# Dyon Production from Near-Extremal Kerr-Newman-(Anti-)de Sitter Black Holes

Chiang-Mei Chen<EMAIL_ADDRESS>Department of Physics, National Central
University, Chungli 32001, Taiwan  Center for High Energy and High Field
Physics (CHiP), National Central University, Chungli 32001, Taiwan  Sang Pyo
Kim<EMAIL_ADDRESS>Department of Physics, Kunsan National University,
Kunsan 54150, Korea

###### Abstract

The near-horizon region of the near-extremal dyonic Kerr-Newman (KN) black
hole in the (A)dS space has a warped AdS geometry, which leads to the exact
solutions of charges in terms of the known function. We explicitly compute the
Schwinger effect for the emission of electric and/or magnetic charges from the
near-extremal KN black hole in the (A)dS space. The emission formula confirms
a universal factorization of the Schwinger formula in the AdS2 and another
Schwinger formula in the two-dimensional Rindler space determined by the
effective temperature and the Hawking temperature with the chemical potentials
of electric and/or magnetic charges and the angular momentum. The emission of
the same species of charges from the KN black hole is enhanced in the AdS
boundary while it is suppressed in the dS boundary. In addition, the dragging
of particles in the KN black hole diminishes the emission of charges in both
AdS and dS spaces. The AdS geometry of near-horizon gives the Breitenloher-
Freedman (BF) bound, within which the stability of dyonic KN black holes is
guaranteed against both the emission of charges and Hawking radiation.

###### pacs:

04.62.+v, 04.70.Dy, 04.50.Gh, 04.40.Nr

##  I Introduction

The Einstein-Maxwell theory has Reissner-Nordström (RN) black holes and Kerr-
Newnman (KN) black holes in the asymptotically flat spacetime. The RN and KN
black hole solutions are also known in the (anti-)de Sitter space (for review
and references, see Griffiths:2012 ). The RN and KN black holes have, in
addition to the mass and angular momentum, one more hair (parameter), the
charge of the hole, and show a richer structure than Schwarzschild and Kerr
black holes. The RN black hole, for instance, has two horizons: the event
horizon and the Cauchy (inner) horizon. When two horizons coincide and form a
degenerate horizon, the RN black hole becomes an extremal black hole and is
described by an AdS geometry near the horizon Kunduri:2008rs . The KN black
hole has two additional hairs (parameters), the angular momentum as well as
the charge. The RN and KN black hole in the (A)dS space has an extra
parameter, the radius of the (A)dS space. Hence, the RN and KN black holes in
the (A)dS space exhibit much richer structure than those in the asymptotically
flat space.

Hawking radiation exhibits part of quantum aspects of black holes
Hawking:1974sw . A puzzling property of Hawking radiation is that the
vanishingly small Hawking temperature for near-extremal black holes
exponentially suppresses Hawking radiation. The black hole thermodynamics is
not apparently obvious for extremal black holes. In the case of charged black
holes in an asymptotically flat space, the electric field on the horizon is
strong enough to discharge the hole’s charge via the Schwinger mechanism
Sauter:1932ab ; Schwinger:1951nm  and in the (A)dS space Belgiorno:2007va ;
Belgiorno:2008mx . Further, the quantum field of a massive charge in the near-
extremal black holes can be solved in terms of known functions, and the
Schwinger emission of charges has been explicitly found for the RN and KN
black holes in the asymptotically flat space Chen:2012zn ; Chen:2014yfa ;
Chen:2016caa ; Chen:2017mnm . Recently, the Schwinger emission from
(near-)extremal RN black holes also was found in the (A)dS space Chen:2020mqs
; Cai:2020trh ; Zhang:2020apg . The AdS boundary compresses the horizon
radius, on which the electric field becomes stronger than that in the
asymptotically flat space. Hence, the Schwinger effect for a fixed mass is
enhanced in the AdS space. In contrast, the dS boundary stretches the horizon
radius and the weakened electric field suppresses the Schwinger effect.

