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# Study of Non-Local Couplings in the f(Q) Gravitational Action: Analysis of Different Forms of f(Q) Gravity

Wen-Xiang Chen

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Wen-Xiang Chen. Study of Non-Local Couplings in the f(Q) Gravitational Action: Analysis of Different Forms of f(Q) Gravity. 2025. hal-05040818

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# Study of Non-Local Couplings in the $f(Q)$ Gravitational Action: Analysis of Different Forms of $f(Q)$ Gravity

Wen-Xiang Chen

Department of Astronomy

School of Physics and Materials Science

GuangZhou University

wxchen4277@qq.com

April 2025

## Abstract

The study of modified gravity theories, particularly $f(Q)$ gravity, has become a significant area of research in theoretical physics, especially in addressing the shortcomings of General Relativity (GR) in the realms of quantum mechanics and cosmology. This paper presents a detailed analysis of various forms of $f(Q)$ gravity, exploring their mathematical structure, the field equations they produce, and their physical implications. We will focus on several well-known functional forms of $f(Q)$, investigate their solutions, and discuss their effects on black holes, cosmological models, superradiance, and thermodynamics.

## 1 Introduction

General Relativity (GR) has long been the cornerstone of gravitational theory, providing profound insights into spacetime curvature and the behavior of matter under gravitational forces. However, it faces challenges in areas such as quantum mechanics, dark energy, dark matter, and singularities. These limitations have sparked the development of alternative theories, one of which is $f(Q)$ gravity. The framework of $f(Q)$ gravity replaces the standard Ricci scalar $R$ in the Einstein-Hilbert action with a general function $f(Q)$ of the non-metricity scalar $Q$. This approach aims to introduce new insights into gravitational interactions, particularly in the context of non-metric geometries, and non-local interactions. As such, it offers a potential avenue for addressing some of the unresolved questions in cosmology and quantum gravity. In this paper, we explore various forms of $f(Q)$ gravity. By analyzing the field equations generated from different functional forms of $f(Q)$, we will derive key results for black hole solutions, examine their cosmological implications, and explore their thermodynamic properties. Additionally, we will explore the superradiance phenomenon within this framework and analyze the potential effects of quantum entanglement and non-local couplings.

Black holes, once believed to be the ultimate absorbers of matter and energy, have evolved in our understanding from purely gravitational objects to quantum systems with intricate thermodynamic properties. This shift in perspective was catalyzed by the pioneering work of
Stephen Hawking in the 1970s, which revealed that black holes are not entirely black but emit thermal radiation due to quantum effects. This radiation, now known as Hawking radiation, has fundamentally transformed our understanding of black hole physics and has opened new pathways in the study of quantum gravity [1–9].

The concept of Hawking radiation arises from quantum field theory in curved spacetime, where particle-antiparticle pairs are generated near the event horizon of a black hole. Under certain conditions, one particle of the pair falls into the black hole while the other escapes, resulting in a net loss of mass and energy from the black hole, observed as thermal radiation. The temperature of this radiation, termed the Hawking temperature, is proportional to the surface gravity of the black hole and inversely proportional to its mass. In classical general relativity, this relationship is straightforward, providing deep insights into black hole thermodynamics. Notably, it suggests that black holes have entropy proportional to the area of their event horizon, a notion first proposed by Jacob Bekenstein.

The Hawking temperature in classical general relativity is given by:

$$T = \frac{1}{4\pi} \left. \frac{d}{dr} (f(r)) \right|_{r=r_+} \qquad (1)$$

where $f(r)$ is the metric function, and $r_+$ denotes the radius of the event horizon. However, the calculation of Hawking temperature becomes substantially more complicated when considering modified theories of gravity, such as $f(Q)$ gravity. These theories extend general relativity by modifying the gravitational action with additional terms that may represent new physical phenomena or quantum corrections to classical gravity. In $f(Q)$ gravity, the gravitational action depends on a function of the non-metricity scalar $Q$, generalizing the curvature and torsion found in other modified gravity theories. As a result, black hole behavior in these frameworks can differ significantly from general relativity, particularly in the thermodynamic context.

One challenge in extending the concept of Hawking temperature to black holes in $f(Q)$ gravity is the complexity of the field equations and the non-trivial nature of black hole solutions in these theories. Traditional methods for calculating Hawking temperature, such as using the surface gravity at the event horizon, may not be directly applicable or may require substantial modifications to accommodate the effects introduced by the $f(Q)$ framework. This is where the RVB (Robson-Villari-Biancalana) method becomes crucial as an alternative.

The RVB method is a topological approach to calculating Hawking temperature, which relates it to the Euler characteristic of the black hole’s horizon. The Euler characteristic is a topological invariant that provides a global description of the event horizon’s geometry, independent of the specific details of the metric. This makes the RVB method particularly valuable in modified gravity theories, where the metric may be highly non-trivial or where standard definitions of surface gravity may not be easily applicable [2, 10–17].

When applying the RVB method to black holes in $f(Q)$ gravity [2], we observe an additional term in the expression for the Hawking temperature. This term does not appear in classical calculations and is a direct consequence of the modifications introduced by the $f(Q)$ theory. Specifically, this term is linked to a residue derived from a contour integral involving a function $f(z)$, which is closely related to the metric or curvature scalar in these gravity models. The presence of this residue suggests that the thermodynamics of black holes in $f(Q)$ gravity differ fundamentally from those in general relativity, with profound implications for our understanding of quantum gravity and the ultimate fate of black holes.

The Hawking temperature in $f(Q)$ gravity is modified by an additional term that can be expressed as: [15–20]
$$T = \frac{1}{4\pi} \left( \frac{2M}{r_+^2} + \alpha \right) \qquad (2)$$

This modification, associated with the residue term, suggests new dependencies on the parameters of the $f(Q)$ model and may lead to observable differences in black hole behavior. The paper [2] systematically applies the RVB method to various black hole solutions within the $f(Q)$ gravity framework, aiming to identify and interpret the physical significance of the residue term. Our results highlight the crucial role of the residue in correcting the standard Hawking temperature, leading to new insights into black hole thermodynamics and potentially opening the door to new phenomena in astrophysical observations.

