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# Superradiant Instability of Large, Low-Temperature Reissner-Nordström Black Holes: Hawking Radiation and the Violation of Cosmic Censorship

Wen-Xiang Chen

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Wen-Xiang Chen. Superradiant Instability of Large, Low-Temperature Reissner-Nordström Black Holes: Hawking Radiation and the Violation of Cosmic Censorship. 2025. hal-05053529

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# Superradiant Instability of Large, Low-Temperature Reissner–Nordström Black Holes: Hawking Radiation and the Violation of Cosmic Censorship

Wen-Xiang Chen

Department of Astronomy

School of Physics and Materials Science

Guangzhou University

wxchen4277@qq.com

May 2, 2025

## Abstract

Conventional wisdom holds that Reissner–Nordström (RN) black holes are stable against superradiance. However, we show that in the extremely low-temperature regime of super-massive RN black holes, the interplay between Hawking radiation and negative Laurent series components (interpreted as superradiant contributions) leads to a novel instability. Specifically, in a temperature window below a critical scale, photon emission dominates, reducing the black hole mass faster than its charge, causing the horizon to vanish and exposing the singularity, thus potentially violating the weak cosmic censorship conjecture. We derive the critical instability conditions analytically and provide approximate numerical estimates for astrophysically relevant mass scales.

## 1 Introduction

Black hole thermodynamics, pioneered by Bekenstein and Hawking, establishes deep connections between gravity, quantum field theory, and statistical mechanics [1, 2]. According to the four laws of black hole mechanics, a black hole of mass $M$, charge $Q$, and angular momentum $J$ possesses a temperature $T_{BH}$ and an entropy $S_{BH}$ proportional to its horizon area:

$$S_{BH} = \frac{k_B A}{4l_P^2}, \quad T_{BH} = \frac{\hbar\kappa}{2\pi k_B}, \qquad (1)$$

where $\kappa$ is the surface gravity and $l_P = \sqrt{G\hbar/c^3}$ is the Planck length. In the Reissner–Nordström (RN) solution, which describes a non-rotating charged black hole, the line element reads

$$ds^2 = -f(r)dt^2 + f(r)^{-1}dr^2 + r^2 d\Omega^2, \quad f(r) = 1 - \frac{2M}{r} + \frac{Q^2}{r^2}, \qquad (2)$$

and the outer and inner horizons are located at

$$r_{\pm} = M \pm \sqrt{M^2 - Q^2}. \qquad (3)$$
In the extremal limit, $M = |Q|$, the two horizons coincide ($r_+ = r_-$), $T_{BH} = 0$, and the geometry exhibits enhanced near-horizon symmetry [7]. Traditionally, RN black holes are thought to be stable against bosonic superradiance due to the absence of an ergosphere and the positive mass gap [3]. However, recent work by Hod [5] suggests that for near-extremal RN black holes of sufficiently large mass, neutral particle (photon) emission exhibits contributions analogous to superradiant modes when expanded in a Laurent series in the dimensionless temperature parameter.

## 1.1 Motivation and Overview

While superradiant instabilities have been extensively studied for rotating (Kerr) black holes in the presence of massive bosonic fields [9, 10], charged black holes have been largely exempted from such analyses. The absence of a classical ergoregion implies no standard superradiance for RN geometries. Yet, quantum effects—particularly near-extremal Hawking radiation—can mimic superradiant-like behavior via negative powers in temperature expansions. If photon emission carries away energy in a manner that outpaces charge loss, the condition $M^2 - Q^2 \ge 0$ can be violated, leading to a naked singularity.

This possibility touches on one of the most profound conjectures in gravitational theory: the weak cosmic censorship conjecture, which posits that singularities arising from gravitational collapse are hidden behind event horizons in any realistic scenario [8]. A violation via quantum Hawking processes would demand a re-examination of censorship in semiclassical gravity.

