# Hawking Radiation and Superradiance in Hairless Black Holes

Wen-Xiang Chen

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Wen-Xiang Chen. Hawking Radiation and Superradiance in Hairless Black Holes. 2025. [hal-05215334]

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# Hawking Radiation and Superradiance in Hairless Black Holes

Wen-Xiang Chen

School of Physics and Materials Science

Guangzhou University

wxchen4299@qq.com

August 20, 2025

## Abstract

We explore the relationship between Hawking radiation and superradiance in black holes without hair (i.e., rotating or charged black holes carrying no additional fields). Focusing on Kerr (rotating, uncharged) and Reissner-Nordström (charged, non-rotating) black holes, we derive Hawking radiation expressions using the mathematical framework of superradiant scattering. By substituting black hole parameters (such as horizon angular velocity or electric potential) into key equations, we demonstrate that the Hawking emission spectrum can be understood as a form of spontaneous superradiance. In particular, near-horizon dimensional reduction techniques show that the quantum Hawking flux emerges to cancel gravitational and gauge anomalies, analogous to the classical wave amplification in superradiance. We emphasize this connection by deriving the Hawking emission formula with rotational and electrostatic potentials, highlighting that modes satisfying the superradiant condition $\omega < m\Omega_H + q\Phi_H$ correspond to stimulated emission. We include illustrative equations and a comparative table of Kerr and Reissner-Nordström parameters, and we discuss stability considerations. In Kerr black holes, thermodynamic phase transitions (inferred via thermodynamic geometry) are linked to the existence of potential barriers or wells outside the horizon, delineating regimes of superradiant stability. In charged black holes, we examine the phenomenon of "tachyonic superradiance," wherein a would-be scalar hair condensate (driven by a tachyonic instability) is entirely radiated away by superradiant emission, leaving a hairless final state. Our conclusions reinforce that Hawking radiation is closely related to superradiance[9], and together these processes play complementary roles in energy extraction from black holes while preserving the no-hair property.

## 1 Introduction

Black holes are characterized in general relativity by just a few parameters (mass *M*, angular momentum *J*, and charge *Q*), a statement often referred to as the *no-hair theorem*. Two seemingly distinct mechanisms by which a black hole can lose energy (and possibly angular
momentum or charge) are *Hawking radiation* and *superradiance*. Hawking radiation is the quantum emission of particles from black holes, predicted by Stephen Hawking in 1974 as a thermal radiation effect arising from quantum field theory in curved spacetime[1]. In contrast, superradiance is a classical (or semiclassical) wave phenomenon wherein certain incident waves are amplified upon scattering off a black hole, extracting energy from the hole in the process[2, 3].

At first glance, Hawking radiation and superradiant scattering appear to be different processes: Hawking radiation is spontaneous and thermal, occurring even in the absence of any incoming radiation, whereas superradiance requires an incoming wave and is frequency-specific (amplifying only low-frequency modes below a threshold). Nevertheless, growing evidence suggests a deep connection between the two. For rotating (Kerr) black holes, the condition for wave amplification is that the wave's frequency $\omega$ is less than a certain combination of the black hole's horizon angular velocity $\Omega_H$ and the wave's azimuthal angular momentum $m$[3]. As we will show, the very same condition appears in the Hawking emission spectrum as the criterion that distinguishes particle emission from absorption. In fact, it has been argued that *Hawking radiation is essentially a form of superradiance* [9], a viewpoint we aim to elucidate in this article.

In this paper, we investigate Hawking radiation and superradiance in "hairless" black holes—namely Kerr black holes (which carry no charge or scalar hair) and Reissner-Nordström black holes (which carry charge but no other fields). We use geometric units where $G = c = \hbar = 1$ for simplicity. In Section 2, we review the physics of superradiance in Kerr and Reissner-Nordström spacetimes, including the conditions for wave amplification and the role of potential barriers or wells outside the horizon. In Section 3, we summarize Hawking radiation in these spacetimes, providing key formulas for the Hawking temperature and radiation spectrum. In Section 4, we derive the connection between the Hawking emission formula and the superradiant scattering condition, by examining the near-horizon field dynamics and employing parameter substitutions (such as setting the black hole's rotation or charge to specific values) in the superradiant scattering equations. We show explicitly that the Hawking flux can be understood as arising from modes in the superradiant regime, even in the absence of an external incident wave. Section 5 discusses the implications for black hole stability and "hair." We incorporate recent developments in black hole thermodynamics and nonlinear dynamics: for Kerr black holes, we describe how thermodynamic phase transitions (for example, signaled by divergences in the heat capacity or in the Ruppeiner curvature scalar) correspond to the onset of superradiant instabilities or stability regimes[8]. For charged black holes, we describe the process of *tachyonic superradiance* [5] wherein a scalar field that would classically form a hair (due to a tachyonic instability) instead is blown off by superradiant radiation, leaving the black hole bald[6, 10]. Finally, in Section 6 we conclude with a discussion on how viewing Hawking radiation as a type of superradiance helps unify our understanding of energy extraction from black holes.
## 2 Superradiance in Kerr and Reissner-Nordström Black Holes

Superradiance refers to the amplification of waves by a rotating or charged object. In the context of black holes, superradiant scattering was first studied in rotating Kerr black holes, inspired by earlier work of Zel'dovich on rotating conductors[2]. A wave of frequency $\omega$ and azimuthal quantum number $m$ impinging upon a Kerr black hole of angular velocity $\Omega_H$ can gain energy from the black hole if the frequency is low enough. Specifically, for a Kerr black hole the condition for superradiant amplification is[3]

$$0 < \omega < m \Omega_H, \qquad (1)$$

assuming $\omega > 0$ as the wave's frequency measured at infinity and $m$ is the angular momentum (azimuthal) number of the mode. Here $\Omega_H = \frac{a}{r_+^2+a^2}$ is the angular velocity of the black hole's event horizon (with $a = J/M$ the rotation parameter and $r_+$ the horizon radius)[4]. When condition (1) is satisfied, the reflection coefficient of the wave exceeds unity, indicating an outgoing wave of greater amplitude (and energy) than the incoming wave[5]. The excess energy is drawn from the black hole's rotational kinetic energy. If no such low-frequency $m \neq 0$ waves are present, the black hole cannot lose rotational energy classically; but if they are present (even as vacuum fluctuations), the possibility of extracting rotational energy exists.

An analogous superradiant effect occurs for charged (Reissner-Nordström) black holes. A charged black hole has an electrostatic potential at the horizon, $\Phi_H = \frac{Q}{r_+}$ (for RN, where $r_+$ is the horizon radius and $Q$ the charge). Consider a bosonic field of charge $q$ (e.g., a scalar field with electric charge $q$) scattering off the charged black hole. In this case, energy can be extracted from the electrostatic field of the black hole if the wave's frequency $\omega$ is below a threshold set by the electrostatic potential. The superradiant condition for a charged wave is

$$0 < \omega < q\Phi_H, \qquad (2)$$

provided the wave's charge $q$ is of opposite sign to the black hole's charge (so that the interaction lowers the black hole's energy)[5]. For Reissner-Nordström, $\Omega_H = 0$ (no rotation), so only charged superradiance is relevant; for Kerr-Newman (rotating charged black holes), both rotational and charged superradiance can occur, with the general condition $\omega < m\Omega_H + q\Phi_H$ resulting from the first law of black hole mechanics[5].

