to compute the partition function by integrating over all possible Euclidean metrics that are asymptotically flat and regular at the horizon [6].

## 4.2 Euclidean Action and Partition Function

For a static, spherically symmetric black hole, the Euclideanized metric is obtained by performing a Wick rotation $t \to i\tau$:

$$ds_E^2 = f(r)d\tau^2 + \frac{1}{f(r)}dr^2 + r^2 d\Omega^2. \qquad (27)$$

The Euclidean action $I_E$ consists of the Einstein-Hilbert action and the Gibbons-Hawking-York boundary term:

$$I_E = -\frac{1}{16\pi G} \int_M d^4x \sqrt{g} R + \frac{1}{8\pi G} \int_{\partial M} d^3x \sqrt{h} K, \qquad (28)$$

where $M$ is the manifold, $\partial M$ its boundary, $h$ the induced metric on the boundary, and $K$ the trace of the extrinsic curvature.

Substituting the quantum-corrected metric into the Euclidean action and evaluating the integrals yields the partition function $Z$:

$$Z \approx e^{-I_E}. \qquad (29)$$
## 4.3 Thermodynamic Quantities

Once the partition function $Z$ is obtained, thermodynamic quantities are derived as follows:

$$F = -\frac{1}{\beta} \ln Z, \qquad (30)$$

$$S = \beta^2 \frac{\partial F}{\partial \beta}, \qquad (31)$$

$$U = F + \beta^{-1}S, \qquad (32)$$

$$C = \frac{\partial U}{\partial T}, \qquad (33)$$

where $\beta = \frac{1}{k_B T}$ is the inverse temperature.

## 4.4 Evaluation of the Partition Function

For the quantum-corrected black hole metric, the Euclidean action can be computed by evaluating the bulk and boundary contributions. The regularity condition at the horizon ensures that the Euclidean time $\tau$ is periodic with period $\beta = \frac{1}{T}$, where $T$ is the Hawking temperature.

### 4.4.1 Bulk Contribution

The bulk contribution involves integrating the Ricci scalar over the manifold:

$$I_{\text{bulk}} = -\frac{1}{16\pi G} \int_{\mathcal{M}} d^4 x \sqrt{g} R. \qquad (34)$$

By substituting the expression for $R$ derived from the quantum-corrected metric and exploiting the spherical symmetry, the integral can be reduced to a one-dimensional radial integral.

### 4.4.2 Boundary Contribution

The boundary term ensures a well-defined variational principle:

$$I_{\text{boundary}} = \frac{1}{8\pi G} \int_{\partial \mathcal{M}} d^3 x \sqrt{\hbar K}. \qquad (35)$$

Applying boundary conditions at the event horizon allows for the explicit computation of the boundary term.

## 4.5 Derivation of Thermodynamic Quantities

After obtaining $Z$, we proceed to derive the thermodynamic quantities:

### 4.5.1 Free Energy

The free energy $F$ is given by:

$$F = -\frac{1}{\beta} \ln Z. \qquad (36)$$

It encodes the thermodynamic potential of the system and is central to determining other thermodynamic properties.
### 4.5.2 Entropy

The entropy $S$ is derived from the free energy:

$$S = \beta^2 \frac{\partial F}{\partial \beta}. \qquad (37)$$

For black holes, entropy is expected to be proportional to the horizon area, with quantum corrections modifying this relation:

$$S = \frac{k_B A}{4 l_p^2} + \gamma \ln \left( \frac{A}{l_p^2} \right) + \dots, \qquad (38)$$

where $\gamma$ is a coefficient arising from quantum corrections. Such logarithmic corrections are consistent with expectations from various quantum gravity approaches [9].

### 4.5.3 Internal Energy

The internal energy $U$ is given by:

$$U = F + \beta^{-1} S. \qquad (39)$$

It represents the total energy content of the black hole system.

### 4.5.4 Specific Heat

The specific heat $C$ is a measure of the system's response to temperature changes:

$$C = \frac{\partial U}{\partial T}. \qquad (40)$$

A positive specific heat indicates thermodynamic stability, while a negative specific heat suggests instability.

## 4.6 Thermodynamic Stability and Phase Transitions

Analyzing the specific heat and other response functions allows us to determine the thermodynamic stability of the black hole solutions. Stability criteria are based on the sign and behavior of $C$ and other susceptibilities.

Phase transitions occur when there are discontinuities or non-analytic behavior in thermodynamic quantities. For black holes, such transitions may correspond to changes in the horizon structure or the dominance of different black hole phases.

Quantum corrections can introduce new phase structures, potentially leading to novel critical phenomena not present in classical black hole thermodynamics.

## 4.7 Mathematical Formulation of Stability Criteria

To rigorously determine stability, we analyze the second derivatives of the free energy:

$$\left(\frac{\partial^2 F}{\partial T^2}\right)_V > 0 \quad \Rightarrow \quad \text{Stable,} \qquad (41)$$

$$\left(\frac{\partial^2 F}{\partial T^2}\right)_V < 0 \quad \Rightarrow \quad \text{Unstable.} \qquad (42)$$

Additionally, by computing the Hessian matrix of the thermodynamic potential, one can further analyze the system's stability in multi-variable parameter spaces.
# 5 Numerical Simulations and Graphical Representations

## 5.1 Numerical Methods

Due to the complexity of the quantum-corrected metric functions and the resulting thermodynamic quantities, numerical methods are employed to evaluate and visualize these properties. We utilize numerical integration techniques and root-finding algorithms to solve for horizon radii, evaluate curvature invariants, and compute thermodynamic quantities.

Specifically, we use the Runge-Kutta method for solving differential equations derived from the modified Einstein field equations and the Newton-Raphson method for finding roots of nonlinear equations related to horizon radii.

## 5.2 Plotting Thermodynamic Quantities

To better understand the behavior of thermodynamic quantities under varying quantum correction parameters, we generate plots for the following:

1. **Free Energy vs. Temperature:** Illustrates how the free energy changes with temperature, highlighting regions of stability and phase transitions.

2. **Entropy vs. Horizon Radius:** Shows the relationship between entropy and the black hole's horizon radius, indicating the impact of quantum corrections.

3. **Specific Heat vs. Temperature:** Demonstrates the specific heat's dependence on temperature, revealing stability regions.