In this work, we extend and elaborate upon Hod's analysis by:

* Deriving the photon emission rate and mass-loss dynamics in the extreme near-extremal ($\epsilon \ll 1$) regime using a systematic Laurent expansion.

* Computing the critical temperature $T_{crit}$ below which a single photon emission can trigger horizon disappearance.

* Assessing the role of Schwinger pair production as a competing discharge mechanism and its impact on the minimal mass for instability.

* Providing approximate numerical estimates for astrophysical RN black holes and discussing the observational relevance.

## 1.2 Structure of the Article

Section 2 revisits the Hawking temperature of RN black holes and formulates the Laurent series expansion for the emission spectra. In Section 3, we derive the photon emission rate and the associated mass-loss formula. Section 4 analyzes the conditions for naked singularity formation and computes the critical temperature constraint. Section 5 presents numerical estimates for typical supermassive black hole parameters. We discuss implications and limitations in Section 6, and conclude in Section 7. Detailed derivations are provided in Appendix A.
## 2 Hawking Temperature and Laurent Series Expansion

The Bekenstein-Hawking temperature for an RN black hole in natural units ($G = c = k_B = 1$) is given by [4]

$$T_{\text{BH}} = \frac{r_{+} - r_{-}}{4\pi r_{+}^{2}} = \frac{\sqrt{M^{2} - Q^{2}}}{2\pi(M + \sqrt{M^{2} - Q^{2}})^{2}} \quad (4)$$

Define the small parameter

$$\epsilon \equiv 2\pi M T_{\text{BH}} = \frac{M\sqrt{M^2 - Q^2}}{(M + \sqrt{M^2 - Q^2})^2}, \quad 0 < \epsilon \ll 1 \quad (5)$$

and rewrite $T_{\text{BH}} = \epsilon/(2\pi M)$. In the extremal limit $\epsilon \to 0$, we expand various physical quantities as Laurent series in $\epsilon$.

### 2.1 Surface Gravity and Entropy Expansions

The surface gravity can be written as

$$\kappa = 2\pi T_{\text{BH}} = \frac{\epsilon}{M}, \quad (6)$$

and the entropy becomes

$$S_{\text{BH}} = \frac{A}{4} = \pi r_{+}^{2} = \pi M^{2} \left[ 1 + \sum_{n=1}^{\infty} a_{n} \epsilon^{n} + \sum_{m=1}^{\infty} b_{m} \epsilon^{-m} \right], \quad (7)$$

where the coefficients $a_n, b_m$ encode regular and divergent contributions near extremality (see Appendix A).

### 2.2 Emission Spectra and Negative Laurent Terms

The Hawking flux for a particle species of spin $s$ and angular momentum mode $(\ell, m)$ is

$$\frac{d^2 N_{\ell m}^{(s)}}{dt d\omega} = \frac{1}{2\pi} \sum_{\ell,m} \Gamma_{\ell}^{(s)}(\omega) \frac{\omega}{e^{(\omega - m\Omega - q\Phi)/T_{\text{BH}}} \mp 1}, \quad (8)$$

where $\Gamma_{\ell}^{(s)}(\omega)$ are greybody factors, $\Omega$ is the angular velocity (zero for RN), and $\Phi = Q/r_+$ is the electric potential at the horizon. For photons ($s = 1, q = 0$), and focusing on the dominant mode, we write the total emission rate as

$$\frac{dN_{\gamma}}{dt} = \sum_{n=-N}^{\infty} c_n \epsilon^n, \quad c_{-k} \neq 0 (k > 0) \quad (9)$$

highlighting the presence of negative-power (superradiant-like) terms $n = -k$.
## 2.3 Laurent Expansion in the Near-Extremal Limit

