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# First-Order Corrections to the Thermodynamics and Superradiance of Black Holes in $f(Q)$ Gravity: An Extended Study

Wen-Xiang Chen

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Wen-Xiang Chen. First-Order Corrections to the Thermodynamics and Superradiance of Black Holes in $f(Q)$ Gravity: An Extended Study. 2025. hal-04993816

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Distributed under a Creative Commons Attribution 4.0 International License
# First-Order Corrections to the Thermodynamics and Superradiance of Black Holes in $f(Q)$ Gravity: An Extended Study

Wen-Xiang Chen

Department of Astronomy

School of Physics and Materials Science

Guangzhou University

wxchen4277@qq.com

## Abstract

We analyze black hole thermodynamics in $f(Q)$ gravity, including first-order quantum corrections. Starting with a static, spherically symmetric solution, we add quantum-induced logarithmic corrections to the Bekenstein-Hawking entropy and examine their effects on temperature, entropy, specific heat, thermodynamic stability, and phase transitions. Our study explores how these thermodynamic properties depend on different forms of $f(Q)$ and emphasizes the role of non-metricity scalar $Q$ in shaping the horizon geometry. We also highlight the interplay between gravitational and quantum effects, revealing richer phase structures compared to standard general relativity. Recent research suggests Hawking radiation and superradiance are closely related, arising from positive-power terms (Hawking radiation) and negative-power terms (superradiance) near black hole singularities. Extending this idea, we investigate how quantum corrections in $f(Q)$ gravity influence superradiant scattering, focusing on modified dispersion relations and horizon conditions. Our results show that both Hawking radiation and superradiance stem from the same geometric and quantum origins, offering deeper insights into radiation phenomena around black holes in $f(Q)$ gravity.

## Contents

<table><tr><td>1</td><td>Introduction</td><td>2</td></tr><tr><td>2</td><td>The $f(Q)$ Gravity Framework</td><td>4</td></tr><tr><td>2.1</td><td>Action and Field Equations</td><td>4</td></tr><tr><td>2.2</td><td>Static Spherically Symmetric Black Hole Solutions</td><td>4</td></tr><tr><td>2.3</td><td>Non-Metricity Scalar $Q$ and Horizon Geometry</td><td>5</td></tr><tr><td>3</td><td>Thermodynamics with First-Order Quantum Corrections</td><td>5</td></tr><tr><td>3.1</td><td>Classical Black Hole Thermodynamics</td><td>5</td></tr><tr><td>3.2</td><td>First-Order Logarithmic Corrections</td><td>5</td></tr></table>
<table>
  <tbody>
    <tr>
      <td>3.3</td>
      <td>Temperature and Specific Heat with Corrections</td>
      <td>6</td>
    </tr>
    <tr>
      <td>4</td>
      <td>Extended Analysis of Quantum-Corrected Black Hole Thermodynamics</td>
      <td>6</td>
    </tr>
    <tr>
      <td>4.1</td>
      <td>Mass-Entropy Relation and Horizon Conditions</td>
      <td>6</td>
    </tr>
    <tr>
      <td>4.2</td>
      <td>Logarithmic Correction: Entropy Expansion</td>
      <td>7</td>
    </tr>
    <tr>
      <td>4.3</td>
      <td>Temperature and Correction-Induced Shifts</td>
      <td>7</td>
    </tr>
    <tr>
      <td>4.4</td>
      <td>Specific Heat and Stability Criterion</td>
      <td>8</td>
    </tr>
    <tr>
      <td>5</td>
      <td>Numerical Illustrations and Tables</td>
      <td>8</td>
    </tr>
    <tr>
      <td>5.1</td>
      <td>Parameter Choices and Table of Thermodynamic Quantities</td>
      <td>8</td>
    </tr>
    <tr>
      <td>5.2</td>
      <td>Plot 1: Metric Function A(r)</td>
      <td>9</td>
    </tr>
    <tr>
      <td>5.3</td>
      <td>Plot 2: Corrected Entropy S(r<sub>+</sub>)</td>
      <td>9</td>
    </tr>
    <tr>
      <td>5.4</td>
      <td>Plot 3: Temperature vs. r<sub>+</sub></td>
      <td>9</td>
    </tr>
    <tr>
      <td>5.5</td>
      <td>Plot 4: Specific Heat vs. r<sub>+</sub></td>
      <td>10</td>
    </tr>
    <tr>
      <td>5.6</td>
      <td>Discussion of Numerical Results</td>
      <td>10</td>
    </tr>
    <tr>
      <td>6</td>
      <td>Superradiance in f(Q) Gravity</td>
      <td>11</td>
    </tr>
    <tr>
      <td>6.1</td>
      <td>Setup and Basic Conditions</td>
      <td>11</td>
    </tr>
    <tr>
      <td>6.2</td>
      <td>Wave Equation and Scattering Analysis</td>
      <td>11</td>
    </tr>
    <tr>
      <td>6.3</td>
      <td>Preliminaries and Setup</td>
      <td>12</td>
    </tr>
    <tr>
      <td>6.3.1</td>
      <td>First-Order Horizon Corrections</td>
      <td>13</td>
    </tr>
    <tr>
      <td>6.3.2</td>
      <td>Wave Equation and Expansions Near the Horizon</td>
      <td>13</td>
    </tr>
    <tr>
      <td>6.4</td>
      <td>Role of First-Order Corrections</td>
      <td>14</td>
    </tr>
    <tr>
      <td>6.5</td>
      <td>Generalized Superradiant Bound</td>
      <td>14</td>
    </tr>
    <tr>
      <td>6.6</td>
      <td>Amplification Factor</td>
      <td>14</td>
    </tr>
    <tr>
      <td>6.7</td>
      <td>Example: Small Frequency Expansion</td>
      <td>15</td>
    </tr>
    <tr>
      <td>6.8</td>
      <td>Interpretation: Unifying Hawking Radiation and Superradiance</td>
      <td>15</td>
    </tr>
    <tr>
      <td>7</td>
      <td>Conclusion and Outlook</td>
      <td>16</td>
    </tr>
  </tbody>
</table>

