Frozen Star Solutions in Gravitational Physics

Updated 2 January 2026

Frozen Star Solutions are horizonless, ultracompact objects defined by anisotropic stress tensors (p₍r₎ = -ρ) that mimic the exterior geometry of classical black holes.
They are constructed by smoothly matching a near-zero interior metric to a Schwarzschild exterior using exotic matter fields or higher-derivative gravity, ensuring regularity.
Observationally, these models feature deep critical horizons and altered oscillatory spectra, providing potential tests for quantum gravity and astrophysical phenomena.

A frozen star, in the modern context of gravitational physics, denotes a class of horizonless, ultracompact astronomical and theoretical objects whose exterior metrics are indistinguishable from those of classical black holes, but which are sustained by matter or effective fields with extreme anisotropic properties—most notably, maximally negative radial pressure or specific higher-derivative interactions. These solutions serve as gravitational black hole mimickers, supported by diversified mechanisms ranging from classical anisotropic fluids, Born-Infeld string fluids, and nontrivial higher-derivative gravities, to nonlinear electrodynamics and solitonic matter fields. Frozen stars offer new avenues for exploring the microscopic and macroscopic structure of compact objects, geometric regularization of the black hole interior, and the avoidance of singularities and event horizons, while preserving classical observational signatures.

1. Construction Principles and Canonical Models

The canonical frozen star arises as a static, spherically symmetric solution of the Einstein equations coupled to an exotic matter sector. The essential requirements are:

The line element is generally taken as

$ds^2 = -f(r)\,dt^2 + \frac{dr^2}{f(r)} + r^2\,d\Omega^2$

with $f(r)$ constant and close to zero in the interior.

The matter supports an anisotropic stress-energy tensor,

$T^\mu{}_\nu = \mathrm{diag}(-\rho(r),\,p_r(r),\,p_\perp(r),\,p_\perp(r))$

with maximally negative radial pressure, $p_r = -\rho$ , and vanishing or controlled tangential pressure, $p_\perp\approx0$ throughout the bulk (Brustein et al., 2021, Brustein et al., 2023).

The resulting geometry is everywhere regular (upon suitable center regularization), and admits a surface at $r=R$ at which $f(R)$ is matched smoothly to the exterior Schwarzschild solution, $f(r) = 1-2GM/r$ for $r>R$ .
In pure-gravity extensions, such as Einsteinian cubic gravity, analogous horizonless, spherically symmetric solutions emerge given only higher-order curvature terms, showing deep "dips" in the metric function at a critical radius and interpolating between a Schwarzschild exterior and a nakedly singular (or sometimes regularized) interior (Wang, 2024).

2. Equations of State, Stress-Energy Sources, and Generalizations

All frozen star models are ultimately characterized by their sourcing fields or fluids. The unifying feature is an equation of state that saturates (or nearly saturates) the radial null energy condition:

Anisotropic fluid models: $p_r+\rho=0$ , $p_\perp=0$ in the core, supporting a degenerate "horizon-everywhere" geometry (Brustein et al., 2021, Brustein et al., 2023, Brustein et al., 2024).
String fluid/Born-Infeld models: The stress-energy tensor is derived from a Born-Infeld-like Lagrangian corresponding to rigid open-string flux tubes. The effective Lagrangian

$\mathcal{L}_\text{string}' = \frac{1}{2\pi\alpha'}\sqrt{-\tfrac12\,\mathcal{K}^{ab}\mathcal{K}_{ab}}$

yields $p_r+\rho=0$ in the bulk and generates the required anisotropic equilibrium (Brustein et al., 2024).

Nonlinear electrodynamics and matter-supported prototypes: Frozen states can arise in Einstein–Bardeen–Dirac, Bardeen–Proca, Bardeen–boson, and Hayward–boson star systems, where critical values of magnetostatic charge and vanishing field frequencies ( $\omega\to0$ ) enforce the development of a critical quasi-horizon (Huang et al., 2023, Zhang et al., 20 Mar 2025, Zhao et al., 19 Feb 2025, Yue et al., 2023, Liu et al., 11 Dec 2025).
Higher-derivative gravity: In theories such as Einsteinian cubic gravity and infinite-tower quasi-topological gravity, frozen stars can form without explicit matter fields, entirely from gravitational self-interactions. The emergence and position of the critical radius are determined by the higher-curvature coupling constants (Wang, 2024, Ma et al., 2024, Chen et al., 31 Dec 2025, Tan et al., 29 Dec 2025).
Extension to "defrosted" configurations: Allowing small deviations from maximal negative radial pressure ( $p_r=-(1-\gamma)\rho$ , $\gamma\ll1$ ) yields "defrosted stars" with a nontrivial spectrum of soft, long-lived fluid or even-parity oscillatory modes, connecting to quantum corrections or string microphysics (Brustein et al., 2023, Brustein et al., 2024).

