Quantum Superradiance on Static Black Hole Space-Times

Quantum superradiance on static black hole space-times

Visakan Balakumar, Elizabeth Winstanley Consortium for Fundamental Physics, School of Mathematics and Statistics, The University of Sheffield, Hicks Building, Hounsfield Road, Sheffield. S3 7RH United Kingdom

Rafael P. Bernar, Lu´ıs C. B. Crispino Faculdade de F´ısica,Universidade Federal do Pará,66075-110, Belém,Pará,Brazil

Abstract We study the quantum analogue of the classical process of superradiance for a massless charged scalar field on a static charged black hole space-time. We show that an “in” vacuum state, which is devoid of particles at past null infinity, contains an outgoing flux of particles at future null infinity. This radiation is emitted in the superradiant modes only, and is nonthermal in nature.

1. Introduction the black hole horizon significantly exceeds the square of the mass/charge ratio of the quantum field [12]. In the classical phenomenon of superradiance, a wave is am- Here we take an alternative perspective, and consider in- plified during a scattering process, resulting in a reflected wave stead a massless scalar field, so that the emission is not expo- with greater amplitude than the incident wave [1]. One man- nentially suppressed. We focus on the construction of quan- ifestation of superradiance is the scattering of low-frequency tum states and the properties of quantum expectation values as bosonic waves on a rotating black hole space-time [2–4]. the charge of the scalar field varies. We consider a massless There is a corresponding process on a static, charged Reissner- charged scalar field minimally coupled to the RN space-time Nordstrom¨ (RN) black hole space-time [5–8], known as “charge geometry and construct natural “in” and “out” vacuum states. superradiance”. A charged scalar field wave is amplified upon Quantum charge superradiance means that these two states are scattering on the RN black hole if its frequency is sufficiently not the same, with the “in” vacuum containing an outgoing flux low. of charged particles far from the black hole. On rotating Kerr black hole space-times, there is a quantum The outline of this letter is as follows. In Sec. 2 we review analogue of the classical superradiance process [9, 10], known the classical process of superradiance for a charged scalar field as Starobinskii-Unruh radiation. The black hole spontaneously on an RN black hole, before studying the quantum analogue of emits particles in those modes which display classical superra- this process in Sec. 3. We define our “in” and “out” vacuum diance. This radiation is in addition to the usual Hawking ra- states, and compute the fluxes of charge and energy emanating diation [11], and is independent of the temperature of the black from the black hole. Our conclusions are presented in Sec. 4. hole. Throughout this letter, the metric has mostly plus signature. We In this paper we study the quantum analogue of classi- use units in which G = c = ~ = 1 and Gaussian units for cal charge superradiance, first studied by Gibbons [12]. As electrodynamic quantities. with Starobinskii-Unruh radiation, a charged black hole spontaneously emits particles in the classically-superradiant modes, arXiv:2010.01630v2 [gr-qc] 7 Nov 2020 resulting in nonthermal emission [13–16]. Much of the lit- 2. Classical superradiance on static black hole space-times erature on this topic to date has focussed on the comparison between quantum charge superradiance and the well-known We consider a massless charged scalar field Φ evolving on Schwinger pair-creation process [17] in a strong electric field the space-time of an RN black hole, which is described by the (see, for example, [13–16, 18–21] for a selection of references following line element considering a charged scalar field on an RN black hole). In par- ds2 = − f (r) dt2 + f (r)−1 dr2 + r2dθ2 + r2 sin2 θ dϕ2, (1) ticular, for a massive quantum field, the emission rate is suppressed by an exponential factor depending on the field mass where the metric function f (r) is given by [13–16], and is negligible unless the electrostatic potential at 2M Q2 f r − , ( ) = 1 + 2 (2) Email addresses: [email protected] (Visakan r r Balakumar), [email protected] (Elizabeth Winstanley), [email protected] (Rafael P. Bernar), [email protected] (Lu´ıs with M being the mass and Q the electric charge of the black C. B. Crispino) hole. If M2 > Q2 (which is the only possibility we consider