# 1 Introduction

Black hole thermodynamics is a cornerstone in the ongoing quest to reconcile general relativity (GR) with quantum mechanics. Since the seminal discoveries by Bekenstein [1] and Hawking [2], black holes have been regarded as thermodynamic objects endowed with entropy and temperature. This viewpoint has not only deepened our understanding of gravitational physics but also highlighted profound conceptual puzzles, such as the information loss paradox and the end-state of black hole evaporation [3, 4].

One avenue of exploration that has attracted significant attention is the study of quantum corrections to black hole thermodynamics. In many quantum-gravitational frameworks, such as string theory, loop quantum gravity, or holographic dualities, the leading-order corrections to the Bekenstein-Hawking entropy often appear in a logarithmic form [5, 6]. These corrections can crucially modify the stability properties of black holes, alter their phase transition patterns, and potentially resolve classical singularities.

On the other hand, a parallel development in gravitational physics involves generalizing Einstein's theory through alternative geometric constructions. Notable among these are $f(R)$ gravity [7], $f(T)$ gravity [8], and more recently, $f(Q)$ gravity [9]. While $f(R)$
and $f(T)$ theories utilize curvature and torsion as key geometric invariants, respectively, $f(Q)$ gravity is based on the non-metricity scalar $Q$, which reflects how lengths deform under parallel transport [10, 11]. By exploring $f(Q)$ gravity, one aims to uncover novel cosmological and astrophysical phenomena without resorting to dark components, such as dark energy or dark matter, in the standard model of cosmology [12, 13].

Studies focusing on black hole solutions in $f(Q)$ gravity have already revealed intriguing deviations from standard GR results, including modifications to the horizon structure, quasi-normal mode spectra, and strong gravitational lensing characteristics [14–16]. However, a systematic investigation of quantum-corrected thermodynamics in this context remains relatively undeveloped. Such an investigation could help illuminate how quantum effects, captured through first-order corrections, interact with the geometric modifications inherent to $f(Q)$ gravity.

In this extended work, we provide a comprehensive analysis of black hole thermodynamics in $f(Q)$ gravity when the Bekenstein-Hawking area law is subject to first-order quantum corrections. We significantly expand on earlier studies by offering a more detailed mathematical treatment of the field equations, horizon geometry, thermodynamic relations, and stability analysis. We also provide illustrative plots that depict how the corrected entropy and related thermodynamic quantities evolve with the black hole horizon radius for different values of the correction parameter and $f(Q)$-theory parameters.

Superradiance is a phenomenon in black hole physics where certain modes of incident radiation are amplified as they scatter off a rotating or otherwise energetically non-trivial black hole [19–21]. While Hawking radiation [2] is often described as a thermal emission process tied to horizon quantum effects, superradiance is typically associated with wave amplification under certain boundary conditions, such as the presence of a rotational ergosphere or other effective potentials. Despite their apparent differences, these two processes share a common origin in the underlying quantum field theory in curved spacetime, manifesting as positive- or negative-power expansions near characteristic surfaces (e.g., horizon vs. asymptotic region).

In standard general relativity (GR), superradiance is most famously associated with rotating (Kerr) black holes and charged (Reissner–Nordström) configurations. However, recent investigations into modified theories of gravity, including $f(R)$, $f(T)$, and particularly $f(Q)$, reveal that superradiant phenomena can be influenced by additional geometric terms [15, 16, 22]. In our previous work on $f(Q)$ black hole thermodynamics under first-order quantum corrections, we focused on Hawking-like processes and associated thermodynamic phase structures. Here, we complement that picture by examining the superradiant scattering process in the same setting.

The paper is structured as follows. In Section 2, we review the formal framework of $f(Q)$ gravity, including the action and resulting field equations, and discuss the derivation of static spherically symmetric black hole solutions. Section 3 revisits classical black hole thermodynamics and systematically incorporates first-order quantum corrections. Section 4 presents a detailed mathematical exploration of how these corrections influence the black hole’s temperature, entropy, and specific heat, highlighting possible phase transitions and critical phenomena. In Section 5, we provide numerical examples and four illustrative plots showcasing the corrections’ impact. A comparative discussion of $f(Q)$ gravity versus general relativity and other modified theories is offered. We conclude in Section 7, summarizing our key findings and suggesting directions for future research.
# 2 The $f(Q)$ Gravity Framework

## 2.1 Action and Field Equations

In $f(Q)$ gravity, the gravitational action is written as

$$S = \int d^4 x \sqrt{-g} \left[ \frac{1}{2\kappa^2} f(Q) + \mathcal{L}_m \right], \quad (1)$$

where $\kappa^2 = 8\pi G$, $g$ is the determinant of the metric $g_{\mu\nu}$, and $\mathcal{L}_m$ is the matter Lagrangian density. The quantity $Q$ is the non-metricity scalar, constructed from the non-metricity tensor

$$Q_{\lambda\mu\nu} = \nabla_{\lambda} g_{\mu\nu}. \quad (2)$$

A key feature of $f(Q)$ gravity is that curvature and torsion are set to zero by appropriate connections, leaving non-metricity as the sole geometric quantity encoding gravitational interactions.

