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Superradiant Amplification by Stars and Black Holes

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Superradiant Amplification by Stars and Black Holes Superradiant amplification by stars and black holes Vitor Cardoso1,2 , Richard Brito1, Jo˜ao L. Rosa1, 1 CENTRA, Departamento de F´ısica, Instituto Superior T´ecnico – IST, Universidade de Lisboa – UL, Avenida Rovisco Pais 1, 1049 Lisboa, Portugal and 2 Perimeter Institute for Theoretical Physics Waterloo, Ontario N2J 2W9, Canada It has long been known that dissipation is a crucial ingredient in the superradiant amplification of wavepackets off rotating objects. We show that, once appropriate dissipation mechanisms are included, stars are also prone to superradiance and superradiant instabilities. In particular, ultra- light dark matter with small interaction cross section with the star material or self-annihilation can trigger a superradiant instability. On long timescales, the instability strips the star of most of its angular momentum. Whether or not stationary hairy configurations exist, remains to be seen. PACS numbers: 04.70.-s,04.80.-y,12.60.-i,11.10.St I. INTRODUCTION long as dissipation is included [23]. The purpose of this work is to compute explicitly the amplification factors when a toy model for dissipation is used, and to show Rotating bodies with internal degrees of freedom – that, within this toy model, stars are superradiantly- where energy can be dumped into – display superradi- unstable against massive scalar-field fluctuations. ance. This means that the scattering of low-frequency Our results make possible the existence of hairy stars, waves extracts rotational energy away from the body and supported solely by superradiance. For dark matter mod- is used to amplify incoming radiation [1–3]. When the els comprised of light degrees of freedom, our results open object is a black hole, dissipation is intrinsically built the door to constrain the interaction cross section with in the problem under the form of a one-way membrane: ordinary matter or the self-annihilation cross-section. the event horizon. Black hole superradiance was in fact shown to exist and to trigger interesting phenomena. These include black hole bombs, once the black hole is II. SETUP surrounded by some confining medium, giving rise to ex- ponentially growing modes [4–8]. The scattering of mas- sive fields belongs to this category: the mass term effec- We are interested in describing ultra-light bosonic de- tively confines the field giving rise to floating orbits [9] grees of freedom, described by a scalar minimally cou- or to superradiant instabilities, which extract rotational pled to gravity, and interacting with a star. Dissipation energy away from the black hole [10–16]. The instabil- can be modelled in several – and complex – ways. For ity leads to a depletion of mass and angular momen- instance, one can let the scalar couple to spacetime de- tum from the black hole, resulting in observationally- grees of freedom, or one can consider the coupling to mi- interesting “holes” in the Regge-plane [17–19]. As a croscopic degrees of freedom, describing the interaction consequence, the observation of supermassive black holes of self-annihilation of – for example – dark matter with alone can impose stringent bounds on the mass of ultra- a star. The description of a dissipative system within light bosons [13, 15]. Finally, in multi-scalar theories or full General Relativity for systems (such as stars) with theories with a complex scalar, superradiance is the re- many internal degrees of freedom is a very complex prob- sponsible for new hairy black hole solutions [20] (whether lem [24]. or not these solutions are generically approached as the Instead, we follow Zeldovich’s seminal work and con- end-state of the superradiant-instability is discussed in sider a toy model for absorption, by modifying the Klein- Ref. [18]). For a Review, we refer the reader to Ref. [3]. Gordon equation inside the star and in a frame co- rotating with the star as [21] Given the rich phenomenology of black hole superradi- ance, one might ask whether those results carry through ∂Φ ✷Φ+ α = µ2Φ . (1) to other self-gravitating objects. Ideal stars do not dis- ∂t play superradiance and none of the associated instabili- ties (except for possible ergoregions instabilities [3]). But Here, µ~ is the mass of the scalar field. The α term is realistic stars do dissipate energy and they have several added to break Lorentz invariance, and describes absorp- channels available for it. The original arguments by Zel- tion on a timescale τ 1/α, in a frame co-rotating with dovich already made clear that any classical system