Instability of the massive Klein-Gordon field on the Kerr spacetime

TL;DR

The paper demonstrates a bound-state instability for a massive scalar field around Kerr black holes arising from superradiance, computed via a continued-fraction method adapted from quasinormal-mode analyses. It shows bound states exist with complex frequencies, and the instability is strongest for the co-rotating l=1, m=1 mode at Mμ ≈ 0.42 and a near-extremal spin, yielding a maximum growth rate τ^{-1} ≈ 1.5×10^{-7} (GM/c^3)^{-1}. The analysis covers Schwarzschild and Kerr cases, revealing hydrogenic behavior in the non-relativistic limit and rotation-induced spectral splitting, with precise upper limits on growth rates across a range of spins. While astrophysical black holes are likely unaffected due to large Mμ, the results have potential relevance for primordial black holes and ultra-light bosons, informing searches for new physics in strong gravity contexts.

Abstract

We investigate the instability of the massive scalar field in the vicinity of a rotating black hole. The instability arises from amplification caused by the classical superradiance effect. The instability affects bound states: solutions to the massive Klein-Gordon equation which tend to zero at infinity. We calculate the spectrum of bound state frequencies on the Kerr background using a continued fraction method, adapted from studies of quasinormal modes. We demonstrate that the instability is most significant for the $l = 1$, $m = 1$ state, for $M μ\lesssim 0.5$. For a fast rotating hole ($a = 0.99$) we find a maximum growth rate of $τ^{-1} \approx 1.5 \times 10^{-7} (GM/c^3)^{-1}$, at $M μ\approx 0.42$. The physical implications are discussed.

Figures (7)

• Figure 1: The frequencies of the lowest $l = 1$ quasinormal modes. The plot shows the QNM frequency as a function of black hole rotation $a$, for a variety of field masses, $\mu = 0$, $0.1$, $0.2$ and $0.3$. The $m = 1$ (right), $m = 0$ (middle) and $m = -1$ (left) branches are shown. The points shown are for the values $a = 0$, $0.1$, $0.2$, $0.3$, $0.4$, $0.5$, $0.6$, $0.7$, $0.8$, $0.9$, $0.95$, $0.99$ and $0.995$.
• Figure 2: Massive scalar $(s=0)$ and spinor $(s=1/2)$ bound state frequencies of the Schwarzschild hole. The upper plots show the real component of energy (i.e. the oscillation frequency), and the bottom plots show the imaginary component (i.e. the decay frequency), as a function of gravitational coupling $M\mu$. The left plots compare the $l =0$ scalar ground state with the $j = 1/2$ spinor ground state. The right plots compare the $l = 1$ (scalar) and the $j = 1/2$ and $j = 3/2$ (spinor) levels.
• Figure 3: The complex frequencies of the lowest-energy Schwarzschild bound states up to $l = 8$. The top plot shows the oscillation frequency $\text{Re}(\omega / \mu)$, and bottom plot shows the decay rate $\text{Im}(\omega / \mu)$, as a function of the mass coupling $GM\mu / \hbar c$.
• Figure 4: Frequency spectrum of the $l =1$, $m = -1 \ldots 1$ bound states at $a = 0.99$. The top plot shows the oscillation frequency $\text{Re}(\omega / \mu)$, and bottom plot shows the damping rate $\text{Im}(\omega / \mu)$,, as a function of mass coupling $M\mu$.
• Figure 5: Bound state frequencies of the $l =1, m = 1$ state for a range of rotation speeds $a$. The top plot shows the oscillation frequency $\text{Re}(\omega / \mu)$, and bottom plot shows the damping rate $\text{Im}(\omega / \mu)$, as a function of mass coupling $M\mu$.