In classical general relativity, the Hawking temperature is directly linked to the surface gravity at the event horizon, which depends on the mass, charge, and angular momentum of the black hole. In $f(Q)$ gravity, the modification to the gravitational action alters the relationship between the metric and surface gravity, resulting in the appearance of the additional term in the Hawking temperature. This term can be interpreted as arising from the non-trivial topology of the modified spacetime, which affects the global properties of the black hole and its thermodynamic behavior.

Our analysis underscores the significance of understanding the interplay between topology and thermodynamics in black hole physics. In modified gravity theories like $f(Q)$ gravity, where the local structure of spacetime may differ significantly from general relativity, topological methods provide a powerful tool to explore these differences and their implications for black hole thermodynamics.

Furthermore, the Hawking temperature suggests that black holes in $f(Q)$ gravity may exhibit novel phenomena not seen in general relativity. For instance, the temperature of radiation emitted by such black holes may have a different dependence on the black hole's parameters, potentially leading to new predictions about black hole evaporation and their ultimate fate. These differences could be detectable in astrophysical observations or in theoretical studies of black hole mergers and gravitational wave signals, providing a potential avenue for testing modified gravity theories.

## 2 Theoretical Framework for $f(Q)$ Gravity

The action for $f(Q)$ gravity is given by: [1, 2, 15–20]

$$S = \int d^4 x \sqrt{-g} f(Q) \qquad (3)$$

where $g$ is the determinant of the metric tensor $g_{\mu\nu}$, and $f(Q)$ is an arbitrary function of the non-metricity scalar $Q$. The non-metricity scalar $Q_{\mu\nu\rho}$ is defined as:

$$Q_{\mu\nu\rho} = \nabla_{\mu} g_{\nu\rho} + \nabla_{\nu} g_{\mu\rho} + \nabla_{\rho} g_{\mu\nu} \qquad (4)$$

The non-metricity tensor describes how the metric tensor varies under parallel transport and is crucial in theories involving non-metric geometries. Unlike in General Relativity, where the connection is torsion-free and metric-compatible, in $f(Q)$ gravity, the connection can be non-metric, which leads to the introduction of additional degrees of freedom in the theory. The field equations are derived by varying the action with respect to the metric tensor $g_{\mu\nu}$, leading to the modified Einstein equations:

$$f'(Q) (\nabla_{\mu} Q_{\nu\rho\lambda} - \nabla_{\nu} Q_{\mu\rho\lambda}) + \cdots = T_{\mu\nu} \qquad (5)$$
where $f'(Q)$ represents the derivative of the function $f(Q)$ with respect to $Q$, and $T_{\mu\nu}$ is the energy-momentum tensor. The specific form of the field equations depends on the choice of $f(Q)$, and these equations dictate the gravitational dynamics in the modified geometry.

# 3 Variation and Lagrangian Analysis of the Modified Action

We will perform a variation and Lagrangian analysis on this modified action.

## 3.1 Form of the Action

The given action is:

$$S = \int d^4 x \sqrt{-g} \left[ \frac{f(Q)}{2} + \frac{g}{2} (\psi^i \partial^\mu \psi^j) (\partial_\mu \psi^i \psi^j) \right] \quad (6)$$

where $f(Q)$ introduces a non-local coupling term, $g$ is a constant, $\psi^i$ is the field, and $Q$ possibly represents a quantity related to gravity (e.g., a physical quantity or a scalar field). The action consists of two terms: the first term is related to the nonlinear features of gravity, and the second term is the dynamical description of some quantum field, which may be a field that describes quantum entanglement.

## 3.2 Derivation of the Lagrangian

Based on the structure of the action, we can extract the Lagrangian:

$$\mathcal{L} = \frac{f(Q)}{2} + \frac{g}{2} (\psi^i \partial^\mu \psi^j) (\partial_\mu \psi^i \psi^j) \quad (7)$$

### 3.2.1 Analysis of the Non-local Coupling Term $f(Q)$

The first term $\frac{f(Q)}{2}$ is related to the nonlinear features of gravity. We assume that $f(Q)$ is a general function of the scalar quantity $Q$, which can be, for example, the Ricci scalar $R$ or some other field-dependent quantity. In the variation procedure, we will need to treat $f(Q)$ with respect to its dependence on the underlying metric or scalar field.

If $Q = f(g_{\mu\nu})$, then the variation of $f(Q)$ with respect to the metric yields:

$$\frac{\delta}{\delta g_{\mu\nu}} \left( \frac{f(Q)}{2} \right) = \frac{1}{2} f'(Q) \frac{\delta Q}{\delta g_{\mu\nu}} \quad (8)$$

Now, if $Q$ is related to the Ricci scalar $R$, for instance, then the variation of the Ricci scalar $R$ with respect to the metric $g_{\mu\nu}$ gives us:

$$\frac{\delta R}{\delta g_{\mu\nu}} = \nabla_{\alpha} \nabla^{\alpha} g_{\mu\nu} - \Box g_{\mu\nu} \quad (9)$$

where $\Box = \nabla^\mu \nabla_\mu$ is the d'Alembertian operator. The full variation of the term $f(Q)$ then involves the derivative of $f(Q)$ and the second-order variations of the metric or scalar field.
### 3.2.2 Variation of the Quantum Field Term

The second term involves the fields $\psi^i$ and $\psi^j$, and their derivatives. We perform the variation of this term with respect to $\psi^i$ and $\psi^j$. Let us rewrite the term as:

$$ \mathcal{L}_{\text{kin}} = \frac{g}{2} (\psi^i \partial^\mu \psi^j) (\partial_\mu \psi^i \psi^j) \quad (10) $$

Now, to vary $\mathcal{L}_{\text{kin}}$, we perform the variation with respect to $\psi^i$:

$$ \frac{\delta \mathcal{L}_{\text{kin}}}{\delta \psi^i} = g \partial_\mu (\psi^j \partial^\mu \psi^j) + g \partial_\mu (\partial^\mu \psi^i \psi^j) \quad (11) $$

This results in the equation of motion for $\psi^i$, which includes second derivatives of the field:

$$ \frac{\partial^2 \psi^i}{\partial x^\mu \partial x^\nu} + \text{other terms} = 0 \quad (12) $$

The variation of $\mathcal{L}_{\text{kin}}$ thus leads to a second-order differential equation for the field $\psi^i$.