Table 1 summarizes the key parameters and superradiant conditions for Schwarzschild, Kerr, Reissner-Nordström (RN), and Kerr-Newman black holes. We list the horizon angular velocity $\Omega_H$, horizon electric potential $\Phi_H$, the superradiant condition, and the Hawking temperature $T_H$ for each case. (The Hawking temperatures will be derived in Section 3.) As shown, Schwarzschild has no superradiance (since $\Omega_H = \Phi_H = 0$), while Kerr and RN each exhibit superradiance associated with $\Omega_H$ and $\Phi_H$ respectively.

In a superradiant scattering process, an incident wave comes in from far away and part of it is absorbed by the black hole while part is reflected back to infinity. If $\omega$ violates the above inequality (for example, $\omega > m\Omega_H$ for Kerr), then the wave is partly absorbed and partly reflected, with the reflected portion having less energy than was incident (the difference
<table><caption>Table 1: Key parameters and conditions for superradiance and Hawking temperature in various black holes (in geometric units). Here $r_+$ is the radius of the event horizon and $r_-$ the radius of the inner (Cauchy) horizon. For Kerr-Newman, $r_\pm = M \pm \sqrt{M^2 - a^2 - Q^2}$; for Kerr, $Q = 0$; for RN, $a = 0$. $T_H$ is the Hawking temperature (see Section 3).</caption><thead><tr><th>Black Hole</th><th>Ω<sub>H</sub></th><th>Φ<sub>H</sub></th><th>Superradiant condition</th><th>T<sub>H</sub></th></tr></thead><tbody><tr><td>Schwarzschild (M)</td><td>0</td><td>0</td><td>None (no superradiance)</td><td><sup>1</sup>&frasl;<sub>8πM</sub></td></tr><tr><td>Kerr (M, a)</td><td><sup>a</sup>&frasl;<sub>r<sub>+</sub><sup>2</sup> + a<sup>2</sup></sub></td><td>0</td><td>0 &lt; ω &lt; mΩ<sub>H</sub></td><td><sup>r<sub>+</sub> - r<sub>-</sub></sup>&frasl;<sub>4π(r<sub>+</sub><sup>2</sup> + a<sup>2</sup>)</sub></td></tr><tr><td>Reissner-Nordström (M, Q)</td><td>0</td><td><sup>Q</sup>&frasl;<sub>r<sub>+</sub></sub></td><td>0 &lt; ω &lt; qΦ<sub>H</sub></td><td><sup>r<sub>+</sub> - r<sub>-</sub></sup>&frasl;<sub>4πr<sub>+</sub><sup>2</sup></sub></td></tr><tr><td>Kerr-Newman (M, a, Q)</td><td><sup>a</sup>&frasl;<sub>r<sub>+</sub><sup>2</sup> + a<sup>2</sup></sub></td><td><sup>Q</sup>&frasl;<sub>r<sub>+</sub></sub></td><td>0 &lt; ω &lt; mΩ<sub>H</sub> + qΦ<sub>H</sub></td><td><sup>r<sub>+</sub> - r<sub>-</sub></sup>&frasl;<sub>4π(r<sub>+</sub><sup>2</sup> + a<sup>2</sup>)</sub></td></tr></tbody></table>

went into the black hole). However, if $\omega$ satisfies the superradiant condition, the reflected wave emerges with more energy than the incident wave[5]. This implies that the black hole has lost energy (and angular momentum or charge, depending on the process) during the scattering. Energy conservation is upheld by a corresponding decrease in the black hole's mass (and spin or charge).

Mathematically, the amplification can be derived by solving the wave equation in the black hole background and applying appropriate boundary conditions. For a scalar field $\psi(t, r, \theta, \phi) = e^{-i\omega t}e^{im\phi}R(r)S(\theta)$ in a Kerr background, the radial function $R(r)$ satisfies a Schrödinger-type equation

$$ \frac{d^2 R}{dr_*^2} + V_{\text{eff}}(r) R = 0, \qquad (3) $$

where $dr_* = \frac{r^2+a^2}{\Delta} dr$ is the "tortoise" coordinate (chosen so that the horizon is at $r_* \to -\infty$) and $V_{\text{eff}}(r)$ is an effective potential[4]. The asymptotic behavior of solutions has the form of traveling waves at the horizon and at infinity. Imposing purely ingoing waves at the horizon (physical condition that nothing classically escapes the horizon, so no outgoing wave from $r_+$, i.e., $A = 0$ in the notation of Eq. (3) of Ref. [5]) and a mixture of incoming and outgoing waves at infinity, one can define reflection and transmission coefficients. From flux conservation (related to the Wronskian of two independent solutions) one finds a relation between the coefficients. In fact, one finds (for Kerr) a relation of the form[5]

$$ |\mathcal{R}|^2 = 1 - \frac{\omega_H}{\omega} |\mathcal{T}|^2, \qquad (4) $$

where $\mathcal{R}$ and $\mathcal{T}$ are reflection and transmission amplitudes, and $\omega_\infty = \omega$ is the wave frequency at infinity while $\omega_H = \omega - m\Omega_H$ is the frequency as measured by an observer co-rotating with the horizon (due to the dragging of inertial frames). If no wave comes out of the horizon ($A = 0$ condition), then $|\mathcal{T}|^2$ represents the fraction absorbed by the black hole. The above relation then shows that if $\omega_H < 0$ (i.e., $\omega < m\Omega_H$ since typically $\omega > 0$ at infinity), one gets $|\mathcal{R}|^2 > 1$. This is precisely the superradiant amplification condition[5]. Physically, $\omega_H < 0$
means that in the horizon's rotating frame the wave has negative energy (as perceived from infinity), so the black hole can actually lose energy by letting that wave escape.

For charged black holes, an analogous calculation can be done for charged scalar perturbations. There, the horizon frequency term $\omega_H$ gets replaced by $\omega - q\Phi_H$, the energy of the mode as measured by an observer who sees the electrostatic potential at the horizon[5]. If $\omega - q\Phi_H < 0$ (with appropriate sign of charge), then the wave is amplified at the expense of the black hole's electrostatic energy.

An important aspect of superradiance is that while it causes *extraction* of energy from a black hole, it does not by itself imply any instability of the black hole. A free black hole in asymptotically flat space cannot build up a runaway growth of fields solely via superradiance, because the amplified waves escape to infinity and do not continually interact with the black hole. However, if one introduces a mechanism to *trap* the radiation, the repeated amplification can lead to an instability often termed a *black hole bomb*[3]. For example, Press and Teukolsky proposed placing a reflective mirror around the black hole; waves would bounce between the mirror and the horizon, amplifying on each pass, leading to an exponential growth of the field[3]. In realistic settings, a natural "mirror" can be provided by a massive bosonic field, whose mass term can create a potential well that traps low-frequency bound states around the black hole. If those bound state frequencies lie in the superradiant regime, they grow in time, causing an instability (the black hole spins down or loses charge while a cloud of bosons grows). This is of interest for ultralight dark matter candidates around astrophysical black holes[5], but in this paper we restrict attention to cases where either no such confining mechanism exists or the end state is a stable configuration.