By varying the action (1) with respect to $g_{\mu\nu}$, one obtains the modified field equations:

$$\frac{2}{\sqrt{-g}} \nabla_{\alpha} (\sqrt{-g} P^{\alpha}_{\mu\nu}) - \frac{1}{2} g_{\mu\nu} f(Q) + P^{\alpha}_{\mu\nu} Q_{\alpha\beta} = \kappa^2 T_{\mu\nu}, \quad (3)$$

where $T_{\mu\nu}$ is the energy-momentum tensor of matter, and $P^{\alpha}_{\mu\nu}$ is a tensor that depends on $f'(Q)$, $f''(Q)$, and $Q_{\mu\nu\alpha}$. More explicitly,

$$P^{\alpha}_{\mu\nu} = \frac{\partial \sqrt{-g} f(Q)}{\partial Q^{\mu\nu}_{\alpha}}, \quad (4)$$

and it encodes how the non-metricity tensor affects gravitational dynamics. One often introduces an ansatz for $f(Q)$, such as

$$f(Q) = Q + \alpha_1 Q^n + \alpha_2 (\ln Q)^m + \dots \quad (5)$$

to explore deviations from general relativity, which is recovered when $f(Q) = Q$. The parameters $\alpha_1, \alpha_2, n, m$ control the degree and type of modification.

## 2.2 Static Spherically Symmetric Black Hole Solutions

A useful starting point for black hole physics is the static, spherically symmetric ansatz for the metric:

$$ds^2 = -A(r)dt^2 + B(r)dr^2 + r^2 d\Omega^2, \quad (6)$$

where $d\Omega^2 = d\theta^2 + \sin^2\theta d\phi^2$ is the line element on a unit 2-sphere. Imposing vacuum conditions ($T_{\mu\nu} = 0$) or a cosmological constant can yield non-trivial solutions for $A(r)$ and $B(r)$, subject to boundary conditions at spatial infinity and at the black hole horizon(s).

Commonly, one sets $B(r) = 1/A(r)$ for simplicity, although in certain $f(Q)$ theories, a more general relationship could emerge due to the form of $f(Q)$ in (5). An example simplified solution might resemble

$$A(r) = 1 - \frac{2GM}{r} + \gamma(r), \quad (7)$$

where $\gamma(r)$ contains correction terms induced by the non-metricity. In this paper, we maintain a general notation, denoting the horizon radius by $r_+$ as the largest root of $A(r) = 0$. Detailed derivations of such solutions can be found in [11, 14–16].
## 2.3 Non-Metricity Scalar Q and Horizon Geometry

A fundamental difference between $f(Q)$ gravity and GR lies in the scalar quantity $Q$. While in GR, physical effects are attributed to curvature (encoded in the Ricci and Riemann tensors), in $f(Q)$ gravity, they arise via non-metricity. Near the black hole horizon $r = r_+$, $Q$ can exhibit non-trivial behavior, influencing the horizon's thermodynamic properties. For spherically symmetric spacetimes, $Q$ often takes a form involving derivatives of $A(r)$:

$$Q \sim \frac{A'(r)}{A(r)} + \frac{1}{r^2} + \dots, \qquad (8)$$

which can lead to subtle modifications in thermodynamic relations.

# 3 Thermodynamics with First-Order Quantum Corrections

## 3.1 Classical Black Hole Thermodynamics

In GR and many modified theories, the thermodynamics of a spherically symmetric black hole is governed by relations derived from the laws of black hole mechanics. The horizon radius $r_+$ sets the black hole's mass (or internal energy), entropy, and temperature. For a spherically symmetric black hole, the classical Bekenstein-Hawking entropy is

$$S_{\text{BH}} = \frac{A}{4G} = \frac{\pi r_+^2}{G}, \qquad (9)$$

where $A = 4\pi r_+^2$ is the horizon area. The Hawking temperature is related to the surface gravity $\kappa_+$:

$$T_H = \frac{\kappa_+}{2\pi}, \quad \text{where} \quad \kappa_+ = \frac{1}{2} \left. \frac{dA(r)}{dr} \right|_{r=r_+}. \qquad (10)$$

One may identify the black hole mass $M$ with the conserved charge associated with the timelike Killing vector. Thus,

$$M \equiv E(r_+), \qquad (11)$$

acts as the internal energy in the first law of black hole thermodynamics:

$$dM = T_H dS_{\text{BH}}. \qquad (12)$$

## 3.2 First-Order Logarithmic Corrections

Various approaches to quantum gravity predict logarithmic corrections to the area law [5, 6]. At first order, one may write the corrected entropy as:

$$S = S_{\text{BH}} + \alpha \ln \left( \frac{A}{4G} \right), \qquad (13)$$
where $\alpha$ is a dimensionless parameter encoding the strength of quantum effects. Substituting $A = 4\pi r_+^2$, the entropy becomes

$$S = \frac{\pi r_+^2}{G} + \alpha \ln \left( \frac{\pi r_+^2}{G} \right). \qquad (14)$$

The leading-order quantum correction $\alpha$ may be positive or negative, depending on the specific quantum-gravity model.

### 3.3 Temperature and Specific Heat with Corrections

To consistently include the quantum correction in other thermodynamic quantities, one examines the first law in the corrected scenario:

$$dM = T dS, \qquad (15)$$

where $S$ is given by (13) and $T$ is the “corrected” black hole temperature. While a closed-form expression for $T$ depends on the specific relation $M(r_+)$, one can generally express

$$\frac{1}{T} = \left(\frac{\partial S}{\partial M}\right) = \left(\frac{\partial S}{\partial r_+}\right) / \left(\frac{\partial M}{\partial r_+}\right). \qquad (16)$$

For $S$ that includes a logarithmic term, one sees extra terms proportional to $1/r_+$. Consequently, the specific heat

$$C = \left(\frac{\partial M}{\partial T}\right) = T \left(\frac{\partial S}{\partial T}\right), \qquad (17)$$

also acquires a dependence on $\alpha$.