## 3.3 Variation Calculation

Now, we proceed with the variation calculation. We consider each term in the Lagrangian and perform the variational derivatives.

1. For the first term $\frac{f(Q)}{2}$, the variation involves computing the derivative of $f(Q)$ with respect to $Q$, which in turn depends on the underlying scalar field or metric:

$$ \frac{\delta \mathcal{L}}{\delta Q} = \frac{1}{2} f'(Q) \delta Q \quad (13) $$

The variation with respect to $Q$ will typically involve an expression like:

$$ \delta Q = \nabla_{\alpha} \nabla^{\alpha} \delta g_{\mu\nu} - \Box \delta g_{\mu\nu} \quad (14) $$

2. For the second term $\frac{g}{2} (\psi^i \partial^\mu \psi^j) (\partial_\mu \psi^i \psi^j)$, we vary the fields $\psi^i$ and $\psi^j$ and use the standard method for field variations. The variation will give the equations of motion for the fields:

$$ \frac{\delta \mathcal{L}}{\delta \psi^i} = g \partial_\mu (\psi^j \partial^\mu \psi^j) \quad (15) $$

This results in the following equation of motion for the fields $\psi^i$:

$$ \frac{\partial^2 \psi^i}{\partial x^\mu \partial x^\nu} + \text{additional terms depending on the interaction} = 0 \quad (16) $$

## 3.4 Quantum Entanglement and Bell's Inequality

By introducing the non-local coupling term $f(Q)$ and the coupling structure of the fields, this modified action has the potential for quantum entanglement. The action is modified to include non-locality due to the coupling of the field to distant regions. The fields $\psi^i$ and $\psi^j$ interact through non-local terms that may lead to quantum entanglement, which is reflected in the non-locality of the variation terms.

Violation of Bell's inequality occurs if the fields exhibit quantum entanglement and non-local correlations. The non-locality arises from the coupling between the fields across different
regions of space-time, as well as the nonlinear features of the gravitational term $f(Q)$. The resulting equations allow for a situation where Bell's inequality is violated, implying the presence of quantum entanglement.

$$ \langle \hat{A}(x) \hat{B}(y) \rangle \neq \langle \hat{A}(x) \rangle \langle \hat{B}(y) \rangle \qquad (17) $$

This equation reflects the violation of Bell's inequality, which can occur due to the non-local couplings in the Lagrangian and the complex interactions between the quantum fields.

# 4 Different Forms of $f(Q)$ Gravity

In this section, we examine several functional forms of $f(Q)$ that are commonly used in the literature. We investigate the resulting field equations, discuss potential solutions for black holes, and analyze their cosmological and thermodynamic implications.

## 4.1 Linear Form: $f(Q) = \alpha Q$

The simplest form of $f(Q)$ is the linear function, which is proportional to the non-metricity scalar:

$$ f(Q) = \alpha Q \qquad (18) $$

where $\alpha$ is a constant. This linear form leads to the action:

$$ S = \int d^4 x \sqrt{-g} \alpha Q \qquad (19) $$

The corresponding field equations are:

$$ \alpha (\nabla_{\mu} Q_{\nu\rho\lambda} - \nabla_{\nu} Q_{\mu\rho\lambda}) = 0 \qquad (20) $$

These equations are relatively simple and describe the behavior of spacetime in a non-metric geometry. The solutions to these equations lead to modifications of the Schwarzschild and Kerr black hole metrics, where the geometry is influenced by the non-metricity scalar. However, this modification does not introduce significant deviations from standard GR in most scenarios.

## 4.2 Implications for Black Hole Solutions

For black hole solutions, the linear form of $f(Q)$ leads to significant corrections to the Schwarzschild and Kerr black hole solutions. These corrections are particularly important for understanding the asymptotic behavior of black holes in modified gravity theories. While these modifications remain small in the weak field limit, they become more significant as we move closer to the event horizon or at large radial distances. In these regimes, the presence of non-metricity, captured by the function $f(Q)$, introduces new terms in the spacetime geometry, affecting both the event horizon and the overall spacetime structure.

In particular, the modified black hole metric can be expressed as:

$$ ds^2 = -f(r)dt^2 + \frac{dr^2}{f(r)} + r^2(d\theta^2 + \sin^2\theta d\phi^2), \qquad (21) $$

where $f(r) = 1 - \frac{2M}{r} + \alpha Q(r) + \beta \frac{Q(r)^2}{r^2} + \gamma \frac{Q(r)^3}{r^3}$. Here, $\alpha$, $\beta$, and $\gamma$ are coupling constants, and $Q(r)$ is a function that depends on the radial coordinate $r$, encoding the non-metricity of
spacetime. The additional terms introduced by α, β, and γ modify the gravitational potential and spacetime curvature in a more complex way than the linear $f(Q)$ model.

These corrections to the black hole metric indicate that the gravitational potential surrounding the black hole is altered by the presence of non-metricity. This change has several physical consequences, including a modification to the trajectory of test particles and photons near the black hole. For example, the bending of light around the black hole, or gravitational lensing, will be altered due to the modified geometry. Similarly, the orbital dynamics of satellites and accretion disks will be affected, leading to potential observable differences in the behavior of matter near the event horizon.

Moreover, these modifications also affect the thermodynamic properties of the black hole. The Hawking temperature, for instance, will be modified due to the change in the black hole's geometry. The entropy of the black hole may also exhibit deviations from the standard Bekenstein-Hawking formula, leading to a new understanding of black hole thermodynamics in modified gravity theories. Additionally, these corrections can influence the black hole's stability, potentially leading to the formation of new types of black hole solutions with different stability properties.

