Schwinger Effect from Near-extremal Black Holes in (A)dS Space

We study the Schwinger effect in near-extremal Reissner-Nordström (RN) black holes with electric and/or magnetic charges in the (Anti-) de Sitter (AdS) space. The formula for the Schwinger effect takes a universal form for near-extremal black holes with the near-horizon geometry of AdS2×S and with the proper radii 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 the AdS (dS) space reinforces the enhancement and suppression.

charges from (near-) extremal dyonic RN-(A)dS black holes. The formula for the mean number of dyon pairs exhibits the universal structure.
The organization of this paper is as follows. In Sec. 2, we study the Schwinger effect in (near-) extremal RN-(A)dS black holes. The near-horizon geometry is still AdS 2 × S 2 , whose effective radii for the AdS 2 space and S 2 are modified by the asymptotic radius of the (A)dS space. The Schwinger effect has a universal structure in terms of the effective temperature for charges and the Hawking temperature, and is suppressed by the dS boundary but enhanced by the AdS boundary. In Sec. III, the Schwinger effect is studied in (near-) extremal dyonic black holes in the (A)dS space. The universal structure for the mean number of dyon pairs shows the Schwinger effect in AdS 2 and quantum electrodynamics (QED) effect of the electromagnetic field in a Rindler space for the surface gravity of the event horizon. The effective temperature is determined by the Unruh temperature for the acceleration of dyons in the electromagnetic field. The Schwinger effect is suppressed in the dS space but enhanced in the AdS space. In Sec. IV, we discuss cosmological and astrophysical implications of the effect of the asymptotic (A)dS boundary on the Schwinger effect.

II. (NEAR-) EXTREMAL RN-(A)DS BLACK HOLES
The action for RN-(A)dS black holes is given by where a cosmological constant Λ = ∓3/L 2 (the upper/lower sign) here and hereafter corresponds to the AdS/dS space, respectively. The metric for an RN-(A)dS black holes is 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 f (r), in general, associated to the causal horizon r − and event horizon r + , and for the dS case a third positive root corresponds to the cosmological horizon r C [18]. Out of three horizons of RN-(A)ds black holes (2), the causal horizon r − and the event horizon r + are made degenerate, M = M 0 , r + = r − = r 0 , to yield an extremal RN-(A)dS black holes, where The cosmological horizon remains outside of the event horizon (r C > r 0 ). By elongating the radial coordinate r = r 0 + ρ and reducing the time coordinate, t = τ / , one has the near-horizon geometry of extremal RN-(A)dS possessing a structure of AdS 2 × S 2 with radii We consider a charged scalar governed by the Klein-Gordon equation, whose solution Φ(τ, ρ, θ, ϕ) = e −iωτ +inϕ R(ρ)S(θ) is separated into the angular and radial equations as By introducing an effective mass and R 2 AdS -rescaled electric forcē the violation of the Breitenlohler-Freedman (BF) bound in AdS 2 space leading to the Schwinger pair production takes the form [19][20][21] The BF bound (µ 2 < 0) of AdS 2 space guarantees the stability against the pair production. The solution to the radial equation is in terms of the Whittaker function [4] According to our earlier result in Ref. [4], the mean number of produced pairs is The mean number (13) of charges from the extremal black hole has a thermal interpretation in terms of the effective temperature Note that the effective temperature (15) is the same as that of charges in a uniform electric field in AdS 2 space, in which T U is the Unruh temperature for the accelerating charge by the electric field and the second term in the square root is related to the BF bound [19,20], which corresponds to the Gibbons-Hawking temperature square in dS 2 space [21]. The factor of two, i.e. T S = 2T U , in the Minkowski spacetime limit (R AdS = ∞) is an ultra-relativistic feature of the Schwinger effect. In general, T S for the AdS space is larger than, and T S for the dS space is the smaller than that for the asymptotically flat space, as shown in Fig. 1. The T S for the (A)dS space approaches that for the asymptotically flat space. The reason for this is that the AdS 2 radius R AdS and the two-sphere radius R S for the dS space are larger than those for the AdS space, see Fig. 2, and thereby the electric field on the horizon is weaker, the Unruh temperature lower for the dS space than those for the AdS space. The asymptotic (A)dS space drastically changes the effective temperature: the higher temperature for the AdS space enhances the Schwinger effect while the lower temperature for the dS space suppresses the Schwinger effect. Remarkably the dS space increases the AdS 2 radius R AdS for BF bound, which implies larger extremal RN black holes stable against the Schwinger effect than the asymptotically flat space and opens possibility for large extremal primordial black holes.
The near-horizon geometry of near-extremal RN black holes is obtained by scaling coordinates r ± = r 0 ± B and parameterizing the mass  AdS and R 2 S in the asymptotically flat space. The yellow (uppermost) curve denotes R AdS and the orange curve above the horizontal line denotes RS in the dS space while the violet curve below the horizontal line denotes R 2 S and the green curve (lowermost) denotes R 2 AdS in the AdS space.
The metric and gauge field for the near-extremal black holes are The Hawking temperature and the chemical potential on the horizon in the metric (17) are then given by while the Hawking temperature in the original metric is T H and suppressed by small due to the rescaling of time.
The radial equation of charged scalar field is The solutions to the radial equation are found in terms of the hypergeometric functions, which lead to the mean number for pair production where The mean number has the following thermal interpretation [22] N The first parenthesis is the Schwinger effect for an extremal black hole. The second curly bracket is expected since the Hawking temperature for a near-extremal black hole does not completely vanish, though small, and the Hawking radiation and the Schwinger effect is intertwined. The surface gravity of the event horizon determines the acceleration of the two-dimensional Rindler space and the effective temperature is the QED effect in the electric field of charged black hole.

III. (NEAR-) EXTREMAL DYONIC RN-(A)DS BLACK HOLES
The second model is the dyonic RN-(A)dS black holes with the electric and magnetic charge Q and P , which have the metric and the potentials for Q and P where the upper(lower) sign in f (r) is for the AdS (dS) space. The magnetic monopole induces a string-like singularity causing dierent choices of the gauge potential: the upper sign is regular in 0 ≤ θ < π/2, and the lower sign in π/2 < θ ≤ π. The field strengths of A andĀ are Hodge dual to each other. The extremal condition, M 0 , and the radius of degenerated horizon, r 0 , are given in (3), with modification of Note that parameters for extremal dyonic RN black holes are obtained by replacing Q 2 + P 2 for Q 2 for extremal RN black holes. By stretching the radial coordinate r = r 0 + ρ but squeezing the time coordinate t = τ / and taking near extremal condition (16), we obtain the near-horizon geometry of near extremal dyonic RN-(A)dS where the radii R AdS and R S are given in Eq. (5) with δ in Eq. (24). The electric and magnetic charges q and p of emitted particles are coupled to the potentials A µ andĀ µ as whose wave function takes the form Φ(τ, ρ, θ, ϕ) = e −iωτ +i[n∓(qP −pQ)]ϕ R(ρ)S(θ), The solution separates into the angular and the radial parts The result is analog with the electric black hole with the modifications The mean number generalizes the formula with zero angular momentum in Ref. [7] to the (A)dS boundary . (31) The Schwinger temperatures, T S andT S , are given in Eq. (15) with a generalized κ in Eq. (30). It should be noted that the mean number (31) for Schwinger pair production has the universal form as Eq. (22) for near-extremal RN black holes: the factorization of Schwinger effect into an AdS 2 and a two-dimensional Rindler space. The charge emission via Hawking radiation is given by the Hawking temperature with chemical potentials for electric and magnetic charges, which is intertangled with the Schwinger term.

IV. CONCLUSION
We have studied the Schwinger effect from (near-) extremal RN black holes with electric and/or magnetic charges in the (A)dS space. It is found that the asymptotic (A)dS boundary drastically changes the AdS 2 and S 2 radii of near-horizon geometry and the effective temperature for Schwinger effect and this boundary effect increases the Schwinger pair production for the AdS space but decreases it for the dS space. The smaller the radius, L, of AdS/dS space is, the larger the enhancement/suppression of the Schwinger effect is. A physical reasoning for this phenomenon is that the AdS asymptotic boundary pushes the event horizon inward and strengthens the electric field on it while the dS boundary pulls the event horizon toward the cosmological horizon and weakens the field on the event horizon. One interesting cosmological implication is that dyonic (near-) extremal RN or KN black holes have larger sizes and longer life times in the dS space than those in the asymptotically flat space and these primordial black holes may be a candidate for dark matter (for constraints for primordial black holes, see Ref. [23]) and their binaries may provide a source for gravitational waves [24]. This issue and other cosmological and astrophysical implications will be addressed in a separate paper.