The rotation of near-extremal KN black hole affects the Schwinger effect via
two opposite factors: decreasing the “inertial mass” of charge due to the
spacetime dragging but increasing the horizon radius. Hence, the rotation of
the black hole lowers the Unruh temperature for the charge acceleration on the
horizon and thereby the effective temperature for the leading Boltzmann factor
for the Schwinger effect Chen:2016caa ; Chen:2017mnm . In fact, the spacetime
dragging around the event horizon affects the radial equation for the charges
reducing the “inertial mass”.

In this paper, we study the Schwinger effect from the near-extremal KN black
holes in the (A)dS space. This model is physically interesting since the
rotation of the black hole reduces the Schwinger effect while the AdS (dS)
boundary increases (decreases) the Schwinger effect. Thus, the rotation
cooperates the suppression in the dS boundary but competes for the enhancement
with the AdS boundary. We show that the Unruh temperature for charge
acceleration and the effective temperature for Schwinger effect are higher in
the AdS space than in the asymptotically flat space while they are lower in
the dS space than in the flat space. We then extend the Schwinger effect to
the emission of dyons with electric and magnetic charges and find a universal
formula for the mean number of produced dyons from the near-extremal KN black
holes in the (A)dS space.

The organization of this paper is as follows. In Sec. II, we study the near-
horizon geometry of near-extremal KN black holes in the (A)dS space. We
parameterize the degenerate horizon and charges by the mass, angular momentum
and the radius of the (A)dS space, and also express the degenerate horizon and
the mass in terms of the charges, angular momentum and the radius of the (A)dS
space. In Sec. III, we analyze the quantum field for dyons in the near-horizon
geometry and find the mean number of produced dyons for the Schwinger effect.
We numerically study the effect of the angular momentum and the radius of the
(A)dS space on the horizon radius, inertial mass, effective temperature as
well as the effective temperature. The BF bound is found for the KN black
holes in the AdS space. In Sec. IV, we discuss the effect of the rotation and
the (A)dS boundary on the Schwinger effect for emission of charges, electric
and/or magnetic charges from the KN black holes.

##  II Extremal dyonic KN Black Holes in (A)dS Space

In this section we express the near-horizon geometry of near-extremal KN black
holes in the (A)dS space. The KN-(A)dS black holes are the most general
stationary solutions, as shown in Fig. 1, where the KN black hole in the
asymptotically flat space and the RN black hole in the (A)dS space are the
limit of the zero (A)dS radius and the zero angular momentum, respectively.
The four-dimensional dyonic KN-(A)dS black holes consisting of four physical
quantities, the mass $M/\Xi^{2}$, angular momentum $J=Ma/\Xi^{2}$, electric
and magnetic charges $Q$ and $P$, have the metric and the gauge potential

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\frac{\Delta_{r}}{\Sigma}\left(dt-\frac{a\sin^{2}\theta}{\Xi}d\varphi\right)^{2}+\frac{\Sigma}{\Delta_{r}}dr^{2}+\frac{\Sigma}{\Delta_{\theta}}d\theta^{2}+\frac{\Delta_{\theta}}{\Sigma}\sin^{2}\theta\left(adt-\frac{r^{2}+a^{2}}{\Xi}d\varphi\right)^{2},$
$\displaystyle A_{[1]}$ $\displaystyle=$ $\displaystyle\frac{Qr-
Pa\cos\theta}{\Sigma}\left(dt-\frac{a\sin^{2}\theta}{\Xi}d\varphi\right)+\frac{P(\cos\theta-\sigma)}{\Xi}d\varphi,$
(1)

where

$\Delta_{r}=(r^{2}+a^{2})\left(1\pm\frac{r^{2}}{L^{2}}\right)-2Mr+Q^{2}+P^{2},\quad\Delta_{\theta}=1\mp\frac{a^{2}}{L^{2}}\cos^{2}\theta,\quad\Sigma=r^{2}+a^{2}\cos^{2}\theta,\quad\Xi=1\mp\frac{a^{2}}{L^{2}}.$
(2)