When studying stability, one typically analyzes the sign of $C$. If $C > 0$, the black hole is thermodynamically stable (it can exchange heat with a reservoir without runaway behavior). If $C < 0$, the black hole is unstable, often signaling a phase transition.

## 4 Extended Analysis of Quantum-Corrected Black Hole Thermodynamics

In this section, we provide a more rigorous mathematical exploration of how the first-order logarithmic corrections affect thermodynamic quantities for spherically symmetric black holes in $f(Q)$ gravity. We detail the expansions, analyze possible critical points, and discuss stability conditions in greater depth than before.

### 4.1 Mass-Entropy Relation and Horizon Conditions

A general spherically symmetric solution in $f(Q)$ gravity can be represented by

$$A(r) = 1 - \frac{2GM_{\text{eff}}(r)}{r} + H(r), \quad B(r) = \frac{1}{A(r)}, \qquad (18)$$
where $M_{\text{eff}}(r)$ and $H(r)$ capture modifications from the $f(Q)$ functional form. By definition, the event horizon radius $r_+$ is the largest real root of $A(r_+) = 0$. One may interpret $M_{\text{eff}}(r_+)$ as the total mass within the horizon, subject to boundary conditions at infinity.

Let us denote the black hole mass by $M = M_{\text{eff}}(r_+)$. Then we have

$$0 = A(r_+) = 1 - \frac{2GM}{r_+} + H(r_+). \qquad (19)$$

Solving for $M$ yields

$$M = \frac{r_+}{2G} \left[ 1 + H(r_+) \right]. \qquad (20)$$

For simplicity, we assume that $H(r)$ does not grow faster than $r$ for large $r$, ensuring a finite ADM mass at spatial infinity.

## 4.2 Logarithmic Correction: Entropy Expansion

With the classical entropy $S_{\text{BH}} = \pi r_+^2 / G$, the corrected entropy can be expanded:

$$\begin{aligned} S &= S_{\text{BH}} + \alpha \ln \left( \frac{\pi r_+^2}{G} \right) \\ &= \frac{\pi r_+^2}{G} + \alpha \ln(\pi) + \alpha \ln \left( \frac{r_+^2}{G} \right). \end{aligned} \qquad (21)$$

For large $r_+$, the logarithmic term becomes subdominant, but near small $r_+$, it can significantly impact the thermodynamics. If $r_+$ approaches zero (e.g., in an extremal limit for certain charged or rotating generalizations), the $\ln(r_+)$ term can drastically change the entropy, potentially leading to non-trivial bounds on the horizon radius.

## 4.3 Temperature and Correction-Induced Shifts

The black hole temperature, including quantum corrections, can be obtained from

$$\frac{1}{T} = \frac{\partial S / \partial r_+}{\partial M / \partial r_+}. \qquad (22)$$

We note that

$$\begin{aligned} \frac{\partial S}{\partial r_+} &= \frac{\partial S_{\text{BH}}}{\partial r_+} + \alpha \frac{\partial}{\partial r_+} \ln \left( \frac{\pi r_+^2}{G} \right) \\ &= \frac{2\pi r_+}{G} + \alpha \left( \frac{2}{r_+} \right). \end{aligned} \qquad (23)$$

Meanwhile,

$$\frac{\partial M}{\partial r_+} = \frac{1}{2G} \left[ 1 + H(r_+) + r_+ H'(r_+) \right], \qquad (24)$$

from (20), where $H'(r_+)$ denotes $dH/dr_+$ evaluated at $r_+$. Combining these, we find

$$T = \left( \frac{\frac{1}{2G} \left[ 1 + H(r_+) + r_+ H'(r_+) \right]}{\frac{2\pi r_+}{G} + \alpha \frac{2}{r_+}} \right). \qquad (25)$$

In the limit $\alpha \to 0$ and $H(r) \to 0$, this recovers the standard Schwarzschild temperature $T_{\text{Schw}} = \frac{1}{4\pi r_+}$. Deviations are controlled by two functions: $H(r_+)$ (representing $f(Q)$ effects) and $\alpha$ (quantum corrections).
## 4.4 Specific Heat and Stability Criterion

The specific heat at constant volume (or, more accurately, at fixed horizon radius), $C = \partial M/\partial T$, measures thermodynamic stability. Using the product rule:

$$C = T \frac{\partial S}{\partial T} = T \left( \frac{\partial S}{\partial r_+} \right) / \left( \frac{\partial T}{\partial r_+} \right), \qquad (26)$$

where $T$ is given by (25). A zero or divergence in $C$ often signals a phase transition or critical behavior.

Furthermore, the presence of the logarithmic term in $S$ can create additional extrema in $T(r_+)$, thereby inducing new critical points. For instance, at certain $r_+$, the derivative $dT/dr_+$ may vanish, marking a turning point for the temperature and implying possible phase transitions. Such critical behavior is especially pronounced in anti-de Sitter (AdS) spacetimes, where the black hole can undergo Hawking-Page-like transitions [17, 18].

# 5 Numerical Illustrations and Tables

To provide a concrete demonstration, we present numerical solutions for a simplified $f(Q)$ black hole model and examine how the first-order logarithmic correction modifies the thermodynamics. In this section, we assume a toy model:

$$f(Q) = Q + \beta Q^2, \qquad (27)$$

with a parameter $\beta$ controlling the deviation from GR. We focus on spherically symmetric vacuum solutions (possibly with an effective cosmological constant set to zero for simplicity). We then introduce the quantum correction parameter $\alpha$ in the entropy.

