Yang-Mills field modified RN black hole and the Strong Cosmic Censorship Conjecture

In this subsection, we study the evolution of scalar field perturbations in the spacetime background of the Yang-Mills field modified RN black hole. For a neutral and massless scalar fieldCardoso et al. (2018), the Klein-Gordon equation can be written as:

$$\frac{1}{\sqrt{-g}}\partial_{\mu}\left(g^{\mu\nu}\sqrt{-g}\,\partial_{\nu}\Phi\right)=0.$$

(6)

For a charged and massive scalar fieldKonoplya and Zhidenko (2013), the Klein-Gordon equation can be written as:

	$\displaystyle\frac{1}{\sqrt{-g}}\partial_{\mu}\left(g^{\mu\nu}\sqrt{-g}\partial_{\nu}\Phi\right)-2iqg^{\mu\nu}A_{\mu}\partial_{\nu}\Phi$		(7)
	$\displaystyle-q^{2}g^{\mu\nu}A_{\mu}A_{\nu}\Phi-m^{2}\Phi=0.$

In four-dimensional spacetime, the mass and charge of the scalar field are represented by the parameters $m$ and $q$, respectively. The electromagnetic potential vector of the black hole can be expressed as $A_{\mu}=(-Q/r,0,0,0)$, where $Q$ is the charge of the black hole and $r$ represents the radial distance. This expression describes the electrostatic potential distribution generated by the black hole at the radial position $r$. Due to the spherical symmetry of spacetime, the scalar field can be expanded in terms of spherical harmonics.

$$\Phi(t,r,\theta,\phi)=\sum_{m^{\prime}\ell}\frac{\varphi(r)}{r}Y_{\ell m^{\prime}}(\theta,\phi)e^{-i\omega t}.$$

(8)

The frequency of the scalar field $\Phi$ is denoted by $\omega$, and $\varphi(r)$ is its corresponding radial wave function. The angular momentum quantum number $l$ only takes non-negative integers, while the magnetic quantum number $m^{\prime}$ is set to zero in this paper. The spherical harmonics $Y_{lm}(\theta,\phi)$ describe the angular distribution characteristics. We focus only on the perturbed radial equation, which can be expressed as:

$$\left(\frac{d^{2}}{dr_{*}^{2}}+\omega^{2}-V(r)\right)\varphi(r)=0.$$

(9)

The tortoise coordinate $dr_{*}$ is defined by the differential equation $dr_{*}=\frac{dr}{f(r)}$, with its range extending from negative infinity at the horizon to positive infinity in space. Its expression can be further given asSarkar et al. (2023):

$$r_{*}=r-\frac{1}{2k_{h_{-}}}\ln\left(\frac{r}{r_{h_{-}}}-1\right)+\frac{1}{2k_{h_{+}}}\ln\left(\frac{r}{r_{h_{+}}}-1\right).$$

(10)

Here, $k_{h_{+}}$ and $k_{h_{-}}$ represent the surface gravities at the event horizon and Cauchy horizon, respectively. Their expressions are given by:

$$k_{h_{+}}=\left|\frac{1}{2}f^{\prime}(r_{h_{+}})\right|,\quad k_{h_{-}}=\left|\frac{1}{2}f^{\prime}(r_{h_{-}})\right|.$$

(11)

The effective potential for a neutral and massless scalar field is given by:

$$V_{\text{eff-1}}(r)=f(r)\left[\frac{\ell(\ell+1)}{r^{2}}+\frac{f^{\prime}(r)}{r}\right].$$

(12)

The effective potential for a charged and massive scalar field becomes:

	$\displaystyle V_{\text{eff-2}}(r)=$	$\displaystyle\,\omega^{2}-f(r)\left(\frac{\ell(\ell+1)}{r^{2}}+\frac{f^{\prime}(r)}{r}\right)$
		$\displaystyle+\frac{q^{2}Q^{2}}{r^{2}}-\frac{2\omega qQ}{r}-m^{2}f(r).$		(13)

