Frozen Stars: Horizonless Black Hole Mimickers

Updated 18 March 2026

Frozen stars are ultracompact, horizonless, and regular objects that mimic black holes by featuring an infinite redshift near a critical shell sourced by anisotropic matter.
Modified gravity and higher-curvature theories yield precise frozen star solutions that avoid singularities and event horizons, offering novel insights into quantum gravity and thermodynamics.
Observationally, frozen stars can be distinguished from true black holes by potential features such as nonzero tidal Love numbers and gravitational-wave echoes from their critical surfaces.

A frozen star refers to a class of ultracompact, horizonless, non-singular (or regulated-singularity) objects in gravitational theory whose external geometry is observationally indistinguishable from that of a black hole, yet whose interior is regular and sourced by highly anisotropic matter. The concept has evolved from early ideas about the slow collapse of massive stars (the "frozen star" as a pre-event-horizon description of Schwarzschild black holes in 20th-century literature), through explicit models in modified and higher-curvature gravity, to contemporary incarnations as precise black-hole mimickers—objects that evade both central singularities and event horizons, and present key implications for quantum gravity, black-hole thermodynamics, and gravitational-wave astrophysics.

1. Historical Context and Etymology

The term “frozen star” originated in the 1960s to describe the appearance, under the classical Schwarzschild solution, of collapsing stars whose surfaces froze in time as observed from infinity: as an infalling surface approaches the gravitational radius $r_s = 2GM/c^2$ , the redshift diverges and signals are emitted ever more slowly, leading to the surface asymptotically "freezing" at the Schwarzschild radius in coordinate time (Trimble, 2014). This interpretation predates and provided the conceptual precursor to modern black-hole physics, where "frozen star" was largely supplanted by the terminology of "event horizon" and "black hole."

From an observational standpoint, the frozen star concept motivated early searches for compact objects in binary systems, utilizing mass-function techniques to infer unseen massive companions—work that led to the eventual discovery of black holes, most notably Cygnus X-1 (Trimble, 2014).

2. Mathematical Structure and Physical Conditions

Frozen stars in modern theory are defined by their metric structure and matter sourcing:

Metric: Typically spherically symmetric, Schwarzschild-like exterior, and an interior metric that is nearly (or exactly) null—inducing infinite or extremely large redshift for all shells at $r < R$ :

$ds^2 = -\varepsilon^2 dt^2 + \varepsilon^{-2} dr^2 + r^2 d\Omega^2, \quad 0 < \varepsilon^2 \ll 1$

with $\varepsilon^2 \rightarrow 0$ reproducing an exact freezing of local clocks (Brustein et al., 2023, Brustein et al., 2021).

Matter Content: The canonical frozen star is sourced by an anisotropic fluid with maximally negative radial pressure $p_r = -\rho$ and vanishing transverse pressure $p_\perp = 0$ everywhere except in thin boundary layers ("transitional" or "crust" regions) (Brustein et al., 2021, Brustein et al., 2024). This equation of state saturates the radial null-energy condition ( $\rho + p_r = 0$ ), thereby evading the Penrose–Hawking singularity theorems and suppressing horizon formation.

Rigorous constructions in Born–Infeld-type models identify the interior fluid as a continuous condensate of open-string flux tubes—the so-called string fluid arising from $D$ -brane decay and tachyon condensation, yielding a precise mapping between the effective stress–energy and the conditions necessary for the frozen geometry (Brustein et al., 2024). The string fluid corresponds physically to BIon-like flux configurations, where the electric field lines stretch from a central point charge to a compensating charge at the radius $R$ .