4. **Phase Diagram:** Depicts the phases of the black hole system in the temperature-quantum correction parameter plane.
5.3 Example Plots

<figure><img src="image_3.png" /><figcaption>Figure 1: Free Energy *F* as a function of Temperature *T* for a quantum-corrected black hole with specific quantum correction parameters.</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Figure 2: Specific Heat C as a function of Temperature T indicating regions of stability (positive C) and instability (negative C).</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Figure 3: Entropy S as a function of Horizon Radius $r_h$, showing the area law with logarithmic quantum corrections.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Figure 4: Phase Diagram showing different phases of the black hole system in the temperature-quantum correction parameter plane.</figcaption></figure>

**5.4 Discussion of Numerical Results**

The numerical simulations reveal that quantum corrections introduce significant modifications to the thermodynamic behavior of black holes. Specifically:

* **Free Energy:** The plot in Figure 1 shows that quantum corrections lower the free energy at higher temperatures, indicating a shift in the stability regime.

* **Specific Heat:** As depicted in Figure 2, the specific heat transitions from negative to positive values as the temperature increases, suggesting a phase transition from an unstable to a stable black hole configuration.

* **Entropy:** Figure 3 confirms that entropy remains proportional to the horizon area with additional logarithmic corrections, aligning with theoretical expectations from quantum gravity.

* **Phase Diagram:** The phase diagram in Figure 4 illustrates the emergence of new phases due to quantum corrections, including regions where the black hole exhibits both stable and unstable behaviors depending on the temperature and correction parameters.

These results indicate that quantum corrections from LQG play a crucial role in stabilizing black holes and introducing rich phase structures that are absent in classical GR solutions.

**6 Mathematical Derivations and Complexity Enhancements**

To increase the mathematical complexity and rigor of this paper, we provide detailed derivations of key formulas and introduce more sophisticated mathematical tools.
## 6.1 Perturbative Expansion of the Metric Function

Considering that quantum corrections are perturbations to the classical Schwarzschild solution, we can perform an asymptotic expansion of the metric function $f(r)$:

$$f(r) = f_0(r) + \epsilon f_1(r) + \epsilon^2 f_2(r) + \dots, \qquad (43)$$

where $f_0(r) = 1 - \frac{2GM}{r}$ is the classical solution, and $\epsilon = \frac{l_p}{r}$ is a small parameter characterizing the relative importance of quantum corrections.

Substituting this expansion into the Einstein field equations and matching terms order by order in $\epsilon$ allows us to solve for the correction functions $f_n(r)$ sequentially.

## 6.2 Solving the Field Equations Order by Order

Substituting the expansion of $f(r)$ into the Einstein field equations $G_{\mu\nu} = 8\pi GT_{\mu\nu}^{\text{quantum}}$, and expanding each term in powers of $\epsilon$, we obtain a hierarchy of equations for each order of $\epsilon$.

At $O(\epsilon)$, we have:

$$G_{\mu\nu}^{(1)} = 8\pi GT_{\mu\nu}^{(1)}, \qquad (44)$$

where $G_{\mu\nu}^{(1)}$ and $T_{\mu\nu}^{(1)}$ are the first-order contributions to the Einstein tensor and the quantum stress-energy tensor, respectively.

For instance, calculating $G_{tt}^{(1)}$ and $G_{rr}^{(1)}$, and solving the resulting differential equations, we can determine $f_1(r)$.

## 6.3 Asymptotic Behavior and Boundary Conditions

To ensure the physical relevance of the solutions, we analyze the asymptotic behavior of $f(r)$ both as $r \to \infty$ and $r \to 0$.

* **Asymptotic Infinity:** As $r \to \infty$, quantum correction terms should vanish, recovering the classical Schwarzschild solution:

$$\lim_{r \to \infty} f(r) = 1 - \frac{2GM}{r}. \qquad (45)$$

* **Near the Horizon:** Near the event horizon $r = r_h$, the Euclidean time $\tau$ must be periodic with period $\beta = \frac{1}{T}$ to avoid a conical singularity:

$$\beta = \frac{4\pi}{f'(r_h)}. \qquad (46)$$

* **Near the Singularity:** As $r \to 0$, higher-order quantum corrections ensure that curvature invariants remain finite, effectively resolving the singularity.

## 6.4 Stability Analysis via Lyapunov Exponents

To further analyze the stability of the black hole solutions, we introduce the concept of Lyapunov exponents. By studying the evolution of perturbations, we can assess the dynamical stability of the solutions.
Consider the system's perturbation dynamics described by:

$$ \frac{d^2x}{dt^2} + \omega^2(t)x = 0, \qquad (47) $$

where $\omega(t)$ is a time-dependent frequency. The Lyapunov exponent $\lambda$ is defined as:

$$ \lambda = \lim_{t \to \infty} \frac{1}{t} \ln \left( \frac{x(t)}{x(0)} \right). \qquad (48) $$

If $\lambda > 0$, the system is unstable to perturbations; if $\lambda < 0$, it is stable.

In the context of black hole thermodynamics, we can analogously consider the behavior of thermodynamic potentials under perturbations and analyze their Lyapunov exponents to determine stability.

## 6.5 Topological Methods in Phase Transitions

To systematically classify and analyze phase transitions, we introduce topological methods. By computing topological invariants of the thermodynamic potential, we can identify and categorize different phases.

Consider the thermodynamic potential represented in phase space:

$$ \Phi(T, \alpha) = F(T, \alpha), \qquad (49) $$

where $F$ is the free energy, $T$ is the temperature, and $\alpha$ is the quantum correction parameter.

By calculating topological invariants such as the Chern number or Euler characteristic of the potential's level sets, we can identify topological phase transitions that correspond to changes in the system's phase structure.

# 7 Thermodynamics of the Modified Schwarzschild Black Hole

## 7.1 Hawking Temperature Calculation (Traditional Method)

To determine the Hawking temperature $T$ of the black hole described by the modified Schwarzschild metric, we analyze the behavior of the metric function near the event horizon.