Preprint submitted to Elsevier November 10, 2020 here), the metric function f (r) given by (2), has two zeros, at A basis of solutions to the radial equation (9) consists of the r = r , where usual “in” and “up” scalar field modes, which have the asymp- ± p 2 2 r± = M ± M − Q . (3) totic forms  in −iωer∗ In this case r+ is the location of the black hole event horizon B e , r∗ → −∞, Xin (r) =  ω` (13a) and r− is the location of the Cauchy horizon. In this paper we ω`  −iωr∗ in iωr∗ e + Aωè , r∗ → ∞, restrict our attention to the region exterior to the event horizon. The dynamics of the scalar field Φ is determined by the field and  up  iωer∗ −iωer∗ → −∞ equation up e + Aωè , r∗ , X (r) = up (13b) µ ω`  iωr∗ DµD Φ = 0, (4) Bωè , r∗ → ∞, where Dµ = ∇µ −iqAµ is the covariant derivative, with Aµ being respectively. The “in” modes correspond to waves incoming the electromagnetic gauge potential Aµ = (A0, 0, 0, 0), where from past null infinity, which are partly reflected back to future null infinity and partly transmitted down the future horizon. Q The “up” modes correspond to waves which are outgoing near A = − , (5) 0 r the past event horizon, partly reflected back down the future horizon and partly transmitted to future null infinity. and we have chosen a constant of integration so that the elec- In addition to the “in” and “up” modes defined above, it is tromagnetic potential vanishes far from the black hole. useful to also consider the time-reverse of these modes, denoted The scalar field modes are of the form “out” and “down” respectively. The radial functions for these e−iωt modes have the asymptotic forms φω`m(t, r, θ, ϕ) = NωXω`(r)Y`m(θ, ϕ), (6) r  in∗ iωr B e e ∗ , r∗, → −∞, Xout(r) = Xin∗(r) =  ω` (14a) where ` = 0, 1, 2,... is the total angular momentum quantum ω` ω`  iωr∗ in∗ −iωr∗ e + Aω` e , r∗ → ∞, number, m = −`, −` + 1, . . . , ` − 1, ` is the azimuthal angular momentum quantum number, ω the frequency of the mode, and N ω is a normalization constant and Y`m(θ, ϕ) is a spherical har-  up∗  −iωer∗ iωer∗ → −∞ monic. We have fixed the normalization of the spherical har- down up∗ e + Aω` e , r∗ , X (r) = X (r) = up∗ (14b) ω` ω`  −iωr∗ monics such that Bω` e , r∗ → ∞, Z respectively. The “out” modes have no flux ingoing at the future Y`m(θ, ϕ)Y`0m0 (θ, ϕ) sin θ dθ dϕ = δ``0 δmm0 . (7) event horizon, while the “down” modes have no outgoing flux at future null infinity. We define the usual “tortoise” coordinate r∗ by To find the normalization constants Nω, we employ the h i dr 1 Klein-Gordon inner product Φ1, Φ2 , defined by ∗ = , (8) dr f (r) Z h ∗ ∗ i √ µ hΦ1, Φ2i = i DµΦ1 Φ2 − Φ1DµΦ2 −g dΣ . (15) Σ in terms of which the radial equation for Xω`(r) takes the form " # This inner product is independent of the choice of Cauchy sur- d2 − V r X r , face Σ over which the integral is performed. Using a Cauchy 2 + eff( ) ω`( ) = 0 (9) dr∗ surface close to the union of the past event horizon and past null infinity for the “in” and “up” modes, and a Cauchy surface where the effective potential Veff(r) is close to the union of the future event horizon and future null infinity for the “out” and “down” modes, we find f (r) qQ2 V r ` (` ) r f 0 r − ω − . eff( ) = 2 + 1 + ( ) (10) r r in/out 1 up/down 1 Nω = √ , Nω = p . (16) 4π |ω| 4π |ω| Near the black hole event horizon, as r → r+ and r∗ → −∞, e → ∞ and at infinity, as r, r∗ , the effective potential Veff, given From the radial equation (9) we can derive the following use- by (10), has the asymptotic values ful Wronskian relations:  2 2 qQ 2 2 up 2 up 2 −ωe = − ω − , r∗ → −∞, ω 1 − Ain = ω Bin , ω 1 − A = ω B , (17) V (r) ∼  r+ (11) ω` e ω` e ω` ω` eff  2 −ω , r∗ → ∞, and where we have defined the quantity in up ωeBω` = ωBω` . (18) qQ From the relations (17) we can observe the phenomenon of ωe = ω − . (12) charge superradiance [5–8] since a scalar field mode for which r+ 2 1.06 The expansion coefficientsa ˆ, bˆ satisfy standard commutation Q/M = 0.8 relations (all other commutators vanish): 1.04 q/M=0.8 h in in† i 0 0 0 q/M=0.64 aˆω`m, aˆω0`0m0 = δ`` δmm δ(ω − ω ), ω > 0, 1.02 q/M=0.48 h in in† i 0 ˆ ˆ 0 0 bω`m, bω0`0m0 = δ`` δmm δ(ω − ω ), ω < 0, 2 q/M=0.32 |