3. Frozen States in Modified and Higher‐Curvature Gravity

Frozen star solutions are ubiquitous in a variety of generalized gravity theories:

Einsteinian Cubic Gravity (ECG): Spherically symmetric, horizonless solutions with central naked singularity ("frozen gravitational stars," FGS) emerge in pure ECG. By tuning parameters (especially the cubic coupling $\lambda$ ), a configuration appears wherein the metric function $f(r)$ approaches zero at a critical radius $r_c$ —yielding an external geometry identical to an extremal Schwarzschild black hole, while the interior supports a diverging but untrapped singularity (Wang, 2024).
Boson Stars with Higher Curvature: In five-dimensional quasi-topological gravity with infinite higher-derivative corrections, enormous coupling strengths enable "frozen" states of boson stars. As the matter field frequency $\omega \to 0$ , the scalar field and energy density are confined within a critical radius $r_c$ , and the periphery of the star is virtually indistinguishable from an extremal black hole's exterior. The full infinite tower regularizes the central singularity (Ma et al., 2024).
Proca Stars: In five-dimensional higher-derivative gravity, vector boson (Proca) stars exhibit frozen states in the $\omega \to 0$ limit, provided the curvature corrections are sufficiently high-order. These objects have no horizon or singularity, but their redshift factor vanishes at a critical surface $r_c$ (Chen et al., 31 Dec 2025).
Gravitational Stars in Non-Polynomial Gravity: For neutron stars in four-dimensional nonpolynomial quasi-topological gravity, increasing the modification parameter $\alpha$ shifts the mass-radius curve upward until a "frozen neutron star" emerges as the endpoint. When $\alpha$ exceeds a threshold, the metric approaches zero extremely close to the stellar surface, creating a critical horizon and yielding an object nearly indistinguishable from a black hole even for realistic equations of state (Tan et al., 29 Dec 2025).
Nonlinear Electrodynamics/Bardeen-Type Regular Black Holes: In models coupling Einstein gravity to nonlinear electrodynamics (Bardeen/Hayward-type) and scalar or spinor fields, critical choices of the charge parameter and matter frequency yield "frozen" boson and Dirac stars. The lightlike character of the radius $r_c$ (frozen shell) is dynamically realized as the matter field frequency $\omega \to 0$ (Yue et al., 2023, Huang et al., 2023, Liu et al., 11 Dec 2025, Huang et al., 19 Mar 2025).

These contemporary instances consistently feature a regular or globally controlled interior, infinite or ultra-large redshift at $r_c$ , no event horizon, and a critical surface that is locally horizon-like for external probes.

4. Thermodynamics, Formation, and Stability

Thermodynamics:

Frozen stars radiate thermally at a temperature and entropy nearly identical to those of black holes of the same mass, as shown by explicit computation using the Euclidean action method of Gibbons and Hawking. The entropy is $S = A/4 + O(\epsilon^2,\lambda)$ , where $A$ is the frozen star's area and the corrections are perturbatively small (Brustein et al., 2023). The thermal spectrum, horizon-fuzziness, and associated off-diagonal elements in the radiation density matrix allow for a Page-curve evolution of entropy, thus ensuring the preservation of unitarity (Brustein et al., 2023).

Dynamical Formation:

It has been demonstrated that the quantum tunneling transition from a collapsing matter shell to a frozen star configuration occurs with unit probability in the Euclidean-instanton framework, owing to the overwhelming microcanonical phase-space of the frozen star interiors (saturating the black-hole area law). This transition replaces event horizon and singularity formation with the nucleation of the frozen configuration (Brustein et al., 4 Aug 2025).

Perturbations and Ultra-stability:

Ideal frozen stars with $p_r = -\rho$ , $p_t = 0$ are shown to be ultra-stable under linear perturbations. All metric and fluid oscillatory modes are frozen out (identically zero) both in the bulk and crust, a direct consequence of the saturated null-energy condition and infinite redshift (Brustein et al., 2021, Brustein et al., 2023). Upon perturbing the equation of state ( $p_r + \rho = \epsilon\rho$ ), ("defrosting" the star), long-lived, nonrelativistic fluid and gravitational modes emerge with frequencies scaling as $\omega \sim \sqrt{\epsilon}/R$ and damping times $\tau \sim 1/\epsilon$ , providing a bridge to quantum string-inspired models and yielding potentially observable gravitational-wave signatures ("echoes") (Brustein et al., 2023, Brustein et al., 2024).

