Frozen Star Gravitational Configurations

• Frozen star refers to ultracompact gravitational configurations with near-zero redshift factors at a finite radius, mimicking black holes without forming true event horizons.
• These models employ diverse matter sectors—such as anisotropic fluids, scalar fields, and higher-curvature corrections—to achieve horizonless yet observationally similar geometries to black holes.
• Observable phenomena, including gravitational-wave echoes, light-ring structures, and accretion signatures, are key to distinguishing frozen stars from conventional black holes.

Frozen star denotes a family of gravitational configurations united by an extreme redshift effect but not by a single underlying model. In the older relativistic-collapse literature, the term referred to the Oppenheimer–Snyder result that a collapsing surface asymptotically approaches the Schwarzschild radius in the external Schwarzschild time coordinate, appearing to “freeze” there (Zhang, 2010). In recent compact-object literature, the same term is used for ultracompact configurations in which the metric redshift factor approaches zero at a finite radius, so that matter and clocks near that radius appear frozen to infinity, while no true event horizon forms. This latter usage includes horizonless solutions in Einstein–Hayward–scalar theory (Yue et al., 2023), anisotropic-fluid and string-fluid constructions (Brustein et al., 2024, Brustein et al., 2021), higher-curvature boson and Proca stars (Ma et al., 2024, Chen et al., 31 Dec 2025), dark-matter-sourced compact stars (Yue et al., 21 Jan 2026), and neutron-star sequences with frozen endpoints (Tan et al., 29 Dec 2025, Tan et al., 11 Sep 2025).

1. Historical and terminological development

The classical “frozen star” paradox originates in the external description of gravitational collapse. In the Oppenheimer–Snyder dust model, the exterior is Schwarzschild and the surface radius satisfies an integral for the Schwarzschild coordinate time that diverges as the radius approaches $R_H=2M$; the comoving observer sees collapse complete in finite proper time, but the external observer never sees the surface cross the Schwarzschild radius (Zhang, 2010). Liu and Zhang generalized this setup to shells of finite thickness falling onto a pre-existing black hole and, crucially, included the effect of changing total gravitating mass. In that treatment the event horizon grows and can swallow shell elements in finite external time, so the enduring “frozen star” picture does not survive once the back-reaction of infalling matter and outer shells is included (Zhang, 2010).

A different historical use appears in Janis–Newman–Winicour spacetimes. There the metric has an infinite-redshift surface at $r=r_0$, with

$$z+1=f(r_e)^{-\alpha/2},\qquad f(r)=1-\frac{r_0}{r},$$

so that clocks near $r_0$ appear frozen to infinity, even though $r=r_0$ is a scalar singularity rather than a regular horizon (Kastor et al., 2016). Cosmological generalizations retain this infinite-redshift property in expanding or contracting backgrounds (Kastor et al., 2016).

Modern compact-object work uses the term in a more specific sense: a frozen star is typically a regular or nonsingular ultracompact object that mimics a black hole from the exterior while remaining horizonless. This suggests that the term has bifurcated into two conceptually distinct usages: a coordinate-time description of collapse, and a class of stationary or limiting black-hole mimickers.

2. Geometric definition and horizon criterion

A common metric ansatz in contemporary frozen-star constructions is

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

or an equivalent Schwarzschild-like form with $f(r)=g(r)$ (Yue et al., 2023, Yue et al., 21 Jan 2026, Tan et al., 29 Dec 2025). The defining geometric feature is that the redshift factor

$g_{tt}(r)=-N(r)\sigma^2(r)$

or $-f(r)$ approaches zero arbitrarily closely at some finite radius $r=r_c$, $r=r_0$0, or $r=r_0$1, without changing sign and without producing a true event horizon (Yue et al., 21 Jan 2026, Tan et al., 29 Dec 2025, Zhang et al., 20 Mar 2025).

In the Einstein–Hayward–scalar model, an event horizon would appear if $r=r_0$2 for some finite $r=r_0$3, and the frozen Hayward–boson-star analysis restricts attention to star-like solutions with $r=r_0$4 for all $r=r_0$5 (Yue et al., 2023). In dark-matter-sourced compact stars, the minimum of $r=r_0$6 approaches zero from above, $r=r_0$7, yet the solution remains horizonless unless one exactly tunes to the regular black-hole case (Yue et al., 21 Jan 2026). In four-dimensional non-polynomial gravities, the frozen endpoint appears when the metric function develops a double root extremely close to the stellar surface, with $r=r_0$8 and $r=r_0$9 (Tan et al., 29 Dec 2025). In five-dimensional higher-derivative Proca stars, the frozen state develops a critical horizon at finite radius $$z+1=f(r_e)^{-\alpha/2},\qquad f(r)=1-\frac{r_0}{r},$$0, where $$z+1=f(r_e)^{-\alpha/2},\qquad f(r)=1-\frac{r_0}{r},$$1 and $$z+1=f(r_e)^{-\alpha/2},\qquad f(r)=1-\frac{r_0}{r},$$2 approach zero, but the configuration contains neither curvature singularities nor event horizons (Chen et al., 31 Dec 2025).