in ω` h up up† i 0 0 0 A q/M=0 aˆ , aˆ 0 0 0 = δ δ δ(ω − ω ), ω > 0, | 1 ω`m ω ` m `` mm e hûp ûp† i 0 b , b 0 0 0 = δ``0 δmm0 δ(ω − ω ), ωe < 0. (20) 0.98 ω`m ω ` m An “in” vacuum state can be defined as the state annihilated by 0.96 thea în/up and bîn/up operators. We denote this state by |ini:

0 0.05 0.1 0.15 0.2 0.25 0.3 in în aˆω`m|ini = 0, ω > 0, bω`m|ini = 0, ω < 0, ω r+ up ûp aˆω`m|ini = 0, ωe > 0, bω`m|ini = 0, ωe < 0. (21) in 2 Figure 1: Reflection coefficient |Aω`| for the “in” mode with ` = 0 as a function of the frequency ω for some values of the scalar field charge q and fixed black The “in” vacuum has no particles incoming from past null in- hole charge Q = 0.8M. Superradiance occurs when |Ain |2 > 1. ω` finity nor outgoing from the past horizon and hence is as empty as possible at past null infinity. To investigate the properties of this state, it is useful to define 2 ωωe < 0 will have |Aω`| > 1, and hence will be reflected with a the time-reverse of the “in” vacuum, namely the “out” vacuum larger amplitude than it had originally. |outi. In order to construct this state, we expand the quantum This effect can also be seen in Fig. 1, where we show the scalar field Φˆ in terms of the “out” and “down” modes (14): in 2 1 reflection coefficient |Aω`| for “in” modes with ` = 0 and qQ > 0. Superradiance occurs for low frequency modes with ∞ ` (Z ∞ Z 0 X X out† in 2 Φˆ = dω aôut φout + dω bˆ φout ω < qQ/r+, when |A | > 1. The wave amplification in this ω`m ω`m ω`m ω`m ω` 0 −∞ process is much larger than that observed in the superradiance `=0 m=−` Z ∞ Z 0 ) of a neutral scalar field on a Kerr black hole [1] (see also Fig. 16 down down ˆdown† down + dωe aˆω`m φω`m + dωe bω`m φω`m , (22) in [22]). 0 −∞ where the expansion coefficients satisfy the standard commutation relations (all other commutators vanish) 3. Quantum superradiance for a charged scalar field h out out† i 0 0 0 aˆω`m, aˆω0`0m0 = δ`` δmm δ(ω − ω ), ω > 0, h out out† i 0 ˆ ˆ 0 0 − We now turn to the quantization of the charged scalar field. bω`m, bω0`0m0 = δ`` δmm δ(ω ω ), ω < 0, We firstly define the two quantum states of interest, and derive h down down† i 0 aˆω`m , aˆ 0 0 0 = δ``0 δmm0 δ(ω − ω ), ωe > 0, quantum superradiance by considering the expectation values ω ` m h down down† i 0 ˆ ˆ 0 0 of the current and stress-energy tensor operators. bω`m , bω0`0m0 = δ`` δmm δ(ω − ω ), ωe < 0. (23) The natural “out” vacuum state to define using this expansion 3.1. “In” and “out” vacuum states of the quantum scalar field is then annihilated by the following aôut/down and bôut/down operators:

