Abstract
Under some conditions, light boson fields grow exponentially around a rotating black hole, called the superradiance instability. We discuss effects of nonlinear interactions of the boson on the instability. In particular, we focus on the effect of the particle production and show that the growth of the boson cloud may be saturated much before the black hole spin is extracted by the boson cloud, while the nonlinear interactions also induce the boson emission. For application, we revisit the superradiant instability of the standard model photon, axion and hidden photon.
Highlights
Under the Kerr metric, the solution to this equation of the form φ ∝ e−iωt+imφ is found, where m is a quantum number corresponds to angular momentum around the rotating axis and φ is the azimuthal angle
We focus on the effect of the particle production and show that the growth of the boson cloud may be saturated much before the black hole spin is extracted by the boson cloud, while the nonlinear interactions induce the boson emission
For GMBHμ 1 [6], where G is the Newton constant, MBH is the black hole mass, a ≡ a/(GMBH) is the dimensionless spin parameter in the range 0 ≤ a ≤ 1 with a being the parameter appearing in the Kerr metric, which is related to the black hole angular momentum JBH through a = JBH/MBH, r+ = GMBH + (GMBH)2 − a2 represents the event horizon and γn m is a numerical constant for the principal quantum number n and orbital angular momentum quantum number
Summary
If there is a bosonic particle, φ, and the mass is the same order as the horizon radius of a Kerr black hole, the rotational energy of the Kerr black hole is efficiently extracted by φ by the superradiant instability. As the boson cloud grows, the nonlinear interactions become important and may affect the superradiant exponential growth. Once the exponential growth of the amplitude stops at φNL, so does the energy/angular momentum extraction rate by the superradiant instability. In such cases, the energy/angular momentum loss rate of the black hole becomes saturated and constant in time. Compared with the free boson superradiant instability, where the loss rate exponentially grows up, the typical time scale needed for a substantial energy/angular momentum extraction becomes significantly lengthen. In order to calculate the black hole spinning down time correctly, we need to estimate the particle creation rate in the boson cloud. Before going into concrete setups, we below summarize some general aspects of the black hole evolution taking account of nonlinear effects
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