A distinct but closely related implementation appears in the anisotropic-fluid frozen-star model of Brustein and collaborators, where the interior is regularized by a small parameter $$z+1=f(r_e)^{-\alpha/2},\qquad f(r)=1-\frac{r_0}{r},$$3 and

$$z+1=f(r_e)^{-\alpha/2},\qquad f(r)=1-\frac{r_0}{r},$$4

so the entire interior has an arbitrarily large but finite redshift and matches to a Schwarzschild exterior at $$z+1=f(r_e)^{-\alpha/2},\qquad f(r)=1-\frac{r_0}{r},$$5 (Brustein et al., 2024, Brustein et al., 2023).

3. Source sectors and model classes

No single matter model defines the frozen-star concept. Recent realizations span anisotropic fluids, bosonic and fermionic fields, nonlinear electromagnetic sectors, string fluids, dark-matter halos, and higher-curvature gravities.

In the anisotropic-fluid constructions, the central equation of state is maximally negative radial pressure,

$$z+1=f(r_e)^{-\alpha/2},\qquad f(r)=1-\frac{r_0}{r},$$6

often together with $$z+1=f(r_e)^{-\alpha/2},\qquad f(r)=1-\frac{r_0}{r},$$7 or perturbatively small deviations from it (Brustein et al., 2021, Brustein et al., 2023, Brustein et al., 2024). In these models Einstein’s equations admit a nearly null interior redshift factor, and the condition $$z+1=f(r_e)^{-\alpha/2},\qquad f(r)=1-\frac{r_0}{r},$$8 is repeatedly identified as the mechanism behind both the near-horizon behavior and the freeze-out of linear fluctuations (Brustein et al., 2021).

The string-fluid realization identifies the matter source with the string fluid resulting from the decay of an unstable $$z+1=f(r_e)^{-\alpha/2},\qquad f(r)=1-\frac{r_0}{r},$$9-brane or brane–antibrane system at the end of open-string tachyon condensation (Brustein et al., 2024). The effective Lagrangian can be recast into a Born–Infeld form, the interior electric displacement obeys

$r_0$0

and the stress tensor has

$r_0$1

inside the star (Brustein et al., 2024).

A second large class couples matter fields to regular-black-hole sectors of Bardeen or Hayward type. The Einstein–Hayward–scalar theory minimally couples a complex, massive, free scalar field to Hayward nonlinear electrodynamics and yields ground and excited frozen Hayward–boson stars (Yue et al., 2023). Closely related Bardeen constructions include frozen Bardeen–boson stars, frozen Bardeen–Dirac stars, and frozen Bardeen–Proca stars, all obtained by adding scalar, Dirac, or Proca matter to the nonlinear electromagnetic background so that an event horizon is disrupted and replaced by a critical surface or critical horizon (Huang et al., 19 Mar 2025, Huang et al., 2023, Zhang et al., 20 Mar 2025).

Further generalizations place the phenomenon in modified gravity and ordinary-matter settings. Four-dimensional non-polynomial gravities produce frozen neutron stars at the end of the neutron-star sequence for the BSk19, SLy4, and AP4 equations of state (Tan et al., 29 Dec 2025). Einstein–nonlinear-electrodynamics models with Bardeen or Hayward magnetic monopoles yield frozen neutron stars at a critical magnetic charge $r_0$2 (Tan et al., 11 Sep 2025). Dark-matter halo models produce horizonless compact stars that approach a frozen state when the condition $r_0$3 is relaxed to broader anisotropic families (Yue et al., 21 Jan 2026). Five-dimensional boson and Proca stars under infinite towers of higher-curvature corrections also admit frozen states when the coupling exceeds a threshold (Ma et al., 2024, Chen et al., 31 Dec 2025).

4. Limiting behavior and representative solution families

The frozen limit usually arises by tuning a control parameter to a critical regime. In boson and Proca stars, the relevant parameter is often the matter frequency $r_0$4. In magnetically charged compact stars, it is frequently a critical magnetic charge. In higher-curvature or non-polynomial gravities, it is a coupling strength together with the central density.