The presence of higher-order corrections in the form of $Q(r)^2$ and $Q(r)^3$ also suggests that black holes in these theories could exhibit unusual behaviors at both short and large distances, offering new insights into the nature of black holes and their interactions with the surrounding spacetime. The study of such black hole solutions could provide crucial information for understanding the role of non-metricity in gravitational systems, especially in the context of modified gravity.

## Calculation of the Event Horizon

Starting from the modified black hole metric:

$$ds^2 = -f(r)dt^2 + \frac{dr^2}{f(r)} + r^2(d\theta^2 + \sin^2\theta d\phi^2) \quad (22)$$

where $f(r)$ is given by:

$$f(r) = 1 - \frac{2M}{r} + \alpha Q(r) + \beta \frac{Q(r)^2}{r^2} + \gamma \frac{Q(r)^3}{r^3} \quad (23)$$

To compute the event horizon, we need to find the root of the equation $f(r) = 0$, i.e., solve the following equation:

$$1 - \frac{2M}{r} + \alpha Q(r) + \beta \frac{Q(r)^2}{r^2} + \gamma \frac{Q(r)^3}{r^3} = 0 \quad (24)$$

### Assuming $Q(r) = r$ for Simplicity

To simplify the calculation, we assume $Q(r) = r$, meaning the charge density (or other field configuration) is proportional to the radius. Then, $f(r)$ becomes:

$$f(r) = 1 - \frac{2M}{r} + \alpha r + \beta \frac{r^2}{r^2} + \gamma \frac{r^3}{r^3} \quad (25)$$

Simplifying further:

$$f(r) = 1 - \frac{2M}{r} + \alpha r + \beta + \gamma \quad (26)$$
To find the event horizon, we set $f(r) = 0$:

$$1 - \frac{2M}{r} + \alpha r + \beta + \gamma = 0 \qquad (27)$$

This is an equation in $r$, which can be solved numerically to obtain $r_+$ (the position of the event horizon).

## Calculation of the Hawking Temperature

The Hawking temperature is given by the formula:

$$T_H = \frac{1}{4\pi} \left( \frac{d}{dr} \left[ \sqrt{-g_{tt}} \right] \right)_{r_+} \qquad (28)$$

In this metric, $g_{tt} = -f(r)$, so:

$$T_H = \frac{1}{4\pi} \left( \frac{d}{dr} [-f(r)] \right)_{r_+} = \frac{1}{4\pi} \left( -\frac{d}{dr} f(r) \right)_{r_+} \qquad (29)$$

Next, we compute $\frac{d}{dr}f(r)$. Using the previously given expression for $f(r)$:

$$f(r) = 1 - \frac{2M}{r} + \alpha r + \beta + \gamma \qquad (30)$$

Taking the derivative with respect to $r$:

$$\frac{d}{dr} f(r) = \frac{2M}{r^2} + \alpha \qquad (31)$$

Thus, the Hawking temperature is:

$$T_H = \frac{1}{4\pi} \left( \frac{2M}{r_+^2} + \alpha \right) \qquad (32)$$

This is the expression for the Hawking temperature.

## Calculation of Other Thermodynamic Quantities

- **Entropy**: The entropy $S$ is related to the area $A$ of the event horizon as:

$$S = \frac{A}{4} \qquad (33)$$

For a spherically symmetric metric, the area of the event horizon is:

$$A = 4\pi r_+^2 \qquad (34)$$

Therefore, the black hole entropy is:

$$S = \pi r_+^2 \qquad (35)$$

- **Energy**: The total energy of the black hole is typically assumed to be $M$, which is the mass of the black hole.

The solution for the event horizon $r_+$ can be obtained numerically from the equation $1 - \frac{2M}{r} + \alpha r + \beta + \gamma = 0$.
The Hawking temperature is:

$$T_H = \frac{1}{4\pi} \left( \frac{2M}{r_+^2} + \alpha \right) \qquad (36)$$

The black hole entropy is:

$$S = \pi r_+^2 \qquad (37)$$

The above derivation is based on the simplified model assuming $Q(r) = r$. Further refinements can be made by solving the equation numerically for more accurate results.

## 4.3 Cosmological Implications

The linear form of $f(Q)$ also significantly modifies the Friedmann equations governing the evolution of the universe. These modifications introduce a new degree of freedom, represented by the term $\alpha Q$, which can have a profound impact on the cosmic expansion rate. At large distances or high redshifts, this form of $f(Q)$ may provide an alternative explanation for the accelerated expansion of the universe, without the need to introduce dark energy. The presence of non-metricity may change the way energy and matter interact on cosmological scales, altering the dynamics of the universe's expansion in ways that cannot be captured by traditional models of gravity.

The modified Friedmann equations, which govern the evolution of the Hubble parameter $H$, are given by:

$$H^2 = \frac{8\pi G}{3} \left( \rho + \frac{\alpha Q}{2} + \beta \frac{Q^2}{r^2} + \gamma \frac{Q^3}{r^3} \right). \qquad (38)$$

Here, $\rho$ is the energy density of the universe, and $Q$ is a function describing the non-metricity. The terms $\alpha$, $\beta$, and $\gamma$ represent new coupling constants that arise from the modified gravitational theory. The inclusion of these additional terms indicates that the universe's expansion rate may be modified in a more complex way than previously thought.

In particular, the term $\alpha Q$ introduces a correction to the cosmic expansion that is linear in the non-metricity. This could lead to an alternative explanation for the accelerated expansion of the universe, with potential implications for the current understanding of dark energy. The higher-order terms, $\beta Q^2$ and $\gamma Q^3$, suggest that non-metricity could have a significant impact on the universe's behavior at both early and late times. For example, during the early universe, the non-metricity could influence the rate of inflation, altering the dynamics of the early universe and providing a new mechanism for explaining cosmic inflation.

Moreover, these modifications may also affect the evolution of matter and radiation in the universe. At high redshifts, the impact of non-metricity could become more pronounced, leading to changes in the way radiation and matter interact, as well as the growth of cosmic structures such as galaxies and clusters of galaxies. This could provide new insights into the formation of large-scale structure and the behavior of the universe during its early stages.