Due to the absorptive nature of the black hole event horizon, all perturbations reaching the horizon are completely absorbed without any reflection. At the event horizon, the perturbations take the form of purely incoming waves, while at infinity, they appear as purely outgoing waves. For a neutral and massless scalar field, the boundary conditions are as follows:

$$\varphi(r)\sim\begin{cases}e^{-i\omega r_{*}},&r_{*}\to-\infty,\\ e^{i\omega r_{*}},&r_{*}\to+\infty.\end{cases}$$

(14)

The boundary conditions for a charged and massive scalar field are:

$$\varphi(r)\sim\begin{cases}e^{-i\left(\omega-\frac{qQ}{r_{h_{+}}}\right)r_{*}},&r_{*}\to-\infty,\\ e^{i\omega r_{*}},&r_{*}\to+\infty.\end{cases}$$

(15)

When the effective potential gradually approaches zero at infinity, the perturbations in this region decay and eventually vanish as the distance increases. Utilizing this property, appropriate boundary conditions can be set, and by solving the perturbation equation, the discrete QNMs frequencies can be obtained. These frequencies reflect the inherent oscillation modes of the system.

In black hole physics, the Strong Cosmic Censorship Conjecture (SCCC)Penrose (1980) is a significant theoretical problem in general relativity. The core claim of the SCCC is that classical solutions of general relativity should have a well-behaved global causal structure and determinism to ensure the global predictability of physical laws. The instability of the Cauchy horizon is key to ensuring that the classical solutions cannot be smoothly extended, and it is also an important criterion for verifying whether the SCCC holds. If the Cauchy horizon is stable, the causal structure inside the black hole would remain smooth and stable, which contradicts the requirements of the SCCC, leading to the conjecture’s failure. Therefore, studying the dynamic evolution and decay characteristics of black holes under perturbations is crucial for verifying the SCCC.

In the mathematical description of black hole perturbations, the commonly used formula to describe the decay of perturbations over time isSingha et al. (2022):

$$\varphi\sim\exp(-Im\omega u)\varphi_{0}.$$

(16)

Here, $Im\omega$ represents the imaginary part of the QNMs frequency, which determines the decay rate of black hole perturbations. The larger the imaginary part, the faster the decay. The parameter $u$ is defined as $u=t-r_{*}$, used to describe the temporal evolution of the perturbations, where $t$ is the time coordinate, and $r_{*}$ is referred to as the ”tortoise coordinate.” Furthermore, under the blueshift effect caused by the strong gravitational field, the change in the scalar field intensity near the horizon can be expressed asSingha et al. (2022):

$$|\varphi_{r_{h}}|^{2}\sim\exp(k_{i}u)|\varphi_{0}|^{2}.$$

(17)

The surface gravity $\kappa_{i}$ is a measure of the gravitational acceleration at the horizon, reflecting the intensity of the gravitational field around the horizon. It can be calculated using the formula $\kappa_{i}=\left|\frac{1}{2}f^{\prime}(r_{i})\right|$, where $f(r)$ is the metric function, and the derivative at $r=r_{i}$ determines the specific value of the surface gravity. Higher surface gravity indicates a stronger gravitational field in the horizon region. In the study of the SCCC, $\kappa_{i}$ and $\text{Im}\,\omega$ are considered key analytical indicators. Whether the SCCC holds depends on which effect dominates, and their ratio can be expressed as:

$$\beta=-\frac{\text{Im}\,\omega}{k_{i}}.$$

(18)

When $\beta<\frac{1}{2}$, the decay effect of the scalar field is weaker than the gravitational blueshift effect, causing the Cauchy horizon of the black hole to become unstable or even break down, thereby satisfying the SCCC. However, when $\beta>\frac{1}{2}$, the decay effect of the scalar field is stronger than the gravitational blueshift effect, and the Cauchy horizon of the black hole remains stable, leading to a violation of the SCCC. Therefore, the SCCC holds when $\beta<\frac{1}{2}$, but it may fail when $\beta>\frac{1}{2}$.