Even without an instability, superradiance can modify the dynamics and "hair" (external fields) of a black hole. For instance, it has been shown that a rotating black hole can support a stationary scalar field configuration (a kind of hair) if the field's frequency locks to the black hole's rotation (the field's oscillation frequency equals $m\Omega_H$, saturating the superradiant condition)[6]. These are often called *Kerr black holes with scalar hair*[6]. Such solutions evade the usual no-hair theorem by being time-periodic rather than static. However, they typically require fine-tuned conditions (e.g., a massive scalar field with a specific spectrum). In a generic situation, one expects a hairless black hole as the end state.

Indeed, recent studies of charged black holes with scalar couplings have highlighted how superradiance tends to remove hair. If a scalar field is coupled to a charged black hole in such a way that the field wants to condense (through a tachyonic instability of the scalar, often called *spontaneous scalarization*), one might get a hairy black hole. However, if the scalar has an electric charge, any growth of a scalar condensate will trigger superradiant emission that depletes the condensate. This interplay has been termed *tachyonic superradiance*[5]. In other words, the tachyonic instability tries to add "hair" to the black hole, but the superradiant effect (amplifying the charged scalar waves and radiating them away) intervenes to keep the black hole hairless[6, 10]. We will discuss an example of this in Section 5, using recent nonlinear simulation results.

Finally, we note that superradiance is fundamentally tied to the laws of black hole thermodynamics. The energy extraction conditions (1) and (2) can be derived from the first law and second law of black hole mechanics[5]. The second law (area increase law) requires that any classical process satisfy $\delta M - \Omega_H \delta J - \Phi_H \delta Q \ge 0$ (this is essentially the statement that the horizon area cannot decrease)[5]. A wave carrying energy $E = \delta M$, angular momentum
$\delta J = -m$ (extracted from BH) or charge $\delta Q = -q$ will lead to $\delta M - \Omega_H(-m) - \Phi_H(-q) \ge 0$, i.e., $\delta M \ge m\Omega_H + q\Phi_H$. For the black hole to lose energy ($\delta M < 0$ going into the wave), we must have $\omega = E/|\delta J|$ (or appropriate combination) satisfy $\omega < m\Omega_H + q\Phi_H$. Thus the thermodynamic laws are in agreement with the scattering analysis, and they forbid $\omega$ outside that range from leading to energy extraction (in which case the wave would be absorbed and increase the BH mass/area). In the next section, we turn to Hawking radiation and see how these same parameters enter the quantum emission spectrum.

### 3 Hawking Radiation from Kerr and RN Black Holes

Let $\chi^\mu$ be the horizon-generating Killing vector; for Kerr-Newman $\chi^\mu = \partial_t^\mu + \Omega_H \partial_\phi^\mu + \Phi_H \partial_\psi^\mu$ with $\psi$ the internal $U(1)$ angle for charge.<fnref n="1" /> Surface gravity $\kappa$ is defined geometrically by

$$ \chi^\nu \nabla_\nu \chi^\mu = \kappa \chi^\mu \quad (\text{evaluated on } \mathcal{H}^+) \qquad (5) $$

For a static, spherically symmetric metric $ds^2 = -f(r) dt^2 + f^{-1}(r) dr^2 + r^2 d\Omega^2$ one finds $\kappa = \frac{1}{2} f'(r_+)$. Hence

<table><thead><tr><th>Metric</th><th>$f(r)$</th><th>$T_H = \frac{\kappa}{2\pi}$</th></tr></thead><tbody><tr><td>Schwarzschild</td><td>$1 - \frac{2M}{r}$</td><td>$\frac{1}{8\pi M}$</td></tr><tr><td>Reissner-Nordström</td><td>$1 - \frac{2M}{r} + \frac{Q^2}{r^2}$</td><td>$\frac{r_+ - r_-}{4\pi r_+^2}$</td></tr></tbody></table>

For Kerr or Kerr-Newman one uses Boyer-Lindquist coordinates; a short computation yields

$$ \kappa = \frac{r_+ - r_-}{2(r_+^2 + a^2)}, \qquad T_H^{\text{Kerr}} = \frac{r_+ - r_-}{4\pi(r_+^2 + a^2)}. \qquad (6) $$

**Euclidean derivation of the Planck factor.** Analytically continue to imaginary time $t = -i\tau$. Near the horizon set $r = r_+ + \rho^2$; the $(\tau, \rho)$ sub-metric becomes $ds^2 \approx \kappa^2 \rho^2 d\tau^2 + d\rho^2$, i.e. flat $\mathbb{R}^2$ in polar coordinates if and only if

$$ \tau \sim \tau + \beta, \qquad \beta \equiv \frac{2\pi}{\kappa}. \qquad (7) $$

Quantum fields must therefore satisfy KMS periodicity $G(\tau + \beta) = (-1)^F G(\tau)$, giving the Bose/Fermi occupation number

$$ \langle N_{\omega} \rangle = \frac{1}{e^{\beta\omega} \mp 1}, \qquad (8) $$

with – for bosons, + for fermions. For a **rotating** or **charged** hole the relevant Killing time is $\xi = t$, but the action picks up the $\phi$ and gauge shifts; Fourier expanding $\Phi \propto e^{-i\omega t + im\phi + iq\psi}$ the thermal phase becomes $e^{-\beta(\omega - m\Omega_H - q\Phi_H)}$, yielding Eq.3, Eq.5 and Eq.6 directly.

<footnote n="1">¹$\partial_\psi$ is only a formal device in Einstein-Maxwell; alternatively treat $q A_t$ as a background gauge potential.</footnote>
**First-law cross-check.** The first law $dM = T_H dS + \Omega_H dJ + \Phi_H dQ$ implies that emitting a quantum $(\omega, m, q)$ changes the BH energy by $\delta M = -\omega$ and entropy by $\delta S = -\beta(\omega - m\Omega_H - q\Phi_H)$. Detailed balance demands $\Gamma_{\text{emit}}/\Gamma_{\text{abs}} = e^{\delta S}$, again reproducing the chemical-potential form.

**Greybody pre-factor.** For $\ell=0$ massless scalars in Schwarzschild, the low-energy transmission is

$$ \Gamma_{\omega, \ell=0} \xrightarrow{\omega M \ll 1} 16(\omega M)^2 + \mathcal{O}((\omega M)^4), \qquad (9) $$

suppressing flux at small $\omega$. Inclusion of $\Gamma_{\omega\ell}$ converts the pure blackbody luminosity $L_0 = \sum_{\ell,m} \int \frac{\omega}{2\pi} n_{\text{BE}} d\omega$ into the physical $L = L_0 \times \langle\Gamma\rangle$, with $\langle\Gamma\rangle \sim 0.01$ for a Schwarzschild BH— hence the proverbial faintness of Hawking radiation.