These corrections to the Friedmann equations imply that future observational data, such as measurements of the cosmic microwave background (CMB), large-scale structure surveys, and supernova observations, could reveal signatures of non-metricity in the early universe. These observations could help to constrain the values of the parameters $\alpha$, $\beta$, and $\gamma$, which would provide a new window into understanding the role of non-metricity in cosmology.

In addition, the modified equations could lead to new predictions for the behavior of the universe at very large scales, such as the formation of cosmic voids, the acceleration of galaxy
clusters, and the evolution of the Hubble parameter at late times. This could lead to a more comprehensive understanding of the universe's expansion and provide a testable framework for modified gravity theories. Furthermore, the influence of non-metricity on the cosmic acceleration could provide a novel approach to explaining the observed phenomena associated with dark energy and dark matter, offering potential alternatives to current cosmological models.

## 4.4 Quadratic Form: $f(Q) = \alpha Q^2$

The quadratic form for $f(Q)$ is a natural extension of the linear form and leads to more complex field equations. The action is:

$$S = \int d^4 x \sqrt{-g} \alpha Q^2 \qquad (39)$$

The field equations are:

$$2\alpha (\nabla_{\mu} Q_{\nu\rho\lambda} - \nabla_{\nu} Q_{\mu\rho\lambda}) Q + \cdots = T_{\mu\nu} \qquad (40)$$

This form introduces non-linear terms that make the gravitational dynamics more sensitive to the curvature of spacetime. These non-linearities are expected to become important in high-energy regimes or large-scale cosmological scenarios.

## 4.5 Implications for Black Hole Solutions

For black holes, the quadratic form of $f(Q)$ leads to more intricate solutions. These solutions include modifications to the event horizon radius, the black hole's entropy, and temperature. The resulting black hole metric deviates significantly from the Schwarzschild or Kerr solutions at large radial distances, with new terms appearing due to the non-linear dependence on $Q$.

The black hole thermodynamics are also affected. The entropy of the black hole is given by:

$$S = \frac{A}{4G} (1 + \alpha Q) \qquad (41)$$

where $A$ is the area of the event horizon. The temperature is similarly modified, and there may be new forms of radiation associated with the modified gravitational field.

## 4.6 Cosmological Implications

The quadratic form of $f(Q)$ has much more profound and intricate implications for cosmology than the linear form. The term $\alpha Q^2$, when included in the framework, introduces a significant modification to the energy-momentum structure, which can lead to substantial changes in the evolution of the cosmos, especially during the early universe's inflationary epoch and the present accelerated expansion. In this extended model, the new energy scale set by $\alpha Q^2$ provides a natural mechanism for altering both the inflationary dynamics and the current acceleration in a way that could explain some of the unresolved problems in modern cosmology, such as the nature of dark energy and the fine-tuning issues in early-universe physics.

During the inflationary phase, the field $Q$ plays a pivotal role in shaping the dynamics of the universe. The quadratic term $\alpha Q^2$ modifies the energy density in such a way that it introduces a new contribution to the total potential governing the inflationary expansion. This correction can alter the slow-roll conditions typically required for sustained inflation. Depending on the magnitude of $\alpha$, the field $Q$ could either act to amplify the expansion by increasing the energy density or to decelerate it by adding an opposing contribution to the total potential. This
sensitivity of the inflationary phase to $\alpha Q^2$ can lead to novel insights into the fine-tuning of inflationary models, potentially providing an alternative explanation for the initial conditions of the universe.

In addition to altering inflation, the quadratic correction has notable consequences for the universe's behavior at later stages, including the current epoch of accelerated expansion. The equation of state for dark energy, which is typically assumed to be constant ($p = -\rho$) for a cosmological constant, could now become a dynamical function due to the presence of $\alpha Q^2$. This modification has the potential to shift the properties of dark energy from the simple cosmological constant to a more complex fluid with an evolving equation of state, which could better explain the observed late-time acceleration of the universe without invoking exotic dark energy models.

The modified Friedmann equations now incorporate the effects of the quadratic correction, reflecting a more complex relationship between the energy density and the expansion rate. The modified first Friedmann equation is given by:

$$H^2 = \frac{8\pi G}{3} (\rho + \alpha Q^2), \qquad (42)$$

where $\rho$ is the total energy density, which includes contributions from matter, radiation, and dark energy. The quadratic term $\alpha Q^2$ modifies this relationship by adding a new component to the energy density, which evolves with time due to the dynamics of $Q$. The influence of $\alpha$ on the energy density could cause a significant departure from the standard cosmological model, especially at high energies and during the transitions between different epochs of cosmic evolution, such as between radiation domination and matter domination.

To further explore the consequences of this quadratic modification, we also derive the modified acceleration equation:

$$\dot{H} + H^2 = -\frac{4\pi G}{3} (\rho + p + 2\alpha \dot{Q} Q), \qquad (43)$$

where $p$ represents the pressure, and $\dot{Q}$ denotes the time derivative of the scalar field $Q$. This equation reveals how the additional $\alpha Q^2$ term influences the rate of change of the Hubble parameter $\dot{H}$, reflecting the dynamical effects of the scalar field. The term $2\alpha \dot{Q} Q$ introduces a coupling between the scalar field's evolution and the cosmic expansion, further complicating the relationship between matter and the geometric properties of spacetime.

An interesting feature of the model is the possibility of a non-trivial contribution to the dark matter sector due to the interaction between $\alpha Q^2$ and the matter fields. This could provide a mechanism for explaining the observed amount of dark matter, as well as its distribution and clustering properties. The presence of the quadratic term could enhance or dampen the growth of cosmic structures, depending on its sign and magnitude, potentially leading to modifications in the power spectrum of the cosmic microwave background (CMB) and large-scale structure surveys.

At late times, the behavior of the universe may be dominated by the $\alpha Q^2$ term, especially if $Q$ evolves to a constant value. In this scenario, the equation of state of the universe could approach the form:

$$p = -\rho + \alpha Q^2, \qquad (44)$$

which implies that the effective dark energy has a different equation of state than that of the cosmological constant. This can have observable consequences for the rate of expansion at
different epochs, influencing parameters such as the deceleration parameter and the transition between deceleration and acceleration in the cosmic expansion.