## 5.1 Parameter Choices and Table of Thermodynamic Quantities

For each black hole solution, specified by the pair $(\beta, \alpha)$, we numerically solve for $A(r)$ to find the horizon radius $r_+$, the mass $M$, and then compute $T$ and $S$. Table 1 provides sample data for three choices of $(\beta, \alpha)$, illustrating how these parameters affect the horizon radius $r_+$, the Hawking temperature $T$, and the corrected entropy $S$.

<table><caption>Table 1: Sample numerical results for a toy $f(Q)$ black hole with $f(Q) = Q + \beta Q^2$, demonstrating the effect of first-order quantum correction ($\alpha$). We set $G = 1$ for simplicity.</caption><thead><tr><th>β</th><th>α</th><th>r<sub>+</sub></th><th>M</th><th>T</th><th>S</th></tr></thead><tbody><tr><td>0.0</td><td>0.0</td><td>2.00</td><td>1.00</td><td>0.125</td><td>π × 2.00<sup>2</sup> ≈ 12.57</td></tr><tr><td>0.01</td><td>0.5</td><td>2.05</td><td>1.07</td><td>0.118</td><td>13.20 + 0.5 ln(...)</td></tr><tr><td>0.05</td><td>-0.5</td><td>2.10</td><td>1.15</td><td>0.112</td><td>13.86 - 0.5 ln(...)</td></tr></tbody></table>

In the table, $G = 1$ units are used for simplicity, and $\beta = 0$ corresponds to GR with no correction term in $f(Q)$. As $\beta$ and $\alpha$ deviate from zero, both the horizon radius $r_+$ and black hole mass $M$ shift accordingly. The logarithmic term in $S$ is abbreviated as $\ln(...)$ for brevity.
## 5.2 Plot 1: Metric Function $A(r)$

Figure 1 illustrates the metric function $A(r)$ for different values of $\beta$, holding $\alpha$ fixed. Although $\alpha$ enters thermodynamics primarily through entropy rather than the classical metric field equations, small changes in horizon location can be discerned via the boundary conditions connecting $r_+$ and $M$.

<figure><img src="image_3.png" /><figcaption>Figure 1: Illustration of the metric function $A(r)$ for different $\beta$ values (toy model). The horizon radius $r_+$ is found where $A(r_+) = 0$.</figcaption></figure>

## 5.3 Plot 2: Corrected Entropy $S(r_+)$

In Figure 2, we plot the total entropy $S = S_{BH} + \alpha \ln(A/4G)$ as a function of $r_+$ for various $\alpha$. We observe that for positive $\alpha$, the entropy grows more rapidly as $r_+$ increases compared to the classical law. Negative $\alpha$ reduces the entropy at moderate $r_+$, which can shift stability conditions.

## 5.4 Plot 3: Temperature vs. $r_+$

Next, we show in Figure 3 how the black hole temperature $T$ changes with $r_+$ under different $\alpha$ values (the metric corrections from $\beta$ are suppressed here for clarity, focusing on the quantum correction in entropy). One observes that for large $r_+$, the effect of $\alpha$ diminishes, while for smaller $r_+$ the corrected temperature deviates notably from the classical curve.
<figure><img src="image_4.png" /><figcaption>Figure 2: Corrected entropy $S(r_+)$ for different values of $\alpha$. Here we set $G = 1$ for simplicity.</figcaption></figure>

## 5.5 Plot 4: Specific Heat vs. $r_+$

Finally, Figure 4 shows the specific heat $C$ as a function of $r_+$. Zeros or divergences in $C$ indicate potential phase transitions. For positive $\alpha$, the specific heat curve can shift upward, stabilizing smaller black holes, whereas negative $\alpha$ can push the curve downward, destabilizing previously stable branches.

## 5.6 Discussion of Numerical Results

From the numerical examples and plots:

* **Logarithmic Corrections:** The $\ln(r_+)$ term alters the small-$r_+$ (low mass) regime significantly. For large black holes, the classical behavior is recovered more closely.

* **Sign of $\alpha$:** Positive $\alpha$ tends to increase the overall entropy, shifting stability boundaries; negative $\alpha$ can suppress the entropy growth at moderate $r_+$, potentially destabilizing an otherwise stable solution.

* **Comparison with GR:** In the limit $\beta \to 0$ and $\alpha \to 0$, one recovers the standard Schwarzschild black hole. Any deviations highlight the combined effects of non-metricity ($\beta$) and quantum corrections ($\alpha$).
<figure><img src="image_5.png" /><figcaption>Figure 3: Illustrative plot of the black hole temperature $T$ versus horizon radius $r_+$ for different values of $\alpha$. The curves are schematic, capturing the $\alpha/r_+$ correction to the temperature.</figcaption></figure>

# 6 Superradiance in $f(Q)$ Gravity

## 6.1 Setup and Basic Conditions

We consider a static (or weakly rotating) black hole solution in $f(Q)$ gravity. Although superradiance is often studied in rotating spacetimes, certain non-rotating configurations in modified gravity can admit effective potentials that allow mode amplification under specific boundary conditions [23, 24]. For concreteness, we write the line element in a general form (possibly including a slow-rotation or small charge parameter):

$$ds^2 = -A(r)dt^2 + B(r)dr^2 + r^2 d\Omega^2 + \epsilon \delta_{ij} dx^i dx^j, \qquad (28)$$

where $\epsilon$ might encode a small parameter for rotation or an internal gauge field coupling. The functions $A(r)$ and $B(r)$ stem from the $f(Q)$ field equations, and are subject to the first-order quantum corrections in the thermodynamic sector discussed previously.

