Light rings as observational evidence for event horizons: long-lived modes, ergoregions and nonlinear instabilities of ultracompact objects

Abstract

Ultracompact objects are self-gravitating systems with a light ring. It was recently suggested that fluctuations in the background of these objects are extremely long-lived and might turn unstable at the nonlinear level, if the object is not endowed with a horizon. If correct, this result has important consequences: objects with a light ring are black holes. In other words, the nonlinear instability of ultracompact stars would provide a strong argument in favor of the "black hole hypothesis," once electromagnetic or gravitational-wave observations confirm the existence of light rings. Here we explore in some depth the mode structure of ultracompact stars, in particular constant-density stars and gravastars. We show that the existence of very long-lived modes -- localized near a second, stable null geodesic -- is a generic feature of gravitational perturbations of such configurations. Already at the linear level, such modes become unstable if the object rotates sufficiently fast to develop an ergoregion. Finally, we conjecture that the long-lived modes become unstable under fragmentation via a Dyson-Chandrasekhar-Fermi mechanism at the nonlinear level. Depending on the structure of the star, it is also possible that nonlinearities lead to the formation of small black holes close to the stable light ring. Our results suggest that the mere observation of a light ring is a strong evidence for the existence of black holes.

Figures (5)

• Figure 1: Examples of the potential governing linear perturbations of a static ultracompact star. The black solid line and the red dashed line correspond to $l=10$ gravitational axial perturbations of a uniform star with $R=2.3 M$ and of a gravastar with $R=2.1M$, respectively.
• Figure 2: Real and imaginary parts of the long-lived modes of a uniform star for different compactness (left panels) and for a gravastar with $R=2.2M$ (right panels). Solid lines are the WKB results, whereas markers show the numerical points (when available) obtained using direct integration or continued fractions. For uniform stars we show gravitational axial modes, whereas for gravastar we show both axial modes (red circles) and gravitational polar modes with $v_s=0.1$ (green squares), where $v_s$ is related to the speed of sound on the shell Pani:2009ss. Note that the modes of a static gravastar become isospectral in the high-compactness regime Pani:2009ss.
• Figure 3: Top panel: gravitational axial eigenfunctions of an ultracompact star for $l=2$ and $l=10$. The radius of the star, $R=2.3M$, is marked by a vertical line. High-$l$ modes correspond to eigenfunctions which are localized near the stable light ring. Middle and bottom panels: time evolution of a scalar Gaussian wavepacket with width $\sigma=4M$ centered at $r_0=6M$ in the background of a constant-density star of radius $R=2.3M$ for $l=2$ and $l=10$. The waveform extracted at $r=0$ (middle panel) and $r=40M$ (bottom panel). Note that the Schwarzschild ringdown phase lasts until $t\sim 60 M$.
• Figure 4: Scalar eigenfunctions of an ultracompact star with $R=2.3M$ for $m=0$ and $l=6,10,20$ (from left to the right). We find that the eigenfunctions have a typical width that scales as $l^{-1}$ in the angular direction and a width in the radial direction that depends on the model used for the star, but typically ranges between $l^{-0.4}-l^{-0.8}$. Therefore, the "aspect ratio" of the perturbation $\sim l^{0.6}-l^{0.2}$ grows in the large-$l$ limit and the perturbation becomes more and more elongated along the radial direction.
• Figure 5: Pictorial description of the nonlinear evolution of a perturbed ultracompact object. The figure represents the equatorial density profile of the object. The solid circumference represents the unperturbed surface, whereas the dashed line represents the stable light ring at its interior. Solid circles represent condensation of nonlinear-growth structures which are the bi-product of the DCF instability. The core is left unperturbed and is now a less compact -- and therefore stable -- configuration. Likewise, the the solid circles are also stable and subsequent time evolution presumably leads to a fall-back on the core. Gravitational radiation, generated during this and subsequent repetitions of this process will lead to loss of mass and possibly a reduction of the star's compactness.