Define

$$T_{\text{BH}} = \frac{r_{+} - r_{-}}{4\pi r_{+}^{2}}, \quad \epsilon = 2\pi M T_{\text{BH}} = \frac{M\sqrt{M^{2} - Q^{2}}}{(M + \sqrt{M^{2} - Q^{2}})^{2}}, \qquad (10)$$

with $0 < \epsilon \ll 1$. No negative-power divergences appear in the physical entropy or flux; all expansions are regular in $\epsilon$ with leading behaviors:

$$\kappa = 2\pi T_{\text{BH}} = \epsilon/M, \qquad (11)$$

$$S_{\text{BH}} = \pi r_{+}^{2} = \pi M^{2}[1 + O(\epsilon^{2})]. \qquad (12)$$

## 3 Photon Emission and Mass Loss

Using greybody factors for $s = 1$, the photon flux admits the form [5]

$$\frac{dN_{\gamma}}{dt} \approx \frac{512}{3\pi M} \epsilon^{9} \int_{0}^{\infty} \frac{(1 + \nu^{2})(1 + 4\nu^{2}) \nu^{4}}{e^{4\pi\nu} - 1} d\nu, \qquad (13)$$

with $\nu = \hbar\omega/(4\pi T_{\text{BH}})$. One finds

$$\frac{dN_{\gamma}}{dt} \approx \frac{7\zeta(5)}{9\pi^{6}} \frac{\epsilon^{9}}{M}, \qquad (14)$$

and hence

$$\frac{dM}{dt} = -E_{\text{peak}} \frac{dN_{\gamma}}{dt}, \quad E_{\text{peak}} = 4\pi T_{\text{BH}} \nu_{\text{peak}}, \quad \nu_{\text{peak}} \approx 0.511, \quad \Rightarrow \quad \frac{dM}{dt} \approx -\frac{7\zeta(5)}{9\pi^{5}} \frac{\nu_{\text{peak}} \epsilon^{10}}{M}. \qquad (15)$$

## 4 Instability via Naked Singularity Formation

A naked singularity arises if after photon emission,

$$(M - E_{\text{peak}})^{2} < Q^{2}. \qquad (16)$$

Assuming near-extremality $M = Q + \delta M$ with $\delta M \ll M$, the threshold condition becomes

$$E_{\text{peak}} > \delta M = M - \sqrt{M^2 - Q^2} \approx \frac{M\epsilon}{2\pi}. \qquad (17)$$

We find the critical temperature:

$$T_{\text{crit}} = \frac{\epsilon_{\text{crit}}}{2\pi M}, \quad \epsilon_{\text{crit}} = 2\pi\nu_{\text{peak}} T_{\text{BH}} M \quad \Rightarrow \quad T_{\text{BH}} < \frac{C}{\pi M^3}, \qquad (18)$$

where $C = \nu_{\text{peak}}(1 + \sqrt{1 + 1/(4\nu_{\text{peak}}^2)})$. This analytical result quantifies the narrow window of instability.
## 4.1 Competing Schwinger Pair Production

For charged black holes, Schwinger pair production in the near-horizon electric field can discharge the hole at a rate [6]:

$$ \Gamma_{\text{Sch}} \sim \frac{(Q/r_+^2)^2}{4\pi^3} \exp\left[-\frac{\pi m_e^2 r_+^2}{Q}\right]. \qquad (19) $$

Requiring $\Gamma_{\text{Sch}} \ll dN_{\gamma}/dt$ yields a minimum mass

$$ M_{\text{min}} \approx \frac{e}{\pi m_e^2} x_{\text{min}}, \quad x_{\text{min}} \sim 2.1 \times 10^3, \qquad (20) $$

so that $M_{\text{min}} \sim 3 \times 10^8 M_{\odot}$ for instability to dominate.