The WKB method (Wentzel-Kramers-Brillouin)Wentzel (1926)Kramers (1926) is an important tool in asymptotic analysis, originally introduced in the field of quantum mechanics to obtain asymptotic solutions to complex wave equations. With its subsequent generalization and development, its applications in black hole research have gradually emerged, particularly demonstrating significant advantages in solving complex potential differential equations. For high-frequency modes, the WKB method can yield extremely accurate results, and by introducing higher-order modified, it further enhances the reliability of numerical solutions. In analyzing the quasinormal mode frequencies (QNMs) of black hole gravitational wave signals, the WKB method has played a significant role, providing strong support for testing general relativity and serving as a groundbreaking theoretical tool for exploring new physical phenomena.

According to Equation (9), the WKB methodIyer and Will (1987)Iyer (1987) is used to derive the frequencies of the QNMs.

$$iK-\left(n+\frac{1}{2}\right)-\Lambda(n)=\Omega(n).$$

(19)

which

$$K=\frac{V_{0}}{\sqrt{2V_{0}^{(2)}}},$$

(20)

$$\Lambda(n)=\frac{1}{\sqrt{2V_{0}^{(2)}}}\left[\frac{(a^{2}+\frac{1}{4})V_{0}^{(4)}}{8V_{0}^{(2)}}-\frac{(60a^{2}+7)}{288}\left(\frac{V_{0}^{(3)}}{V_{0}^{(2)}}\right)^{2}\right],$$

(21)

	$\displaystyle\Omega(n)=$	$\displaystyle\,\frac{n+\frac{1}{2}}{2V_{0}^{(2)}}\left[\frac{5(188a^{2}+77)}{6912}\left(\frac{V_{0}^{(3)}}{V_{0}^{(2)}}\right)^{4}-\frac{(100a^{2}+51)}{384}\frac{(V_{0}^{(3)})^{2}V_{0}^{(4)}}{(V_{0}^{(2)})^{3}}\right]$
		$\displaystyle+\frac{n+\frac{1}{2}}{2V_{0}^{(2)}}\left[\frac{(68a^{2}+67)}{2304}\left(\frac{V_{0}^{(4)}}{V_{0}^{(2)}}\right)^{2}+\frac{(28a^{2}+19)}{288}\frac{V_{0}^{(3)}V_{0}^{(5)}}{(V_{0}^{(2)})^{2}}-\frac{(4a^{2}+5)}{288}\frac{V_{0}^{(6)}}{V_{0}^{(2)}}\right],$		(22)

In the above formula, $V_{0}^{(k)}=\left.\frac{d^{k}V}{dx^{k}}\right|_{r=r_{0}}$ represents the $k$-th derivative of the effective potential at $r=r_{0}$, and $a=n+\frac{1}{2}$. Therefore, according to Equation (19), we can obtain:

	$\displaystyle\omega^{2}={}$	$\displaystyle\left[V_{0}+\left(-2V_{0}^{(2)}\right)^{1/2}\tilde{\Lambda}(n)\right]$
		$\displaystyle-i\left(n+\frac{1}{2}\right)\left(-2V_{0}^{(2)}\right)^{1/2}\left[1+\tilde{\Omega}(n)\right].$		(23)

In the above expression

$$\tilde{\Lambda}=\frac{\Lambda}{i},\quad\tilde{\Omega}=\frac{\Omega}{\left(n+\frac{1}{2}\right)}.$$

(24)

The $\omega$ in Equation (23) is the result derived using the first-order WKB method. For the related programs of higher-order WKB methods, one can refer to the work of Konoplya et al.Konoplya et al. (2019), which provides a detailed description of the application of higher-order WKB formulas and Padé approximation techniques. Equation (19) is used to analyze physical systems governed by the Schrödinger-like equation and quasinormal boundary conditions, playing a key role particularly in studying quantum resonance phenomena at the top of one-dimensional potential barriers. In the following sections, the key numerical results obtained through the WKB method and their physical significance will be presented.