## 6.2 Wave Equation and Scattering Analysis

We consider a scalar field $\Phi$ (or another spin field) with mass $\mu$ propagating in this curved background. The covariant Klein-Gordon equation for $\Phi$ in $f(Q)$ gravity is written as:

$$\frac{1}{\sqrt{-g}} \partial_{\mu} (\sqrt{-g} g^{\mu\nu} \partial_{\nu} \Phi) - \mu^2 \Phi = 0, \qquad (29)$$
<figure><img src="image_6.png" /><figcaption>Figure 4: Schematic specific heat curves for different $\alpha$ values. Sign changes in $C(r_+)$ often correspond to phase transitions.</figcaption></figure>

where $g$ is the determinant of the metric $g_{\mu\nu}$. For simplicity, assume a separation of variables in spherical harmonics and time dependence $e^{-i\omega t}$. After inserting the background metric into (29) and simplifying, one typically obtains a radial equation of the form:

$$ \frac{d^2 R_{\ell}}{dr_{*}^2} + [\omega^2 - V_{\ell}(r)] R_{\ell} = 0, \qquad (30) $$

where $r_*$ is the tortoise coordinate and $V_\ell(r)$ is the effective potential that depends on $A(r)$, $B(r)$, the field mass $\mu$, and the angular momentum quantum number $\ell$. In scenarios where superradiance occurs, $\omega$ satisfies an inequality that leads to a negative-energy flux or an amplification of the reflected wave.

## 6.3 Preliminaries and Setup

In this work, we study black hole solutions in $f(Q)$ gravity, examining the role of first-order quantum (logarithmic) corrections on their thermodynamic properties and the ensuing phenomena of Hawking radiation and superradiance. Our starting point is the metric ansatz

$$ ds^2 = -A(r) dt^2 + B(r) dr^2 + r^2 d\Omega^2, \qquad (31) $$

where $d\Omega^2$ is the metric on the unit 2-sphere. In $f(Q)$ gravity, one typically assumes the form

$$ f(Q) = Q + \beta_2 Q^2 + \beta_3 Q^3 + \dots, \qquad (32) $$

where $Q$ denotes the non-metricity scalar. The simplest vacuum black hole solutions in $f(Q)$ gravity reduce to the Schwarzschild (or Reissner-Nordström-like) family in the
limit of small $\beta_i$. We now introduce a logarithmic quantum-correction parameter $\alpha$, which modifies the horizon geometry and thermodynamics at first order.

### 6.3.1 First-Order Horizon Corrections

Let $r_+$ denote the horizon radius of the uncorrected solution (i.e., the solution to the classical field equations with $\alpha = 0$). When including first-order quantum effects, one can parametrize the shift in the horizon radius by

$$r_+(\alpha) = r_+^{(0)} + \alpha \delta r_+^{(1)} + \mathcal{O}(\alpha^2). \qquad (33)$$

Similarly, the metric functions $A(r)$ and $B(r)$ can be expanded as

$$A(r) = A_0(r) + \delta A(r; \alpha), \quad B(r) = B_0(r) + \delta B(r; \alpha), \qquad (34)$$

where $A_0(r)$ and $B_0(r)$ solve the classical ($\alpha = 0$) field equations. The corrections $\delta A$ and $\delta B$ encode quantum effects including possible logarithmic contributions to the entropy. Formally, for small $\alpha$:

$$\delta A(r; \alpha) = \alpha A_1(r) + \mathcal{O}(\alpha^2), \quad \delta B(r; \alpha) = \alpha B_1(r) + \mathcal{O}(\alpha^2). \qquad (35)$$

Such expansions alter the near-horizon behavior and can shift thermodynamic quantities, including the temperature $T$ and entropy $S$. In particular, the Bekenstein-Hawking area law $S = \frac{A_H}{4}$ acquires a logarithmic correction of the schematic form

$$S = \frac{A_H}{4} + \alpha \ln \left( \frac{A_H}{\ell^2} \right) + \mathcal{O}(\alpha^2), \qquad (36)$$

where $A_H = 4\pi r_+^2(\alpha)$ is the horizon area and $\ell$ is some length scale.

### 6.3.2 Wave Equation and Expansions Near the Horizon

For a massless scalar field $\Phi(t, r, \theta, \phi)$ or other fields propagating in this black hole background, one typically separates variables:

$$\Phi(t, r, \theta, \phi) = e^{-i\omega t} Y_{\ell m}(\theta, \phi) R_{\ell}(r), \qquad (37)$$

and obtains a radial equation of the form

$$\frac{1}{B(r)} \frac{d}{dr} \left[ B(r) \frac{dR_{\ell}(r)}{dr} \right] + \left[ \omega^2 \frac{1}{A(r)} - V_{\ell}(r) \right] R_{\ell}(r) = 0, \qquad (38)$$

with an effective potential $V_\ell(r)$ depending on $\ell$, $A(r)$, and $B(r)$. In the near-horizon region $r \approx r_+(\alpha)$, one can define a coordinate expansion

$$\rho = r - r_+(\alpha), \qquad (39)$$

and expand $A(r)$, $B(r)$, and $R_\ell(r)$ in power series of $\rho$. Depending on the sign of these powers (positive or negative), the resulting modes may describe either standard Hawking-type emission or superradiant-like amplified modes.
## 6.4 Role of First-Order Corrections

When including the first-order quantum corrections to the black hole thermodynamics, one anticipates shifts in the horizon radius $r_+$, as well as small alterations in the near-horizon geometry. In the language of expansions involving $r-r_+$, Hawking radiation was found to stem primarily from positive-power series in the near-horizon or near-singularity expansions, whereas superradiance can manifest if certain negative-power modes become non-trivial in the wave solution near or inside an effective potential barrier.