3. Critical Horizon, Metric Structure, and Observational Indistinguishability

A defining feature of the frozen star is the emergence of a "critical horizon": a surface at radius $r_c$ or $r_h$ where the metric components $-g_{tt}$ and $g^{rr}$ approach (but do not attain) zero. This behavior is generic across diverse models:

As relevant parameters (mass, charge, frequency, or couplings) approach criticality, $f(r)$ develops an arbitrarily deep minimum at $r_c$ , signifying an effective event horizon.
For all values of relevant couplings (e.g., $\lambda$ in ECG), the critical horizon $r_h(\lambda)$ coincides numerically with the Schwarzschild radius $r_s=2GM$ for a given mass (Wang, 2024).
From an external observer's viewpoint, the metric exterior to $r_c$ is Schwarzschild or Reissner–Nordström, and all classical observables—light deflection, ringdown, photon spheres—are indistinguishable from an extremal black hole of the same mass and charges (Brustein et al., 2021, Zhang et al., 20 Mar 2025, Wang, 2024, Chen et al., 31 Dec 2025, Tan et al., 29 Dec 2025).
Internally, processes are "frozen" due to the arbitrarily large redshift at $r_c$ , effectively trapping infalling matter and suppressing observable dynamical response.

4. Singularities, Regularization, and Core Structure

The treatment of the central region is model-dependent:

Regularization: In fluid-based and BIon models, the central $1/r^2$ energy density divergence is manually regularized via a smooth transition to a finite, analytic density core, preserving the ultrastable, horizon-free nature of the solution (Brustein et al., 2023, Brustein et al., 2021). Spacetimes with string fluid support, for example, employ a central regularization region $0<r<\eta$ matched to the "frozen" interior.
Curvature in higher-derivative models: In some pure-gravity frozen stars (e.g., ECG), the solution possesses a naked singularity at $r=0$ ; however, the singularity is shielded by the critical surface and does not affect external measurements (Wang, 2024). In higher-rank gravity corrections or infinite-tower theories, frozen stars can be globally regular (Ma et al., 2024, Chen et al., 31 Dec 2025).
Matter-supported models: For Dirac and bosonic matter (e.g., in Bardeen–Dirac, Bardeen–boson, Proca, and Hayward–boson stars), the matter field is confined within $r_c$ , decaying exponentially outside, and the solutions may be everywhere regular in field variables and curvature invariants (Huang et al., 2023, Zhao et al., 19 Feb 2025, Yue et al., 2023, Liu et al., 11 Dec 2025).

5. Thermodynamics, Quantum Formation, and Information Aspects

Frozen stars share key thermodynamic and quantum characteristics with true black holes:

Entropy and temperature: Using Euclidean-action techniques, the entropy of a frozen star is $S=A/4$ up to $1/M$ corrections, and thermal radiation is Planckian with temperature $T=1/(8\pi M)$ , matching Hawking’s expressions for the same mass (Brustein et al., 2023, Brustein et al., 4 Aug 2025).
Formation by tunneling: The quantum probability for a collapsing shell of matter to tunnel into a frozen star is unity up to negligible corrections, due to the vanishing Euclidean action difference between the Minkowski and frozen-star true vacuum states. The process is mediated by an instanton associated with the thin transition layer between the interior and Schwarzschild exterior, yielding a path for regular black hole mimickers to form dynamically (Brustein et al., 4 Aug 2025).
Information and Page curve: The finite thickness of the critical surface and the associated horizon-width corrections introduce suppressed but nonzero off-diagonal elements in the particle density matrix, inducing a natural Page-time turnover consistent with unitary quantum evolution (Brustein et al., 2023).