The “in” and “up” modes (13) form an orthonormal basis of out ôut aˆω`m|outi = 0, ω > 0, bω`m|outi = 0, ω < 0, field modes. The “in” modes (13a) have positive norm when aˆdown|outi = 0, ω > 0, bˆdown|outi = 0, ω < 0. (24) ω > 0, while the “up” modes (13b) have positive norm when ω`m e ω`m e ˆ ωe > 0. We therefore write the quantum field Φ as the following The “out” vacuum is as empty as possible at future null infin- mode sum ity, and also contains no particles ingoing at the future event horizon. X∞ X` (Z ∞ Z 0 ˆ in in în† in Φ = dω aˆω`mφω`m + dω bω`mφω`m `=0 m=−` 0 −∞ 3.2. Observables Z ∞ Z 0 ) We are interested in whether the “in” and “out” vacua are, + dω aûp φup + dω bûp† φup . (19) e ω`m ω`m e ω`m ω`m in fact, identical. Since they have been defined in such a way 0 −∞ that the “out” vacuum is the time-reverse of the “in” vacuum, 1 ˆ ˆ † the expectation value of the scalar field condensate 2 hΦΦ + † 1 | in |2 | up |2 Φˆ Φˆ i will be the same in both states. We therefore consider the The Wronskian relations imply that the quantities Aω` and Aω` are equal. expectation values of the scalar field current and stress-energy 3 out in∗ down up∗ tensor, which, being tensor operators, will be able to distinguish Since Xω` = Xω` and Xω` = Xω` , the mode contributions between the two states. t,k k k k jω`, ttt,ω`, trr,ω` and tθθ,ω` are the same for the “out” modes as The scalar field current operator Jˆµ is given by they are for the “in” modes, and the same for the “down” modes iq † † as for the “up” modes. Therefore the expectation values of the Jˆµ = − Φˆ † DµΦˆ + DµΦˆ Φˆ † − Φˆ DµΦˆ − DµΦˆ Φˆ , 16π corresponding components of the current and stress-energy ten- (25) sor are identical in the “in” and “out” vacuum states. We there- and the stress-energy tensor operator Tˆµν takes the form fore focus our attention on the remaining components, namely ˆr ˆ r the fluxes hJ i and hTt i. 1 † † † Tˆ = D Φˆ D Φˆ + D Φˆ D Φˆ + D Φˆ D Φˆ µν 4 µ ν ν µ ν µ † ρσ † † 3.3. Fluxes of energy and charge +DµΦˆ DνΦˆ − gµνg DρΦˆ DσΦˆ + DσΦˆ DρΦˆ . (26) Expectation values of the current operator in any quantum Expectation values of the current and stress-energy tensor oper- state are conserved [24]: ators in the “in” and “out” vacuum states can be written as sums ˆµ over combinations of the field modes ∇µhJ i = 0. (29) q X∞ Z ∞ h i hin|Jˆµ|ini = dω (2` + 1) jµ,in + jµ,up , For static states as considered here, this gives 32π2 ω` ω` `=0 −∞ K q X∞ Z ∞ h i hJˆri = − , (30) hout|Jˆµ|outi = dω (2` + 1) jµ,out + jµ,down , r2 32π2 ω` ω` `=0 −∞ ∞ where K is a constant whose value depends on the quantum 1 X Z ∞ h i hin|Tˆ |ini = dω (2` + 1) tin + tup , state under consideration. Physically, K is the flux of charge µν 16π µν,ω` µν,ω` `=0 −∞ from the black hole. When K has the same sign as the black ∞ 1 X Z ∞ h i hole charge Q, the black hole is losing charge. h | ˆ | i out down out Tµν out = dω (2` + 1) tµν,ω` + tµν,ω` , (27) Since there is a background electromagnetic field, expecta- 16π −∞ `=0 tion values of the stress-energy tensor are not conserved [24], where the nonzero components of the mode contributions to the but instead satisfy expectation values are (see [23] for details) µ µ ∇ hTˆµνi = 4πFµνhJˆ i, (31) t,k 1 2 2 qQ j = − Nk Xk (r) ω − , ω` r2 f (r) ω ω` r where Fµν is the background electromagnetic field strength. For  Xk∗ r  Xk r  r,k k 2  ω`( ) d  ω`( ) static states on an RN black hole space-time, the t-component j = − f (r) N =    , ω` ω  r dr  r  of this equation can be integrated to give (" 2 # 2 1 qQ ` (` + 1) f (r) 2 tk = Nk ω − + Xk (r) L 4πQK tt,ω` ω r2 r r4 ω` hTˆ ri − , t = 2 + 3 (32) 2 r r  Xk (r)  2 d  ω`   + f (r)   , L dr  r   where is another constant depending on the particular quantum state under consideration. Physically, L is the flux of en-  k∗  k  qQ 2  Xω`(r) d  Xω`(r) ergy from the black hole and L > 0 corresponds to a loss of tk = − 2 ω − Nk =    , tr,ω` r ω  r dr  r  energy by the black hole. (" 2 # 2 1 qQ ` (` + 1) 2 From the mode contributions to the expectation values (28), k k k ∗ up∗ trr,ω` = Nω ω − − Xω`(r) out in down f (r)2r2 r r4 f (r) and using the properties Xω` = Xω` and Xω` = Xω` , we have the results  k  2 d  X (r)  +  ω`   ,    ˆr ˆr ˆ r ˆ r dr r  hout|J |outi = −hin|J |ini, hout|Tt |outi = −hin|Tt |ini, (33) ( 2 2 1 qQ 2 tk = Nk ω − Xk (r) which are to be expected since the “out” vacuum is the time re- θθ,ω` ω f (r) r ω` verse of the “in” vacuum. It is therefore sufficient to study these 2  Xk (r)  2 d  ω`   expectation values in the “in” vacuum state. It is proven in [23] − f (r)r   , (28) dr  r   that these components of the current and stress-energy tensor do not require renormalization, which simplifies the computations k k 2 with tϕϕ,ω` = tθθ,ω` sin θ and k = in, up, out, down labels the greatly. specific mode contribution. The symbol = denotes the imagi- We first consider the form of the expectation values hin|Jˆr|ini ˆ r nary part. and hin|Tt |ini as r → ∞. Using the form of the modes (13), we 4 find, as r → ∞, the following leading order behaviour