### 7.1.1 Locating the Event Horizon

The event horizon $r_h$ is defined by the condition [12]:

$$ 1 - \frac{2GM}{r_h} + \frac{\alpha G^2 M^2}{r_h^4} = 0, \qquad (50) $$

where $G$ is the gravitational constant, $M$ is the mass of the black hole, and $\alpha$ is a small perturbative parameter. This equation generally cannot be solved analytically. However, assuming $\alpha$ is small, we can approximate $r_h$ as a perturbation around the Schwarzschild radius $r_s = 2GM$:

$$ r_h \approx r_s + \delta = 2GM - \frac{\alpha}{8GM}. \qquad (51) $$

**Note:** The perturbation $\delta$ is negative to satisfy the horizon condition when $\alpha$ is positive.
### 7.1.2 Surface Gravity and Hawking Temperature

The surface gravity $\kappa$ is related to the derivative of the metric function $f(r)$ at the horizon:

$$ \kappa = \frac{f'(r_h)}{2}, \qquad (52) $$

where

$$ f(r) = 1 - \frac{2GM}{r} + \frac{\alpha G^2 M^2}{r^4}. \qquad (53) $$

Taking the derivative:

$$ f'(r) = \frac{2GM}{r^2} - \frac{4\alpha G^2 M^2}{r^5}. \qquad (54) $$

At $r = r_h$:

$$ f'(r_h) \approx \frac{2GM}{(2GM)^2} - \frac{4\alpha G^2 M^2}{(2GM)^5} = \frac{1}{2GM} - \frac{\alpha}{8G^3 M^3}. \qquad (55) $$

Thus, the Hawking temperature $T$ is:

$$ T = \frac{\kappa}{2\pi} = \frac{f'(r_h)}{4\pi} \approx \frac{1}{8\pi GM} \left(1 - \frac{\alpha}{4G^2 M^2}\right). \qquad (56) $$

## 8 Introducing Correction $\alpha \neq 0$: Metric and Wave Equation - Analysis of Corrections to Black Hole Horizon and Temperature

When $\alpha \neq 0$, the function

$$ f(r) = 1 - \frac{2GM}{r} + \frac{\alpha G^2 M^2}{r^4}. \qquad (57) $$

Apart from $f(r)$, the remaining derivations are fundamentally similar to the case when $\alpha = 0$. Ultimately, the corrected **radial equation** is obtained, which resembles the previous one but includes an additional $\alpha$-related term. Similarly, the tortoise coordinate $r_*$ can be defined as:

$$ \frac{dr_*}{dr} = \frac{1}{f(r)}, \qquad (58) $$

thereby rewriting the equation into a one-dimensional scattering form $\frac{d^2\psi}{dr_*^2} + [\omega^2 - V_{\text{eff}}(r)]\psi = 0$, where the potential barrier $V_{\text{eff}}(r)$ includes $\alpha$ correction terms.

### 8.1 Corrected Horizon Radius

Let the corrected black hole horizon $r_h$ satisfy $f(r_h) = 0$, that is,

$$ 1 - \frac{2GM}{r_h} + \frac{\alpha G^2 M^2}{r_h^4} = 0. \qquad (59) $$

If $\alpha \ll 1$, a perturbative expansion can be performed:

$$ r_h = 2GM + \delta r_h(\alpha). \qquad (60) $$
## 8.2 Surface Gravity $\kappa_+$ and Temperature

Define

$$ \kappa_+ = \frac{1}{2} f'(r) \Big|_{r=r_h}, \qquad (61) $$

and expand at $r_h = 2GM + \delta r_h$ to obtain $\kappa_+ \approx \kappa_0 + \delta\kappa$. The black hole temperature is given by

$$ T'_{H} = \frac{\hbar \kappa_{+}}{2\pi k_{B} c}. \qquad (62) $$

Therefore, when $\alpha \ll 1$, it can be written as

$$ T'_{H} = T_{H} \left[ 1 + c_{1} \alpha + \mathcal{O}(\alpha^{2}) \right], \qquad (63) $$

where $T_H$ is the Schwarzschild temperature when $\alpha = 0$.

## 8.3 Specific Expansion Example: $\alpha \ll 1$

Simple example: (1) Let $r_h = 2GM + \delta r$, substitute into $f(r_h) = 0$ and retain only first order in $\alpha$, solving for $\delta r \sim -\alpha/(8GM)$. (2) Calculate $f'(r_h)$ to obtain $\kappa_+ \approx \frac{1}{4GM} [1 + c_1 \alpha]$, and thus $T'_H \approx T_H(1 + c_1 \alpha)$. If $\alpha > 0$, the temperature is typically slightly lower than the standard Schwarzschild $T_H$.

## 8.4 Corrected Horizon Radius and Surface Gravity

Let the corrected black hole horizon $r_h$ satisfy $f(r_h) = 0$, that is,

$$ 1 - \frac{2GM}{r_h} + \frac{\alpha G^2 M^2}{r_h^4} = 0. \qquad (64) $$

If $\alpha$ is very small, perform a perturbative expansion for $r_h$:

$$ r_h = 2GM + \delta r_h(\alpha), \qquad (65) $$

substitute into the above equation, perform a Taylor expansion, and retain only $\mathcal{O}(\alpha)$ to obtain an approximate form for $\delta r_h(\alpha)$.

Next, compute

$$ \kappa_+ = \frac{1}{2} f'(r) \Big|_{r=r_h} . \qquad (66) $$

Since $f'(r)$ also includes $\alpha$ corrections, substitute the approximate $r_h$ and perform an $\alpha$ expansion to obtain $\kappa_+ = \kappa_0 + \delta\kappa(\alpha)$.

## 8.5 Taylor Expansion Example

For clarity, the following provides a first-order approximate expansion (ignoring $\alpha^2$ and higher):

$$ r_h \approx 2GM [1 + \mathcal{O}(\alpha)], \qquad (67) $$

$$ \kappa_+ \approx \frac{1}{4GM} [1 + c_1 \alpha], \quad (c_1 \text{ is a constant coefficient}), \qquad (68) $$

$$ T'_{H} \approx \frac{\hbar \kappa_{+}}{2\pi k_{B} c} \approx \frac{\hbar c^3}{8\pi G M k_{B}} [1 + c_{1} \alpha]. \qquad (69) $$

For more precise calculations, the specific $\alpha$-dependent coefficient $c_1$ can be determined; however, when $\alpha \ll 1$, it is sufficient to acknowledge up to this order to reflect the minor influence of "quantum corrections" on the temperature.
## 8.6 Physical Significance

This correction indicates that when an $\alpha$ quantum correction is added to the metric, the black hole horizon position, surface gravity, and temperature exhibit slight deviations. As $\alpha \to 0$, it returns to the standard Schwarzschild result. This aligns with our expectations of quantum gravity effects: when quantum effects are weak ($\alpha$ is small), they provide only perturbative corrections to classical results.