Here the upper (lower) sign corresponds to the AdS (dS) space and $\sigma=\pm
1$ denotes a different choice of the Dirac string. The associated
thermodynamic quantities, such as the Hawking temperature, entropy, horizon
angular velocity and chemical potential are given by, respectively,

$\hat{T}_{\mathrm{H}}=\frac{r_{+}}{4\pi(r_{+}^{2}+a^{2})}\left(1-\frac{a^{2}+Q^{2}+P^{2}}{r_{+}^{2}}\pm\frac{a^{2}}{L^{2}}\pm\frac{3r_{+}^{2}}{L^{2}}\right),\quad\hat{S}_{\mathrm{BH}}=\frac{\pi(r_{+}^{2}+a^{2})}{\Xi},\quad\hat{\Omega}_{\mathrm{H}}=\frac{a\Xi}{r_{+}^{2}+a^{2}},\quad\hat{\Phi}_{\mathrm{H}}=\frac{Qr_{+}}{r_{+}^{2}+a^{2}}.$
(3)

Figure 1: Chart of charged black holes.

The extremal limit with a degenerate horizon at $r_{0}$ can be achieved by
requiring the conditions $\Delta_{r}(r_{0})=\Delta_{r}^{\prime}(r_{0})=0$ that
identify the relation of physical parameters. We introduce two convenient ways
to express the physical quantities: first in terms of mass and then in terms
of charges. In the dS space the degenerated horizon $r_{0}=r_{+}=r_{-}$ is
given, in terms of mass, by

$r_{0}=\sqrt{\frac{2}{3}}\left(1-\frac{a^{2}}{L^{2}}\right)^{1/2}L\sin\frac{\vartheta}{3},\quad
Q^{2}+P^{2}=\frac{2}{3}\left(1-\frac{a^{2}}{L^{2}}\right)^{2}L^{2}\sin^{2}\frac{\vartheta}{3}\cos\frac{2\vartheta}{3}-a^{2},$
(4)

where

$\sin\vartheta=3\sqrt{\frac{3}{2}}\left(1-\frac{a^{2}}{L^{2}}\right)^{-3/2}\frac{M}{L},\qquad
0\leq\vartheta\leq\frac{\pi}{2}.$ (5)

There is an upper limit for the mass and its maximum value corresponds to
$\theta=\pi/2$. A simple way to give the associated physical quantities in AdS
space is analytically continuing $L\to-iL,\vartheta\to i\zeta$, or
equivalently flipping the sign of the cosmological constant, leading to

$r_{0}=\sqrt{\frac{2}{3}}\left(1+\frac{a^{2}}{L^{2}}\right)^{1/2}L\sinh\frac{\zeta}{3},\quad
Q^{2}+P^{2}=\frac{2}{3}\left(1+\frac{a^{2}}{L^{2}}\right)^{2}L^{2}\sinh^{2}\frac{\zeta}{3}\cosh\frac{2\zeta}{3}-a^{2},$
(6)

where

$\sinh\zeta=3\sqrt{\frac{3}{2}}\left(1+\frac{a^{2}}{L^{2}}\right)^{-3/2}\frac{M}{L}.$
(7)

The parametrization of the extremal KN black holes, in terms of charges,
gives111There is another degenerate limit in the dS space, namely the Nariai
branch $r_{0}=r_{+}=r_{C}$, in terms of mass,
$r_{0}=\sqrt{\frac{2}{3}}\left(1-\frac{a^{2}}{L^{2}}\right)^{1/2}L\sin\left(\frac{\pi}{3}-\frac{\vartheta}{3}\right),\quad
Q^{2}+P^{2}=\frac{2}{3}\left(1-\frac{a^{2}}{L^{2}}\right)^{2}L^{2}\sin^{2}\left(\frac{\pi}{3}-\frac{\vartheta}{3}\right)\cos\left(\frac{2\pi}{3}-\frac{2\vartheta}{3}\right)-a^{2},$
or in terms of charges,
$r_{0}^{2}=\frac{L^{2}}{6}\left(\sqrt{\left(1-\frac{a^{2}}{L^{2}}\right)^{2}-\frac{12(a^{2}+Q^{2}+P^{2})}{L^{2}}}+\left(1-\frac{a^{2}}{L^{2}}\right)\right).$
The corresponding near-horizon geometry becomes warped $dS_{2}\times S^{2}$
since the signatures of $d\tau$ and $d\rho$ are flipped.