The Prony methodDubinsky (2024)Berti et al. (2007) is a dynamic signal analysis tool that precisely extracts features such as frequency and decay rates by decomposing complex signals into a superposition of exponentially decaying sinusoidal waves. It is particularly suitable for nonlinear decaying signals and has been successfully applied to the analysis of black hole QNMs frequenciesLiu et al. (2024a)Liu et al. (2024b)Yang et al. (2023). The Prony method does not require initial parameter guesses and is computationally efficient, but it is sensitive to noise. In high-noise environments, it is often combined with methods such as the matrix pencil method or singular value decomposition to improve accuracy. ReferenceGundlach et al. (1994) proposes an efficient time-domain integration method for analyzing the master wave equation (9). By introducing the null coordinates $du=dt-dr_{*}$ and $dv=dt+dr_{*}$, Equation (9) can be reformulated in the following form:

$$\left(4\frac{\partial^{2}}{\partial u\partial v}+V(u,v)\right)\varphi(u,v)=0.$$

(25)

Using the finite difference scheme, the data at position $N$ can be derived from the data at positions $W$, $E$, and $S$, thereby expressing the above equation in the following form:

$$\varphi(N)=\varphi(W)+\varphi(E)-\varphi(S)-\frac{h^{2}}{8}V(S)\left(\varphi(W)+\varphi(E)\right).$$

(26)

In the above formula, the coordinates of the points are defined as $S=(u,v)$, $W=(u+h,v)$, $E=(u,v+h)$, and $N=(u+h,v+h)$. Using the integration method of Equation (26), the value of the wave function $\varphi$ can be calculated based on the region defined by the initial conditions $u=u_{0}$ and $v=v_{0}$. Through this calculation process, the properties of the QNMs can be further analyzed, and their frequencies can be extracted. For this purpose, the late-time signal at a specific $r_{*}$ can be expressed as a superposition of multiple QNMs:

$$\varphi(t)=\sum_{j=1}^{p}C_{j}e^{-i\omega_{j}t}.$$

(27)

Here, $C_{j}$ and $\omega_{j}$ represent the amplitude and complex frequency of the mode, respectively. To facilitate the numerical computation and processing of the signal, the time $t$ is converted into a discrete form $t=nh$ (where $h$ is the time step and $n$ is an integer). This operation allows the continuous signal to be expressed as a discrete sequence, enabling better utilization of numerical methods for operations and supporting the storage and analysis of the signal. On this basis, the form of the discrete signal $x_{n}$ can be obtained:

$$x_{n}=\varphi(nh)=\sum_{j=1}^{p}C_{j}e^{-i\omega_{j}nh}=\sum_{j=1}^{p}C_{j}z_{j}^{n}.$$

(28)

Next, we use the known $x_{n}$ to construct a polynomial $A(z)$ such that the modal roots $z_{j}$ of the signal become its roots. Through these roots, the corresponding complex frequencies $\omega_{j}$ can be further calculated. To achieve this, the polynomial function $A(z)$ can be defined in the following form:

$$A(z)=\prod_{j=1}^{p}(z-z_{j})=\sum_{m=0}^{p}a_{m}z^{p-m},\quad a_{0}=1.$$

(29)

Thus, the signal satisfies

$$\sum_{m=0}^{p}a_{m}x_{n-m}=\sum_{m=0}^{p}a_{m}\sum_{j=1}^{p}C_{j}z_{j}^{n-m}=\sum_{j=1}^{p}C_{j}z_{j}^{n-p}A(z_{j})=0.$$

(30)

From this, we can obtain

$$\sum_{m=1}^{p}a_{m}x_{n-m}=-x_{n}.$$

(31)

By determining the coefficients $a_{m}$ of the polynomial $A(z)$, the roots $z_{j}$ of the equation $A(z)=0$ can be solved. Subsequently, the frequencies of each mode can be further calculated using the following formula:

$$\omega_{j}=\frac{i}{h}\ln(z_{j}).$$

(32)

The Prony method has demonstrated excellent applicability in the study of black hole QNMs, particularly playing a crucial role in the testing of the SCCC. By accurately capturing the decay behavior of perturbation signals, this method enables researchers to efficiently extract the frequencies of QNMs, providing significant assistance in gaining deeper insights into the internal structure and dynamical evolution of black holes. The following sections will present the relevant computational results in detail.