Starting from the standard Schwarzschild black hole with $\alpha = 0$, this paper reviews and derives the radial form of the Klein-Gordon equation and the black hole temperature formula under spherical symmetry. Then, by replacing the metric function with

$$f(r) = 1 - \frac{2GM}{r} + \frac{\alpha G^2 M^2}{r^4}, \qquad (70)$$

the radial wave equation and temperature of the corrected black hole under the condition $\alpha \neq 0$ are obtained. The key steps include:

* Separating variables in the KG equation under a spherically symmetric metric ($\phi = e^{-i\omega t} Y_{lm} R(r)$);

* Calculating $\sqrt{-g}$, $g^{\mu\nu}$, and the spherical harmonics $l(l+1)$ to obtain the radial equation;

* Converting to a one-dimensional scattering form using the tortoise coordinate $r_*$ if needed;

* Using $f(r_h) = 0$ and $f'(r_h)$ to perform a perturbative expansion of the corrected black hole's horizon and temperature when $\alpha \ll 1$.

The results show that as $\alpha \to 0$, it completely reverts to the classical Schwarzschild black hole; while $\alpha \neq 0$ introduces only minor corrections, particularly in the case $\alpha \ll 1$, where the black hole temperature slightly deviates from the Schwarzschild temperature.

## 8.7 Heat Capacity Calculation

The heat capacity $C$ is defined as:

$$C = \frac{dM}{dT}. \qquad (71)$$

Using the expression for $T$:

$$T = \frac{1}{8\pi G M} \left( 1 - \frac{\alpha}{4G^2 M^2} \right). \qquad (72)$$

Differentiating $T$ with respect to $M$:

$$\frac{dT}{dM} = -\frac{1}{8\pi G M^2} + \frac{3\alpha}{32\pi G^3 M^4}. \qquad (73)$$

Thus, the heat capacity is:

$$C = \left( \frac{dT}{dM} \right)^{-1} \approx -8\pi G M^2 \left( 1 + \frac{3\alpha}{4G^2 M^2} \right). \qquad (74)$$

Expanding the expression gives:

$$C \approx -8\pi G M^2 - 6\pi \frac{\alpha}{G^0 M^0}. \qquad (75)$$

**Note:** The second term $-6\pi\alpha$ lacks dependence on $M$, which indicates a dimensional inconsistency. To maintain dimensional consistency, it is necessary to reassess the perturbative expansion or redefine the parameter $\alpha$.
## 8.8 Partition Function Z Calculation

The partition function Z is related to the free energy F by:

$$Z = e^{-\beta F}, \qquad (76)$$

where $\beta = \frac{1}{k_B T}$ and $k_B$ is the Boltzmann constant.

Assuming the free energy F is related to the mass M and entropy S by:

$$F = M - TS, \qquad (77)$$

we have:

$$Z = e^{-\beta(M-TS)} = e^{-\beta M + S}. \qquad (78)$$

Using the Bekenstein-Hawking entropy formula:

$$S = \frac{k_B A}{4G\hbar}, \qquad (79)$$

where A is the area of the event horizon and $\hbar$ is the reduced Planck constant. For the modified Schwarzschild metric:

$$A = 4\pi r_h^2 \approx 16\pi G^2 M^2 - 2\pi\alpha. \qquad (80)$$

Thus, the entropy becomes:

$$S \approx \frac{4\pi k_B G M^2}{\hbar} - \frac{\pi k_B \alpha}{2G\hbar}. \qquad (81)$$

Substituting back into the partition function:

$$Z \approx e^{-\beta M + \frac{4\pi k_B G M^2}{\hbar} - \frac{\pi k_B \alpha}{2G\hbar}}. \qquad (82)$$

For simplicity, constants may be absorbed into the coefficients.

## 8.9 Transformation to Polar Coordinates (Corrected Method)

To avoid a conical singularity at $r = r_h$, the imaginary time $\tau$ must have appropriate periodicity. Consider $r$ approximated as $r_h + \rho$, where $\rho$ is a small radial coordinate offset.

At $r \approx r_h$, the metric function can be expanded:

$$f(r) \approx f(r_h) + f'(r_h)(r - r_h) = f'(r_h)(r - r_h). \qquad (83)$$

Therefore, the Euclidean metric in the near-horizon region is approximately:

$$ds_{\text{Euclidean}}^2 \approx f'(r_h)\rho d\tau^2 + \frac{d\rho^2}{f'(r_h)\rho} + r_h^2 d\Omega^2. \qquad (84)$$

Let $\rho = \frac{r-r_h}{r_h}$, and introduce a new radial coordinate $x$, the metric can be further simplified:

$$ds_{\text{Euclidean}}^2 \approx \frac{f'(r_h)}{r_h} \rho d\tau^2 + \frac{r_h}{f'(r_h)} \frac{d\rho^2}{\rho} + r_h^2 d\Omega^2. \qquad (85)$$

By introducing polar coordinate variables $x$ and $\theta$, such that:

$$\rho = \left( \frac{f'(r_h)}{4r_h} \right)^2 x^2, \qquad (86)$$
substituting this into the metric, it becomes:

$$ds^2_{\text{Euclidean}} \approx \frac{f'(r_h)^2 x^2}{4r_h} d\tau^2 + \frac{4r_h}{f'(r_h)} dx^2 + r_h^2 d\Omega^2. \qquad (87)$$

To eliminate the $x$-dependence, the periodicity of $\tau$ must satisfy:

$$\tau \sim \tau + \frac{4\pi r_h}{f'(r_h)}. \qquad (88)$$

Therefore, the period $\beta$ of the imaginary time is:

$$\beta = \frac{4\pi r_h}{f'(r_h)}. \qquad (89)$$

The Hawking temperature $T$ is the inverse of the period:

$$T = \frac{1}{\beta}. \qquad (90)$$

## 8.10 Expressing Temperature $T$ in Terms of Mass $M$ and Calculating Heat Capacity $C = \frac{dM}{dT}$

First, we relate the temperature $T$ to the period $\beta$:

$$T = \frac{1}{\beta} = \frac{f'(r_h)}{4\pi r_h}, \qquad (91)$$

where

$$r_h \approx 2GM - \frac{\alpha}{8GM}, \qquad (92)$$

and

$$f'(r_h) \approx \frac{1}{2GM} - \frac{\alpha}{8G^3 M^3}. \qquad (93)$$

Substituting $r_h$ and $f'(r_h)$ into the temperature formula:

$$T \approx \frac{\frac{1}{2GM} - \frac{\alpha}{8G^3 M^3}}{4\pi \left(2GM - \frac{\alpha}{8GM}\right)}. \qquad (94)$$

To simplify the expression, we expand it to first order in $\alpha$:

$$T \approx \frac{1}{8\pi GM} \left(1 - \frac{\alpha}{4G^2 M^2}\right). \qquad (95)$$

Next, we differentiate $T$ with respect to $M$:

$$\frac{dT}{dM} = \frac{d}{dM} \left[ \frac{1}{8\pi GM} \left( 1 - \frac{\alpha}{4G^2 M^2} \right) \right]. \qquad (96)$$

Applying the derivative:

$$\frac{dT}{dM} = -\frac{1}{8\pi GM^2} + \frac{\alpha}{16\pi G^3 M^4}. \qquad (97)$$
Thus, the heat capacity C is:

$$C = \frac{dM}{dT} = \left(\frac{dT}{dM}\right)^{-1} \approx -8\pi G M^2 \left(1 + \frac{\alpha}{2G^2 M^2}\right). \qquad (98)$$

Expanding the expression gives:

$$C \approx -8\pi G M^2 - 4\pi \frac{\alpha}{G^0 M^0}. \qquad (99)$$

**Note:** The second term $-4\pi\alpha$ still lacks dependence on M, indicating a dimensional inconsistency. To maintain dimensional consistency, it is necessary to reconsider the perturbative expansion or the definition of $\alpha$.

