Schwinger Effect from Near-Extremal Black Holes in (A)Ds Space

PHYSICAL REVIEW D 101, 085014 (2020)

Schwinger effect from near-extremal black holes in (A)dS space

Chiang-Mei Chen * Department of Physics, National Central University, Chungli 32001, Taiwan and Center for High Energy and High Field Physics (CHiP), National Central University, Chungli 32001, Taiwan † Sang Pyo Kim Department of Physics, Kunsan National University, Kunsan 54150, Korea and Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China

(Received 10 February 2020; accepted 2 April 2020; published 24 April 2020)

We study the Schwinger effect in near-extremal Reissner-Nordström black holes with electric and/or magnetic charges in (anti-)de Sitter (AdS) space. The formula for the Schwinger effect takes a universal 2 form for near-extremal black holes with the near-horizon geometry of AdS2 × S and with the proper radii 2 for the AdS2 space and the two-sphere S , regardless of the asymptotically flat or (A)dS space. The asymptotic AdS boundary enhances and the dS boundary suppresses the Schwinger effect, and the small radius of AdS (dS) space reinforces the enhancement and suppression.

DOI: 10.1103/PhysRevD.101.085014

I. INTRODUCTION black holes [6], and dyonic KN black holes [7] (for review and references, see Ref. [8]). The Schwinger fermion Quantum fluctuations spontaneously create particle pairs production has also been studied for charged black holes from the vacuum unless there exists an external mechanism in (A)dS space [9–11]. preventing annihilation of pairs. The event horizon of a Primordial black holes have been studied in cosmology black hole casually separates pairs into the interior and the and astrophysics [12–14]. It was argued that dynonic exterior, and the black hole emits all species of particles [1]. extremal black holes are stable against Hawking radiation A sufficiently strong electric field separates charged pairs [15]. Black holes of Planck scale are stable quantum and accelerates them in opposite directions, the so-called mechanically and can be candidates for dark matter [16]. Schwinger pair production [2,3]. The Hawking radiation The Schwinger effect from (near-)extremal RN and dyonic and Schwinger effect are consequences of the nonpertur- RN black holes in (anti-)de Sitter space is interesting bative effects of quantum field theory. The Hawking radiation and Schwinger effect both act as theoretically and cosmologically since the asymptotic the emission channels of charged pairs from charged black (A)dS boundary affects the near-horizon geometry of holes. Hawking radiation obeying the Bose-Einstein or (near-)extremal black holes [17]. In particular, the expand- Fermi-Dirac distribution with the Hawking temperature ing early universe described by dS space drastically emits charges with a Coulomb chemical potential on the changes the property and evolution of charged black holes. horizon. Though the Hawking radiation from (near-) We study the effect of the asymptotic (A)dS boundary on extremal black holes is exponentially suppressed due to Schwinger pair production in (near-)extremal RN black vanishing Hawking temperature, the Schwinger mecha- holes. The extremal RN-(A)dS black holes are obtained by nism triggers the emission of charges. The Schwinger effect degenerating the event horizon and the causal horizon has been extensively studied for (near-)extremal Reissner- while keeping the cosmological horizon separate from the Nordström (RN) black holes [4,5], Kerr-Newman (KN) black hole horizon. The near-horizon region of (near-) extremal RN-(A)dS black holes has the geometry of 2 AdS2 × S , whose symmetry allows explicit solutions of *[email protected] a charged field in terms of confluent hypergeometric † [email protected] function. The asymptotic (A)dS boundary makes the effective radii of the AdS2 and the two-sphere being Published by the American Physical Society under the terms of different from that of (near-) extremal RN black holes the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to in the asymptotically at space [4,5]. The formula for the author(s) and the published article’s title, journal citation, the mean number for pair production has a universal and DOI. Funded by SCOAP3. structure in terms of the effective temperature for charge