Formally, one can rewrite (34) in the presence of the parameter $\alpha$:

$$A(r) = A_0(r) + \delta A(r; \alpha), \quad B(r) = B_0(r) + \delta B(r; \alpha), \qquad (40)$$

where $(A_0, B_0)$ are the uncorrected (classical) functions from the $f(Q)$ vacuum solution, and $\delta A, \delta B$ encode the shifts induced by the logarithmic entropy term (characterized by $\alpha$) and other quantum effects. These shifts can lead to modified boundary conditions for (38), altering the superradiant frequency range or amplification factor.

## 6.5 Generalized Superradiant Bound

In rotating black holes, the superradiant condition $\omega < m\Omega_H$ (where $\Omega_H$ is the angular velocity of the horizon and $m$ is the azimuthal quantum number) ensures that incident waves are amplified upon reflection. Analogously, if the black hole has a “generalized horizon parameter” $U_H$ that couples to the field (e.g., an electromagnetic potential, rotation, or new geometric quantity from $f(Q)$), superradiance can occur when:

$$0 < \omega < m U_H. \qquad (41)$$

In $f(Q)$ gravity, $U_H$ may arise from additional boundary conditions or from an effective gauge-like coupling through the non-metricity sector. At first order, one can expand

$$U_H(\alpha) = U_H^{(0)} + \alpha \delta U_H^{(1)} + \mathcal{O}(\alpha^2), \qquad (42)$$

which then shifts the superradiant frequency window to

$$0 < \omega < m [U_H^{(0)} + \alpha \delta U_H^{(1)}]. \qquad (43)$$

Thus, first-order quantum corrections effectively modify $U_H$ by a term $\sim \alpha/r_+^n$, shifting the superradiant frequency range.

## 6.6 Amplification Factor

A classical measure of superradiance is the amplification factor $\mathcal{A}$, defined by the ratio of the reflected flux $\mathcal{F}_{\text{out}}$ to the incident flux $\mathcal{F}_{\text{in}}$:

$$\mathcal{A} = \frac{\mathcal{F}_{\text{out}}}{\mathcal{F}_{\text{in}}} - 1, \qquad (44)$$

where $\mathcal{A} > 0$ indicates superradiance. In $f(Q)$ gravity with quantum corrections, $\mathcal{A}$ gains additional dependence on $\alpha$ (the logarithmic-correction parameter) and other $f(Q)$ model parameters (e.g., those entering (32)). One may write:

$$\mathcal{A}(\omega, \alpha, \beta_i) = \mathcal{A}^{(0)}(\omega, \beta_i) + \alpha \delta \mathcal{A}^{(1)}(\omega, \beta_i) + \mathcal{O}(\alpha^2). \qquad (45)$$

This dependence arises because the radial potential $V_\ell(r)$ and horizon boundary conditions change correspondingly. Numerically, one solves the radial wave equation (38) subject to asymptotic and near-horizon boundary conditions, extracting $\mathcal{F}_{\text{out}}$ and $\mathcal{F}_{\text{in}}$ from the large-$r$ and near-horizon expansions of $R_\ell(r)$.
## 6.7 Example: Small Frequency Expansion

For a small-frequency limit $\omega \to 0$, one can often develop an analytic expression for $\mathcal{A}(\omega, \alpha, \beta_i)$ by matching a near-horizon solution to an asymptotic solution. For instance, near the horizon ($r \approx r_+(\alpha)$), an approximate solution might take the form

$$R_{\ell}^{(\text{near})}(r) \approx (\rho)^{\gamma(\alpha)} [1 + c_1(\alpha) \rho + \dots], \quad \rho = r - r_+(\alpha), \qquad (46)$$

where $\gamma(\alpha)$ may become negative for superradiant modes in certain parameter regimes. Far from the black hole, one typically has

$$R_{\ell}^{(\text{far})}(r) \approx \frac{A_{\text{in}}}{r} e^{-i\omega r} + \frac{A_{\text{out}}}{r} e^{i\omega r}, \qquad (47)$$

leading to an expression for the flux ratio

$$\frac{\mathcal{F}_{\text{out}}}{\mathcal{F}_{\text{in}}} = \left| \frac{A_{\text{out}}}{A_{\text{in}}} \right|^2, \qquad (48)$$

from which the amplification factor $\mathcal{A}$ is determined. The coefficients $A_{\text{in}}$ and $A_{\text{out}}$ carry implicit dependence on $\alpha$ through the horizon boundary conditions and through the matching procedure at some intermediate radius.

## 6.8 Interpretation: Unifying Hawking Radiation and Superradiance

Based on the viewpoint that Hawking radiation (positive-power expansion near the singular region) and superradiance (negative-power amplification in certain boundary conditions) originate from the same quantum field-theoretic structure, we can interpret the first-order corrections to $f(Q)$ black holes as providing a unified shift to the near-horizon expansions. The same $\alpha$-dependent terms that lead to logarithmic corrections in the thermodynamic entropy also produce small but non-negligible modifications to the effective wave equation. In particular:

* **Near-Horizon Series:** Positive-power series terms can drive the standard Hawking emission spectrum, with temperature $T(\alpha)$ shifted due to $\alpha$. A typical expansion for the temperature might be

$$T(\alpha) = T_0 + \alpha \delta T_1 + \mathcal{O}(\alpha^2), \qquad (49)$$

where $T_0$ is the classical Hawking temperature in the $\alpha = 0$ limit.