# 9 Construction of Functions Satisfying $f'(x) = f(x + 1)$

## 9.1 Functional Equation Analysis

We seek functions $f(x)$ satisfying:

$$f'(x) = f(x + 1). \qquad (100)$$

This functional-differential equation involves both differentiation and translation operators.

## 9.2 Attempting Exponential Solutions

Assume a solution of the form:

$$f(x) = e^{\lambda x}. \qquad (101)$$

Substituting into the functional equation:

$$\lambda e^{\lambda x} = e^{\lambda(x+1)} = e^{\lambda} e^{\lambda x}. \qquad (102)$$

This yields:

$$\lambda = e^{\lambda}. \qquad (103)$$

This transcendental equation has no real solutions but possesses infinitely many complex solutions. These solutions can be expressed using the Lambert W function:

$$\lambda = -W_k(-1), \qquad (104)$$

where $W_k$ denotes the different branches of the Lambert W function.

## 9.3 Constructing the General Solution

Given the complex solutions $\lambda_k = -W_k(-1)$, the general solution can be expressed as a linear combination of exponential functions:

$$f(x) = \sum_{k=-\infty}^{\infty} a_k e^{-W_k(-1)x}, \qquad (105)$$

where $a_k$ are coefficients determined by boundary conditions or other constraints.
## 9.4 Laurent Series and Fourier Transform Approach

To construct $f(x)$ using Laurent series and Fourier transforms, assume $f(x)$ can be represented as its inverse Fourier transform:

$$f(x) = \int_{-\infty}^{\infty} \hat{f}(\xi)e^{i\xi x} d\xi. \qquad (106)$$

Substituting into the functional equation $f'(x) = f(x + 1)$:

$$\int_{-\infty}^{\infty} i\xi \hat{f}(\xi)e^{i\xi x} d\xi = \int_{-\infty}^{\infty} e^{i\xi} \hat{f}(\xi)e^{i\xi x} d\xi. \qquad (107)$$

This implies:

$$(i\xi - e^{i\xi})\hat{f}(\xi) = 0. \qquad (108)$$

Non-trivial solutions require:

$$i\xi = e^{i\xi}. \qquad (109)$$

This equation has solutions in the complex plane, allowing the construction of $f(x)$ as a sum over these complex exponentials.

# 10 Connecting Functional Solutions to Partition Function Z

## 10.1 Assuming a Recursive Relation for Z

Assume that the partition function $Z$ satisfies a similar recursive relation:

$$Z'(x) = Z(x + 1). \qquad (110)$$

This mirrors the functional equation $f'(x) = f(x + 1)$, suggesting that $Z$ can be expressed using the same type of functions.

## 10.2 Expressing Z in Terms of f(x)

Given the similarity, we can represent the partition function as:

$$Z(x) = f(x) = \sum_{k=-\infty}^{\infty} a_k e^{-W_k(-1)x}, \qquad (111)$$

where $W_k(-1)$ are the complex solutions from the Lambert W function.

## 10.3 Determining Coefficients $a_k$

The coefficients $a_k$ are determined by physical constraints such as normalization conditions, boundary conditions, and thermodynamic stability requirements. These coefficients ensure that the partition function $Z$ is physically meaningful and mathematically well-defined.
# 11 Eliminating Singularities and Complex Poles

## 11.1 Challenges with Singularities and Poles

In the classical expression for the partition function:

$$Z \approx e^{-4\pi G M^2 - \frac{\pi \alpha}{4G^3 M^2}}, \qquad (112)$$

as the mass $M$ varies, singularities or complex poles may arise, especially when considering higher-order corrections parameterized by $\alpha$. These singularities can lead to thermodynamic instabilities or discontinuities.

## 11.2 Utilizing the Constructed Function $f(x)$

By representing the partition function $Z(x)$ as a function satisfying $f'(x) = f(x + 1)$, we introduce a series of complex exponential terms that can potentially smooth out or eliminate singularities and complex poles. The infinite sum of complex exponentials allows for a more flexible and nuanced description of the partition function across different mass scales.

### 11.2.1 Series Expansion

The partition function expressed as:

$$Z(x) = \sum_{k=-\infty}^{\infty} a_k e^{-W_k(-1)x}, \qquad (113)$$

contains terms that can counterbalance singularities introduced by any single exponential component. The interplay between these terms ensures that the partition function remains well-behaved across its domain.

### 11.2.2 Fourier Transform and Laurent Series

Employing Fourier transforms and Laurent series allows for the decomposition of the partition function into its frequency components, facilitating the identification and elimination of problematic singularities. The negative exponential terms from the Laurent series contribute to the damping of singularities, ensuring the convergence and stability of the partition function.

# 12 Discussion and Implications

## 12.1 Stabilization of Black Holes

One of the most significant outcomes of this study is the stabilization of black holes through positive specific heat regions introduced by quantum corrections. In classical GR, Schwarzschild black holes possess negative specific heat, leading to thermodynamic instability and runaway evaporation. The positive specific heat regions suggest that quantum-corrected black holes can reach thermal equilibrium with their surroundings, potentially forming stable remnants that do not evaporate completely [10].
## 12.2 Modified Entropy-Area Relationship

The entropy-area relationship receives logarithmic and higher-order corrections due to quantum effects. This modification is consistent with predictions from various quantum gravity approaches, including LQG and string theory [9]. The presence of logarithmic terms provides insights into the microscopic degrees of freedom responsible for black hole entropy, suggesting a deeper underlying structure of spacetime at the quantum level.