In 2007, Arkani-Hamed et al. published a groundbreaking paper on the Weak Gravity Conjecture (WGC)Arkani-Hamed et al. (2007). In recent years, research on the WGC has garnered increasing attentionHarlow et al. (2023), with related discussions and explorations deepening continually. The WGC is a theoretical hypothesis about the effects of quantum gravity, which asserts that the strength of gravitational interactions should not exceed that of other fundamental interactions, expressed as $F_{\text{gravity}}\leq F_{\text{any}}$. By imposing constraints on specific physical models, the WGC provides an important approach to understanding the effects of quantum gravity in the low-energy regime. Furthermore, the WGC requires that, in a $U(1)$ gauge theory, there exists at least one object that satisfies the following condition:

$$\frac{|q|}{m}\geq\left.\frac{|Q|}{M}\right|_{\text{ext}}.$$

(33)

In the above expression, $q$ represents the charge of the object, $m$ represents its mass, and $\left.\frac{|Q|}{M}\right|_{\text{ext}}$ denotes the charge-to-mass ratio of an extremal black hole. This relation suggests that if an object’s charge-to-mass ratio is smaller than this extremal limit, it may lead to unstable physical phenomena, thereby affecting the existence of black holes.

The WGC ensures that gravity remains sufficiently strong in physical theories by comparing the strength of gauge forces and gravity. This simple yet profound idea has had far-reaching impacts across multiple areas of modern fundamental physics, including cosmology, particle physics, black holes, and scattering amplitudes. In the study of the SCCC, the WGC plays a critical role. Specifically, when considering black holes in extremal states, the WGC ensures compliance with the SCCC by constraining the charge-to-mass ratio of charged and massive scalar fields. More concretely, the WGC offers a new perspective for exploring the SCCC and provides conditions under which extremal black holes adhere to the conjecture. This novel approach opens up new directions for investigating the validity and applicability of the SCCC in different physical scenarios.

Although the WGC has triggered extensive research and led to variants such as the “tower WGC”Montero et al. (2016) and “sublattice WGC,”Heidenreich et al. (2017)Heidenreich et al. (2019) each proposing different constraints in various physical contexts, no version has yet been rigorously proven mathematically. Existing studies, while supportive of certain versions, still face challenges such as the lack of precise $O(1)$ factorsCheung and Remmen (2014)Andriolo et al. (2020) or reliance on unverified assumptions. Therefore, while the WGC has profound significance in theoretical physics, it remains a conjecture requiring further validation and development. The specific computational results will be presented in detail in the following sections.

This chapter examines the applicability and validity of the SCCC by analyzing the perturbative behavior of the RN black hole modified by a Yang-Mills field. A neutral and massless scalar field is introduced in the study, and the WKB method and Prony method are employed to compute the QNMs frequencies, evaluating the dynamical stability of the black hole system. Ultimately, this paper provides solid numerical support for the verification of the SCCC by exploring the perturbative characteristics of the black hole. The following table presents the perturbation decay rate and the ratio of gravitational blueshift effects, $\beta$, under different parameters $p$ and Yang-Mills field charge $q_{YM}$ for the Yang-Mills field modified RN black hole. Under the assumptions of electromagnetic charge $Q=1$ and mass $M=1$, the data are calculated using the WKB method and the Prony method. To further investigate the physical implications of these results, we will conduct a detailed analysis and discussion based on the data in the table. Specifically, we will focus on whether these numerical results align with the expectations of the SCCC, thereby verifying the validity of the SCCC in the background of the Yang-Mills field modified RN black hole. By comparing the QNMs frequencies obtained from different methods, we can comprehensively evaluate the dynamical stability and perturbation characteristics of the black hole system. This analysis not only helps to confirm the validity of the SCCC but also reveals the potential impact of the Yang-Mills field on black hole perturbative behavior.