2470-0010=2020=101(8)=085014(7) 085014-1 Published by the American Physical Society CHIANG-MEI CHEN and SANG PYO KIM PHYS. REV. D 101, 085014 (2020) pffiffiffi pffiffiffi acceleration and Hawking temperature with modifications Lð3 δÞ δ L δ M0 ¼ pffiffiffi ;r0 ¼ pffiffiffi ; due to the effect of the asymptotic (A)dS boundary. We then 3 6 6 investigate the pair production of dyons with electric and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi magnetic charges from (near-)extremal dyonic RN-(A)dS δ ¼ð 1 12Q2=L2 − 1Þ: ð3Þ black holes. The formula for the mean number of dyon pairs exhibits the universal structure. The cosmological horizon remains outside of the event The organization of this paper is as follows. In Sec. II,we horizon (rC >r0). By elongating the radial coordinate study the Schwinger effect in (near-)extremal RN-(A)dS r ¼ r0 þ ϵρ and reducing the time coordinate, t ¼ τ=ϵ, 2 black holes. The near-horizon geometry is still AdS2 × S , one has the near-horizon geometry of extremal RN-(A)dS, 2 whose effective radii for AdS2 space and S are modified by the asymptotic radius of (A)dS space. The Schwinger effect ρ2 R2 Q ds2 ¼ − dτ2 þ AdS dρ2 þ R2dΩ2;A¼ − ρdτ; has a universal structure in terms of the effective temper- R2 ρ2 S 2 R2 ature for charges and the Hawking temperature, and is AdS S ð Þ suppressed by the dS boundary but enhanced by the AdS 4 boundary. In Sec. III, the Schwinger effect is studied in 2 (near-)extremal dyonic black holes in (A)dS space. The possessing a structure of AdS2 × S with radii universal structure for the mean number of dyon pairs 2δ 2δ shows the Schwinger effect in AdS2 and the quantum 2 L 2 2 L R ¼ ;R¼ r0 ¼ : ð5Þ electrodynamics (QED) effect of the electromagnetic field AdS 6ð1 δÞ S 6 in Rindler space for the surface gravity of the event horizon. The effective temperature is determined by the Unruh From Eq. (3), we find another expression temperature for the acceleration of dyons in the electro- rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffi 2 1 3 magnetic field. The Schwinger effect is suppressed in dS L 27 M 27 M0 = R ¼ pffiffiffi 1 þ 0 þ space but enhanced in AdS space. In Sec. IV, we discuss S 6 2 L2 2 L cosmological and astrophysical implications of the effect of rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffi 2 −1 3 27 M 27 M0 = the asymptotic (A)dS boundary on the Schwinger effect. − 1 þ 0 þ ; ð6Þ 2 L2 2 L II. (NEAR-)EXTREMAL RN-(A)dS BLACK HOLES for AdS space, and The action for RN-(A)dS black holes is given by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffi Z 27 2 27 1=3 pffiffiffiffiffiffi 6 ¼ − pLffiffiffi 1 − M0 þ M0 ¼ 4 − − μν ð Þ RS i 2 i S d x g R 2 FμνF ; 1 6 2 L 2 L L rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffi 2 −1 3 27 M 27M0 = 2L ϑ Λ ¼∓ 3 2 − 1 − 0 þ i ¼ pffiffiffi sin ; where a cosmological constant =L (the upper/ 2 L2 2 L 6 3 lower sign) here and hereafter corresponds to AdS/dS rffiffiffiffiffi space, respectively. The metric for a RN-(A)dS black −1 27M0 ϑ ¼ sin ; 0 ≤ ϑ ≤ π=2; hole is 2 L ð Þ 2 7 2 2 dr 2 2 ds ¼ −fðrÞdt þ þ r dΩ2; fðrÞ for dS space. The black hole radius takes the large L limit 2M Q2 r2 ð Þ¼1 − þ 2 4 6 f r 2 2 ; M0 M0 M0 r r L R ¼ M0 1 ∓ 2 þ 12 ∓ 96 S L L L Q A ¼ dt; ð2Þ 8 r M0 þ O : ð8Þ L where M and Q are the mass and charge of the black hole and L is the (A)dS radius. There are two positive roots of The radius of AdS2 and the charge are fðrÞ—in general, associated with the causal horizon r− and — sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi event horizon rþ and for the dS case a third positive root 2 2 2 R RS corresponds to the cosmological horizon rC [18]. Out of ¼ S ¼ 1 3 ð Þ RAdS ;QRS : 9 1 6ðRSÞ2 L three horizons of RN-(A)ds black holes (2), the causal L horizon r− and the event horizon rþ are made degenerate, M ¼ M0, rþ ¼ r− ¼ r0, to yield an extremal RN-(A)dS We consider a charged scalar governed by the Klein- black hole, where Gordon equation,