∞ Z max{ qQ ,0} q X r+ ω r up 2 6 r2 Tˆr hin|Jˆ |ini ∼ − dω (2` + 1) B , h t i 3 2 qQ ω` 64π r min{ ,0} |ω| 2 r `=0 r+ e r Jˆ (34a) 4 h i i i r qQ r t

∞ ˆ

{ } ˆ Z max ,0 2 2 J r+ T h 1 X ω 2 h

up 2 r 2 ˆ r hin|Tt |ini ∼ − dω (2` + 1) B . r 2 2 qQ ω` + 2 16π r |ω| +

min{ ,0} 0 r `=0 r+ e r 6 6 10

(34b) 10 2 − These are clearly nonzero, and thus the “in” and “out” vacuum 4 states are not the same. Using (30, 32), we find the constants K − and L to be 6 − 1 2 3 4 5 6 7 8 9 10 ∞ Z max{ qQ ,0} q X r+ ω 2 r/r+ K dω ( ` ) Bup , = 3 2 + 1 ω` (35a) 64π min{ qQ ,0} |ω| `=0 r+ e 2 ˆr 2 ˆ r Figure 2: Expectation values of r J and r Tt for the “in” vacuum state for ∞ ` Z max{ qQ ,0} q = Q = 0.8M. The quantity r2hJˆri is constant and negative. The quantity X X r+ 2 1 ω up 2 2 ˆ r L dω ( ` ) B . r hTt i is negative for sufficiently large values of r but positive close to the = 2 2 + 1 ω` (35b) 16π min{ qQ ,0} |ω| event horizon. `=0 m=−` r+ e Both the expectation values (34) involve sums over just the superradiant “up” modes with ωωe < 0. The nonzero expecta- r tion value hin|Jˆ |ini corresponds to an outgoing flux of charge 3.5 as seen by a static observer at a fixed value of the radial coor- ˆ r 3 dinate r r+, while the nonzero expectation value hin|Tt |ini represents an outgoing flux of energy as seen by that static ob- 2.5 i r ˆ server. This is precisely the phenomenon of quantum superra- J h

2 2 diance [12]. The charged black hole spontaneously emits parti- r + r

cles in the superradiant modes. 6 1.5 10

The fluxes (35) contain a nonthermal distribution of parti- − cles, which is present even for extremal black holes for which 1 the Hawking temperature vanishes. Since we are considering a 0.5 massless charged scalar field, there is no exponential suppres- 0 sion of the flux, as seen in the massive case [13–16]. 0 0.5 1 1.5 2 2.5 3 3.5 To calculate numerical values for the expectation values, the q/M

in/up 2 transmission coefficients Bω` are computed by integrating Figure 3: Expectation value of the conserved quantity −r2hJˆri for the “in” vac- the radial equation (9) to obtain the radial modes. These can uum state as a function of the scalar field charge q, with Q = 0.8M. also be inserted directly into the mode sums associated with the ˆr ˆ r expectation values of J and Tt , given in (27), as a check of our numerical results. In Figs. 2–4 we display the components 2 ˆr 2 ˆ r r hin|J |ini and r hin|Tt |ini for Q = 0.8M and a selection of 8 positive values of the scalar field charge q. q/M = 0.32 2 ˆr 6 In Fig. 2, we see that, as expected, r hJ i is a constant −K q/M = 0.48 (30, 35a). Fig. 3 shows the value of K as a function of the 4 q/M = 0.64 i r scalar field charge q for fixed black hole charge Q = 0.8M. t ˆ T

h 2 q/M = 0.8 From (35a), the ﬂux of charge K always has the same sign as 2 r 2 + the black hole charge Q, so that the black hole discharges due to r 0 6

quantum superradiance. As q increases, it can be seen in Fig. 3 10 2 that K increases rapidly. − 2h | ˆ r| i Fig. 4 shows the behaviour of r in Tt in as the scalar field 4 2 ˆ r − charge q varies. From (32), as r → ∞, the quantity r hin|Tt |ini approaches a constant −L. In Fig. 4 we see that L is always 6 − 1 2 3 4 5 6 7 8 9 10 positive (as may be anticipated from (35b)), corresponding to a r/r+ loss of energy by the black hole. The constant L also increases as the scalar field charge q increases for fixed black hole charge. 2 ˆ r Figure 4: Expectation value r hTt i for the “in” vacuum state for selected values ˆ r Close to the horizon, the expectation value hTt i is positive, of the scalar field charge q, with Q = 0.8M. due the second term in (32) and the fact that QK > 0. This 5 10 5 × −

is in contrast to the situation for Starobinskii-Unruh radiation ˆ r 2.3 from a Kerr black hole [25], for which hTt i has the same sign everywhere outside the event horizon. Therefore, at the event horizon, we find a flux of ingoing rather than outgoing energy. 2.2 Furthermore, the magnitude of this ingoing flux at the horizon increases as the magnitude of the scalar field charge increases. r + h ˆ i /r 2.1 The expectation value Tt vanishes when r = r0, where 0 r 4πQK r = . (36) 0 L 2