Additionally, this model introduces a rich dynamical structure that could potentially address other cosmological puzzles, such as the horizon problem, the flatness problem, and the structure formation problem. The interaction between the quadratic correction and the scalar field $Q$ could lead to novel signatures in the cosmic background radiation, the distribution of galaxies, and the growth of large-scale cosmic structures. These effects could be tested through precision measurements in upcoming galaxy surveys, CMB experiments, and gravitational wave astronomy.

Thus, the quadratic form of $f(Q)$ not only modifies the cosmic evolution in the early universe but also provides a more flexible and dynamic framework for understanding the current acceleration of the universe. The introduction of a new energy scale via the term $\alpha Q^2$ could offer new insights into the nature of dark energy, dark matter, and the fundamental physics governing the evolution of the cosmos, suggesting possible avenues for future observational and theoretical studies.

## 4.7 Superradiance in $f(Q)$ Gravity

Superradiance is a phenomenon that occurs in rotating black holes, where waves can be amplified as they scatter off the black hole’s event horizon. In $f(Q)$ gravity, the modified geometry affects the conditions for superradiance. Specifically, the function $f(Q)$ influences the energy levels of the black hole and could alter the threshold for superradiance.

The superradiance condition for a rotating black hole is: [1, 2, 15–20]

$$ \omega > \frac{m}{r_+} \qquad (45) $$

where $\omega$ is the frequency of the incident wave, $m$ is the azimuthal quantum number, and $r_+$ is the event horizon radius. In modified $f(Q)$ gravity, the event horizon radius is affected by the non-metricity, thus changing the superradiance threshold.

## 4.8 Superradiance Stability Analysis in $f(Q)$ Gravity

Superradiance stability in $f(Q)$ gravity is a crucial aspect of understanding the behavior of perturbations around rotating black holes. In particular, we focus on the evolution of these perturbations and investigate the growth rates of superradiant modes, which are solutions to the wave equation that describe energy extraction from the black hole. The stability of superradiance is dictated by the interplay between the frequency of the incident wave, the black hole’s angular momentum, and the geometry of the event horizon. In $f(Q)$ gravity, where the non-metricity of spacetime is introduced, these effects can be significantly altered.

### Wave Equation in $f(Q)$ Gravity

To begin the stability analysis, we consider the wave equation for a scalar field $\Phi$ in the spacetime of a rotating black hole in $f(Q)$ gravity:

$$ (\nabla^\mu \nabla_\mu - \mu^2) \Phi = 0 \qquad (46) $$

where $\nabla^\mu \nabla_\mu$ is the d’Alembertian operator, $\mu$ is the mass of the scalar field, and $\Phi$ is the perturbation scalar. The presence of non-metricity in $f(Q)$ gravity alters the structure of the d’Alembertian operator, modifying the propagation of the perturbations. Specifically, in $f(Q)$
gravity, the modified Ricci curvature tensor $Q_{\mu\nu}$ introduces corrections to the spacetime geometry, which directly influence the wave equation.

For $f(Q)$ gravity, the wave equation can be written in the form:

$$ (\nabla^\mu \nabla_\mu - \mu^2 + \delta f(Q)) \Phi = 0 \qquad (47) $$

where $\delta f(Q)$ is a correction term arising due to the non-metricity scalar field that modifies the spacetime geometry and dynamics of the field.

### Impact of Non-Metricity on Superradiant Modes

The effects of non-metricity in $f(Q)$ gravity introduce additional terms to the dispersion relation of the perturbation, modifying the conditions under which superradiance occurs. Superradiance refers to the phenomenon where a rotating black hole can amplify the energy of a scalar field in certain conditions. This amplification is governed by the resonance between the field's frequency and the rotational frequency of the black hole.

The superradiance condition, typically derived from the Penrose process, is modified in $f(Q)$ gravity. In standard General Relativity, superradiance occurs when the frequency of the perturbation satisfies the condition:

$$ \omega > \frac{m}{r_+} \qquad (48) $$

where $\omega$ is the frequency of the perturbation, $m$ is the azimuthal quantum number, and $r_+$ is the radius of the event horizon. In the presence of $f(Q)$ gravity, this condition is modified due to the changes in the spacetime geometry induced by the non-metricity. The modified condition for superradiance stability in $f(Q)$ gravity becomes:

$$ \omega > \frac{m}{r_+} (1 + \delta f(Q)) \qquad (49) $$

where $\delta f(Q)$ represents the perturbative correction due to the non-metricity scalar. This correction depends on the specific form of the function $f(Q)$ and can either enhance or suppress the superradiance instability, depending on the nature of the modification to the geometry.

### Stability of Superradiant Modes in $f(Q)$ Gravity

To investigate the stability of superradiant modes, we need to analyze the growth rate of these modes under the influence of the $f(Q)$ gravity corrections. For rotating black holes, superradiance instability occurs when the incident wave's frequency falls within a certain band, such that the wave is amplified by the black hole. The growth rate of the superradiant instability is governed by the imaginary part of the frequency $\omega$, which can be expressed as:

$$ \gamma = \operatorname{Im}(\omega) \qquad (50) $$

where $\gamma$ represents the instability growth rate. The presence of $f(Q)$ gravity modifies the dispersion relation for the perturbations, leading to changes in the imaginary part of $\omega$. This can either stabilize or destabilize the superradiant modes, depending on the form of the non-metricity corrections.

### Correction to the Angular Momentum of the Black Hole

One of the key features of $f(Q)$ gravity is the modification of the angular momentum of the black hole due to the non-metricity of spacetime. In general relativity, the angular momentum of the black hole is given by:

$$ J = M r_+ $$
where M is the mass of the black hole, and $r_+$ is the radius of the event horizon. In $f(Q)$ gravity, the modified geometry introduces a correction to the angular momentum, which affects the critical condition for superradiance. This modification can be written as:

$$J_{\text{eff}} = J(1 + \delta f(Q)) \qquad (51)$$

where $J_{\text{eff}}$ is the effective angular momentum in $f(Q)$ gravity. This correction has profound implications for the superradiance condition and can lead to different stability behaviors based on the form of $f(Q)$.