085014-2 SCHWINGER EFFECT FROM NEAR-EXTREMAL … PHYS. REV. D 101, 085014 (2020) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi μ μ 2 ð∇μ − iqAμÞð∇ − iqA ÞΦ − m Φ ¼ 0; ð10Þ m¯ 1 κ=R2 T ¼ ¼ T þ T2 − ;T¼ AdS ; S 2πðκ −μÞ U U 4π2 2 U 2π ¯ RAdS m whose solution sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m¯ 1 T¯ ¼ ¼ T − T2 − : ð Þ Φðτ ρ θ φÞ¼ −iωτþinφ ðρÞ ðθÞðÞ S 2πðκ þ μÞ U U 4π2 2 19 ; ; ; e R S 11 RAdS is separated into the angular and radial equations as Note that the effective temperature (19) is the same as that of charges in a uniform electric field in AdS2 space, in 1 2 ∂ ð θ∂ Þ − n − ð þ 1Þ ¼ 0 ð Þ which T is the Unruh temperature for the accelerated θ sin θS 2 l l S ; 12 U sin θ sin θ charge by the electric field, and the second term in the square root is related to the BF bound [19,20], which 4 ð ρ − ω 2Þ2 corresponds to the Gibbons-Hawking temperature squared 2 RAdS qQ RS 2 2 ∂ρðρ ∂ρRÞþ − R m ¼ 2 4 ρ2 AdS in dS2 space [21]. The factor of 2, i.e., TS TU, in the RS ¼ ∞ Minkowski spacetime limit (RAdS ), is an ultrarelativ- R2 − AdS ð þ 1Þ ¼ 0 ð Þ istic feature of the Schwinger effect. In general, TS for AdS 2 l l R : 13 RS space is larger, and TS for dS space is smaller than that for the asymptotically flat space, as shown in Fig. 1. The TS for 2 (A)dS space approaches that for asymptotically flat space. By introducing an effective mass and RAdS-rescaled electric force The reason for this is that the AdS2 radius RAdS and the two-sphere radius RS for dS space are larger than those for AdS space (see Fig. 1); thereby, the electric field on the 2 2 lðl þ 1Þ 1=4 2 qQ m¯ ¼ m þ þ ; κ ¼ R ; ð14Þ horizon is weaker, and the Unruh temperature lower for dS R2 R2 AdS R2 S AdS S space than those for AdS space. The asymptotic (A)dS space drastically changes the effective temperature: the the violation of the Breitenlohler-Freedman (BF) bound in higher temperature for AdS space enhances the Schwinger AdS2 space leading to the Schwinger pair production takes effect, while the lower temperature for dS space suppresses – the form [19 21] the Schwinger effect. Remarkably, dS space increases the AdS2 radius R for the BF bound, which implies larger μ2 ¼ κ2 − 2 ¯ 2 ≥ 0 ð Þ AdS RAdSm : 15 extremal RN black holes that are stable against the Schwinger effect compared to asymptotically flat space; 2 The BF bound (μ < 0)ofAdS2 space guarantees the this opens the possibility for large extremal primordial stability against the pair production. The solution to the black holes. radial equation is in terms of the Whittaker function [4] The dependence of TS and radii RS, RAdS with respect to M0 is given in Fig. 2. The effect of the cosmological 2ωR2 constant becomes more significant, enhanced in AdS and ¼ ð Þþ ð Þ ¼ AdS ð Þ R c1Miκ;iμ z c2Miκ;−iμ z ;zi ρ : 16 suppressed in dS, when M0 is larger. The small M0 threshold in TS reflects the BF bound in (15). The Schwinger effect is actually in accordance with the unstable According to our earlier result in Ref. [4], the mean modes of AdS2 in near-horizon geometry. For the asymp- number of produced pairs is totically dS spacetimes, there is an upper bound of M0 such that the Schwinger mechanism is terminated. This corre- sinhð2πμÞ N ¼ expðπμ − πκÞ sponds to the limit when the black hole horizon RS coshðπκ þ πμÞ approaches the cosmological horizon, namely δ ¼ 1 or −2πðκ−μÞ − −2πðκþμÞ ϑ ¼ π=2 in (7), and then the R diverges. ¼ e e ð Þ AdS −2πðκþμÞ : 17 The near-horizon geometry of near-extremal RN black 1 þ e holes is obtained by scaling coordinates r ¼ r0 ϵB and parametrizing the mass The mean number (17) of charges from the extremal black hole has a thermal interpretation 2 RS M ¼ M0 þðϵBÞ : ð20Þ − ¯ − ¯ ¯ 2 m=TS − m=TS 2R N ¼ e e ð Þ AdS − ¯ ¯ ; 18 1 þ e m=TS The metric and gauge fields for the near-extremal black in terms of the effective temperature holes are