$r_{0}=\frac{L\sqrt{\delta}}{\sqrt{6}},\qquad
M_{0}=\frac{L\left[3\left(1\pm\frac{a^{2}}{L^{2}}\right)\pm\delta\right]\sqrt{\delta}}{3\sqrt{6}},$
(8)

where

$\delta=\pm\left(\sqrt{\left(1\pm\frac{a^{2}}{L^{2}}\right)^{2}\pm\frac{12(a^{2}+Q^{2}+P^{2})}{L^{2}}}-\left(1\pm\frac{a^{2}}{L^{2}}\right)\right).$
(9)

As discussed in Chen:2020mqs , the horizon radius decreases in the AdS space
and increases in the dS space. The rotation provides an additional repulsive
centrifugal force for produced particles which increases the horizon radius in
both dS and AdS spaces, see Fig. 2. Moreover, in AdS space, the contributions
from $a$ and $L$ are competitive so that their dominance switches at almost
the same value of $r_{0}$ and $L$ corresponding to the U-turn (convexity) in
the Unruth temperature.

Figure 2: Horizon radius $r_{0}$ in AdS (blue) and dS (green) with parameters
$Q^{2}+P^{2}=12$.

The near-horizon geometry of $r_{+}=r_{-}=r_{0}$ can be obtained by applying
the following transformation

$\varphi\to\varphi+\hat{\Omega}_{\mathrm{H}}\,t,\qquad r\to
r_{0}+\epsilon\,\rho,\qquad
t\to\frac{r_{0}^{2}+a^{2}}{\Delta_{0}}\,\frac{\tau}{\epsilon},\qquad M\to
M_{0}+\frac{\Delta_{0}}{2r_{0}}B^{2}\epsilon^{2},$ (10)

and then taking $\epsilon\to 0$ limit. Finally, the near-horizon geometry of
KN-(A)dS black hole is given by

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\frac{\Sigma_{0}}{\Delta_{0}}\left(-(\rho^{2}-B^{2})d\tau^{2}+\frac{d\rho^{2}}{\rho^{2}-B^{2}}+\frac{\Delta_{0}}{\Delta_{\theta}}d\theta^{2}\right)+\frac{(r_{0}^{2}+a^{2})^{2}\Delta_{\theta}\sin^{2}\theta}{\Sigma_{0}}\left(\frac{d\varphi}{\Xi}+\frac{2ar_{0}\,\rho\,d\tau}{(r_{0}^{2}+a^{2})\Delta_{0}}\right)^{2},$
(11) $\displaystyle A_{[1]}$ $\displaystyle=$
$\displaystyle-\frac{Q(r_{0}^{2}-a^{2}\cos^{2}\theta)-2Par_{0}\cos\theta}{\Sigma_{0}\Delta_{0}}\rho\,d\tau-\frac{Qar_{0}\sin^{2}\theta-P(r_{0}^{2}+a^{2})\cos\theta+\sigma
P\Sigma_{0}}{\Sigma_{0}\Xi}d\varphi,$

where

$\Delta_{0}=1\pm\frac{a^{2}}{L^{2}}\pm\frac{6r_{0}^{2}}{L^{2}},\qquad\Sigma_{0}=r_{0}^{2}+a^{2}\cos^{2}\theta.$
(12)

The dual gauge potential is

$\bar{A}_{[1]}=-\frac{P(r_{0}^{2}-a^{2}\cos^{2}\theta)+2Qar_{0}\cos\theta}{\Sigma_{0}\Delta_{0}}\rho\,d\tau-\frac{Par_{0}\sin^{2}\theta+Q(r_{0}^{2}+a^{2})\cos\theta-\sigma
Q\Sigma_{0}}{\Sigma_{0}\Xi}d\varphi.$ (13)

##  III Schwinger Emission of Dyons

In this section, we are going to briefly summarize the computation of the
production rate for the Schwinger effect occurring in the near-horizon region
of near-extremal Kerr-Newman (KN) black holes in (A)dS space. The detailed
calculation can be found in Chen:2016caa ; Chen:2017mnm  for the case with
vanishing cosmological constant. The Klein-Grodon equation for a charged
complex scalar field $\Psi$

$\left(\nabla_{\mu}-iqA_{\mu}-ip\bar{A}_{\mu}\right)\left(\nabla^{\mu}-iqA^{\mu}-ip\bar{A}^{\mu}\right)\Psi-m^{2}\Psi=0,$
(14)

with the following ansatz

$\Psi(\tau,\rho,\theta,\varphi)=\exp\left(-i\omega\tau+i\frac{n-\sigma(qP-
pQ)}{\Xi}\varphi\right)R(\rho)S(\theta),$ (15)

can be separated to the angular part

$\displaystyle\frac{1}{\sin\theta}\partial_{\theta}(\Delta_{\theta}\sin\theta\,\partial_{\theta}S)-\left(\frac{[n\Sigma_{0}+(qQ+pP)ar_{0}\sin^{2}\theta-(qP-
pQ)(r_{0}^{2}+a^{2})\cos\theta]^{2}}{(r_{0}^{2}+a^{2})\Delta_{\theta}\sin^{2}\theta}-m^{2}a^{2}\sin^{2}\theta-\lambda_{l}\right)S=0,$
(16)

and the radial part

$\displaystyle\Delta_{0}\partial_{\rho}\left((\rho^{2}-B^{2})\partial_{\rho}R\right)+\left(\frac{[\omega(r_{0}^{2}+a^{2})\Delta_{0}-(qQ+pP)(r_{0}^{2}-a^{2})\rho+2nar_{0}\rho]^{2}}{(r_{0}^{2}+a^{2})^{2}\Delta_{0}(\rho^{2}-B^{2})}-m^{2}(r_{0}^{2}+a^{2})-\lambda_{l}\right)R=0.$
(17)

The effective mass in the radial equation, proportional to the constant term
in the last parentheses, is

$m_{\mathrm{eff}}^{2}=m^{2}-\frac{[2nar_{0}-(qQ+pP)(r_{0}^{2}-a^{2})]^{2}}{(r_{0}^{2}+a^{2})^{3}\Delta_{0}}+\frac{\lambda_{l}}{r_{0}^{2}+a^{2}},$
(18)

and the BF bound, a stability condition in the near-horizon AdS space i.e.
$m_{\mathrm{eff}}^{2}\geq-1/4L_{\mathrm{AdS}}^{2}$, is

$m_{\mathrm{eff}}^{2}\geq-\frac{\Delta_{0}}{4(r_{0}^{2}+a^{2})}.$ (19)

The pair production, corresponding to the unstable modes, occurs when the BF
is violated Pioline:2005pf ; Kim:2008xv . As plotted in Fig. 3, it is clear
that the pair production always occurs in dS space, but for AdS there is a
threshold of $L$ which increases as rising $a$ (see Table 1).

$a$ |  $L$
---|---
$0$ | $1.5749$
$0.4$ | $2.1085$
$0.8$ | $4.1391$
$1.2$ | $9.2893$
Table 1: Critical values of $a$ and $L$ of the BF bound in AdS:
$Q^{2}+P^{2}=12,qQ+pP=6,m=1,\lambda_{l}=0,n=0$. Figure 3: The plot of BF bound
$m_{\mathrm{eff}}^{2}+\frac{\Delta_{0}}{4(r_{0}^{2}+a^{2})}$ for AdS (blue)
and dS (green) with parameters $Q^{2}+P^{2}=12,qQ+pP=6,m=1,\lambda_{l}=0,n=0$.
The BF bound can be saturated, i.e. positive region, only in AdS.

The solution of radial equation is given by the hypergeometric functions
Chen:2016caa ; Chen:2017mnm  with the generalized parameters

$\displaystyle\tilde{\kappa}=\frac{\omega}{B},\qquad\kappa=\frac{(qQ+pP)(r_{0}^{2}-a^{2})-2nar_{0}}{(r_{0}^{2}+a^{2})\Delta_{0}},\qquad\mu=\sqrt{\kappa^{2}-\frac{m^{2}(r_{0}^{2}+a^{2})}{\Delta_{0}}-\frac{\lambda_{l}}{\Delta_{0}}-\frac{1}{4}}.$
(20)

The violation of the BF bound leads to real value of $\mu$ which implies the
existence of pair production. According the analysis in Chen:2016caa ;
Chen:2017mnm , the mean number of Schwinger effect is remarkably given by a
universal formula

$\displaystyle\mathcal{N}=\frac{\mathrm{e}^{-2\pi(\kappa-\mu)}-\mathrm{e}^{-2\pi(\kappa+\mu)}}{1+\mathrm{e}^{-2\pi(\kappa+\mu)}}\times\frac{1-\mathrm{e}^{-2\pi(\tilde{\kappa}-\kappa)}}{1+\mathrm{e}^{-2\pi(\tilde{\kappa}-\mu)}}.$
(21)

The exponentials can be rewritten as the Boltzmann factors in thermodynamics
with

$2\pi(\kappa-\mu)=\bar{m}/T_{\mathrm{eff}},\qquad
2\pi(\kappa+\mu)=\bar{m}/\bar{T}_{\mathrm{eff}},$ (22)

in which the associated “inertial mass” $\bar{m}$, the Unruh temperature and
the effective temperatures for the Schwinger effect are defined as

$\displaystyle\bar{m}^{2}=m^{2}+\frac{\lambda_{l}}{r_{0}^{2}+a^{2}}+\frac{\Delta_{0}}{4(r_{0}^{2}+a^{2})},\qquad
T_{\mathrm{U}}=\frac{\Delta_{0}\kappa}{2\pi\bar{m}(r_{0}^{2}+a^{2})},$
$\displaystyle
T_{\mathrm{eff}}=T_{\mathrm{U}}+\sqrt{T_{\mathrm{U}}^{2}-\frac{\Delta_{0}}{4\pi^{2}(r_{0}^{2}+a^{2})}},\qquad{\bar{T}}_{\mathrm{eff}}=T_{\mathrm{U}}-\sqrt{T_{\mathrm{U}}^{2}-\frac{\Delta_{0}}{4\pi^{2}(r_{0}^{2}+a^{2})}}.$
(23)

Note that the inertial mass increases in the AdS space while it decreases in
the dS space, and the rotation reduces its value, as shown in Fig. 4.

Figure 4: Inertial mass $\bar{m}$ in AdS (blue) and dS (green) with parameters
$Q^{2}+P^{2}=12,qQ+pP=6,m=1,\lambda_{l}=0,n=0$.

The Unruh and effective temperatures are shown in Fig. 5. Both temperatures
basically are inversely proportional to $r_{0}$ since the electric field is
stronger when the horizon is closer to the origin. Generally the angular
momentum reduces these temperatures via enlarging the horizon. However, in AdS
space, the dominant effect by $a$ or $L$ switches at almost the same value of
$L$, which causes the U-turns in the Unruh temperature. Moreover, the $r_{0}$
can reduce to a value closed to $a$, then the factor $r_{0}^{2}-a^{2}$
(assuming $n=0$) significantly suppresses the value of $\kappa$ and,
accordingly the Unruh temperature. The effective temperature is physically
meaningful only in the situation that pair production occurs, i.e. violating
the BF bound, otherwise it does not take real values anymore. The left
endpoints of the effective temperature are just the thresholds of the BF
bound.