6. Defrosted Stars, Perturbations, and Astrophysical Observability

While strict frozen stars are "bald"—i.e., possess no classical normal mode excitations—defrosted variants, with small but finite deviation from maximal negative pressure, admit nontrivial oscillatory modes:

The fundamental oscillation frequency scales as $\Re\,\omega\sim\sqrt{\gamma}$ , and the damping time as $\tau\sim1/\gamma^2$ , with $\gamma$ encoding the deviation from strict $p_r=-\rho$ (Brustein et al., 2024, Brustein et al., 2023).
The spectrum is discrete and parametrically soft (sound speed $v_s^2\sim\gamma$ ), distinct from standard neutron star or black hole quasinormal modes.
Such spectral features could inform observational discriminants in gravitational wave signals, electromagnetic emission, and accretion dynamics if frozen stars exist in nature.
In models with matter content (e.g., Bardeen–Dirac, Proca, or boson stars), frozen branches exhibit double photon spheres and altered light ring structures, affecting shadow phenomenology and orbital stability (Huang et al., 2023, Zhao et al., 19 Feb 2025, Zhang et al., 20 Mar 2025, Yue et al., 2023, Liu et al., 11 Dec 2025, Brihaye et al., 11 Jul 2025).

7. Extensions: Model Diversity, Cosmological Generalizations, and Open Questions

Frozen star solutions have been identified and studied across a broad array of gravitational, matter, and gauge field theories:

Gravity sector: Einsteinian cubic gravity, infinite-tower higher-derivative gravities, and non-polynomial gravities all support frozen stars with coupling-dependent critical radius and mass (Wang, 2024, Ma et al., 2024, Chen et al., 31 Dec 2025, Tan et al., 29 Dec 2025).
Matter sector: Nonlinear electrodynamics (e.g., Bardeen, Hayward), scalar and spinor fields (free and self-interacting), and Proca fields allow for continuous interpolation between regular horizonless stars and frozen configurations through parameter control (magnetic charge, field frequency) (Huang et al., 2023, Zhang et al., 20 Mar 2025, Zhao et al., 19 Feb 2025, Yue et al., 2023, Liu et al., 11 Dec 2025).
Cosmological extensions: Generalized dimensional reductions allow the construction of time-dependent, cosmological frozen star analogs in JNW–brane backgrounds and related scalar cosmologies (Kastor et al., 2016).
Astrophysical relevance: Frozen neutron stars constitute universal, causally consistent endpoints in non-polynomial gravity, regardless of equation of state, and are not yet excluded by NICER or GW170817 constraints (Tan et al., 29 Dec 2025).

Open questions remain regarding dynamical (nonlinear) stability, detailed gravitational wave signatures, and the degree to which quantum, string-theoretic, or semi-classical corrections alter the classical picture.

References:

(Brustein et al., 2021) "Black holes as frozen stars"
(Brustein et al., 2023) "Black holes as frozen stars: Regular interior geometry"
(Brustein et al., 2024) "Frozen stars: Black hole mimickers sourced by a string fluid"
(Wang, 2024) "Frozen gravitational stars in Einsteinian cubic gravity"
(Brustein et al., 2023) "Thermodynamics of frozen stars"
(Brustein et al., 4 Aug 2025) "Formation of Frozen Stars from collapsing matter by tunneling"
(Brustein et al., 2023) "Defrosting frozen stars: spectrum of internal fluid modes"
(Brustein et al., 2024) "Defrosting frozen stars: spectrum of non-radial oscillations"
(Huang et al., 2023) "Frozen Bardeen-Dirac stars and light ball"
(Zhao et al., 19 Feb 2025) "Non-Topological Soliton Bardeen Boson Stars and Their Frozen States"
(Yue et al., 2023) "Frozen Hayward-boson stars"
(Zhang et al., 20 Mar 2025) "Spherically symmetric horizonless solutions and their frozen states in Bardeen spacetime with Proca field"
(Liu et al., 11 Dec 2025) "Frozen solitonic Hayward-boson stars in Anti-de Sitter Spacetime"
(Brihaye et al., 11 Jul 2025) "Frozen states of charged boson stars"
(Ma et al., 2024) "Frozen boson stars in an infinite tower of higher-derivative gravity"
(Chen et al., 31 Dec 2025) "Proca stars and their frozen states in an infinite tower of higher-derivative gravity"
(Tan et al., 29 Dec 2025) "Frozen Neutron Stars in Four-Dimensional Non-polynomial Gravities"
(Kastor et al., 2016) "Building Cosmological Frozen Stars"