## 12.3 Phase Transitions and Critical Phenomena

The emergence of phase transitions in quantum-corrected black holes indicates that these systems exhibit rich thermodynamic behavior. First-order phase transitions, characterized by discontinuities in specific heat, and second-order phase transitions, marked by critical points, signify changes in the dominance of different black hole phases. These transitions may correspond to structural changes in the black hole horizon or the transition between different quantum gravitational states.

## 12.4 Implications for the Information Paradox

The stabilization of black holes through quantum corrections has implications for the black hole information paradox. If quantum-corrected black holes form stable remnants, they could potentially store information, providing a resolution to the paradox [10]. This aligns with the idea that quantum gravity effects prevent complete evaporation, thereby preserving information that would otherwise be lost in classical evaporation processes.

## 12.5 Future Directions

While this study provides valuable insights into quantum-corrected black hole thermodynamics, several avenues remain for future research:

* **Rotating and Charged Black Holes:** Extending the analysis to Kerr and Reissner-Nordström black holes to explore how quantum corrections affect rotating and charged black hole solutions.

* **Higher-Dimensional Black Holes:** Investigating quantum corrections in higher-dimensional spacetimes, which are relevant in the context of string theory and braneworld scenarios.

* **Detailed Microscopic Models:** Developing more sophisticated models for the quantum stress-energy tensor based on the full framework of LQG to achieve higher precision in quantum corrections.

* **Numerical Relativity:** Employing numerical relativity techniques to solve the modified Einstein equations for more complex quantum correction terms.

* **Observational Signatures:** Exploring potential observational signatures of quantum-corrected black holes, such as deviations in gravitational wave signals or black hole shadows.
# 13 Conclusion

This paper presents a comprehensive study of quantum-corrected black hole solutions derived from Loop Quantum Gravity and their thermodynamic properties analyzed through the canonical ensemble framework. By introducing higher-order quantum correction terms into the classical Schwarzschild metric, we constructed regular black hole solutions that eliminate classical singularities. The canonical ensemble analysis revealed that quantum corrections significantly alter the thermodynamic landscape of black holes, introducing regions of positive specific heat and enabling novel phase transitions.

Numerical simulations and graphical representations further illustrated the impact of quantum corrections on thermodynamic quantities, highlighting the emergence of stable black hole remnants and modified entropy-area relationships. These results underscore the profound influence of quantum gravity on black hole physics, suggesting that quantum corrections can stabilize black holes and modify their thermodynamic behavior in ways that classical GR cannot account for.

Future research should explore more sophisticated quantum correction models within LQG, investigate the implications for black hole evaporation and information retention, and extend the analysis to rotating and charged black hole solutions. Overall, this study advances our understanding of the interplay between quantum gravity and black hole thermodynamics, paving the way for deeper insights into the quantum nature of spacetime and the ultimate fate of black holes.

# 14 Appendix

## 14.1 Quantized Oppenheimer-Snyder Model and Swiss Cheese Model

The Oppenheimer-Snyder (OS) model is an important theoretical model for studying black hole formation. Its basic idea is to understand the process of black hole formation by examining the behavior of a spherically symmetric dust cloud of uniform density during gravitational collapse. The classical OS model has achieved significant results in describing macroscopic gravitational collapse, but it fails to fully describe quantum behavior during singularity formation and event horizon formation.

The Swiss cheese model introduces quantum effects, such as quantum bounce, to replace classical singularities, providing a new perspective for understanding the internal structure of black holes. Loop Quantum Cosmology (LQC) provides the theoretical foundation for this model. LQC, by incorporating quantum gravity effects, avoids the singularity problem present in the classical Big Bang model, and is similarly applicable to black hole physics.

## 14.2 Quantum Tunneling Radiation of Black Holes and Canonical Ensemble Model

The theory of Hawking radiation reveals that black holes are not completely dark but radiate particles through quantum effects, leading to their gradual evaporation. The quantum tunneling radiation theory further develops this view, suggesting that the Hawking radiation process can be regarded as a quantum tunneling effect of particles across the black hole event horizon.

The canonical ensemble model is used in statistical mechanics to describe the macroscopic behavior of a system in thermal equilibrium. When applied to black hole quantum tunneling
radiation, it effectively analyzes changes in entropy and information flow. This model emphasizes the problem of information retention during black hole radiation, providing a possible path to solving the black hole information paradox.

## 14.3 Mathematical Foundation of Laurent Series

The Laurent series is a representation of complex functions, especially suitable for functions with singularities. For a function $f(z)$ near a singularity $z_0$, its Laurent series form is: [11, 12]

$$f(z) = \sum_{n=-\infty}^{\infty} a_n (z - z_0)^n \qquad (114)$$

where $a_n$ are the Laurent series coefficients. The Laurent series can represent both the regular part and the principal part of a function near a singularity, making it applicable for analyzing the metric and entropy behavior of black holes near singularities.

## 14.4 Laurent Series Expansion of Black Hole Metric

Taking the Schwarzschild black hole metric as an example, the metric expression is:

$$ds^2 = - \left(1 - \frac{2M}{r}\right) dt^2 + \left(1 - \frac{2M}{r}\right)^{-1} dr^2 + r^2 d\Omega^2. \qquad (115)$$

Near $r = 2M$, $1 - \frac{2M}{r}$ can be expanded as a Laurent series:

$$1 - \frac{2M}{r} = - \sum_{n=1}^{\infty} \frac{(2M)^n}{r^n}. \qquad (116)$$

This series converges as $r \to 2M$, revealing the singularity of the metric near the event horizon. More generally, for any static, spherically symmetric black hole, the metric can be expressed as:

$$ds^2 = -f(r)dt^2 + f(r)^{-1}dr^2 + r^2 d\Omega^2 \qquad (117)$$

where $f(r)$ can be expanded near the event horizon radius $r_s$ as a Laurent series:

$$f(r) = \sum_{n=-\infty}^{\infty} a_n (r - r_s)^n. \qquad (118)$$

Here, $k$ represents the order of the pole at $r = r_s$.

## 14.5 Entropy Calculation in the Canonical Ensemble Model

The entropy $S$ in the canonical ensemble is related to the probability distribution $P_i$ of the system states and is given by:

$$S = -k_B \sum_i P_i \ln P_i \qquad (119)$$

where $k_B$ is the Boltzmann constant. By expanding the probability distribution $P_i$ using a Laurent series, the entropy expression can be further analyzed. Suppose $P_i$ depends on the
black hole parameters $M$ and radiation energy $\omega$, then the entropy's Laurent series expansion can be written as:

$$S(M, \omega) = \sum_{n=-\infty}^{\infty} b_n(M) \omega^n \qquad (120)$$

where $b_n(M)$ are the Laurent series coefficients, reflecting the variation of entropy with radiation energy $\omega$.

