Ergoregion instability in bosonic stars: scalar mode structure, universality, and weakly nonlinear effects

Ultracompact spinning horizonless spacetimes with ergoregions are subject to the ergoregion instability. We systematically investigate the instability of a massless scalar field in a variety of rapidly spinning Proca stars and boson stars using WKB-, frequency-, and time-domain methods. We find universal features in the mode structure: the onset of the instability is signaled by a zero-mode, the mode frequencies and growth rates are related by a simple scaling relation in the small-frequency limit (as found for Kerr-like objects), the mode frequencies approach the orbital frequency of counter-rotating stably trapped null geodesics in the eikonal limit, and for each unstable azimuthal mode only a finite number of overtones and polar modes are exponentially growing. The e-folding times are as short as $τ\sim 10^4 M$ (in terms of the spacetime’s ADM mass $M$). Interestingly, we find a near universal relationship between the frequencies and growth rates across all bosonic stars and also compared with Kerr-like objects. Furthermore, we show that weakly nonlinear backreaction of the instability induces a shift in growth rates as well as emission of gravitational waves; we find evidence that these effects lead to an amplification of the unstable process. This suggests that strongly nonlinear interactions are important during the gravitational saturation of the instability.

💡 Research Summary

This paper presents a comprehensive investigation into the ergoregion instability within rapidly rotating, ultracompact bosonic stars, focusing on boson stars (BSs) and Proca stars (PSs). Ergoregions, regions where no stationary observers can exist, develop in such horizonless spacetimes and are known to trigger linear instabilities for massless fields, as established by Friedman. The instability mechanism involves the creation of negative-energy states within the ergoregion, compensated by the emission of positive energy to infinity, leading to exponential growth of field perturbations.

The authors employ a multi-method approach to analyze the instability of a massless scalar test field on fixed bosonic star backgrounds. They compare results from: (1) a WKB method valid in the eikonal limit (large azimuthal number m), (2) a frequency-domain direct integration method that assumes slow rotation to separate the angular dependence, and (3) fully numerical time-domain simulations which are the most accurate but computationally expensive. This cross-validation highlights the limitations of the approximate methods; for instance, the WKB method can overestimate growth rates by orders of magnitude for low m, while the frequency-domain method, though efficient, may deviate from time-domain results due to its slow-rotation assumption.

The core of the work involves applying these methods to several distinct families of rotating bosonic star solutions (Axionic, KKLS, Solitonic BSs, and PSs), all possessing ergoregions. The analysis reveals universal characteristics in the structure of the unstable scalar modes:

• The onset of instability is marked by the appearance of a zero-frequency mode (ω_R ≈ 0).
• In the small-frequency limit, the growth rate ω_I and real frequency ω_R obey a simple scaling relation: ω_I M ∝ |ω_R M|^{2ℓ+1}.
• In the eikonal limit (ℓ = m → ∞), the mode frequencies ω_R approach ℓω_-, where ω_- is the orbital frequency of counter-rotating, stably trapped null geodesics (the light ring).
• For each unstable azimuthal mode m, only a finite number of accompanying polar modes (ℓ ≥ m) and radial overtones (n) are exponentially growing.
• The e-folding timescales can be as short as τ ∼ 10^4 M, comparable to or shorter than those found for Kerr-like compact objects. A particularly striking finding is a near-universal relationship between the frequencies and growth rates, which holds not only across the different bosonic star families but also when compared to unstable modes around Kerr-like objects. This suggests the instability’s fundamental properties are dictated more by the spacetime’s geometry (the presence of an ergoregion and light ring) than by the specifics of the matter sourcing it.

Moving beyond linear theory, the paper explores weakly nonlinear effects that become relevant as the unstable field amplitude grows. When the field’s energy and angular momentum become significant, their gravitational backreaction slightly modifies the background spacetime. This backreaction induces a shift in the unstable mode’s frequency and growth rate and opens an additional energy-loss channel via gravitational wave emission. Intriguingly, for the case of a Proca star, the analysis finds evidence that these weakly nonlinear effects can amplify the instability, increasing its growth rate. This indicates that the linear instability may develop even more violently in the early nonlinear regime than predicted.

The conclusion emphasizes that these findings strongly suggest strongly nonlinear interactions will be crucial in determining the final, gravitational saturation of the ergoregion instability—whether it leads to collapse, dispersal, or a transition to a different stable configuration. The work thus lays essential groundwork and provides motivation for future fully nonlinear numerical relativity simulations to unravel the ultimate fate of unstable spinning bosonic stars.