In the Einstein–Hayward–scalar system, each radial excitation $r_0$5 has a branch of solutions labeled by $r_0$6. For $r_0$7, the mass–frequency curve is multivalued and spirals as in ordinary boson stars. As $r_0$8, the spiral opens, $r_0$9, and a single-valued branch extends to $r=r_0$0 (Yue et al., 2023). In that limit the scalar profile develops a very steep wall at a critical radius $r=r_0$1, $r=r_0$2, and $r=r_0$3, yet no true horizon forms (Yue et al., 2023). Numerically, the Hayward case gives

$r=r_0$4

and, remarkably, the critical radius $r=r_0$5 and critical mass $r=r_0$6 at $r=r_0$7 are independent of the radial node number $r=r_0$8 for fixed $r=r_0$9 (Yue et al., 2023).

Bardeen-based models show analogous limiting behavior with model-specific details. Frozen Bardeen–Dirac stars exist for $$ds^2=-N(r)\sigma^2(r)\,dt^2+\frac{dr^2}{N(r)}+r^2d\Omega^2,$$0, and for $$ds^2=-N(r)\sigma^2(r)\,dt^2+\frac{dr^2}{N(r)}+r^2d\Omega^2,$$1 the Dirac construction gives $$ds^2=-N(r)\sigma^2(r)\,dt^2+\frac{dr^2}{N(r)}+r^2d\Omega^2,$$2; the metric outside the critical horizon tracks that of an extremal Bardeen black hole, while the interior remains nonsingular (Huang et al., 2023). In the Bardeen–Proca case, sufficiently large magnetic charge allows $$ds^2=-N(r)\sigma^2(r)\,dt^2+\frac{dr^2}{N(r)}+r^2d\Omega^2,$$3, $$ds^2=-N(r)\sigma^2(r)\,dt^2+\frac{dr^2}{N(r)}+r^2d\Omega^2,$$4, $$ds^2=-N(r)\sigma^2(r)\,dt^2+\frac{dr^2}{N(r)}+r^2d\Omega^2,$$5, and $$ds^2=-N(r)\sigma^2(r)\,dt^2+\frac{dr^2}{N(r)}+r^2d\Omega^2,$$6 inside $$ds^2=-N(r)\sigma^2(r)\,dt^2+\frac{dr^2}{N(r)}+r^2d\Omega^2,$$7, while the ADM mass remains finite and the Noether charge tends to zero (Zhang et al., 20 Mar 2025). In the Bardeen–boson-star null-geodesic analysis, the frozen configurations are the $$ds^2=-N(r)\sigma^2(r)\,dt^2+\frac{dr^2}{N(r)}+r^2d\Omega^2,$$8 solutions whose metric becomes very similar to that of black holes near the critical horizon, while remaining fully horizonless and possessing complete null geodesics (Huang et al., 19 Mar 2025).

In non-polynomial gravity, increasing the modification parameter $$ds^2=-N(r)\sigma^2(r)\,dt^2+\frac{dr^2}{N(r)}+r^2d\Omega^2,$$9 enlarges both neutron-star radius and mass, and a frozen state emerges when the metric functions approach zero extremely close to the stellar surface (Tan et al., 29 Dec 2025). The allowed observational window is roughly $f(r)=g(r)$0, and within this range frozen-star solutions appear for BSk19, SLy4, and AP4 (Tan et al., 29 Dec 2025). In the model $f(r)=g(r)$1, the quoted thresholds are

$f(r)=g(r)$2

above which frozen neutron stars appear within the causal range of central densities (Tan et al., 29 Dec 2025). In dark-matter-sourced compact stars, the frozen regime is reached when the minimum of $f(r)=g(r)$3 approaches zero from above; for the Einasto-profile Case I, the critical densities are $f(r)=g(r)$4 for $f(r)=g(r)$5, $f(r)=g(r)$6 for $f(r)=g(r)$7, and $f(r)=g(r)$8 for $f(r)=g(r)$9 (Yue et al., 21 Jan 2026).

5. Perturbations, thermodynamics, and formation

One prominent claim in the anisotropic-fluid frozen-star program is that the exact frozen configuration is dynamically bald. In the matched interior–crust–exterior construction with $g_{tt}(r)=-N(r)\sigma^2(r)$0, linearized analysis gives $g_{tt}(r)=-N(r)\sigma^2(r)$1, $g_{tt}(r)=-N(r)\sigma^2(r)$2, and, after imposing the inner boundary conditions, $g_{tt}(r)=-N(r)\sigma^2(r)$3, so that every metric perturbation and every matter perturbation vanishes identically at linear order (Brustein et al., 2021). This is the strongest available formulation of “freezing out” in the perturbative sense.