085014-3 CHIANG-MEI CHEN and SANG PYO KIM PHYS. REV. D 101, 085014 (2020)

R TS S,AdS 0.585 13.0

0.580 12.5 0.575

0.570 12.0 0.565

0.560 11.5

0.555 L L 20 40 60 80 100 120 20 40 60 80 100 120 pffiffiffiffiffi FIG. 1. (Left panel) The effective temperature TS against L with Q ¼ 12, q ¼ π, m ¼ 1, l ¼ 0 fixed. The green (upper) and yellow (lower) curves are the effective temperature for extremal RN black holes in AdS and dS spaces, respectively, while the bluepffiffiffiffiffi horizontal line is that in asymptotically flat space. (Right panel) AdS2 radius RAdS and the two-sphere radius RS against L with Q ¼ 12, q ¼ π, m ¼ 1, l ¼ 0 fixed. The horizontal line denotes both RAdS and RS in the asymptotically flat space. The yellow (uppermost) curve denotes RAdS, and the orange curve above the horizontal line denotes RS in dS space; the violet curve below the horizontal line denotes RS, and the green curve (lowermost) denotes RAdS in AdS space. ρ2 − 2 2 4 ð ρ − ω 2Þ2 2 B 2 RAdS 2 2 2 2 2 RAdS qQ RS 2 2 ds ¼ − dτ þ dρ þ R dΩ ; ∂ρ½ðρ − B Þ∂ρRþ − R m R2 ρ2 − B2 S 2 R4 ρ2 − B2 AdS AdS S Q 2 ¼ − ρ τ ð Þ RAdS A 2 d : 21 − lðl þ 1Þ R ¼ 0: ð23Þ R 2 S RS

The Hawking temperature and the chemical potential on the The solutions to the radial equation are found in terms of horizon in the metric (21) are then given by the hypergeometric functions, which leads to the mean number for pair production B QB T ¼ ; Φ ¼ − ; ð22Þ H 2πR2 H R2 ð2πμÞ ðπκ˜ − πκÞ AdS S N ¼ sinh sinh coshðπκ þ πμÞ coshðπκ˜ − πμÞ while the Hawking temperature in the original metric is −2πðκ−μÞ −2πðκþμÞ −2πðκ˜−κÞ ϵ ϵ e − e 1 − e TH and suppressed by small due to the rescaling of time. ¼ × ; ð24Þ The radial equation of the charged scalar field is 1 þ e−2πðκþμÞ 1 þ e−2πðκ˜−μÞ