5. Observational Properties and Astrophysical Significance

Black Hole Mimicry:

For all observables accessible to an external observer (photon sphere, gravitational lensing, quasinormal ringing, gravitational-wave emission outside $r_c$ ), frozen stars are virtually indistinguishable from extremal Schwarzschild or Reissner–Nordström black holes. The external spacetime is Schwarzschild (or appropriate regular-BH analog) down to a shell of Planckian (or model-dependent) width above the gravitational radius (Wang, 2024, Brustein et al., 2021, Brustein et al., 2023).

Light Rings and Geodesic Structure:

Frozen stars systematically sustain light rings (unstable photon orbits) at or just above $r_c$ , and for certain parameter regimes additional stable inner light rings, meeting the ultra-compact object (UCO) criteria. Inside $r_c$ , photon propagation is almost frozen, leading to "light ball" structures where light takes arbitrarily long coordinate time to traverse the interior (Huang et al., 19 Mar 2025, Huang et al., 2023).

Potential Observational Discriminants:

While frozen stars can in principle be distinguished from true black holes via:

The finite (non-zero) tidal Love numbers, absent for black holes.
GW echoes or modified late-time ringdown, reflecting at or near $r_c$ rather than being absorbed.
Tiny Planck-suppressed or model-dependent deviations in external field observables. However, current electromagnetic and GW observations do not rule out frozen stars within allowed parameter ranges (Tan et al., 29 Dec 2025).

6. Connections to String Theory and Quantum Gravity

Frozen stars naturally realize a classical macroscopic geometry dual to quantum collapsed-polymer models: the maximally negative radial pressure ( $p_r = -\rho$ ) is the classical imprint of the stringy equation of state ( $p_r = +\rho$ ) with vanishing entropy density, produced by string condensates near the Hagedorn temperature (Brustein et al., 2021, Brustein et al., 2024). The matter Lagrangian is cast into Born–Infeld form, consistent with the dynamics of fundamental string fluid arising from tachyon condensation and flux-tube formation. This structurally knits together the microphysical (string/brane) rationale for regular interiors and maximal entropy with the emergent black-hole-like macroscopic geometry (Brustein et al., 2024).

The thermodynamic and quantum information properties of frozen stars reproduce the Bekenstein–Hawking area law, Hawking-like thermal radiation, and unitary Page-curve semiclassical evolution (Brustein et al., 2023). The frozen star thus satisfies the requirements for a black hole alternative, avoiding both the information-loss problem and the breakdowns associated with curvature singularities.

7. Variants and Generalizations

The frozen star framework encompasses a wide range of solutions:

Regular black-hole mimickers in modified gravity
Frozen boson and Proca stars in higher dimensions and higher-derivative gravity (Ma et al., 2024, Chen et al., 31 Dec 2025)
Frozen Bardeen-Dirac stars and Hayward-type configurations with nonlinear electrodynamics (Yue et al., 2023, Huang et al., 2023, Liu et al., 11 Dec 2025)
Cosmological frozen stars, including the Janis–Newman–Winicour (JNW) class, which retain the divergent redshift behavior at the "frozen" surface but with varying scalar charge and dynamical extensions (Kastor et al., 2016)
General Lorentz ether theory models of frozen stars that regularize black hole formation by introducing preferred coordinate fields and halting collapse above the would-be horizon, evading traditional black hole no-hair and causal structure theorems (Schmelzer, 2010)

A key unifying element is the appearance of critical surfaces or shells supporting large (often infinite) redshift, absence of true event horizons, and physical regularity or controlled singularity within a modified or extended gravity framework.

Key References: