Dynamically Stable Ergostars

• Dynamically stable ergostars are compact, rapidly rotating stars with an ergoregion but no event horizon, characterized by unique spacetime properties.
• Their formation critically depends on the equation of state and rotation law, with models emphasizing stiff nuclear matter and strangeon matter in achieving stability.
• GRMHD simulations reveal that while ergostars can produce outflows, their jets are less relativistic than those from black hole systems, offering distinct astrophysical signatures.

A dynamically stable ergostar is a compact, rapidly rotating, self-gravitating star whose spacetime possesses an ergoregion—the region where the timelike Killing vector becomes spacelike and stationary observers cannot exist—while lacking an event horizon. The existence and properties of such ergostars connect deeply with the relativistic structure of neutron stars, the nature of matter at supranuclear densities, and relativistic jet formation in high-energy astrophysics. Detailed investigation reveals that dynamically stable ergostars arise only in specific regions of the stellar parameter space, are highly sensitive to the equation of state (EOS), and may have significant astrophysical implications.

1. Definition and Fundamental Properties

An ergostar is defined as a rotating, axisymmetric compact star with an interior or exterior region—the ergoregion—where the norm of the stationary Killing vector field $\xi^\mu = (\partial_t)^\mu$ becomes positive:

$g_{tt} > 0.$

The ergosurface, marking the boundary of this zone, is given by $g_{tt}=0$. Inside, all worldlines must co-rotate, and no static (zero angular momentum) observer can remain at rest with respect to infinity. Unlike the Kerr black hole, whose ergoregion lies outside a horizon, the ergostar’s ergoregion is bounded entirely within the stellar matter or partially abuts the exterior, but no event horizon is present (Tsokaros et al., 2019, 2002.01473, Ruiz et al., 2020, Xia et al., 5 Jan 2026).

The general form of the stationary, axisymmetric line element employed in constructing these models is:

$$ds^2 = -e^{\gamma+\rho}\,dt^2 + e^{2\alpha}(dr^2 + r^2 d\theta^2) + e^{\gamma-\rho} r^2 \sin^2\theta (d\phi - \omega dt)^2$$

where the metric coefficients depend on $(r,\theta)$ only. The appearance of the ergoregion requires sufficiently rapid rotation and a high degree of compaction, which depends sensitively on both the EOS and the rotation law.

2. Equations of State and Rotational Configurations

The presence and stability of ergostars are strongly dependent on the choice of EOS and the rotation law:

• Causal-core nuclear EOSs: Maximally stiff prescriptions enforce the sound speed $c_s^2 = 1$ (in geometrized units) above a critical matching density. For example, the ALF2_cc EOS adopts the causal core for $$\rho_0 \geq 2.7 \times 10^{14}\ \mathrm{g\ cm^{-3}}$$, enabling increased compaction and high mass support (Tsokaros et al., 2019, 2002.01473).
• Piecewise-polytropic and SLy-based causal EOSs: Piecewise-polytropic representations cap the low-density regime, with a causal core at higher densities (SLycc1, SLycc2, SLycc4). This permits systematic study of how ergoregion formation is linked to variations in stiffness and surface density (2002.01473, Ruiz et al., 2020).
• Strangeon matter EOS: Recent models employ a Lennard–Jones-type potential to describe self-bound, quark-clustered strangeon matter. These EOSs support a broad domain of stable ergostars even under uniform rotation, contrasting with nuclear models where only strong differential rotation yields stability (Xia et al., 5 Jan 2026).

Rotational Laws

• Uniform rotation ($\Omega = \mathrm{const}$): For standard nuclear EOSs, uniformly rotating models rarely permit ergoregions and, when present, are typically unstable except under exotic EOSs such as strangeon matter (Xia et al., 5 Jan 2026, 2002.01473).
• Differential rotation (“$j$-constant” law): Given by $$\Omega(r,\theta) = \Omega_c [1 + (r\sin\theta/A)^2]^{-1}$$, this law allows for steep rotation gradients, enabling ergoregion formation for a wider set of EOSs. Mild differential rotation ($\hat{A}^{-1} = R_e/A \sim 0.1$–$0.2$) creates a robust stable ergostar band (Tsokaros et al., 2019, 2002.01473).

3. Dynamical Stability Analysis and Parameter-Space Mapping

Dynamical stability is a key concern for ergostar viability:

• Turning-point method: Along constant-$J$ (angular momentum) sequences, the maxima of gravitational mass $M(\rho_c;J)$ mark the onset of secular (and nearly dynamical) instability. Models to the left of this line in the $M$–$\rho_c$ plane are dynamically stable (Xia et al., 5 Jan 2026).
• Dynamical simulations: General-relativistic hydrodynamics and magnetohydrodynamics (GRMHD) simulations confirm the existence of dynamically stable ergostars under both unperturbed and perturbed initial data. For instance, models with ALF2_cc EOS and mild differential rotation remain dynamically stable beyond $30\,P_c$ (rotational periods), with $\langle |\Delta\rho_c/\rho_c| \rangle < 1\%$ and nonaxisymmetric mode amplitudes at noise level ($\sim10^{-6}$) (Tsokaros et al., 2019, Ruiz et al., 2020).
• Parameter-space mapping: Systematic scans over central density, EOS parameters, and rotation law reveal that dynamically stable ergostars exist only in a narrow band: high compactness ($C \simeq 0.3$), moderate $T/|W| \sim 0.2$–$0.3$, and for EOSs with a significant surface density (2002.01473).