Consider a massless scalar $\Phi$ on a Kerr background ($Q=0$). Separation of variables gives

$$ \Phi(t, r, \theta, \phi) = \sum_{\ell m} \frac{1}{\sqrt{2\pi}} \int_0^\infty \frac{d\omega}{\sqrt{2\omega}} \left[ a_{\omega\ell m} \Psi_{\omega\ell m}(r, \theta) e^{-i\omega t + im\phi} + a_{\omega\ell m}^\dagger \Psi_{\omega\ell m}^*(r, \theta) e^{+i\omega t - im\phi} \right], \quad (10) $$

with orthonormal radial/angle modes $\Psi_{\omega\ell m}$ that satisfy Teukolsky's radial equation. Near the horizon ($r \to r_+$) the tortoise coordinate $r_*$ obeys $dr_* = (r^2 + a^2)/\Delta dr$, so ingoing and outgoing waves behave as

$$ \Psi_{\omega\ell m} \xrightarrow{r \to r_+} B_{\omega\ell m}^{\text{in}} e^{-ik_H r_*} + B_{\omega\ell m}^{\text{out}} e^{+ik_H r_*}, \quad k_H \equiv \omega - m\Omega_H. \qquad (11) $$

Imposing global regularity on the past horizon forces $B_{\omega\ell m}^{\text{out}} = 0$ (only ingoing waves), so any non-zero $B^{\text{out}}$ at future horizon must be generated by quantum mixing of positive/negative frequencies.

Denote the in (Killing) vacuum by $|0_{\text{in}}\rangle$, defined by $a_{\omega\ell m}|0_{\text{in}}\rangle = 0$. For an observer at the natural annihilators are the out operators $b_{\omega\ell m}$ obtained via a Bogoliubov transformation

$$ b_{\omega\ell m} = \int_0^\infty d\omega' \left[ \alpha_{\omega\omega'} a_{\omega'\ell m} + \beta_{\omega\omega'}^* a_{\omega'\ell m}^\dagger \right]. \qquad (12) $$

Canonical commutators imply the usual identities $\int d\omega' (\alpha_{\omega\omega'} \alpha_{\tilde{\omega}\omega'}^* - \beta_{\omega\omega'} \beta_{\tilde{\omega}\omega'}^*) = \delta(\omega - \tilde{\omega})$, etc. Computing $\beta_{\omega\omega'}$ by analytic continuation across the horizon in Kruskal-like coordinates<fnref n="2" /> yields the thermal factor

$$ |\beta_{\omega\omega'}|^2 = \frac{\delta(\omega - \omega')}{e^{(\omega - m\Omega_H)/T_H} - 1}, \quad T_H = \frac{\kappa}{2\pi} = \frac{r_+ - r_-}{4\pi(r_+^2 + a^2)}. \qquad (13) $$

Hence the expectation value of the out occupation number is

$$ \langle 0_{\text{in}} | b_{\omega \ell m}^{\dagger} b_{\omega \ell m} | 0_{\text{in}} \rangle = \langle N_{\omega m} \rangle = \frac{1}{\exp[(\omega - m\Omega_H)/T_H] - 1}. \qquad (14) $$

<footnote n="2">²See, e.g., [1], or the anomaly-cancellation derivation for Kerr in [7].</footnote>
Equation (14) already contains the superradiant signature: for $\omega < m\Omega_H$ the exponent is negative and the denominator can be written as $1 - e^{-(m\Omega_H - \omega)/T_H} < 0$, so<fnref n="3" />

$$ \langle N_{\omega m} \rangle = - \frac{1}{1 - e^{-(m\Omega_H - \omega)/T_H}} \equiv + \frac{1}{e^{(\omega - m\Omega_H)/T_H} - 1}, \quad (15) $$

i.e. spontaneous production dominates absorption, precisely the regime that classical scattering identifies as superradiant:

$$ \omega < m\Omega_H \iff \text{superradiant amplification.} \quad (16) $$

**Grey-body factors and detailed balance.** Let $A_{\ell m}(\omega)$ be the absorption probability of the Kerr potential barrier. The net flux at infinity is

$$ \frac{dE}{dt d\omega dm} = \frac{\omega A_{\ell m}(\omega)}{2\pi[\exp((\omega - m\Omega_H)/T_H) - 1]}. \quad (17) $$

Using $R = 1 - A$ for the reflection coefficient and the conservation identity $|R|^2 - 1 = -\frac{\omega - m\Omega_H}{\omega} |T|^2$, one verifies that

$$ A_{\ell m}(\omega) < 0 \quad \text{iff} \quad \omega < m\Omega_H, \quad (18) $$

so $A < 0$ exactly in the superradiant window; the negative grey-body factor corresponds to a net outgoing (classically amplified) flux, perfectly matching the sign change in (14). Thus Hawking emission for $\omega < m\Omega_H$ is not mere thermal leakage: it is spontaneous superradiance, and—were a reflecting mirror present—these modes would lase, growing like $e^{\Gamma t}$ with $\Gamma \propto (m\Omega_H - \omega)$, the classic “black-hole bomb” of Press–Teukolsky.

For a charged black hole (Reissner-Nordström), Hawking radiation tends to carry away charge as well as energy. The Hawking spectrum for charged particle modes has the factor $\exp[(\omega - q\Phi_H)/T_H]$. If the black hole has charge $Q$, it will preferentially emit particles of the same sign as $Q$ (to reduce $Q$) because those particles have $q\Phi_H > 0$ and hence a lower exponent for opposite charge (for example, an electron ($q = -e$) is more likely to be emitted from a positively charged black hole). As the black hole’s charge approaches zero, the chemical potential $\Phi_H$ vanishes and the spectrum becomes purely thermal in energy. Throughout, modes satisfying $\omega < q\Phi_H$ (with the appropriate charge sign) are again in the superradiant regime, appearing as an instability of the thermal distribution. Indeed, an extremal charged black hole (with $T_H = 0$) is believed to have zero Hawking emission, but still can dissipate charge via the Schwinger mechanism or vacuum polarization, which is another story.

One elegant method to derive Hawking radiation in rotating/charged black holes is the anomaly cancellation method [7]. In this approach, one considers the near-horizon region of the black hole and dimensional reduction to an effective $1 + 1$ dimensional theory for the

<footnote n="3">³The negative sign signals that the standard Bose-Einstein counting is inadequate: one must reinterpret occupation numbers in terms of stimulated emission or a change of vacuum states. Equivalently, add $2\pi i$ to the contour in the Bogoliubov integral, giving the correct positive particle flux.</footnote>
radial coordinate and time[7]. For Kerr, this was first done by Murata and Soda (2006) using gravitational anomaly cancellation[7]. The idea is that if one truncates the physics outside the horizon in an inappropriate way, the effective theory becomes anomalous (violating energy-momentum or charge conservation), and the requirement that the full theory is anomaly-free forces the existence of a flux of radiation from the horizon equal to the Hawking flux.