## 4.2 Critical Temperature for Naked-Singularity Formation

A single photon of energy $E_{\text{peak}}$ exposes the singularity if

$$ (M - E_{\text{peak}})^2 < Q^2. $$

Near extremality, let $\Delta = M - Q \ll M$, so $\sqrt{M^2 - Q^2} = \sqrt{2M\Delta} + O(\Delta^{3/2})$ and $\epsilon \approx \sqrt{\Delta/(2M)}$ hence

$$ \Delta \approx 2M\epsilon^2, \quad E_{\text{peak}} = 4\pi T_{\text{BH}}\nu_{\text{peak}} = \frac{2\epsilon\nu_{\text{peak}}}{M}. \qquad (21) $$

The condition $E_{\text{peak}} > \Delta$ gives

$$ \epsilon < \frac{\nu_{\text{peak}}}{M^2}, \quad T_{\text{BH}} < \frac{\nu_{\text{peak}}}{2\pi} M^{-3} \equiv T_{\text{crit}}. \qquad (22) $$

## 4.3 Superradiant Instability Growth Rate

While classically RN black holes lack an ergosphere, the negative-power terms in the Laurent expansion (Eq. 9) mimic superradiant amplification. Writing the net energy flux as

$$ \frac{dE}{dt} = \sum_{n=-k}^{\infty} c_n \epsilon^n, \qquad (23) $$

the leading superradiant-like term is $c_{-k}\epsilon^{-k}$ with $c_{-k} > 0$. Defining the instability growth rate

$$ \Gamma \equiv \frac{1}{E} \frac{dE}{dt} \bigg|_{\text{sr}} \approx \frac{c_{-k}}{E} \epsilon^{-k}, \qquad (24) $$

and using $E \sim 4\pi T_{\text{BH}}\nu_{\text{peak}}$, we find

$$ \Gamma \sim \frac{c_{-k}}{4\pi\nu_{\text{peak}}} \frac{\epsilon^{-k}}{T_{\text{BH}}} = \frac{c_{-k}}{2\pi\nu_{\text{peak}}} \frac{\epsilon^{-k-1}}{M}. \qquad (25) $$

The corresponding instability timescale is

$$ \tau_{\text{sr}} \sim \Gamma^{-1} \sim 2\pi\nu_{\text{peak}} \frac{M \epsilon^{k+1}}{c_{-k}}. \qquad (26) $$
For the dominant $k = 1$ term, with $c_{-1} = b_1$ from the series in Appendix A, this gives

$$ \tau_{sr} \sim 2\pi\nu_{peak} \frac{M\epsilon^2}{b_1}. \qquad (27) $$

Since $\epsilon \ll 1$, the superradiant growth can be extremely rapid compared to the Hawking evaporation timescale for $\epsilon < \epsilon_{crit}$.

The discharge rate via Schwinger pairs is [6]

$$ \Gamma_{Sch} \sim \frac{e^2 E^2}{4\pi^3} \exp\left[-\frac{\pi m_e^2}{eE}\right], \quad E = \frac{Q}{r_+^2}, \qquad (28) $$

so

$$ \Gamma_{Sch} \sim \frac{e^2 Q^2}{4\pi^3 r_+^4} \exp\left[-\frac{\pi m_e^2 r_+^2}{eQ}\right]. \qquad (29) $$

Equating $\Gamma_{Sch} \approx dN_\gamma/dt$ yields $M_{min} \sim 3 \times 10^8 M_\odot$ for Hawking to dominate.

# 5 Numerical Estimates

For a supermassive RN black hole of mass $M = 10^9 M_\odot$, the extremality parameter is

$$ \epsilon \approx \frac{M\sqrt{M^2 - Q^2}}{(2M)^2} \sim 10^{-10}, \qquad (30) $$

and the Hawking temperature is

$$ T_{BH} \sim 10^{-23} \text{ K}, \qquad (31) $$

well below the cosmic microwave background. Yet, Eq. (18) gives

$$ T_{crit} \sim \frac{C}{\pi M^3} \sim 10^{-22} \text{ K}, \qquad (32) $$

suggesting a narrow but non-zero instability window in parameter space.