From the data in Tables 1, 2 and 3, and FIG 2, it can be observed that in the Yang-Mills field modified RN black hole, under the conditions of black hole mass $M=1$ and electromagnetic charge $Q=1$, and with the parameter $p$ fixed, as the Yang-Mills field charge $q_{YM}$ gradually increases, the RN black hole transitions from initially violating the SCCC to eventually complying with it. More importantly, as $q_{YM}$ further increases, the likelihood of the extreme RN black hole adhering to the SCCC increases significantly. This result indicates that the Yang-Mills field introduces new spacetime geometric properties to the classical RN black hole. It not only alters the relative position of the event horizon and the Cauchy horizon but also significantly affects the behavior of scalar field perturbations on the RN black hole and its compliance with the SCCC. Combined with the results in Figure 1, it can be seen that the impact of the Yang-Mills field on the RN black hole under extreme conditions is particularly significant. Even when the RN black hole is in an extreme state, as the Yang-Mills field charge $q_{YM}$ increases, the black hole’s state can gradually stabilize, ultimately making it satisfy the requirements of the SCCC. This finding reveals the profound influence of the Yang-Mills field on the horizon structure of black holes, providing a concrete example of how an RN black hole can transition from an extreme state to a stable state. It also offers critical theoretical support for the completeness and consistency of general relativity.

This section explores the perturbation characteristics of a charged and massive scalar field near the Yang-Mills field modified RN black hole, aiming to evaluate its impact on the validity of the SCCC. Considering the fine-structure constant $e^{2}/c\hbar\simeq 1/137$ and the fact that the charge $Q$ of the black hole is usually very large, for a scalar field with a slight charge, its charge $q$ would satisfy the condition $qQ\gg 1$. To avoid the Schwinger effectHod (1999) imposing constraints on the black hole’s electric field, the electric field strength of the black hole must be much lower than the critical value for Schwinger discharge, requiring the condition $Q/r_{h_{+}}^{2}\ll m^{2}/q$, where $r_{h_{+}}$ represents the radius of the event horizon, and $m$ and $q$ are the mass and charge of the scalar field, respectively. Thus, the constraint relationship $m^{2}r_{h_{+}}^{2}\gg qQ\gg 1$ can be derived. Based on the research by Jafar Saeed Noori Gashti et al.Sadeghi et al. (2023), the following constraint conditions can be introduced to simplify the analysis of the QNM perturbations of the charged, massive scalar field near the Yang-Mills field modified RN black hole:

$$m^{2}r_{h_{+}}^{2}\gg l(l+1);\quad m^{2}r_{h_{+}}^{2}\gg 2k_{r_{h_{+}}}r_{h_{+}}.$$

(34)

By introducing two approximation conditions, Equation (34) simplifies the effective potential (13) of the charged and massive scalar field, thereby significantly reducing the complexity of calculating QNMs.

To study the linear dynamical properties of the charged and massive scalar field near the horizon of the Yang-Mills field modified RN black hole, the imaginary part of its QNMs needs to be calculated. First, the perturbative characteristics of the region are analyzed using the radial potential (13), and the electric potential in the region (34) is treated as the effective potential. Then, the imaginary part of the perturbative QNMs near the black hole horizon is calculated using the WKB method. Assuming that the Yang-Mills field modified RN black hole has a maximum effective potential at the point $r_{0}$ near the event horizon, the position of this maximum effective potential can be determined using Equation (13) and the condition $V^{\prime}(r_{0})=0$Sadeghi et al. (2023):

$$r_{0}=\frac{q^{2}Q^{2}}{qQ\omega-m^{2}r_{h_{+}}^{2}k_{r_{h_{+}}}}.$$

(35)

Using Equations (13), (20), (21), (IV.1), (34), and (35)., the corresponding results can be derived:

$$K\simeq\frac{k_{r_{h_{+}}}^{2}m^{4}r_{h_{+}}^{4}qQ}{2f_{r_{0}}\left(k_{r_{h_{+}}}m^{2}r_{h_{+}}^{2}-qQ\omega\right)^{2}},$$