Figure 5: Unruh temperature $T_{\mathrm{U}}$ and effective temperature
$T_{\mathrm{eff}}$ in AdS (blue) and dS (green) with parameters
$Q^{2}+P^{2}=12,qQ+pP=6,m=1,\lambda_{l}=0,n=0$.

Finally, we obtain the remarkable thermal interpretation for the Schwinger
effects Kim:2015kna

$\displaystyle\mathcal{N}=\underbrace{\Biggl{(}\frac{\mathrm{e}^{-\bar{m}/T_{\mathrm{eff}}}-\mathrm{e}^{-\bar{m}/\bar{T}_{\mathrm{eff}}}}{1+\mathrm{e}^{-\bar{m}/\bar{T}_{\mathrm{eff}}}}\Biggr{)}}_{\textrm{Schwinger
effect in
AdS${}_{2}$}}\times\Biggl{\\{}\mathrm{e}^{\bar{m}/T_{\mathrm{eff}}}\underbrace{\Biggl{(}\mathrm{e}^{-\bar{m}/T_{\mathrm{eff}}}\frac{1-\mathrm{e}^{-(\omega+q\Phi_{\mathrm{H}}+p\bar{\Phi}_{H}+n\Omega_{H})/T_{\mathrm{H}}}}{1+\mathrm{e}^{-\bar{m}/T_{\mathrm{eff}}}\mathrm{e}^{-(\omega+q\Phi_{\mathrm{H}}+p\bar{\Phi}_{H}+n\Omega_{H})/T_{\mathrm{H}}}}\Biggr{)}}_{\textrm{Schwinger
effect in Rindler${}_{2}$}}\Biggr{\\}}.$ (24)

The emission formula has a universal factorization of the Schwinger formula in
the AdS2 Cai:2014qba  and another Schwinger formula in the two-dimensional
Rindler space Gabriel:1999yz . Here, the thermodynamic quantities in the near-
horizon geometry reduce to

$T_{H}=\frac{B}{2\pi},\qquad\Omega_{H}=\frac{2ar_{0}B}{(r_{0}^{2}+a^{2})\Delta_{0}},\qquad\Phi_{H}=-\frac{Q(r_{0}^{2}-a^{2})B}{(r_{0}^{2}+a^{2})\Delta_{0}},\qquad\bar{\Phi}_{H}=-\frac{P(r_{0}^{2}-a^{2})B}{(r_{0}^{2}+a^{2})\Delta_{0}},$
(25)

which are the near extremal limit $r_{+}=r_{0}+\epsilon B$ of the
corresponding quantities in (3):

$\hat{T}_{\mathrm{H}}\to\frac{B}{2\pi}\frac{\Delta_{0}\epsilon}{r_{0}^{2}+a^{2}},\quad\hat{\Omega}_{\mathrm{H}}\to\frac{a\Xi}{r_{0}^{2}+a^{2}}-\frac{2ar_{0}B}{(r_{0}^{2}+a^{2})\Delta_{0}}\frac{\Xi\Delta_{0}\epsilon}{r_{0}^{2}+a^{2}},\quad\hat{\Phi}_{\mathrm{H}}\to\frac{Qr_{0}}{r_{0}^{2}+a^{2}}-\frac{Q(r_{0}^{2}-a^{2})B}{(r_{0}^{2}+a^{2})\Delta_{0}}\frac{\epsilon\Delta_{0}}{r_{0}^{2}+a^{2}}.$
(26)

The limit is as follows: the first term in $\hat{\Omega}_{\mathrm{H}}$ is
eliminated by transformation $\varphi\to\varphi+\hat{\Omega}_{H}t$, and the
first term in $\hat{\Phi}_{\mathrm{H}}$ is discarded when taking the near
horizon limit (it is a divergent constant). The factor
$\frac{\Delta_{0}\epsilon}{r_{0}^{2}+a^{2}}$ is simply due to the rescaling of
time, and $\Xi$ in $\hat{\Omega}_{\mathrm{H}}$ is from the $\varphi$
dependence in the ansatz of the scalar field.