TS RS,AdS

0.50 100

0.10

0.05 10

M0 M0 0.5 1 5 10 50 5 10 20 50

FIG. 2. (Left panel) The effective temperature TS against M0 with q ¼ π, m ¼ 1, l ¼ 0 fixed. The blue (middle) curve is the effective temperature in the asymptotically flat space ðL ¼ ∞Þ. The green (uppermost) and purple (above the blue) curves are TS with L ¼ 60 and L ¼ 120, respectively, in AdS space while the red (below the blue) and yellow (bottommost) curves are TS with L ¼ 120 and L ¼ 60, respectively, in dS space. (Right panel) AdS2 radius RAdS and the two-sphere radius RS against M0 with L ¼ 60, q ¼ π, m ¼ 1, l ¼ 0 ¼ fixed. The blue curve is RAdS RS in the asymptotically flat space. The red (uppermost) curve is RAdS, and the yellow (above the blue) curve is RS in dS space; the green (below the blue) curve is RS, and the purple (bottommost) curve is RAdS in AdS space.

085014-4 SCHWINGER EFFECT FROM NEAR-EXTREMAL … PHYS. REV. D 101, 085014 (2020) where Note that parameters for extremal dyonic RN black holes are obtained by replacing Q2 þ P2 for Q2 for extremal RN R2 qΦ ωR2 ω ¼ þ ϵρ κ ¼ AdS ¼ − H κ˜ ¼ AdS ¼ ð Þ black holes. By stretching the radial coordinate r r0 2 qQ 2π ; 2π : 25 ¼ τ ϵ RS TH B TH but squeezing the time coordinate t = and taking the near-extremal condition (20), we obtain the near-horizon The mean number has the following thermal interpreta- geometry of near-extremal dyonic RN-(A)dS, tion [22] 2 2 2 − ¯ − ¯ ¯ 2 ρ − B 2 R 2 2 2 e m=TS − e m=TS ds ¼ − dτ þ AdS dρ þ R dΩ ; N ¼ 2 ρ2 − 2 S 2 − ¯ ¯ RAdS B 1 þ e m=TS |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Q Schwinger effect in AdS2 A ¼ − ρdτ þ Pðcos θ ∓ 1Þdφ; R2 −ðωþqΦ Þ=T S ¯ − ¯ 1 − e H H m=TS m=TS ð Þ P × e e − ¯ −ðωþ Φ Þ : 26 ¯ ¼ − ρ τ − ð θ ∓ 1Þ φ ð Þ 1 þ e m=TS e q H =TH A 2 d Q cos d ; 29 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} RS Schwinger effect in Rindler2 δ The first set of parentheses contains the Schwinger effect where the radii RAdS and RS are given in Eq. (5), with in for an extremal black hole. The terms in curly brackets are Eq. (28). The electric and magnetic charges q and p of μ ¯ μ expected since the Hawking temperature for a near- emitted particles are coupled to the potentials A and A as extremal black hole does not completely vanish, though ¯ μ μ ¯ μ 2 small, and the Hawking radiation and the Schwinger effect ð∇μ − iqAμ − ipAμÞð∇ − iqA − ipA ÞΦ − m Φ ¼ 0; are intertwined. The surface gravity of the event horizon determines the acceleration of the two-dimensional Rindler ð30Þ space, and the effective temperature is the QED effect in the electric field of a charged black hole. whose wave function takes the form