(36)

$\displaystyle\Lambda(n)\simeq\frac{k_{r_{h_{+}}}^{2}m^{4}\left[17-60\left(n+\frac{1}{2}\right)^{2}\right]r_{h_{+}}^{4}+2k_{r_{h_{+}}}m^{2}\left[36\left(n+\frac{1}{2}\right)^{2}-7\right]qQr_{h_{+}}^{2}\omega f_{r_{0}}}{16qQ\left(qQ\omega-3k_{r_{h_{+}}}m^{2}r_{h_{+}}^{2}\right)^{2}},$

(37)

$\displaystyle\Omega(n)\simeq\frac{15k_{r_{h_{+}}}^{4}m^{8}\left[148\left(n+\frac{1}{2}\right)-41\right]r_{h_{+}}^{8}+12k_{r_{h_{+}}}^{3}m^{6}\left[121-420\left(n+\frac{1}{2}\right)^{2}\right]qQr_{h_{+}}^{6}\omega(-\left(n+\frac{1}{2}\right)Q^{3}q^{3}f_{r_{0}}^{2})}{64q^{5}Q^{5}\left(k_{r_{h_{+}}}m^{2}r_{h_{+}}^{2}-qQ\omega\right)^{4}}.$

(38)

Subsequently, it is necessary to derive the specific form of the QNMs in the system to further explore the validity of the SCCC. For this purpose, $\text{Im}(\omega)$ can be calculated by combining Equations(19), (36), (37), and (38).

	$\displaystyle\omega\simeq\frac{qQ}{r_{h_{+}}}-\frac{2k_{r_{h_{+}}}m^{2}r_{h_{+}}^{2}}{qQ}\left[1-\frac{14400}{11644}\left(\frac{n+\frac{1}{2}}{qQ}f_{r_{0}}\right)^{4}\right]$		(39)
	$\displaystyle-i\left[4f_{r_{0}}k_{r_{h_{+}}}\left(n+\frac{1}{2}\right)\frac{m^{2}r_{h_{+}}^{2}}{q^{2}Q^{2}}\left(1-\frac{34qQf_{r_{0}}^{4}}{11644}\right)\right].$

Thus, the relationship between the imaginary part of the QNMs and the surface gravity of the event horizon can be expressed as:

$$\beta=\frac{-\text{Im}(\omega)}{k_{r_{h_{+}}}}\simeq 2f_{r_{0}}\frac{m^{2}r_{h_{+}}^{2}}{q^{2}Q^{2}}\left[1-\frac{34qQf_{r_{0}}^{4}}{11664}\right].$$

(40)

Since $r_{0}$ is very close to the event horizon $r_{h_{+}}$, it follows that $f_{r_{0}}\ll 1$. When $\beta<1/2$, the SCCC will hold, and the condition $\frac{m^{2}r_{h_{+}}^{2}}{q^{2}Q^{2}}<1$ further ensures the validity of the SCCC. Therefore, the condition for the SCCC to hold can be derived from Equation (40) asSadeghi et al. (2023):

$$\frac{q}{m}\geq\frac{r_{h_{+}}}{Q}.$$

(41)

If in Equation (40), $qQ$ satisfies $qQ<2\sqrt{f_{r_{0}}}\,mr_{h_{+}}$, it is possible for $\beta>1/2$ to occur, which would lead to the SCCC no longer being valid. Next, we will refer to the previously derived method to study the metric equation (5) in this paper. When the metric function (5) of the black hole satisfies the relations $f(r)=0$ and $f^{\prime}(r)=0$, the extremal expressions for the mass $M$ and electromagnetic charge $Q$ of the Yang-Mills field modified RN black hole can be obtained as:

$$M=\frac{-2Q_{YM}\,p\,r^{3}+2Q_{YM}\,r^{3}+r^{4p+1}}{r^{4p}}.$$

(42)

$$Q=\sqrt{\frac{-4Q_{YM}\,p\,r^{4}+3Q_{YM}\,r^{4}+r^{4p+2}}{r^{4p}}}.$$

(43)