The charge-mass ratio, $\sqrt{Q^{2}+P^{2}}/(M/\Xi^{2})$, in the dS space is
shown in Fig. 6. Note that there is an upper limit of $M$ in dS space
corresponding the triple degeneracy $r_{+}=r_{-}=r_{C}$.

Figure 6: Charge-mass ratio in dS.

##  IV Conclusion

We have studied the emission of electric and/or magnetic charges from the
near-extremal dyonic KN black holes in the (A)dS space. The dyonic KN black
holes in the (A)dS space have the limits: (i) the dyonic KN black holes in the
asymptotically flat space when the (A)dS radius is infinite, (ii) the dyonic
RN black holes in the (A)dS space when the rotation of the hole vanishes and
(iii) the KN black holes in the (A)dS space when the magnetic charge is zero
or the magnetic KN black holes in the (A)dS space when the electric charge is
zero. Using the AdS geometry of near-horizon region, we have found the
solutions of the dyons in terms of the hypergeometric function and obtained
the Schwinger effect by computing the mean number of the emitted dyons.

We have shown that the emission formula still has a universal factorization of
the Schwinger formula in the AdS2 and another Schwinger formula in the two-
dimensional Rindler space determined by the effective temperature and the
Hawking temperature with the chemical potentials of electric and magnetic
charges and angular momentum. The emission formula recovers the formulae for
RN or KN black holes in the asymptotically flat space and also the formula for
RN black holes in the (A)dS space. Further, we have shown that there exists a
parameter region, known as the Breitenloher-Freedman bound for the stability
of dyonic KN black holes in the AdS space against both the emission of charges
and Hawking radiation.

We have observed that the Schwinger effect is affected by the (A)dS boundary
and the rotation of the black hole as follows:

  * •

The boundary AdS space compresses the horizon radius while the dS space
stretches the radius, which in turn strengthens or weakens the electric field
on the horizon where the pair are mostly produced. This holds for RN black
holes or KN black holes in the boundary (A)dS space. Hence, the Schwinger
effect for a fixed angular momentum is enhanced in AdS space and suppressed in
dS space.

  * •

The rotation of near-extremal KN black hole decreases the “inertial mass” of a
charge since the dragging of particles around the event horizon lightens the
effective mass along the radial direction, but increases the horizon radius
and reduces the electric field on the horizon. For a large radius of the (A)dS
space, the decreasing electric field affects more than the decreasing inertial
mass in the charge acceleration for the Unruh temperature. Hence, for a large
radius of the (A)dS space, the angular momentum decreases the Unruh and
effective temperatures and thus, suppresses the Schwinger effect. However, for
a small radius of the AdS boundary, as the angular momentum changes, there is
a crossover of the horizon radius at almost the same value of the AdS radius:
below the crossover the horizon radius decreases as the angular momentum while
above the crossover the horizon radius increases as the angular momentum. The
competitive effect of the crossover of the horizon radius and the decreasing
inertial mass results in the convexity of the Unruh and effective
temperatures.

  * •

In the AdS space the KN black hole has a BF bound that depends on the angular
momentum and provides a threshold for the Schwinger effect in AdS space. The
radius of the AdS space for the BF bound increases as the angular momentum
increases. In the dS space the charge to mass ratio has an upper bound due to
the triple degeneracy of horizons: the Cauchy horizon, black hole horizon and
cosmological horizon. The ratio decreases as the angular momentum increases
for small mass of the black hole but approaches the same value until the upper
bound.

###### Acknowledgements.

The work of C.M.C. was supported by the National Science Council of the R.O.C.
(Taiwan) under the grants MOST 109-2112-M-008-010 and 110-2112-M-008-009. The
work of S.P.K. was supported in part by National Research Foundation of Korea
(NRF) funded by the Ministry of Education (2019R1I1A3A01063183).

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