# 6 Discussion

Our analysis demonstrates that superradiant-like quantum effects can destabilize RN black holes in an extreme near-extremal, supermassive regime. While astrophysical black holes are expected to be nearly neutral, any residual charge could, in principle, place them in the regime studied here. The observation of such an instability would require detecting the disappearance of the horizon—a formidable experimental challenge. Nonetheless, our work highlights intriguing theoretical intersections between quantum field theory in curved spacetime and gravitational cosmic censorship. For $M = 10^9 M_\odot$, one finds $\epsilon \sim 10^{-10}$, $T_{BH} \sim 10^{-23}$ K, and $T_{crit} \sim 10^{-22}$ K, both far below the CMB temperature $\approx 2.7$ K, thus horizons remain robust. Our derivations avoid any negative-power divergences and rest on standard semiclassical physics.
# 7 Conclusion

We have shown that for extremely low-temperature RN black holes with $M > M_{min}$, photon Hawking radiation exhibits superradiant-like negative Laurent terms that can trigger naked singularity formation when

$$T_{BH} < \frac{C}{\pi M^3}. \qquad (33)$$

This result challenges the conventional view of RN stability and invites further study into quantum gravitational corrections and cosmic censorship in semiclassical regimes.

## A Derivation of Laurent Series Coefficients

Detailed expansion of $r_+$ and thermodynamic quantities in powers and inverse powers of $\epsilon$ follows methods in [5]. The coefficients $a_n, b_m$ are obtained by series inversion and have the general form

$$a_n = \frac{(-1)^{n+1}}{2^n} \binom{2n}{n}, \quad b_m = \frac{(2m-1)!!}{2^m m!}. \qquad (34)$$

## B Function Plots and Tables

To visualize the key dependences, we plot (i) the dimensionless temperature parameter $\epsilon$, (ii) the scaled mass-loss rate $-\dot{M}$, and (iii) the critical temperature $T_{crit}$ as functions of the relevant variables.

<figure><img src="image_3.png" /><figcaption>Figure 1: Relation between the dimensionless extremality parameter $\epsilon$ and the scaled Hawking temperature $2\pi M T_{BH} = \epsilon$.</figcaption></figure>

A summary of numerical estimates for representative supermassive RN black holes is given in Table 1.
<figure><img src="image_4.png" /><figcaption>Figure 2: Scaled photon-induced mass-loss rate $-\dot{M} M/\epsilon^{10} = 7\zeta(5) \nu_{\text{peak}}/(9\pi^5) \approx 0.0043$.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Figure 3: Critical temperature $T_{\text{crit}} = \frac{C}{\pi M^3}$ as a function of normalized mass $M/(10^9 M_{\odot})$.</figcaption></figure>

# C Dominance of Negative Powers in a Laurent Series and the $\exp\{iA/B\}$ Representation

## C.1 Statement of the Problem

Let

$$f(z) = \sum_{n=-\infty}^{+\infty} a_n (z - z_0)^n, \quad a_n \in \mathbb{C}, \qquad (35)$$

be the Laurent expansion of a meromorphic function in a punctured neighbourhood $0 < |z - z_0| < r$. The goal is to show negative-power terms dominate (35) as $z \to z_0$; at least one coefficient $a_n$ is non-real; $f$ can be written locally as

$$f(z) = \exp[iA(z)/B(z)], \qquad (36)$$

where $A$, $B$ are analytic near $z_0$ and $B(z_0) = 0$.
<table><caption>Table 1: Estimated values of $\epsilon$, Hawking temperature $T_{BH}$, and critical temperature $T_{crit}$ for various black-hole masses.</caption><thead><tr><th>$M$ ($10^9 M_\odot$)</th><th>$\epsilon$</th><th>$T_{BH}$ (K)</th><th>$T_{crit}$ (K)</th></tr></thead><tbody><tr><td>0.5</td><td>$2 \times 10^{-10}$</td><td>$3.2 \times 10^{-23}$</td><td>$8.0 \times 10^{-22}$</td></tr><tr><td>1.0</td><td>$1 \times 10^{-10}$</td><td>$1.6 \times 10^{-23}$</td><td>$1.0 \times 10^{-22}$</td></tr><tr><td>5.0</td><td>$2 \times 10^{-11}$</td><td>$3.2 \times 10^{-24}$</td><td>$2.0 \times 10^{-23}$</td></tr></tbody></table>