EOS Type	Rotation Law	Stability Domain
Causal-core ALF2	Mild differential	Robust, $A/R_e \sim 0.2$
SLycc1	Mild diff./Uniform	Reduced, fine-tuning needed
Strangeon matter	Uniform	Broad, high stability

4. Magnetohydrodynamic Effects and Jet Launching

The interaction of ergoregions with strong magnetic fields is central to the ergostar hypothesis as an alternative engine for short gamma-ray bursts (sGRBs):

• GRMHD simulations: Simulations show that magnetized ergostars (with poloidal fields $B_p \sim 10^{15}\,$G) can launch mildly relativistic outflows, but only achieve maximum Lorentz factors $\Gamma_L \lesssim 3$, far below the $\Gamma_L \gtrsim 20$–100 required for sGRB jets (Ruiz et al., 2020). The force-free parameter $B^2/(8\pi\rho_0) \lesssim 10$, indicating that the jets do not reach the strongly magnetized, Poynting-dominated regime seen in Blandford–Znajek outflows from black holes.
• Jet comparison (NS/ergostar/BH-disk): Both normal hypermassive neutron stars and ergostars produce wide-opening, slowly accelerated magnetic funnels, while BH-disk remnants generate narrower, more relativistic jets ($\Gamma_L \gtrsim 100$) due to the presence of a horizon allowing efficient BZ extraction (Ruiz et al., 2020).
• Collapse and post-collapse properties: Strongly magnetized ergostars may undergo MRI-driven collapse, forming a black hole with spin $a/M_{BH}\simeq0.93$ and a small remnant disk ($\sim4\%$ of initial baryonic mass) (Ruiz et al., 2020).

5. Equation of State, Compactness, and Ergoregion Morphology

• Compactness threshold: Ergoregions first appear at a compaction $C = M/R_e \simeq 0.3$ for ALF2_cc and similar EOSs. This threshold necessitates both a stiff core and sufficient rotational kinetic support (Tsokaros et al., 2019).
• Ergoregion geometry: In dynamically stable models, the ergoregion tends to occupy an oblate, toroidal shell around the rotational axis, with the ergosurface forming a “doughnut” of $g_{tt} > 0$ within the stellar interior. Typical inner and outer radii in the equatorial plane are $r_{in} \simeq 7\,$km and $r_{out} \simeq 9\,$km (Tsokaros et al., 2019).
• Strangeon ergostars: The existence of a self-bound surface and enhanced EOS stiffness allows strangeon-matter ergostars to access stable ergoregion domains even for uniform rotation, with mass bounds $M_{\rm min–ergo}\sim2.8$–$3.0\,M_\odot$ and $M_{\rm max–ergo}\sim3.3$–$4.0\,M_\odot$ (Xia et al., 5 Jan 2026).

6. Extractable Rotational Energy and Astrophysical Implications

• Extractable energy estimates: The maximum energy release along constant-baryon-mass, decreasing-$J$ evolutionary paths is

$$\Delta E = M_{\rm initial}(J_i, M_b) - M_{\rm final}(J_f, M_b)$$

For strangeon ergostars, $\Delta E_{\rm max} \simeq (0.005$–$0.05) M_\odot$ ($10^{52}$–$5\times10^{52}$ erg), ample to power canonical sGRB outflows (Xia et al., 5 Jan 2026).

• Comparative Poynting luminosity: The outgoing EM luminosity $L_{\rm EM}\sim10^{53}$ erg s$^{-1}$ is comparable for ergostars, normal neutron stars, and BH-disk remnants, highlighting that observations of $L_{\rm EM}$ alone do not distinguish these possibilities in the “mass-gap” range $M\sim3$–$5\,M_\odot$ (Ruiz et al., 2020).
• sGRB progenitor viability: For conventional nuclear EOSs, ergostars do not yield highly relativistic, force-free jets required for sGRBs, making the presence of an event horizon crucial. However, in the strangeon scenario, robust ergoregions and large $\Delta E$ may revitalize the ergostar engine hypothesis provided efficient energy extraction mechanisms operate (Xia et al., 5 Jan 2026).

7. Current Limitations and Prospects

Dynamically stable ergostars are favored in models featuring a finite-density surface (self-bound stars) and mild rather than extreme differential rotation. Nuclear EOSs with only moderate stiffness yield a limited or nonexistent stable ergostar domain unless specifically tuned. By contrast, strangeon-matter stars possess a broad and robust stable ergostar region even for uniform rotation.

The central open issues include:

• Determining the real EOS of supranuclear matter, which directly constrains the astrophysical relevance of ergostars.
• Assessing realistic post-merger evolution, including secular processes (magnetic braking, gravitational radiation, viscosity) and the competition between differential-rotation damping and jet formation timescales (Tsokaros et al., 2019, 2002.01473).
• Disentangling gravitational-wave and electromagnetic signals from ergostar and black-hole remnant scenarios in light of multi-messenger detections.

The discovery and modeling of dynamically stable ergostars has dispelled the notion that ergostars are generically unstable, and has established firm links between the microphysics of dense matter and the macroscopic appearance of relativistic outflows and gravitational-wave sources (Tsokaros et al., 2019, 2002.01473, Ruiz et al., 2020, Xia et al., 5 Jan 2026).