Here, we choose the parameter $p=\frac{1}{2}$ because it satisfies the energy and causality conditions in general relativity, significantly alters the spacetime structure of the original RN black hole, and ensures the related problems remain tractable. Compared to other values of $p$, the choice $p=\frac{1}{2}$ does not lead to overly complex or difficult-to-understand solutions. Instead, it provides a concise and meaningful analytical solution, particularly when describing the event horizon radius $r_{h_{+}}$ and the Cauchy horizon radius $r_{h_{-}}$. From equation (5), we can obtain the expressions for these two horizon radii as:

$$r_{h_{\pm}}=\frac{M\pm\sqrt{M^{2}-Q^{2}(1+Q_{YM})}}{1+Q_{YM}}$$

(44)

In (44), to avoid the appearance of a naked singularity and to establish a clear relationship between the two horizons, the following conditions must be satisfied: $Q_{YM}>-1$ and $0<Q/M<\sqrt{\frac{1}{1+Q_{YM}}}$ Gómez et al. (2023). Therefore, from equations (43) and (44), the expression for the total charge $Q_{\text{all}}$ of the Yang-Mills field modified RN black hole in the extremal case can be derived as:

$$Q_{\text{all-exe}}^{2}=Q^{2}(1+Q_{YM})=Q^{2}\left(1-\frac{\sqrt{2}}{2}q_{YM}\right)=r^{2}\left(1-\frac{\sqrt{2}}{2}q_{YM}\right)^{2}$$

(45)

Substituting the obtained extremal total charge expression $Q_{\text{all-exe}}$ (45) into equation (40), we can further obtain:

$$\beta\simeq 2f_{r_{0}}\frac{m^{2}}{q^{2}}\left(1-\frac{\sqrt{2}}{2}q_{YM}\right)^{-2}\left[1-\frac{34qf_{r_{0}}^{4}r_{h_{+}}}{11664}\left(1-\frac{\sqrt{2}}{2}q_{YM}\right)\right]$$

(46)

According to the WGC, we can obtain the condition under which the Yang-Mills field modified RN black hole respects the SCCC as:

$$\frac{q}{m}>\left(1-\frac{\sqrt{2}}{2}q_{YM}\right)^{-1}$$

(47)

Therefore, when $\frac{q}{m}$ satisfies equation (47), then $\beta<\frac{1}{2}$, and the SCCC will certainly be respected. When $\frac{q}{m}$ satisfies:

$$2\sqrt{f_{r_{0}}}\left(1-\frac{\sqrt{2}}{2}q_{YM}\right)^{-1}>\frac{q}{m}.$$

(48)

then $\beta>\frac{1}{2}$, and the SCCC is violated.

In extremal black holes, since the event horizon and the Cauchy horizon nearly coincide and the surface gravity approaches zero, the perturbation of neutral and massless scalar fields cannot effectively accumulate energy through the gravitational blueshift effect. Its dynamical behavior is dominated by decay effects, ultimately causing the perturbation to decay rapidly, thereby maintaining the stability of the Cauchy horizon and violating the SCCC. In contrast, the dynamics of charged and massive scalar fields are more complex. Under the conditions satisfying the WGC, although the weakened surface gravity limits the strength of the gravitational blueshift effect and the damping effect in the initial stage may suppress the amplitude growth, a large charge-to-mass ratio $\frac{q}{m}$ allows the scalar field to accumulate energy near the Cauchy horizon through electromagnetic coupling. When the local energy density rapidly grows due to the coupling effect and exceeds a critical value, it may trigger a divergence in spacetime curvature, thereby destroying the Cauchy horizon and supporting the validity of the SCCC. It should be noted that this result strongly depends on parameters such as the black hole’s mass, charge, and the initial conditions of the perturbation. The specific mechanism still requires further verification through numerical simulations or theoretical analysis.