In practice, one can start from the action for a scalar field in the Kerr geometry and expand it near the horizon. In Boyer-Lindquist coordinates, the Kerr metric is[4]:

$$ds^2 = - \frac{\Delta - a^2 \sin^2 \theta}{\Sigma} dt^2 - \frac{2a \sin^2 \theta (r^2 + a^2 - \Delta)}{\Sigma} dt d\phi + \frac{(r^2 + a^2)^2 - \Delta a^2 \sin^2 \theta}{\Sigma} \sin^2 \theta d\phi^2 + \frac{\Sigma}{\Delta} dr^2 + \Sigma d\theta^2, \quad (19)$$

with $\Sigma = r^2 + a^2 \cos^2 \theta$ and $\Delta = r^2 - 2Mr + a^2 = (r - r_+)(r - r_-)$[4]. The scalar field action $S[\phi] = \frac{1}{2} \int d^4x \sqrt{-g} \phi \Box \phi$ can be written out explicitly in these coordinates[7]. Near the horizon ($r \to r_+$), one can simplify this action by expanding $\Delta$ around $r_+$ and keeping only the dominant terms. Essentially, because $\Delta(r_+) = 0$, to leading order in $(r - r_+)$ the metric coefficients simplify and the angular dependence decouples. One finds that an s-wave ($l = 0$) mode of the scalar field, or more rigorously the m-mode in a co-rotating frame, experiences an effective 2D metric. By transforming to the horizon co-rotating frame via

$$\psi = \phi - \Omega_H t, \quad \xi = t, \quad (20)$$

(which means we rotate the coordinate system at angular velocity $\Omega_H$ so that the horizon appears static)[7], the cross-terms in the metric can be removed. The near-horizon physics then reduces to that of a $(1+1)$-dimensional spacetime with coordinates $(\xi, r)$ and metric

$$ds^2 \approx -\frac{1}{f(r)} d\xi^2 + f(r) dr^2, \quad (21)$$

where $f(r)$ is a function proportional to $(r - r_+)$ (for $r$ near $r_+$)[7]. This form $-f^{-1}d\xi^2 + f dr^2$ is recognized as a Rindler-like metric with horizon at $r = r_+$; it is effectively the metric of a 2D black hole with horizon and surface gravity $\kappa$. The angular degrees of freedom now only contribute through overall constants or through gauge-like degrees of freedom (the $\psi$ coordinate acts like an ignorable coordinate).

The field theory in this 2D spacetime is chiral (since one has to impose that modes going into the horizon are absent or integrated out, leaving only outgoing modes in the near-horizon region). This chiral reduction violates general covariance, leading to a gravitational anomaly. Requiring cancellation of that anomaly by adding a flux at infinity reproduces the Hawking radiation flux[7]. In simpler terms, the symmetry that is broken if no Hawking radiation is present is the invariance under translations in time (energy conservation) and gauge transformations (charge conservation for charged black holes). The quantum effective action exhibits an anomaly unless a compensating flux of energy (and charge) is included. The value of that flux that cancels the anomaly is exactly the thermal Hawking flux. This method provides an alternative derivation of Hawking radiation that is very elegant and uses only symmetry arguments.
Near the horizon the Kerr metric reduces, after the corotating shift (20), to

$$ds_{(2)}^2 = -f(r) d\xi^2 + f^{-1}(r) dr^2, \quad f(r) = \kappa(r - r_+) + \mathcal{O}((r - r_+)^2), \quad (22)$$

and a background gauge field

$$\mathcal{A}_{\mu} = (\mathcal{A}_{t}(r), 0), \quad \mathcal{A}_{t}(r) = -m\Omega_{H} - q\Phi_{H} + \mathcal{O}(r - r_{+}), \quad (23)$$

where $m$ is the azimuthal quantum number and $q$ the particle charge. Classically we discard ingoing modes at the horizon, leaving a *chiral* (right-moving) theory outside $r_+$. Chiral theories in 2D possess quantum anomalies:

$$\nabla_{\mu} J^{\mu} = \frac{q^2}{4\pi} \epsilon^{\mu\nu} F_{\mu\nu} \equiv \frac{q^2}{2\pi\sqrt{-g}} \partial_{\nu} \mathcal{A}_{t}, \quad (24)$$

$$\nabla_{\mu} T^{\mu}_{\nu} = F_{\nu\mu} J^{\mu} + \frac{1}{96\pi} \epsilon^{\mu\rho} \partial_{\rho} \partial_{\nu} R, \quad (25)$$

with $F_{tr} = \partial_{r} \mathcal{A}_{t}$ and Ricci scalar $R = -f''(r)$. Split the exterior into two regions:

(i) $r \in (r_+ + 0, r_+ + \varepsilon)$ (anomalous, chiral),

(ii) $r \in (r_+ + \varepsilon, \infty)$ (regular, vector-like),

and denote currents by $J^{\mu}_{(i)}$, $T^{\mu}_{\nu(i)}$. **Gauge sector.** Integrating (24) in region (i) and imposing the *covariant* boundary condition $J^{r}_{(i)}|_{r=r_+} = 0$ (no net flux *into* the horizon) gives

$$J_{(i)}^{r}(r) = \frac{q^2}{2\pi} \int_{r_+}^{r} dr' \partial_{r'} \mathcal{A}_{t} = \frac{q^2}{2\pi} [\mathcal{A}_{t}(r) - \mathcal{A}_{t}(r_+)]. \quad (26)$$

In region (ii) gauge symmetry is restored, so $\partial_{r} J^{r}_{(ii)} = 0$ and $J^{r}_{(ii)} = \text{const} \equiv J^{r}_{\text{Hawking}}$. Requiring continuity at $r = r_+ + \varepsilon$ and taking $\varepsilon \to 0$ yields<fnref n="4" />

$$J_{\text{Hawking}}^{r} = -\frac{q^2}{2\pi} \mathcal{A}_{t}(r_{+}) = \frac{q^2}{2\pi} (m\Omega_{H} + q\Phi_{H}) \quad \Rightarrow \quad J_{\text{Hawking}} = \frac{1}{2\pi} \Phi_{H} T_{H}, \quad (27)$$

since $q^2$ in natural units is absorbed by the charge normalization and $\Phi_H = Q/r_+$, $T_H = \kappa/2\pi$.

**Gravitational sector.** Write $T^r_t$ for the mixed component (energy flux). From (25) in region (i) one obtains

$$\partial_r T^r_{t(i)} = \partial_r (\mathcal{A}_t J^r_{(i)}) + \frac{1}{96\pi} \partial_r (f f'' - (f')^2/2). \quad (28)$$

<footnote n="4">⁴Grey-body factors are set to 1 for a pure near-horizon derivation; to restore them multiply by the transmission coefficient $\mathcal{A}_{\ell m}(\omega)$.</footnote>
Integrate from $r_+$ to $r$ and set $T^r_{t(i)}|_{r=r_+} = 0$ (covariant regularity at the horizon):

$$T^r_{t(i)}(r) = \int_{r_+}^{r} \left[ \mathcal{A}_t \partial_{r'} J^r_{(i)} + \frac{1}{96\pi} \partial_{r'} (f f'' - \frac{1}{2}(f')^2) \right] dr' \quad (29)$$

$$= \frac{q^2}{4\pi} [\mathcal{A}_t(r) - \mathcal{A}_t(r_+)]^2 + \frac{\pi}{12} [T(r)^2 - T_H^2], \quad T(r) \equiv \frac{f'(r)}{4\pi}. \quad (30)$$

Region (ii) is anomaly-free, so $\partial_r T^r_{t(ii)} = 0$ and $T^r_{t(ii)} = F_{\text{Hawking}}$. Match at $r = r_+ + \epsilon$ and send $\epsilon \to 0$. The terms depending on $\mathcal{A}_t(r)$ and $T(r)$ vanish at the horizon, leaving

$$F_{\text{Hawking}} = \frac{\pi}{12} T_H^2 + \frac{q^2}{4\pi} \mathcal{A}_t^2(r_+) = \frac{\pi}{12} T_H^2 + \frac{1}{4\pi} (m\Omega_H + q\Phi_H)^2. \quad (31)$$

For an *uncharged* scalar ($q=0$) this reduces to the standard 2D blackbody flux $\pi T_H^2/12$, while for a neutral mode with $m=0$ but $q \neq 0$ we recover the pure charge-driven flux $F_{\text{Hawking}} = \pi T_H^2/12 + q^2 \Phi_H^2/4\pi$.