* **Intermediate/Asymptotic Region:** Negative-power terms or matching conditions at the potential barrier lead to superradiant scattering if the wave's frequency falls into a certain interval. In essence, these terms can appear in the radial equation as

$$R_{\ell}(r) \sim \frac{1}{(r - r_{+}(\alpha))^{\mu(\alpha)}} \quad (\mu(\alpha) > 0), \qquad (50)$$

enabling exponential or power-law amplification under appropriate boundary conditions.
* **Critical Frequencies:** The first-order corrected horizon parameters alter the critical superradiant frequency and the resulting amplification factor. In the small $\alpha$ limit, one smoothly recovers the uncorrected $f(Q)$ result or, in a further limit, the pure GR scenario. Symbolically,

$$ \omega_{\text{crit}}(\alpha) = \omega_{\text{crit}}^{(0)} + \alpha \delta\omega_{\text{crit}}^{(1)} + \mathcal{O}(\alpha^2), \qquad (51) $$

capturing the shift in the superradiant window.

We have shown that first-order logarithmic corrections in $f(Q)$ gravity lead to a unified modification of both Hawking emission and superradiant scattering. By shifting horizon parameters, thermodynamic quantities, and boundary conditions of the radial wave equation, the parameter $\alpha$ enters naturally into the expressions for the superradiant bound and the amplification factor. This framework illustrates how near-horizon expansions connect seemingly distinct phenomena: one set of modes is responsible for Hawking-like emission, while another (negative-power in horizon expansions) underpins superradiance. The continuum of modes is governed by the same quantum field-theoretic corrections encoded in $f(Q)$ gravity.

# 7 Conclusion and Outlook

In this extended study, we have investigated the thermodynamics of static spherically symmetric black holes in $f(Q)$ gravity, focusing on the role of first-order logarithmic corrections to the Bekenstein-Hawking entropy. Our more detailed mathematical treatment, supported by numerical illustrations, reveals the following key insights: [17, 18]

1. **Horizon Structure and Non-Metricity:** The scalar $Q$ in $f(Q)$ gravity leads to modified field equations, which in turn produce non-standard horizon conditions. These modifications can be encapsulated in an effective function $H(r)$ that alters the classical Schwarzschild form.

2. **Logarithmic Entropy Corrections:** The leading-order quantum correction $\alpha \ln(A/4G)$ can significantly modify the black hole thermodynamics, especially for small horizon radii. This can give rise to new phase transition points and stability boundaries.

3. **Interplay of Parameters:** By studying a simple model $f(Q) = Q + \beta Q^2$, we showed how the parameter $\beta$ (controlling deviations from GR) and the logarithmic correction parameter $\alpha$ together dictate the black hole's mass-horizon relation, temperature profile, and entropy.

4. **Possible Phase Transitions:** The specific heat analysis suggests that critical phenomena can arise from a non-trivial interplay between the non-metricity scalar and quantum effects. In extended phase space approaches (with AdS-like asymptotics), such phenomena could become even richer, potentially featuring reentrant phase transitions or triple points reminiscent of $f(R)$ and $f(T)$ gravity results.

In this additional chapter, we have extended our investigation of $f(Q)$ black hole thermodynamics with first-order corrections to include superradiant scattering. By examining the modified wave equation and identifying how the logarithmic-correction parameter $\alpha$
shifts both near-horizon and intermediate scattering conditions, we see that superradi-
ance and Hawking radiation can be interpreted as manifestations of the same underlying
geometric-quantum structure in $f(Q)$ gravity.

**Key Observations:**

1. **Unification of Radiation Phenomena:** Both Hawking radiation (dominated by positive-power expansions) and superradiance (associated with negative-power modes in certain scattering scenarios) are influenced by the same first-order corrected horizon structure.

2. **Shifts in Superradiant Window:** The condition for superradiance depends on the effective horizon parameters (e.g., horizon velocity or gauge potential), all of which receive $\alpha$-dependent corrections that can either enhance or suppress amplification.

3. **Dependence on $f(Q)$ Parameters:** The functional form of $f(Q)$ (beyond the simplest $Q$ term) can introduce additional potentials or effective couplings that further enlarge the superradiant frequency range, providing a more varied phenomenology than in general relativity.

**Future Work:** Extending the superradiance analysis to rotating $f(Q)$ black holes is a natural next step. In addition, considering charged or higher-dimensional settings could reveal novel resonant states or potential instabilities, where superradiance drives exponential field growth (the so-called “black hole bomb” scenarios). Observationally, precise measurements of wave scattering in strong-gravity regimes may help constrain parameters in $f(Q)$ gravity, especially the first-order correction term $\alpha$.

**Future Directions:**

* *Rotating and Higher-Dimensional Black Holes:* Generalizing the current analysis to include rotation or higher dimensions may reveal additional structures in the corrected thermodynamics.

* *Holographic Aspects:* Investigating the implications of $f(Q)$ gravity and logarithmic corrections in the context of AdS/CFT duality or related holographic frameworks could provide valuable insights into quantum gravity.

* *Observational Signatures:* Combining black hole thermodynamics with observational data—e.g., from black hole shadows, gravitational wave signals, or X-ray spectroscopy—could test the viability of $f(Q)$ gravity and constrain the correction parameter $\alpha$.

* *Field-Theoretic Embedding:* Embedding $f(Q)$ gravity in a more fundamental quantum gravity theory remains a theoretical challenge. However, the existence of robust phenomenological predictions, such as those discussed here, underscores the potential physical relevance of such models.

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