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 Par´a,66075-110, Bel´em,Par´a,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 spon- taneously 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 d'2; (1) ticular, for a massive quantum field, the emission rate is sup- pressed 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 8 in −i!er∗ In this case r+ is the location of the black hole event horizon >B e ; r∗ ! −∞; Xin (r) = < !` (13a) and r− is the location of the Cauchy horizon. In this paper we !` > −i!r∗ in i!r∗ :e + A!`e ; r∗ ! 1; restrict our attention to the region exterior to the event horizon. The dynamics of the scalar field Φ is determined by the field and 8 up > i!er∗ −i!er∗ ! −∞ equation up <>e + A!`e ; r∗ ; X (r) = up (13b) µ !` > i!r∗ DµD Φ = 0; (4) :B!`e ; r∗ ! 1; where Dµ = rµ −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 fu- ture 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−i!t modes have the asymptotic forms φ!`m(t; r; θ; ') = N!X!`(r)Y`m(θ; '); (6) r 8 in∗ i!r >B e e ∗ ; r∗; ! −∞; Xout(r) = Xin∗(r) = < !` (14a) where ` = 0; 1; 2;::: is the total angular momentum quantum !` !` > i!r∗ in∗ −i!r∗ :e + A!` e ; r∗ ! 1; number, m = −`; −` + 1; : : : ; ` − 1; ` is the azimuthal angu- lar momentum quantum number, ! the frequency of the mode, and N ! is a normalization constant and Y`m(θ; ') is a spherical har- 8 up∗ > −i!er∗ i!er∗ ! −∞ monic. We have fixed the normalization of the spherical har- down up∗ <>e + A!` e ; r∗ ; X (r) = X (r) = up∗ (14b) !` !` > −i!r∗ monics such that :B!` e ; r∗ ! 1; Z respectively. The “out” modes have no flux ingoing at the future Y`m(θ; ')Y`0m0 (θ; ') sin θ dθ d' = 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 N!, we employ the h i dr 1 Klein-Gordon inner product Φ1; Φ2 , defined by ∗ = ; (8) dr f (r) Z h ∗ ∗ i p µ hΦ1; Φ2i = i DµΦ1 Φ2 − Φ1DµΦ2 −g dΣ : (15) Σ in terms of which the radial equation for X!`(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( ) !`( ) = 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 − ! − : eff( ) = 2 + 1 + ( ) (10) r r in=out 1 up=down 1 N! = p ; N! = p : (16) 4π j!j 4π j!j Near the black hole event horizon, as r ! r+ and r∗ ! −∞, e ! 1 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: 8 2 2 qQ 2 2 up 2 up 2 >−!e = − ! − ; r∗ ! −∞; ! 1 − Ain = ! Bin ; ! 1 − A = ! B ; (17) V (r) ∼ < r+ (11) !` e !` e !` !` eff > 2 :>−! ; r∗ ! 1; and where we have defined the quantity in up !eB!` = !B!` : (18) qQ From the relations (17) we can observe the phenomenon of !e = ! − : (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 iny i 0 0 0 q/M=0.64 aˆ!`m; aˆ!0`0m0 = δ`` δmm δ(! − ! ); ! > 0; 1.02 q/M=0.48 h in iny i 0 ˆ ˆ 0 0 b!`m; b!0`0m0 = δ`` δmm δ(! − ! ); ! < 0; 2 q/M=0.32 | in ω` h up upy i 0 0 0 A q/M=0 aˆ ; aˆ 0 0 0 = δ δ δ(! − ! ); ! > 0; | 1 !`m ! ` m `` mm e hˆup ˆupy i 0 b ; b 0 0 0 = δ``0 δmm0 δ(! − ! ); !e < 0: (20) 0.98 !`m ! ` m An “in” vacuum state can be defined as the state annihilated by 0.96 thea ˆin=up and bˆin=up operators.