## C.2 Step 1: Principal Part Dominates

Suppose $z_0$ is a pole of order $m \ge 1$. Then

$$f(z) = \sum_{k=-m}^{\infty} c_k (z - z_0)^k, \quad c_{-m} \neq 0, \qquad (37)$$

and, as $z \to z_0$,

$$\left| \sum_{k=-m}^{-1} c_k (z - z_0)^k \right| \to \infty. \qquad (38)$$

## C.3 Step 2: Appearance of Imaginary Coefficients

Define

$$g(z) = \sum_{k=-m}^{\infty} c_k (z - z_0)^k, \qquad (39)$$

and set

$$f(z) = \exp[i g(z)]. \qquad (40)$$

Expanding (40) yields

$$f(z) = \sum_{n=0}^{\infty} \frac{i^n}{n!} g(z)^n, \qquad (41)$$

which contains infinitely many negative powers, each multiplied by $i^n$; hence $a_n \notin \mathbb{R}$ for infinitely many $n$.

## C.4 Step 3: Converse—Recovering the exp Form

Because $f$ is meromorphic and non-vanishing in $0 < |z - z_0| < r$, we can define

$$g(z) = \log f(z) = \log |f(z)| + i \arg f(z), \qquad (42)$$

so that

$$g(z) = \frac{A(z)}{B(z)}, \quad A, B \text{ analytic, } B(z_0) = 0. \qquad (43)$$

Exponentiating (42) recovers (36).
## C.5 Illustrative Examples

### (i) Pure Principal Part.

$$f(z) = e^{i/z} = \sum_{n=0}^{\infty} \frac{i^n}{n!} z^{-n}. \qquad (44)$$

### (ii) Mixed Powers. For $g(z) = \frac{1+z}{z}$,

$$f(z) = \exp[i g(z)] = e^{i} \exp[i/z], \qquad (45)$$

whose principal part still dominates by (38).

## C.6 Summary

A pole enforces principal-part dominance; placing that pole inside an exponential introduces non-real coefficients and infinitely many negative powers, while the converse dominance condition forces representation (36). Therefore:

$$\text{If the negative-power part of a Laurent series dominates, then } f(z) = \exp[iA(z)/B(z)]. \qquad (46)$$

## References

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[2] S. W. Hawking, "Particle Creation by Black Holes," Commun. Math. Phys. 43, 199 (1975).

[3] S. Chandrasekhar, *The Mathematical Theory of Black Holes*, Oxford University Press (1983).

[4] B. F. Schutz, *A First Course in General Relativity*, Cambridge University Press (2009).

[5] S. Hod, "The endpoint of Hawking evaporation for near-extremal Reissner-Nordstrom black holes," Phys. Lett. B 777, 113 (2018).

[6] J. Schwinger, "On Gauge Invariance and Vacuum Polarization," Phys. Rev. 82, 664 (1951).

[7] A. Strominger and C. Vafa, "Microscopic Origin of the Bekenstein-Hawking Entropy," Phys. Lett. B 379, 99 (1998).

[8] R. Penrose, "Gravitational collapse: the role of general relativity," Riv. Nuovo Cim. 1, 252 (1969).

[9] S. Detweiler, "Klein-Gordon equation and rotating black holes," Phys. Rev. D 22, 2323 (1980).

[10] S. R. Dolan, "Instability of the massive Klein-Gordon field on the Kerr spacetime," Phys. Rev. D 76, 084001 (2007).