Φðτ; ρ; θ; φÞ¼e−iωτþi½n∓ðqP−pQÞφRðρÞSðθÞ: ð31Þ III. (NEAR-)EXTREMAL DYONIC RN-(A)dS BLACK HOLES The solution separates into the angular and the radial parts The second model includes the dyonic RN-(A)dS black holes with the electric and magnetic charges Q and P, with the metric and potentials 1 ½n − ðqP − pQÞ cos θ2 ∂θðsin θ∂θSÞ − − λ S ¼ 0; sin θ sin2 θ dr2 ds2 ¼ −fðrÞdt2 þ þ r2dΩ2; ð32Þ fðrÞ 2 2M Q2 þ P2 r2 fðrÞ¼1 − þ ; r r2 L2 R4 ½ðqQ þ pPÞρ − ωR22 ∂ ½ðρ2 − 2Þ∂ þ AdS S ρ B ρR 4 2 2 ¼ Q þ ð θ ∓ 1Þ φ R ρ − B A dt P cos d ; S r 2 2 2 R ¯ P − R m − AdS λ R ¼ 0: ð33Þ A ¼ dt − Qðcos θ ∓ 1Þdφ; ð27Þ AdS R2 r S ð Þ where the upper (lower) sign in f r is for AdS (dS) space. The result is analogous to an electric black hole with the The magnetic monopole induces a stringlike singularity modifications causing different choices of the gauge potential: the upper sign is regular in 0 ≤ θ < π=2, and the lower sign is ¯ R2 qΦ þ pΦ¯ PB π=2 < θ ≤ π. The field strengths of A and A are Hodge κ ¼ AdS ð þ Þ¼− H H Φ¯ ¼ − 2 qQ pP 2π ; H 2 : dual to each other. The extremal condition, M0, and the RS TH RS radius of the degenerated horizon, r0, are given in (3), with ð34Þ a modification of qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 The mean number generalizes the formula with zero δ ¼ 1 12ðQ þ P Þ=L − 1 : ð28Þ angular momentum in Ref. [7] to the (A)dS boundary

085014-5 CHIANG-MEI CHEN and SANG PYO KIM PHYS. REV. D 101, 085014 (2020) − ¯ − ¯ ¯ m=TS − m=TS Schwinger effect, and this boundary effect increases the N ¼ e e − ¯ ¯ Schwinger pair production for AdS space but decreases it 1 þ e m=TS |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} for dS space. The smaller the radius L of AdS/dS space is, Schwinger effect in AdS2 the larger the enhancement/suppression of the Schwinger −ðωþ Φ þ Φ¯ Þ 1 − e q H p H =TH effect is. A physical reasoning for this phenomenon is that m=T¯ S −m=T¯ S × e e − ¯ −ðωþ Φ þ Φ¯ Þ : the AdS asymptotic boundary pushes the event horizon 1 þ e m=TS e q H p H =TH |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} inward and strengthens the electric field on it, while the dS Schwinger effect in Rindler2 boundary pulls the event horizon toward the cosmological ð35Þ horizon and weakens the field on the event horizon. One interesting cosmological implication is that dyonic (near-) ¯ The Schwinger temperatures, TS and TS, are given in extremal RN or KN black holes are larger and have longer Eq. (19) with a generalized κ in Eq. (34). It should be noted lifetimes in dS space than those in asymptotically flat that the mean number (35) for Schwinger pair production space; these primordial black holes may be candidates for has the universal form as Eq. (26) for near-extremal RN dark matter (for constraints for primordial black holes, black holes: the factorization of the Schwinger effect into see Ref. [23]), and their binaries may provide a source for an AdS2 and a two-dimensional Rindler space. The gravitational waves [24]. This issue and other cosmological charge emission via Hawking radiation is given by the and astrophysical implications will be addressed in a Hawking temperature with chemical potentials for electric separate paper. and magnetic charges, which is intertangled with the Schwinger term. ACKNOWLEDGMENTS S. P. K. would like to thank the National Central IV. CONCLUSION University for warm hospitality. The work of C. M. C. We have studied the Schwinger effect from (near-) was supported by the Ministry of Science and Technology extremal RN black holes with electric and/or magnetic of the R.O.C. under Grant No. MOST 108-2112-M-008- charges in (A)dS space. It is found that the asymptotic (A) 007. The work of S. P. K. was supported in part by the 2 dS boundary drastically changes the AdS2 and S radii of National Research Foundation of Korea (NRF) funded by near-horizon geometry and the effective temperature for the the Ministry of Education (2019R1I1A3A01063183).

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