Quantum Treatment of Black Hole Superradiance

TL;DR

The paper delivers a canonical quantum treatment of Kerr BH superradiance by quantizing a massive scalar field in the Kerr background, revealing a complete set of normalizable modes comprising both continuous and discrete spectra. It introduces a pseudo-particle number counting the outside-barrier cloud, showing exponential growth independent of the initial state, while energy conservation is maintained via negative-energy contributions inside the ergoregion. The authors connect this quantum picture to Hawking radiation (via the Unruh state for continuous modes), discuss adiabatic backreaction and mode vanishing at critical boundaries, and examine level transitions under self-interactions, offering a coherent framework that unifies growth, radiation, and backreaction in a quantum setting. The results establish a robust quantum foundation for black hole superradiance and point to future work on dynamical collapse spacetimes and other field types.

Abstract

Rotating black holes can form dense boson clouds through superradiant instability, making Kerr black holes a powerful probe of ultralight massive bosons. Previous studies of black hole superradiance have often treated bosonic fields classically, leaving open questions about how particles are produced and how the clouds grow over time. In this work, we canonically quantize a massive scalar field around a Kerr black hole, providing a fully quantum description of black hole superradiance. We show that the evolution of the particle number in the cloud, as well as the energy and angular momentum of the scalar field, can be consistently explained within the standard framework of quantum field theory in curved spacetime. Furthermore, we prove that the growth of the cloud occurs independently of the choice of initial state. We also explore several phenomena related to a massive scalar field in a rotating black hole spacetime, including Hawking radiation, adiabatic backreaction on the black hole spin, and the direction of level transitions in the presence of self-interactions of the field. Our analysis provides a consistent quantum-mechanical perspective that includes all these phenomena.

Figures (3)

• Figure 1: The Penrose diagram of Kerr BH on the equatorial plane. Region I is the universe we live while region IV is the parallel universe. $\mathcal{H}^\pm$ is the future/past outer horizon, $\mathcal{I}^\pm$ is the future/past null infinity and $i^\pm$ is the future/past timelike infinity. The wavy line corresponds to the singularity.
• Figure 2: All four kinds of modes in the region I of Kerr BH Penrose diagram shown in Fig. \ref{['fig:KerrBHPenrose']}. The "in" and "up" modes represent the wave ingoing and upcoming from $i^-$ and $\mathcal{H}^-$ respectively, while the "out" and "down" modes represent the wave outgoing and down to $i^+$ and $\mathcal{H}^+$ respectively.
• Figure 3: Radial profile $R_{211}(r)$ of the fundamental mode. The BH and field parameters are set as $a/M = 0.99, \mu M = 0.42$. The blue line denotes the real part of $R_{211}$ while the orange line denotes the imaginary part.