Introduction
Two key characteristics of black holes (BHs) are the $\textit{event horizon}$ and the $\textit{ergoregion}$. The former represents the "surface of no return", i.e. the boundary of the region of spacetime we cannot communicate with (at least in classical theory), while the latter is a region where there are no timelike static observers and all trajectories (timelike or null) must rotate in the direction of rotation of the BH (frame-dragging). For stationary, rotating spacetimes the existence of an event horizon implies the existence of an ergoregion, but the opposite is not true. Ergoregions are associated with two important astrophysical processes which are both related to the extraction of energy from a rotating BH: First, as described by Penrose, since the energy of a particle as seen by an observer at infinity can be negative inside the ergoregion, energy extraction is possible through a simple decay. Second, the powering of relativistic jets can occur through the Blandford-Znajek process. Although according to the membrane paradigm, jet formation is associated with the BH horizon, Komissarov pointed out that the threading of the ergoregion by magnetic field lines and the subsequent twisting of them due to frame dragging is all that is necessary for the energy creation of a relativistic jet, while a horizon is not. Preliminary force-free numerical simulations of ergostars using the Cowling approximation confirm this hypothesis.
A stationary, asymptotically flat spacetime possesses a timelike Killing vector that asymptotically corresponds to time translations. This vector inside an ergoregion tips over and becomes spacelike, making the conserved total energy of a freely moving particle there negative with respect to the asymptotic observer. A nonaxisymmetric perturbation that radiates positive energy at infinity will make the negative energy in the ergoregion even more negative in order for the conservation of energy to be satisfied. This will lead to a cascading instability that was first discovered by Friedman and recently was put on a rigorous footing by Moschidis. It belongs to the class of "rotational dragging instabilities" whose most famous member is the so-called Chandrasekhar-Friedman-Schutz (CFS) instability (induced by gravitational-radiation) valid for any rotating star, irrespective of its rotation rate. In this paper we call stars that contain ergoregions $\textit{ergostars}$.
The fact that the ergoregion instability was considered "secondary" was not only due to the scarcity of rotating star models exhibiting such behavior, but equally importantly, due to its very long $\textit{secular}$ ($\gtrsim$ gravitational radiation) timescale for instabilities. Although the existence of ergoregions in rotating stars has been questioned, they were found by a number of authors since the first work of Wilson, who employed a compressible equation of state (EoS), differential rotation, and an assumed density distribution. Butterworth and Ipser and more recently Ansorg, Kleinwachter, and Meinel constructed self-consistent, rapidly rotating, incompressible stars containing ergoregions. A larger parameter space was investigated by Komatsu, Eriguchi, and Hachisu (KEH) who presented self-consistent solutions with a polytropic EoS and differential rotation, reaching all the way up to the most extreme toroidal configurations ($R_p/R_e=0$, where $R_p,\ R_e$ are the polar and equatorial radii, respectively).
We have reconstructed a sample of these ergostars and have built some new ones. We have then used these models as initial data in our relativistic evolution code. Simulations were performed on the Blue Waters supercomputer at UIUC, the Extreme Science and Engineering Discovery Environment (XSEDE), and the NASA High-End Computing (HEC) Program at Ames Research Center. All equilibrium models are constructed with the Cook-Shapiro-Teukolsky (CST) code, and subsequently evolved with the ILLINOIS GRMHD adaptive-mesh-refinement code, which employs the Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formulation of the Einstein's equations.