We note that r0 > 0 for all Q , 0. For all values of q, Q studied, we find that r0 > r+. Fig. 4 seems to indicate that r0 1.9 is independent of the scalar field charge q, for fixed black hole 0 0.5 1 1.5 2 2.5 3 3.5 charge Q. However, there is a slight variation as one can see q/M in Fig. 5. For fixed Q, we find that r increases with q up to a 0 h ˆ ri ˆ r Figure 5: Radial position r0, where the expectation value Tt vanishes, for the saturation point. As q → 0, the expectation value hTt i vanishes “in” vacuum state, as a function of the scalar field charge q, with Q = 0.8M. everywhere outside the black hole, which means r0 is not well- defined in this limit. This is reflected in a loss of accuracy in the numerical estimation of r0 for small values of q. expectation values of the components of scalar field current h ˆ ri The fact that the energy flux Tt has opposite signs close and stress-energy tensor operators in the “in” and “out” vacua to and far from the black hole is reminiscent of the notion of are the same, except for the fluxes of charge and energy. The an “effective” ergosphere [7, 26, 27]. Inside the effective ergo- fact that these fluxes are different means that these two vacuum sphere, the energy of a charged particle can be negative as seen states are not identical. Computing the fluxes is comparatively by an observer at infinity. The presence of the effective ergo- straightforward as these components of the current and stress- sphere enables a classical process of charge and energy extrac- energy tensor operators do not require renormalization [23]. In tion from a charged black hole, namely the charged analogue order to investigate the properties of the “in” and “out” vacua in of the Penrose process. In this process, a particle orbiting the more detail, we would need to examine the other components black hole splits into two other particles with charges of oppo- of these operators, which would require renormalization. site sign. The particle with charge of the same sign as the black hole charge escapes to infinity, whereas the other falls into the Neither the “in” nor “out” vacua considered here are empty black hole, thereby effectively stealing charge from it. For a at both future and past null infinity, unlike the Boulware vac- massless charged particle, the effective ergosphere has outer- uum [28] defined on a Schwarzschild black hole. It is known most radius given by that, as a consequence of quantum superradiance, there is no state empty at both future and past null infinity on a Kerr black q 2 2 2 4 2 hole [25]. Examining whether or not such a state exists for a M + M − Q + q Q /pϕ r , charged scalar field on a charged black hole space-time would ergo = 2 2 2 (37) 1 − q Q /pϕ be an interesting extension of our work here, to which we plan to return in the near future [23]. where pϕ is the particle angular momentum. We note that rergo depends weakly on the scalar field charge q, in analogy with the weak dependence of r0 on q seen in Fig. 5. Acknowledgments 4. Conclusions In this letter we have studied the quantum analogue of classical superradiance for a charged scalar field on an RN black hole V.B. thanks STFC for the provision of a studentship sup- space-time. For a massive field whose Compton wavelength is porting this work, and the Universidade Federal do Para,´ significantly smaller than the size of the black hole, this process for hospitality while this work was in progress. The work is exponentially suppressed [13–16, 20], so we have studied a of R.P.B. and L.C.B.C. is financed in part by Coordenaçao˜ massless scalar field, for which this exponential suppression is de Aperfeiçoamento de Pessoal de N´ıvel Superior (CAPES, absent. We have computed the expectation values of the fluxes Brazil) - Finance Code 001 and by Conselho Nacional de of charge and energy from the black hole for an “in” vacuum Desenvolvimento Cient´ıfico e Tecnologico´ (CNPq, Brazil). state which is as empty as possible at past null infinity. The The work of E.W. is supported by the Lancaster-Manchester- superradiant emission is nonthermal in nature and such that the Sheffield Consortium for Fundamental Physics under STFC black hole loses both charge and mass. grant ST/P000800/1. 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