Nonlinear evolution of the ergoregion instability: Turbulence, bursts of radiation, and black hole formation

TL;DR

This work investigates the nonlinear evolution of the ergoregion instability in horizonless, rapidly spinning compact objects (ergostars) by numerically evolving a massless vector field on a spinning boson star background within full general relativity. Using axisymmetric Einstein-Maxwell-Klein-Gordon dynamics with a linearly unstable $m=1$ vector mode, it demonstrates that nonlinear backreaction accelerates the instability, induces large-amplitude bursts of gravitational and vector radiation, and triggers a gravitational cascade transferring energy to higher-$\ell$ vector modes. The eventual outcome is the collapse of the ergostar into a rapidly spinning black hole with $a/M\approx 0.95$, with the bursts showing frequencies and decay rates close to the quasinormal modes of the final Kerr black hole. These results imply horizonless ultra-compact objects are prone to collapse under ergoregion instability and suggest distinctive gravitational-wave signatures that could aid in distinguishing black-hole mimickers in observations.

Abstract

Spacetimes with an ergoregion that is not connected to a horizon are linearly unstable. While the linear regime has been studied in a number of settings, little is known about the nonlinear evolution of this ergoregion instability. Here, we investigate this by numerically evolving the unstable growth of a massless vector field in a rapidly spinning boson star in full general relativity. We find that the backreaction of the instability causes the star to become more gravitationally bound, accelerating the growth, and eventually leading to black hole formation. During the nonlinear growth phase, small scale features develop in the unstable mode and emitted radiation as nonlinear gravitational interactions mediate a direct turbulent cascade. The gravitational wave signal exhibits bursts, akin to so-called gravitational wave echoes, with increasing amplitude towards black hole formation.

Figures (9)

• Figure 1: Linear frequencies $\omega_R$ and growth rates $\omega_I$ of the most unstable $m=1$ massless vector test field configurations along the family of BS solutions with $\sigma=0.2$ and $\tilde{m}=3$. Error bars indicate the uncertainty of our numerical methods. The size of the ergoregion increases with $\mu M_0$. These are compared to the corresponding quantities for a massless scalar test field on this family of spacetimes obtained in Ref. Siemonsen:2025wib.
• Figure 2: Evolution of the probe field's angular momentum $J_A$, for three initial amplitudes. The curves are aligned in time at $t_0$, defined (roughly) as the peak of $|J_A|$. The dashed line indicates test field growth starting from $-J^0_A/J_0=0.004$.
• Figure 3: The unstable field's angular momentum density $\rho_J$ at a few select times during the strongly nonlinear phase around $t_0$. The solid black lines indicate where $g_{tt}=0$ (the ergosurface, when the system is stationary with respect to $t$). Gray lines show surfaces with $|\Phi|/\max|\Phi(t=0)|=10^{-1}$ (dotted) and $10^{-3}$ (dashed). For visual clarity, we rescale the angular momentum density in a few panels (as indicated by the labels). The apparent horizon's interior is approximately indicated by the black area. The $z$-axis is the symmetry axis.
• Figure 4: (left) The gravitational wave emission (top panel) and massless vector emission (bottom panel) for several different polar modes $\ell$ of the (spin-weighted) spherical harmonic decomposition of the Newman-Penrose scalar $\Psi_4$ and massless vector projection $\chi$ on a coordinate sphere of radius $r$, respectively. Recall, $\Psi_4$ contains only $m=0$ modes and $\chi$ only $m=1$ modes. In both insets, we also compare the burst decay timescales with the gravitational and electromagnetic quasi-normal modes of a Kerr black hole with spin $a/M=0.95$Berti:2005ysBerti:2009kk. The frequencies roughly match those shown here for all $\ell$. The time $t-t_0$ an apparent horizon is found is indicated in gray (and labeled "BH"). (right) The spectrum of (time-averaged) massless vector modes $\chi_\ell$ during the peak of burst, where "Burst 0" labels the last such burst in the sequence prior to black hole formation.
• Figure 5: The light ring structure of the BS solution considered throughout the main text: $R$ is the star's radius (as defined in Ref. Siemonsen:2020hcg), $r^u_{\rm polar}$ ($r^s_{\rm polar}$) is the radial coordinate at which unstable (stable) polar light rings intersect the equatorial plane, $H_\pm$ are the effective potentials for equatorial null geodesics (defined in the text), and the gray shaded area is the coordinate size of the ergoregion in the equatorial plane.