## Analysis of Various $f(Q)$ Gravity Models

Different forms of $f(Q)$ gravity models lead to distinct modifications of the spacetime geometry and the resulting superradiance stability. Below, we explore several common models of $f(Q)$ gravity:

1. Linear $f(Q)$ Gravity: In the case of a linear function $f(Q) = Q$, the non-metricity is directly proportional to the curvature. This leads to a straightforward modification of the wave equation:

$$(\nabla^{\mu}\nabla_{\mu} - \mu^2 + \delta Q)\Phi = 0$$

The stability of superradiant modes in this model is expected to be weakly modified compared to General Relativity, with small corrections to the superradiance threshold due to the linear nature of $f(Q)$.

2. Exponential $f(Q)$ Gravity: In the exponential case, $f(Q) = e^{\alpha Q}$, the non-metricity has a more significant impact on the spacetime geometry. The perturbative corrections $\delta f(Q)$ grow exponentially, leading to potentially larger modifications of the superradiance condition. The stability analysis in this model requires solving a more complex wave equation, with the possibility of enhanced superradiance instability for certain values of $\alpha$.

3. Power-Law $f(Q)$ Gravity: For a power-law function $f(Q) = Q^n$, where $n$ is a constant, the non-metricity modifies the spacetime geometry in a more nonlinear fashion. The corrections $\delta f(Q)$ lead to more intricate modifications of the superradiance stability, potentially leading to either enhanced or suppressed instability depending on the value of $n$. This requires a detailed perturbation analysis to determine the precise behavior of the superradiant modes.

4. Quadratic $f(Q)$ Gravity: A quadratic model of $f(Q) = \beta Q^2$, where $\beta$ is a constant, introduces quadratic corrections to the geometry. This form is particularly interesting as it leads to significant changes in the black hole's structure, with possible effects on the growth rate of superradiant instability. The stability analysis involves higher-order terms in the perturbative expansion, and the superradiance instability could either be suppressed or enhanced depending on the sign and magnitude of $\beta$.

## Final Stability Condition

In summary, the stability of superradiance in $f(Q)$ gravity is determined by the balance between the frequency of the perturbation and the modified geometry induced by the non-metricity scalar. The final condition for superradiance stability in $f(Q)$ gravity can be expressed as:

$$\omega > \frac{m}{r_+} (1 + \delta f(Q)) \qquad (52)$$

This condition suggests that the superradiance instability could either be enhanced or suppressed by the presence of the $f(Q)$ term, depending on the specific form of the function and the properties of the black hole. The detailed behavior of the superradiance growth rate and its stability must be studied further by solving the perturbation equations in the context of the specific $f(Q)$ model under consideration.
Thus, the introduction of $f(Q)$ gravity not only modifies the geometry of the black hole but also alters the fundamental behavior of superradiant modes, which could lead to new insights into the stability of black holes in modified gravitational theories.

# 5 Thermodynamic Geometry Analysis

## 5.1 Temperature Calculation

The temperature of the black hole can be calculated from the metric function $f(r)$, where

$$f(r) = 1 - \frac{2M}{r} + \alpha r + \beta + \gamma \quad (53)$$

The temperature $T$ is given by:

$$T = \frac{1}{4\pi} \left. \frac{d}{dr} (f(r)) \right|_{r=r_+} \quad (54)$$

First, we compute the derivative of $f(r)$:

$$\frac{d}{dr} f(r) = \frac{2M}{r^2} + \alpha \quad (55)$$

Thus, the temperature is:

$$T = \frac{1}{4\pi} \left( \frac{2M}{r_+^2} + \alpha \right) \quad (56)$$

## 5.2 Entropy Calculation

The entropy $S$ of the black hole is related to the area of the event horizon:

$$A = 4\pi r_+^2 \quad (57)$$

Thus, the entropy $S$ is:

$$S = \frac{A}{4G} = \frac{r_+^2}{G} \quad (58)$$

## 5.3 Thermodynamic Geometry

The thermodynamic geometry is defined through the Gibbs potential $F = U - TS$. We can compute the thermodynamic geometric tensor $G$ as:

$$G = \frac{\partial^2 F}{\partial r_+^2} \quad (59)$$

This can be computed after solving for the Gibbs potential, depending on the specific expression for the internal energy $U$.

The heat capacity $C$ is defined as:

$$C = \left( \frac{\partial U}{\partial T} \right)_{r_+} \quad (60)$$
For a black hole, the internal energy $U$ is given by the mass $M$, so:

$$U = M \qquad (61)$$

The Gibbs potential $F$ is given by:

$$F = M - TS \qquad (62)$$

Using the expressions for $T$ and $S$, we substitute into the Gibbs potential:

$$F = M - \left( \frac{1}{4\pi} \left( \frac{2M}{r_+^2} + \alpha \right) \right) \frac{r_+^2}{G} \qquad (63)$$

The heat capacity can then be calculated by taking the derivative:

$$C = \left( \frac{\partial M}{\partial T} \right)_{r_+} \qquad (64)$$

## 5.4 Conditions for Phase Transition

The phase transition occurs when the heat capacity $C$ changes sign, indicating a change in stability. If the heat capacity $C$ is negative, the system is unstable, signaling a phase transition. This condition can be determined by solving:

$$\left( \frac{\partial T}{\partial r_+} \right) = 0 \qquad (65)$$

This gives the critical temperature and radius at which the system undergoes a phase transition. In this case, the black hole undergoes a first-order phase transition, similar to a liquid-gas transition.

## 5.5 Divergence of Thermodynamic Geometry and Phase Transition

The divergence of the thermodynamic geometric tensor $G$ often corresponds to critical points where phase transitions occur. The divergence or singularities in $G$ signal the presence of a first-order phase transition. These can be computed from the second derivatives of the Gibbs potential, or by analyzing the heat capacity and its critical points.

# 6 Conclusion

This paper presents a detailed analysis of various forms of $f(Q)$ gravity. We have explored the mathematical structure of different functional forms of $f(Q)$, derived the corresponding field equations, and discussed their implications for black hole solutions, cosmological models, and thermodynamics. Our analysis shows that different choices of $f(Q)$ lead to distinct modifications of the gravitational dynamics, with significant implications for both black hole physics and cosmology. Future work should focus on deriving exact solutions for black holes and cosmological models and exploring potential observational signatures of these modifications.
# 7 Appendix

The given action is:

$$S = \int d^4 x \sqrt{-g} \left[ \frac{f(Q)}{2} + \frac{g}{2} (\psi^i \partial^\mu \psi^j) (\partial_\mu \psi^i \psi^j) \right] \quad (66)$$

where the first term $\frac{f(Q)}{2}$ is the non-local coupling term, and the second term $\frac{g}{2} (\psi^i \partial^\mu \psi^j) (\partial_\mu \psi^i \psi^j)$ involves the dynamics of the field.

## 7.1 Extracting the Lagrangian

From the form of the action, we can extract the Lagrangian:

$$\mathcal{L} = \frac{f(Q)}{2} + \frac{g}{2} (\psi^i \partial^\mu \psi^j) (\partial_\mu \psi^i \psi^j) \quad (67)$$

## 7.2 Variation of the First Term: Non-local Coupling Term $\frac{f(Q)}{2}$

For the first term $\frac{f(Q)}{2}$, we need to perform variation with respect to $Q$. Assume that $Q$ is a quantity related to a scalar field or gravitational field (e.g., Ricci scalar). To perform the variation, we first need to know the specific form of $f(Q)$.

**Variation Process:** - If $Q$ is a scalar field, the variation typically involves the Laplacian operator or a corresponding operator in the gravitational background. The variation steps are as follows:

1. Perform the variation of $\frac{f(Q)}{2}$, that is, compute $\delta(\frac{f(Q)}{2})$.

2. Assuming $f(Q)$ is a known function, we get:

$$\frac{\delta}{\delta Q} \left( \frac{f(Q)}{2} \right)$$

3. This variation will yield an equation describing the dynamics of $Q$ related to gravity or physical quantities.

## 7.3 Variation of the Second Term: Quantum Field Dynamics

The second term involves the field $\psi^i$ and its derivatives $\partial_\mu \psi^j$, which describe the field dynamics. We need to perform variation with respect to $\psi^i$ and $\psi^j$.

**Variation Process:**

1. **Preliminary Assumptions:** Both $\psi^i$ and $\psi^j$ are fields, so we will use the variational principle to compute their variation. The expression for the variation is as follows:

$$\delta \left( \frac{g}{2} (\psi^i \partial^\mu \psi^j) (\partial_\mu \psi^i \psi^j) \right)$$

where $\delta$ denotes the variation operator.

2. **Step-by-step calculation:** - Vary $\psi^i$ and $\psi^j$:

$$\delta (\psi^i \partial^\mu \psi^j) = \delta \psi^i \partial^\mu \psi^j + \psi^i \partial^\mu \delta \psi^j$$

- Vary $\partial_\mu \psi^i \psi^j$:

$$\delta (\partial_\mu \psi^i \psi^j) = \partial_\mu \delta \psi^i \psi^j + \partial_\mu \psi^i \delta \psi^j$$
3. **Obtain the equation of motion:** By substituting the variations into the Lagrangian, we get the equation of motion for $\psi^i$ and $\psi^j$. For example:

$$ \frac{\delta \mathcal{L}}{\delta \psi^i} = 0 $$

This yields the equation of motion for the field $\psi^i$, which typically involves second derivatives of the field.

## 7.4 Deriving the Variational Equations

By applying the standard variation method, we vary each term in the Lagrangian. Setting the variation to zero will yield the field equations.

1. Variation of the first term $\frac{f(Q)}{2}$: - The variation will involve the variation of $f(Q)$ with respect to $Q$ (assuming $Q$ is a scalar field or a gravitational-related quantity). - If $Q$ is a scalar field, its variation will involve the Laplacian operator on $Q$ or the corresponding operator in the gravitational background.

2. Variation of the second term $\frac{g}{2}(\psi^i \partial^\mu \psi^j)(\partial_\mu \psi^i \psi^j)$: - The variation will give the equation of motion for $\psi^i$, usually involving second derivatives of the field.

## 7.5 Quantum Entanglement and Bell's Inequality

By introducing the non-local coupling term $f(Q)$ and the coupling structure of the fields, this modified action has the potential for quantum entanglement. Non-locality is introduced by coupling the field to other regions, while quantum entanglement is reflected in the interactions between $\psi^i$ and $\psi^j$, especially in the mutual influence between different regions.

**Violation of Bell's Inequality:** - The violation of Bell's inequality requires the system to exhibit quantum entanglement and non-locality. - Due to the non-locality of this action and the nonlinear features of gravity, situations may arise where Bell's inequality is violated.

## 7.6 Summary

By performing a variation and Lagrangian analysis on this action, we can obtain the corresponding equations, where the non-local coupling term and the dynamical terms of the fields work together to provide a potential quantum entanglement mechanism, which may lead to violations of Bell's inequality. The specific equations and solutions depend on the precise form of $f(Q)$ and $\psi^i$, as well as how $Q$ and $\psi^i$ are related.
<table><caption>Table 1: Summary of different forms of $f(Q)$ gravity and their implications.</caption><thead><tr><td>Form of f(Q)</td><td>Modified Field Equations</td><td>Implications</td></tr></thead><tbody><tr><td>Linear αQ</td><td>α (∇<sub>μ</sub>Q<sub>νρλ</sub> − ∇<sub>ν</sub>Q<sub>μρλ</sub>) = 0</td><td>Small corrections to Schwarzschild black holes</td></tr><tr><td>Quadratic αQ<sup>2</sup></td><td>2α (∇<sub>μ</sub>Q<sub>νρλ</sub> − ∇<sub>ν</sub>Q<sub>μρλ</sub>) Q + ...</td><td>Significant modifications to black hole thermodynamics</td></tr></tbody></table>

<figure><img src="image_4.png" /><figcaption>Figure 1: Plot showing the quadratic form $f(Q) = αQ^2$ with $α = 1$.</figcaption></figure>

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