with the star. The constant∼ 1/α has dimensions of time, or dissipation will be prone to superradiant scattering [1, 21] length in geometrical units. Contact with particle physics (see also Ref. [22] who studied the correspondence be- can be made by considering the particle associated with tween superradiance and tidal friction on viscous Newto- the scalar field and identifying 1/α with the mean free nian anisotropic stars). In fact, stars in full General Rel- path ℓ of the particle inside the star. In other words, for a ativity were recently shown to display superradiance, as particle with cross section σ to decay into other particles 2 via interaction with say a neutron of mass mN , then when transforming to a frame at rest with the star, and using (1). α = nσ , (2) where n is the nucleon number density in the star. One can also express the dissipation parameter α with the III. SUPERRADIANT AMPLIFICATION help of the star’s total mass M, in the dimensionless com- FACTORS OF MASSLESS FIELDS bination GM ρσ -2 Mα = 2 10 c mN -3 10 R=50M ρ M σ -4 10 0.92 15 −3 −41 2 . (3) ∼ 10 Kg m M⊙ 10 cm -5 -1 10 2 | -6 Thus, the dimensionless combination can be of order one in 10 A R=10M -7 for interesting dark matter candidates [25–27] (and ref- / | 10 2 erences therein). The constant α could also be used sim- | -8 out 10 A ilarly to describe dark matter self-annihilation. | -9 10 R=5M It is easy to see, following Zeldovich, that if the fre- -10 10 numerical quency in the accelerated frame is ω and the field behaves -11 analytical −iωt+imϕ 10 as e then in the inertial frame the azimuthal -12 coordinate is ϕ = ϕ′ Ωt, and hence the frequency is 10 0 0.002 0.004 0.006 0.008 0.010 ω ω′ = ω mΩ. In other− words, the effective damping pa- M rameter−αω′ becomes negative in the superradiant regime and the medium amplifies – rather than absorbing– radi- FIG. 1. Superradiant amplification of a scalar field by a rotat- ation [1, 3, 21]. This procedure can be used to derive the ing star, where dissipation is modeled through equation (1). wave equation of a scalar field in a rotating black hole Here, Mα =0.1, MΩ=0.01. We find that the amplification background (up to first order in rotation), starting from factor scales with the dissipation parameter α for small Mα. that of a scalar field in a Schwarzschild geometry. As we For non-compact, slowly-rotating stars, the amplification fac- will shortly see, it also allows to predict the amplification tor (solid lines) is well described by the analytic result (8), factors of scalar fields in a Kerr background. Thus, while shown as a dashed curve in the plot. only a phenomenological approach to the problem, the inclusion of the Lorentz-violating term is well motivated To solve the Klein-Gordon equation (1), we use the and consistent with known results. In Appendix A we following ansatz for the field, work out another model for dissipation, inspired in the theory behind dissipation in fluids [28]. The results are Ψ(r) Φ(t, r, ϑ, ϕ)= e−iωt+imϕP (cos ϑ) , (6) perfectly consistent with the toy model (1) adopted in r l the main body of this work. As is clear from the previous discussion, superradi- where Pl are Legendre functions. To solve the equation ance requires only rotation and dissipation. The par- inside the star, we first transform to a frame co-rotating ticular background geometry or material is only relevant with the star (which amounts to replacing ϕ ϕ Ωt, or ω ω mΩ), impose regularity at the center→ and− then to quantitatively describe the phenomena. We will focus → − exclusively on backgrounds describing a constant density, match the radial function and its derivative. For now we perfect fluid star in General Relativity as a proxy for the focus on zero field mass, µ = 0. At large distances, the generic case: solution takes the form 2 iωr −iωr 2 2Φ 2 dr 2 2 Ψ= Aoute + Aine , (7) ds = e dt + + r dΩ , (4) − 1 2m(r)/r − where the subscripts denote an out-going and an in-going with piece (due to the time dependence of the ansatz for the 4π 1 2Mr2/R3 1 2M/R field). Thus, the asymptotic solution allows one to frame m = ρr3 , P = ρ − − − , 2 3 this in a context of a scattering experiment. We define 3 3 1 2M/R 1 2Mr /R ! 2 2 p − − p− the amplification factor as Aout / Ain 1. Positive 3 1 | | | | − eΦ = 1 2M/R p1 2Mr2/R3p. (5) amplification factors correspond to superradiant ampli- 2 − − 2 − fication from the star. Negative factors correspond to Here, M,Rp are the star’sp mass and radius, and are related overall absorption by the star. 4π 3 to the density ρ via M = 3 ρR . Our numerical results are summarized in Fig. 1 for Notice that we take an exact general-relativistic solu- Mα = 0.1, MΩ = 0.01.
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