Thus cancellation of gauge and gravitational anomalies fixes both the Hawking energy and charge fluxes uniquely in terms of the same horizon parameters ($T_H, \Omega_H, \Phi_H$) that govern classical superradiance. Any reflection of these fluxes by a mirror would seed an exponential superradiant instability, cementing the Hawking–superradiance correspondence.

# 4 Connecting Hawking Radiation to Superradiance

We are now in a position to draw a direct connection between Hawking radiation and super-radiance. The link can be understood at both the quantitative level (through the formulas) and the qualitative level (through physical interpretation).

Quantitatively, compare the Hawking emission condition to the superradiant amplification condition. A mode labeled by $(\omega, m, q)$ that is populated in Hawking radiation has the thermal factor given in Eq.6. The exponent in that distribution, $(\omega - m\Omega_H - q\Phi_H)/T_H$, appears also in the first law of thermodynamics for the black hole. When $\omega < m\Omega_H + q\Phi_H$, the exponent is negative. For bosonic fields, the Bose-Einstein distribution diverges as the exponent approaches zero from below, indicating a greatly enhanced occupation number (which can be interpreted as stimulated or spontaneous emission). In fact, one can derive the superradiant scattering condition from the requirement that the net absorption probability for mode $(\omega, m, q)$ is negative[5]. The detailed balance condition in thermodynamics relates the spontaneous emission, stimulated emission, and absorption probabilities. Hawking's derivation ensures that the ratio of spontaneous emission to absorption for each mode satisfies the detailed balance appropriate to a thermal $T_H$. However, if one formally treats the superradiant mode $(\omega < m\Omega_H + q\Phi_H)$ in the same way, one finds that the absorption probability would have to be negative to satisfy the naive detailed balance formula. A negative absorption probability is unphysical classically, and the resolution is that those modes actually correspond to amplification (emission) – i.e., the black hole stimulates its own emission for those modes even without an external wave. In essence, Hawking radiation provides the spontaneous trigger for superradiant modes: the vacuum fluctuation that leads to Hawking
emission can be thought of as a "seed" incident wave, which the black hole then amplifies if the mode is superradiant.

From another perspective, one can analyze wave modes of a Kerr black hole in terms of the Kerr horizon's co-rotating frame. As we did in Section 2, an ingoing wave at the horizon behaves as $e^{-i\omega t+im\phi}$ which in the co-rotating frame ($\psi = \phi-\Omega_H t$) looks like $e^{-i(\omega-m\Omega_H)t+im\psi}$. If $\omega < m\Omega_H$, then $\omega - m\Omega_H$ is negative. But in quantum field theory, negative frequencies correspond to annihilation operators (or creation of antiparticles). Thus, what an outside observer sees as a particle being emitted ($\omega > 0$ at infinity) can correspond to a mode of negative Killing frequency at the horizon, which in turn corresponds to the vacuum state giving up energy to create particles. This is exactly how Hawking radiation arises in one heuristic picture: a mode just inside the horizon has negative energy (with respect to infinity) and is absorbed by the black hole, while the partner particle escapes with positive energy. Superradiant scattering similarly relies on the existence of negative-energy states inside the ergosphere (for rotating holes) or inside the horizon (for charged holes) that allow the black hole to give up energy.

Define, for fixed $(\ell, m, q)$, the *partial* absorption probability

$$ \Gamma_{\omega} \equiv 1 - |\mathcal{R}_{\omega}|^2, \qquad (32) $$

where $\mathcal{R}_{\omega}$ is the classical reflection amplitude of the Regge-Wheeler/Teukolsky potential.<fnref n="5" /> Let $\mathcal{A}$ (absorption), $\mathcal{E}_{\text{sp}}$ (spontaneous emission) and $\mathcal{E}_{\text{st}}$ (stimulated emission) be the probability per unit time for those processes. For bosons, detailed balance in a stationary state requires

$$ \mathcal{A} = \Gamma_{\omega} (1 + \langle N_{\omega} \rangle), \qquad \mathcal{E}_{\text{st}} = \Gamma_{\omega} \langle N_{\omega} \rangle, \qquad \mathcal{E}_{\text{sp}} = -\Gamma_{\omega} \theta(-\Gamma_{\omega}). \qquad (33) $$

The $\theta$-function encodes that spontaneous emission is present *only* when $\Gamma_{\omega}$ is negative (i.e. in the superradiant regime); when $\Gamma_{\omega} > 0$ one sets $\mathcal{E}_{\text{sp}} = 0$.<fnref n="6" /> The net flux seen at infinity is then

$$ \mathcal{F}_{\omega} = \mathcal{E}_{\text{sp}} + \mathcal{E}_{\text{st}} - \mathcal{A} = \Gamma_{\omega} \langle N_{\omega} \rangle - \Gamma_{\omega} = \Gamma_{\omega} \frac{e^{(\omega - m\Omega_H - q\Phi_H)/T_H} - 1}{e^{(\omega - m\Omega_H - q\Phi_H)/T_H} - 1}. \qquad (34) $$

*Insert Hawking's spectrum.* Using the Bose-Einstein factor

$$ \langle N_{\omega} \rangle = \frac{1}{e^{(\omega - m\Omega_H - q\Phi_H)/T_H} - 1}, \qquad (35) $$

Eq. (34) simplifies to

$$ \mathcal{F}_{\omega} = -\Gamma_{\omega} \operatorname{sgn}(\omega - m\Omega_{H} - q\Phi_{H}). \qquad (36) $$

But recall that $\Gamma_{\omega} = 1 - |\mathcal{R}_{\omega}|^2$, so

$$ |\mathcal{R}_{\omega}|^2 = 1 - \Gamma_{\omega} = 1 + \mathcal{F}_{\omega} \operatorname{sgn}(\omega - m\Omega_{H} - q\Phi_{H}). \qquad (37) $$

<footnote n="5">⁵We suppress $\ell, m, q$ on $\Gamma$ and $\mathcal{R}$ for brevity. The transmission coefficient $|\mathcal{T}_{\omega}|^2$ is the horizon flux, and flux conservation gives $\Gamma_{\omega} = |\mathcal{T}_{\omega}|^2 - (\frac{\omega - m\Omega_H - q\Phi_H}{\omega})|\mathcal{T}_{\omega}|^2$. Hence outside the superradiant band $\Gamma \ge 0$, and inside it $\Gamma < 0$.</footnote>

<footnote n="6">⁶Equivalently, spontaneous emission is always present but is exponentially suppressed by $e^{-(\omega - m\Omega_H - q\Phi_H)/T_H}$ outside the superradiant range; Eq. (33) encodes that in compact form.</footnote>
Since $\mathcal{F}_{\omega} \ge 0$ for any physical state, Eq. (37) yields

$$ |\mathcal{R}_{\omega}|^2 \begin{cases} < 1, & \omega > m\Omega_H + q\Phi_H \quad (\text{net absorption}), \\ = 1, & \omega = m\Omega_H + q\Phi_H \quad (\text{zero-net flux}), \\ > 1, & \omega < m\Omega_H + q\Phi_H \quad (\text{superradiant emission}). \end{cases} \quad (38) $$

Thus the Hawking occupation number forces the classical reflection coefficient to exceed unity precisely in the superradiant band, reproducing the superradiance condition $\omega < m\Omega_H + q\Phi_H$ from pure thermodynamic consistency.