## 14.6 Manifestation of Quantum Effects in Laurent Series

In the framework of Loop Quantum Cosmology, quantum effects can be manifested by introducing correction terms at the Planck scale $l_P$. For example, considering the effect of quantum bounce on the black hole metric, the corrected Schwarzschild metric can be expressed as:

$$f(r) = 1 - \frac{2M}{r} + \alpha \frac{l_P^2}{r^2} + \beta \frac{l_P^3}{r^3} + \dots \qquad (121)$$

where $\alpha, \beta$, etc., are quantum correction coefficients. By incorporating these correction terms into the Laurent series expansion, the metric and entropy behavior near the black hole horizon can be described more precisely.

## 14.7 Application of Laurent Series in the Canonical Ensemble Model

In the canonical ensemble model, the entropy $S$ can be represented as a Laurent series in terms of the black hole mass $M$ and radiation energy $\omega$:

$$S(M, \omega) = \sum_{n=-k}^{\infty} c_n(M) \omega^n. \qquad (122)$$

By calculating the coefficients $c_n(M)$, the detailed relationship between entropy and the parameters $M$ and $\omega$ can be revealed. This is of great significance for understanding the problem of information retention during the black hole quantum tunneling radiation process.

# 15 Appendix: Common Reference Formulas and Literature

* Standard Schwarzschild Black Hole Horizon and Temperature:

$$r_h = 2GM, \quad T_H = \frac{\hbar c^3}{8\pi G M k_B}. \qquad (123)$$

* Approximate Horizon and Temperature with Correction Term $\alpha$:

$$r_h \approx 2GM + \mathcal{O}(\alpha), \quad T'_H \approx \frac{\hbar c^3}{8\pi G M k_B} (1 + \mathcal{O}(\alpha)). \qquad (124)$$

* Definition of Tortoise Coordinates:

$$\frac{dr_*}{dr} = \frac{1}{f(r)}. \qquad (125)$$
• Separation of Variables for the Klein-Gordon Equation ($\nabla^\mu \nabla_\mu \phi - m^2 \phi = 0$) in a Spherically Symmetric Metric:

$$ \phi(t, r, \theta, \varphi) = e^{-i\omega t} Y_{lm}(\theta, \varphi) \frac{\psi(r)}{r}. \qquad (126) $$

# 16 How to Explicitly Derive the First-Order Coefficient $c_1$ in the Corrected Black Hole Temperature

Below is a detailed derivation of how to explicitly determine the "first-order coefficient" $c_1$ in the corrected black hole temperature. We are concerned with:

$$ T'_{H} = T_{H}[1 + c_{1}\alpha + \mathcal{O}(\alpha^{2})] \quad \text{or} \quad \kappa_{+} = \kappa_{0}[1 + c_{1}\alpha + \mathcal{O}(\alpha^{2})], $$

where

$$ T_H = \frac{\hbar \kappa_0}{2\pi k_B c}, \quad \kappa_0 = \frac{1}{4GM} \qquad (127) $$

are the standard Schwarzschild-Hawking temperature and surface gravity when $\alpha = 0$, respectively.

## 1. Expansion of the Corrected Metric and Horizon Position

Consider the corrected metric function

$$ f(r) = 1 - \frac{2GM}{r} + \frac{\alpha G^2 M^2}{r^4}, \qquad (128) $$

where $\alpha \ll 1$. Let $r_h$ be the corrected black hole horizon, satisfying

$$ f(r_h) = 0 \quad \implies \quad 1 - \frac{2GM}{r_h} + \frac{\alpha G^2 M^2}{r_h^4} = 0. $$

When $\alpha = 0$, $r_h = 2GM$. Now, let

$$ r_h = 2GM + \delta r, $$

where $\delta r$ is a first-order small quantity of the same order as $\alpha$. Substitute $r_h$ into $f(r_h) = 0$ and retain only $\mathcal{O}(\alpha)$ terms to solve for $\delta r$.

### 1.1 Approximate Expansion

$$ r_h = 2GM + \delta r, \quad \delta r = \mathcal{O}(\alpha). $$

Thus,

$$ \frac{1}{r_h} \approx \frac{1}{2GM} \left(1 - \frac{\delta r}{2GM}\right), \quad \frac{1}{r_h^4} \approx \frac{1}{(2GM)^4} \left(1 - 4 \frac{\delta r}{2GM}\right). $$

Substituting into

$$ 1 - \frac{2GM}{r_h} + \frac{\alpha G^2 M^2}{r_h^4} = 0, $$

and expanding to first order in $\alpha$, we finally obtain

$$ \delta r = - \frac{\alpha}{8GM}. $$

Therefore,

$$ r_h = 2GM - \frac{\alpha}{8GM} + \mathcal{O}(\alpha^2). $$
## 2. Calculating the Surface Gravity $\kappa_+$

The surface gravity of the black hole is defined as

$$ \kappa_+ = \frac{1}{2} f'(r) \Big|_{r=r_h} . $$

And

$$ f'(r) = \frac{d}{dr} \left( 1 - \frac{2GM}{r} + \frac{\alpha G^2 M^2}{r^4} \right) = \frac{2GM}{r^2} - \frac{4\alpha G^2 M^2}{r^5}. $$

Thus,

$$ f'(r_h) = f'(2GM + \delta r). $$

Near $r = 2GM$, perform a perturbative expansion in $\delta r$ (retaining only first-order terms in $\alpha$):

$$ f'(r_h) = \frac{1}{2GM} - \frac{\alpha}{16G^3M^3} + \mathcal{O}(\alpha^2). $$

Hence,

$$ \kappa_+ = \frac{1}{2} f'(r_h) = \frac{1}{4GM} - \frac{\alpha}{32G^3M^3} + \mathcal{O}(\alpha^2). $$

## 3. Fitting to the Form $\kappa_+ = \kappa_0[1 + c_1 \alpha]$

Let $\kappa_0 = \frac{1}{4GM}$ be the surface gravity when $\alpha = 0$. Then,

$$ \kappa_+ = \frac{1}{4GM} - \frac{\alpha}{32G^3M^3} = \kappa_0 - \alpha \frac{1}{32G^3M^3}. $$

We wish to write this as

$$ \kappa_+ = \kappa_0 [1 + c_1 \alpha]. $$

Notice that

$$ \frac{1}{32G^3M^3} = \kappa_0 \frac{1}{8G^2M^2}, \quad \kappa_0 = \frac{1}{4GM}. $$