When the equation of state is perturbed away from maximal negativity, the star “defrosts.” In the Cowling treatment of internal fluid modes, a small parameter $g_{tt}(r)=-N(r)\sigma^2(r)$4 yields a leading-order dispersion relation

$g_{tt}(r)=-N(r)\sigma^2(r)$5

so the real frequencies vanish in the strict frozen limit $g_{tt}(r)=-N(r)\sigma^2(r)$6 (Brustein et al., 2023). With even-parity metric perturbations included, the defrosting parameter $g_{tt}(r)=-N(r)\sigma^2(r)$7 produces non-radial oscillations whose sound speed is non-relativistic and proportional to $g_{tt}(r)=-N(r)\sigma^2(r)$8, while the lifetime is parametrically long and proportional to $g_{tt}(r)=-N(r)\sigma^2(r)$9 (Brustein et al., 2024). These results were presented as a starting point for calculating the spectrum of emitted gravitational waves from an excited frozen star (Brustein et al., 2024).

Thermodynamically, frozen stars have been argued to reproduce black-hole behavior to leading order. Using a Euclidean-action calculation, the entropy is

$-f(r)$0

and the temperature is

$-f(r)$1

so both are perturbatively close to the Bekenstein–Hawking values (Brustein et al., 2023). The same work estimated a transition probability $-f(r)$2 for a collapsing shell to tunnel into a frozen star, and then argued that the phase-space factor $-f(r)$3 makes the total formation probability of order unity (Brustein et al., 2023). A later tunneling analysis sharpened that conclusion, stating that the transition probability is unity, up to negligible corrections, because the Euclidean-action difference is approximately zero and the determinant prefactor is unity (Brustein et al., 4 Aug 2025).

6. Exterior mimicry, observables, and unresolved issues

Across the modern literature, frozen stars are presented as black-hole mimickers. In the string-fluid model, the exterior is exactly Schwarzschild and for $-f(r)$4 it is stated to be indistinguishable from Schwarzschild under Event Horizon Telescope bounds (Brustein et al., 2024). In frozen Hayward, Bardeen, boson, Dirac, Proca, and higher-curvature models, the exterior geometry is repeatedly said to coincide with, or become nearly indistinguishable from, the corresponding extremal black hole outside the critical radius (Yue et al., 2023, Huang et al., 2023, Chen et al., 31 Dec 2025).

The proposed discriminants are correspondingly subtle. Several models predict light-ring structures resembling black holes but not identical to them. Frozen Bardeen–Proca stars have two light rings, one stable inside and one unstable outside the critical horizon, and these move further apart as $-f(r)$5 decreases (Zhang et al., 20 Mar 2025). Frozen Bardeen–Dirac stars also have an outer unstable light ring, but inside the critical horizon the photon velocity becomes very close to zero, leading to a “light ball” (Huang et al., 2023). In frozen Bardeen–boson stars, photons undergo sharp deflections near the critical horizon, travel nearly straight inside it, and can spend a very long coordinate time in the interior as seen from infinity (Huang et al., 19 Mar 2025).

Other proposed observables include gravitational-wave echoes, late-time ringdown modifications, nonzero tidal deformability, and accretion signatures. The dark-matter compact-star model states that frozen stars mimic black-hole shadows, ringdown spectra, and strong lensing, yet may exhibit reflectivity, echoes, and slight shifts in quasinormal frequencies; it also states that linearized axial perturbations are stable for viable parameter ranges (Yue et al., 21 Jan 2026). The non-polynomial-gravity neutron-star analysis identifies late-time ringdown, echoes, tidal deformability $-f(r)$6, and accretion-disk and iron-$-f(r)$7 diagnostics as possible tests, while emphasizing that the frozen configurations remain allowed by present NICER and GW170817 constraints (Tan et al., 29 Dec 2025).

Several central issues remain open in the cited literature. The five-dimensional higher-derivative Proca-star work states explicitly that linear stability remains to be fully studied (Chen et al., 31 Dec 2025). The frozen-neutron-star analysis likewise notes that a rigorous stability analysis under radial perturbations is essential and that electromagnetic or gravitational-wave imprints remain to be thoroughly investigated (Tan et al., 11 Sep 2025). This suggests that the current status of frozen stars is strongest at the level of existence proofs, limiting geometries, and kinematical phenomenology, and less settled at the level of full dynamical selection, nonlinear stability, and observational identifiability.