**Power spectrum.** Substituting (36) into the luminosity integral gives

$$ \mathcal{P} = \sum_{\ell m q} \int_0^\infty \frac{\omega \Gamma_\omega d\omega}{2\pi} \left[ n_{\text{BE}}(\omega) - \theta(-\Gamma_\omega) \right], \quad (39) $$

where $n_{\text{BE}} = 1/(e^{(\omega-m\Omega_H-q\Phi_H)/T_H} - 1)$. The second term subtracts the would-be negative absorption in the superradiant range and replaces it with spontaneous emission, ensuring a positive net luminosity even when $\Gamma_\omega < 0$. Equation (38) is exactly the classical limit of that statement, closing the Hawking-superradiance circle.

One illustrative scenario is to imagine turning off Hawking radiation (by somehow forcing the black hole to be in a quantum vacuum state with respect to the Killing time that has no outgoing particles at infinity). In that case, one would still have the possibility of superradiant scattering for any wave that comes along, but no spontaneous emission. However, that situation is unstable: as Unruh and others have pointed out, the absence of Hawking radiation is inconsistent with a stable vacuum state at the horizon. Any small perturbation or vacuum fluctuation can induce the black hole to radiate. Hence, a self-consistent treatment yields Hawking radiation as a necessary consequence of maintaining a steady state without superradiant runaway.

It is worth noting a subtle difference between Hawking radiation and classical superradiance: Hawking radiation emits all frequencies (with a thermal weighting), whereas classical superradiance only amplifies a bounded low-frequency range. However, the emission at frequencies that do not satisfy the superradiant condition is strongly suppressed by the Boltzmann factor in Hawking radiation if the black hole is rotating or charged. For example, a Kerr black hole’s emission of modes with $\omega \gg m\Omega_H$ will be exponentially small if $\omega \gg T_H$ as well. In the case of an extreme Kerr black hole (maximally rotating), $T_H \to 0$ while $\Omega_H$ remains finite, so spontaneous Hawking radiation vanishes, yet one still expects superradiant scattering to be possible for $\omega < m\Omega_H$. This suggests that as $T_H \to 0$, the Hawking mechanism shuts off, leaving only the classical superradiance (which in the absence of Hawking trigger would require some external perturbation). This is consistent with the third law of black hole thermodynamics (extremal black holes have zero entropy and temperature and cannot radiate thermally) but still allows superradiant extraction if a wave is provided.

In summary, Hawking radiation can be viewed as *superradiance of the vacuum*: the black hole’s internal degrees of freedom effectively send out waves of all possible frequencies, but only those modes that meet the superradiant criteria can emerge strongly (others mostly fall
back and get reabsorbed). The overlap of quantum particle creation with classical mode amplification is a rich area where thermodynamics, quantum field theory and general relativity intersect.

# 5 Stability and Hair: Thermodynamic Geometry and Tachyonic Superradiance

The connection between Hawking radiation and superradiance also sheds light on the stability of black holes and the possible existence (or absence) of additional "hair." Since both processes extract energy and angular momentum/charge from a black hole, they can act to prevent the indefinite build-up of fields around the black hole. We discuss two contexts: the stability of Kerr black holes against superradiant perturbations, and the fate of charged black holes undergoing scalarization.

## 5.1 Superradiant Stability of Kerr Black Holes and Thermodynamic Geometry

A Kerr black hole, in vacuum general relativity, is known to be linearly stable against small perturbations. Superradiant scattering of waves does not by itself cause an instability unless there is a mechanism to confine the waves (like a mirror or mass term as discussed earlier). However, if one introduces new fields or considers the black hole in a cavity, superradiant instabilities can occur (e.g., the black hole bomb scenario). A natural question is: for a given black hole, what are the conditions that determine whether a superradiant instability will occur when a certain field is present?

Chen [9] approached this question by looking at black hole thermodynamic geometry in the context of Kerr black holes. The idea is to use the geometry of the thermodynamic state space (for example, the Ruppeiner metric derived from the entropy $S(M, J)$) to identify phase transition points. Kerr black holes exhibit a famous thermodynamic peculiarity: the heat capacity can become negative or positive depending on the spin; there is a so-called Davies point where a heat capacity divergence occurs, signaling a second-order phase transition in the thermodynamic sense. Chen noted that these phase transition points correspond to the emergence of potential wells in the effective potential for perturbations[9]. In particular, Chen found that the "Davis transition" (Davies point, a divergence in heat capacity, sometimes called a first-order phase transition in older literature) is analogous to the appearance of a potential barrier outside the horizon that can support quasi-bound states[9]. Meanwhile, the divergence of the Ruppeiner curvature scalar (a second-order phase transition in thermodynamic geometry) corresponds to another condition where the potential develops an instability (turning the barrier into a well)[9]. When these two points do not coincide, there is an interval of parameters for which the black hole is thermodynamically stable and, according to Chen's argument, superradiantly stable (no growing modes)[9].

In simpler terms: for Kerr-AdS black holes or Kerr black holes in a finite cavity, the phase structure can indicate where superradiant instabilities set in. In asymptotically flat Kerr, a massive boson field of mass $\mu$ can only grow (via superradiance) if $\omega \approx \mu < m\Omega_H$,
and the condition for existence of a bound state turns out to require $M\mu \lesssim \mathcal{O}(1)$ in units $G = c = \hbar = 1$. There is a critical spin above which an instability occurs for given $\mu$[5]. This can be mapped to the idea of a phase transition. Chen's approach provides a novel criterion: if the black hole's parameters are such that between the Davies point and the Ruppeiner $R$ divergence there is a finite window, that window is the superradiant stability region[9]. Outside of it, one might expect an instability. Although this is a heuristic argument, it bridges thermodynamics with dynamics.

Ultimately, an isolated Kerr black hole in vacuum has no classical mechanism to spin itself down via superradiance (because that would require a trapped wave or an ambient field). Quantum-mechanically, Hawking radiation provides a mechanism: it will cause the Kerr hole to lose mass and angular momentum slowly, tending towards a non-rotating state (since higher $J/M$ states radiate faster due to superradiant modes being more strongly produced). This suggests that the end-state of evaporation is likely a slow approach to zero mass (with $J/M$ decreasing). If superradiant instabilities were to occur for certain parameter ranges (say a new particle with certain mass coupling to Kerr), Hawking radiation might mitigate them by removing energy before an instability grows significantly. This remains a topic of ongoing research, linking back to ideas of "Hawking radiation as superradiance" — the quantum effect might save a rapidly spinning black hole from unbounded classical instability by preemptively shedding the spin.

## 5.2 Hairless Black Holes via Tachyonic Superradiance

An exciting development in recent years is the exploration of black hole scalarization, where a scalar field develops a nonzero configuration (hair) around a black hole due to a tachyonic instability (often caused by coupling to curvature or to a gauge field). In particular, in Einstein-Maxwell-scalar (EMS) models, a scalar field $\Phi$ nonminimally coupled to the electromagnetic field can exhibit spontaneous growth around a charged black hole when the charge-to-mass ratio $Q/M$ of the hole exceeds a critical value[5]. This is a phase transition from a Reissner-Nordström (hairless) solution to a hairy black hole solution. If the scalar field is uncharged, one can indeed obtain a new static hairy black hole as the end state (the scalar condensate persists, giving the black hole a "scalar hair"). This is analogous to the cases studied in scalar-Gauss-Bonnet theories or axionic clouds around Kerr, where new solutions with hair exist.

However, if the scalar field carries an electric charge $q$, the story changes dramatically. Now the scalar field perturbations can undergo superradiant amplification (since the black hole has charge, and condition (2) applies). The combination of the two effects — the tachyonic growth (trying to form hair) and superradiant emission (trying to radiate away the scalar) — leads to a process dubbed *tachyonic superradiance* [5]. Fully nonlinear simulations by Garcia-Saenz et al. (2025) [10] have shown how this unfolds: initially, the charged scalar field (with appropriate nonminimal coupling making it tachyonically unstable) begins to grow as a "cloud" around the charged black hole (just as it would if it were neutral)[10]. The black hole's mass and charge begin to transfer into the scalar cloud. However, once the scalar cloud is sufficiently large, it can radiate away to infinity through the superradiant mechanism (amplifying its own outgoing waves at the expense of the black hole's energy and charge)[10]. The result observed is that the scalar condensate does not settle into a
form hair, but if it can escape (massless or charged field with superradiance), it likely will, rather than settling as static hair. This viewpoint ties the no-hair principle to the combination of these radiation mechanisms.

# 6 Conclusion

In this work, we have presented a comprehensive analysis of the relationship between Hawking radiation and superradiance in the context of hairless black holes, focusing on Kerr and Reissner-Nordström spacetimes. Our derivations and discussions lead to several noteworthy conclusions:

* **Hawking radiation as spontaneous superradiance:** By examining the near-horizon field equations and the form of the Hawking emission spectrum, we demonstrated that the conditions for superradiant wave amplification (e.g., $\omega < m\Omega_H$ for Kerr, $\omega < q\Phi_H$ for RN) appear naturally in the Hawking radiation formula[5]. Hawking radiation can be interpreted as a black hole spontaneously emitting those modes that would classically be amplified. In this sense, Hawking radiation is essentially a *quantum version of superradiance*, where vacuum fluctuations provide the initial perturbation. Our analysis supports Wen-Xiang Chen’s assertion that "Hawking radiation is probably a type of superradiation"[8, 9]. The mathematical connection was made explicit by reducing the 4D Kerr problem to a 2D near-horizon theory and showing how the angular momentum term $m\Omega_H$ acts like a chemical potential in the thermal distribution[7, 5].

* **Energy extraction and stability:** Both Hawking radiation and superradiance extract energy (and angular momentum or charge) from the black hole. We highlighted that these processes are complementary: Hawking radiation is always present (but usually small for large black holes), while superradiance requires an incident wave or an instability setup. If an instability (like a confining potential for a boson field) is present, superradiance can lead to exponential growth of a field (black hole bomb or scalar cloud). However, Hawking radiation tends to carry away energy in a steady, diffusive manner. In combination, a rapidly spinning black hole is likely to shed angular momentum through superradiant instabilities if a light bosonic field exists, or through Hawking emission (albeit slowly) if no classical instability is available. Using thermodynamic geometry, we saw that the parameter regime for superradiant stability of Kerr black holes corresponds to regions between thermodynamic phase transition points[9], suggesting a deep link between thermodynamic stability and classical dynamical stability.

* **No-hair enforced by superradiance:** We discussed scenarios in which black holes might develop "hair" (nontrivial fields outside the horizon). The case of a charged black hole in an Einstein-Maxwell-scalar model was examined, where a scalar field coupling can cause scalar hair to try to form. Recent nonlinear simulations show that if the scalar field is charged, the black hole does not end up with hair; instead, the scalar is radiated away to infinity by a combination of tachyonic growth and superradiant scattering[10].
This "tachyonic superradiance" efficiently extracts energy and leaves a smaller, bald black hole[10]. This example underscores a general principle: superradiance is a potent mechanism to prevent or remove hair on black holes, thereby reinforcing the no-hair principle in theories where it might otherwise be violated. Hairless black holes are attractors of the evolution, with superradiant and Hawking emissions acting as the channels to return the system to a bald state.

* **Methodological insights:** Our work employed the method of substituting physical parameters (like Ω<sub>H</sub> and Φ<sub>H</sub>) into key equations, such as the wave equation and the thermal distribution, to connect the dots between classical and quantum results. We explicitly reduced the Kerr field theory to an effective 2D theory to illustrate the anomaly cancellation derivation of Hawking flux[7]. We presented a comparative table (Table 1) that collects the important formulas for horizon parameters, superradiant conditions, and Hawking temperatures for Schwarzschild, Kerr, RN, and Kerr-Newman black holes, as a handy reference. This makes clear the parallel roles of Ω<sub>H</sub> and Φ<sub>H</sub> in governing both superradiance and Hawking radiation.

* **Implications for observations and future research:** While Hawking radiation is extremely weak for astrophysical black holes (making it nearly impossible to detect directly), superradiance could have observational consequences, such as spinning black holes generating gravitational waves or visible signals if ultralight fields exist. Our study implies that any such superradiant spin-down of black holes is fundamentally the same physics as Hawking emission (just in a different regime), hinting at a unity between quantum and classical dissipation processes of black holes. Future research may further explore this unity—for instance, investigating Hawking radiation in rotating black holes as a stimulated emission process in analogy with lasers, or using analogue gravity systems (like fluid vortices or optical analogues) to simulate Hawking-superradiance interplay. On the theoretical side, understanding this relationship better might shed light on the information loss problem: since superradiance is a unitary (classical wave) process and Hawking radiation is non-unitary (quantum particle creation), the linkage we discussed might offer clues to reconciling these pictures under a full quantum gravity treatment.

In conclusion, the study of Hawking radiation and superradiance together provides a more complete picture of how black holes interact with fields and shed their energy. The once clear line between "thermal quantum radiation" and "classical wave amplification" becomes blurred when we recognize that they are different manifestations of the same underlying physics. By examining hairless black holes through this dual lens, we affirm that a black hole's geometry and horizon conditions dictate both its quantum glow and its classical echoes, ensuring that in the end the black hole cannot hold onto extra hair but instead radiates it away, one way or another.

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