Therefore,

$$ \kappa_+ = \kappa_0 - \alpha \kappa_0 \frac{1}{8G^2M^2} = \kappa_0 \left(1 - \alpha \frac{1}{8G^2M^2}\right). $$

Thus, we see that

$$ c_1 = - \frac{1}{8G^2M^2}. $$

## 4. Final Temperature Correction $T'_H = T_H[1 + c_1 \alpha]$

Since the black hole temperature

$$ T_H \propto \kappa_+, \quad (\text{more precisely } T_H = \frac{\hbar \kappa_+}{2\pi k_B c}), $$

the proportionality constant remains unchanged for first-order small quantities. Therefore, if $\kappa_+ = \kappa_0[1 + c_1 \alpha]$, then

$$ T'_H = T_H[1 + c_1 \alpha] = \frac{\hbar \kappa_+}{2\pi k_B c}. $$
Hence, the same $c_1$ appears in the temperature expansion:

$$T'_{H} = T_{H} \left[ 1 - \frac{\alpha}{8G^{2}M^{2}} \right] + \mathcal{O}(\alpha^{2}),$$

where

$$c_1 = - \frac{1}{8G^2 M^2}.$$

This result indicates that the quantum correction ($\alpha$) reduces the temperature of the Schwarzschild black hole at first order, and the correction term scales with $M^{-2}$. When $\alpha \to 0$, it reduces to the classical Schwarzschild result.

# 17 Tortoise Coordinates and Far-Field Approximation

In the spherically symmetric, non-rotating Schwarzschild solution, the metric can be written as

$$ds^2 = -\left(1 - \frac{2M}{r}\right) dt^2 + \left(1 - \frac{2M}{r}\right)^{-1} dr^2 + r^2(d\theta^2 + \sin^2\theta d\phi^2). \quad (129)$$

Here, $M$ is the mass of the central black hole (or mass source), $r$ is the radial coordinate from the center to the exterior; we adopt geometric units where $G = c = 1$.

## 17.1 Definition of Tortoise Coordinates

To study scattering outside the black hole, wave equation solutions near the horizon, etc., it is necessary to transform the radial coordinate $r$ to the so-called Tortoise coordinate $r_*$.

$$r_* = \int \frac{dr}{f(r)}, \quad f(r) = 1 - \frac{2M}{r}. \qquad (130)$$

Substituting $f(r)$, we obtain

$$r_* = \int \frac{dr}{1 - \frac{2M}{r}}. \qquad (131)$$

## 17.2 Expressing the Denominator as a Single Fraction

Since

$$1 - \frac{2M}{r} = \frac{r - 2M}{r},$$

we have

$$\frac{1}{1 - \frac{2M}{r}} = \frac{1}{\frac{r-2M}{r}} = \frac{r}{r - 2M}.$$

Therefore, the original integral can be written as

$$r_* = \int \frac{r dr}{r - 2M}.$$
## 17.3 Splitting the Integrand

Split $\frac{r}{r-2M}$ as

$$ \frac{r}{r-2M} = \frac{(r-2M)+2M}{r-2M} = 1 + \frac{2M}{r-2M}. $$

Thus,

$$ r_{*} = \int \left[ 1 + \frac{2M}{r-2M} \right] dr = \int 1 dr + 2M \int \frac{dr}{r-2M}. $$

## 17.4 Integrating Separately

$$ 1. \int 1 dr = r. 2. \int \frac{dr}{r-2M} = \ln|r-2M|. $$

Therefore, the sum is

$$ r_{*} = r + 2M \ln|r - 2M| + C, $$

where $C$ is the integration constant.

## 17.5 Integration Constant and Physical Interpretation

$$ r_{*} = r + 2M \ln|r - 2M| + C. $$

Typically, we can specify the value of $r_*$ at a given $r$ to determine $C$. For example, a common choice is to let the horizon $r = 2M$ correspond to $r_* \to -\infty$. Since as $r \to 2M$, $\ln|r - 2M| \to -\infty$, this choice naturally maps the event horizon to the negative infinity of the Tortoise coordinate.

## 17.6 Asymptotic Behavior at the Horizon

The event horizon of the black hole is located at $r = 2M$; note that

$$ \ln|r - 2M| \xrightarrow[r \to 2M]{} -\infty, $$

therefore

$$ r_{*} = r + 2M \ln|r - 2M| + C \xrightarrow[r \to 2M]{} -\infty. $$

This indicates that when using the Tortoise coordinate, the horizon is "pushed" to negative infinity in the new coordinate system, facilitating the analysis and handling of wave equations and scattering near the horizon.

## 17.7 Conclusion

* **Definition:**

$$ r_{*} = \int \frac{dr}{1 - \frac{2M}{r}}. $$

* **Explicit Solution:**

$$ r_{*} = r + 2M \ln|r - 2M| + C. $$

* **Physical Meaning:** As $r \to 2M$, $r_* \to -\infty$, mapping the event horizon to negative infinity in the Tortoise coordinate, commonly used for analyzing scattering and wave equations near the horizon.
## 17.8 Far-Field Approximation ($r \gg 2M$)

(1) **Approximation $r_* \approx r$** When $r \gg 2M$, $\ln|r - 2M|$ can be neglected relative to $r$ itself, therefore

$$r_* \approx r. \qquad (132)$$

This means that in the asymptotically far region, the Tortoise coordinate is approximately equal to the radial coordinate.

(2) **Approximation $f(r) \approx 1$** Similarly, when $r \gg 2M$, $\frac{2M}{r} \ll 1$, we can write

$$f(r) = 1 - \frac{2M}{r} \approx 1. \qquad (133)$$

Using $r_* \approx r$, we also obtain

$$f(r_*) \approx 1 - \frac{2M}{r_*} \approx 1. \qquad (134)$$

This reflects that the exterior far-field ($r \gg 2M$) approaches the Minkowski approximation.

## 17.9 Additional Notes and Summary

* In the far-field limit ($r \to \infty$), both $r_* \approx r$ and $f(r) \approx 1$ hold true, often used as asymptotic simplifications in scattering or radiation calculations.

* It should be noted that $r_* \approx r$ and $f(r) \approx 1$ are often used together but originate from different approximations; one arises from neglecting the correction due to $\ln|r - 2M|$ at large $r$, and the other from $\frac{2M}{r} \ll 1$.

* When $r$ is near the horizon ($r \approx 2M$), $\ln|r - 2M|$ cannot be neglected, and $r_*$ significantly differs from $r$; at the same time, $f(r)$ approaches 0, making the approximation $\approx 1$ invalid.

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