5D Proca Star: Frozen States in Higher-Derivative Gravity

• The paper demonstrates that five-dimensional Proca stars are self-gravitating, horizonless configurations realized with complex massive spin-1 fields under higher-derivative gravity corrections.
• It reveals that under critical curvature conditions, these systems evolve into regular ‘frozen star’ states that achieve extremal compactness without curvature singularities or event horizons.
• The analysis provides practical insights into the dynamics of conserved Noether charges and mass, linking numerical solutions to theoretical predictions in higher-dimensional gravity.

A five-dimensional Proca star is a self-gravitating, horizonless, and stationary configuration of a complex massive spin-1 field (Proca field) in five-dimensional spacetime, realized within gravity theories that include higher-curvature (quasi-topological) corrections. These solutions generalize the concept of four-dimensional boson stars to higher-dimensional gravity coupled to vector fields and are characterized by the absence of curvature singularities, the presence of a conserved Noether charge, and relativistic compactness approaching extremal black hole limits under certain conditions. Notably, under sufficiently high curvature corrections, these systems admit "frozen star" states—regular, extremally compact, ultracompact objects whose exteriors are observationally indistinguishable from extremal five-dimensional black holes yet contain no event horizon or central singularity (Chen et al., 31 Dec 2025).

1. Model Construction and Action

Five-dimensional Proca stars are realized in a theoretical framework comprising five-dimensional gravity augmented by an infinite tower of higher-order quasi-topological curvature terms, minimally coupled to a complex Proca field $A_\mu$ with mass %%%%1%%%%. The action takes the form:

$$S = \int d^5x \sqrt{-g} \left[ R + \sum_{n=2}^{n_\text{max}} \alpha_n \mathcal{L}_{(n)} \right] - \int d^5x \sqrt{-g} \left( \frac{1}{4} F_{\mu\nu}\bar{F}^{\mu\nu} + \frac{1}{2} \mu^2 \bar{A}_\mu A^\mu \right),$$

where $F=dA$, the overbar denotes complex conjugation, and $\mathcal{L}_{(n)}$ represents the unique order-$n$ quasi-topological curvature invariant. For $n_\text{max}=1$ and vanishing $\alpha_n$, the action reduces to five-dimensional Einstein–Proca theory. The line element adopted for static, spherically symmetric configurations is

$$ds^2 = -\sigma(r)^2 N(r) dt^2 + \frac{dr^2}{N(r)} + r^2 d\Omega_3^2,$$

and the Proca field ansatz is harmonic in time,

$A = [F(r) dt + i\,H(r) dr]\,e^{-i\omega t}.$

2. Field Equations and Boundary Conditions

Variation with respect to the metric functions and Proca components yields four coupled ordinary differential equations (ODEs), involving the metric variables $N(r)$, $\sigma(r)$, and Proca field functions $F(r)$, $H(r)$. Defining $\psi(r) = (1 - N(r))/r^2$ and $$h(\psi) = \psi + \sum_{n=2}^{n_{\max}} \alpha_{n-1} \psi^n$$, the system is given by:

• $H(\omega^2 - \mu^2 N \sigma^2)/\omega - F' = 0$
• $$\omega F/(N^2 \sigma^2) + H[3/r + (N'/N + \sigma'/\sigma)] + H' = 0$$
• $$(8\pi/3) H \mu^2 r \left[ F^2/(N^2 \sigma) + H^2 \sigma \right] + \sigma' h'(\psi) = 0$$
• $$(8\pi/3) H r^3 [\mu^2 F^2 + N(H^2(\omega^2 + \mu^2 N \sigma^2) - 2\omega H F' + F'^2)]/(N \sigma^2) + (r^4 h(\psi))' = 0$$

Boundary conditions enforce regularity at the origin ($N(0) = 1$, $\sigma(0)$ finite, $F'(0) = 0$, $H(0) = 0$) and asymptotic flatness ($N \rightarrow 1$, $\sigma \rightarrow 1$, $F \rightarrow 0$, $H \rightarrow 0$) as $r \rightarrow \infty$. The ADM mass $M$ and Noether charge $Q$ are extracted from the asymptotic metric expansion and the time component of the conserved current, respectively.

3. Solution Space: Classical Proca Stars and Critical Behavior

Numerical integration of the field equations reveals several regimes associated with the truncation of the curvature expansion:

• Einstein–Proca Case ($n=1$): For $n_{\max} = 1$, Proca stars exist for $\omega_\text{min}<\omega<\mu$, forming spiral trajectories in the $(M, Q)$ plane as functions of $\omega$. As $\omega \to \mu$, both $M$ and $Q$ approach finite, nonzero values.
• Gauss–Bonnet Case ($n=2$): For $n=2$, the qualitative behavior mirrors the Einstein limit at small $\alpha$. However, above a critical coupling $\alpha_c \approx 1.14$, solutions can be extended to $\omega \to 0$. In this limit, field amplitudes and the energy density diverge at the origin: $F(r) \sim F_0(\omega) \to \infty$ as $\omega \to 0$, and $\rho(r) \propto |F_0|^2 / r^4$ diverges for small $r$.
• Higher-Order Cases ($n \geq 3$, including $n \to \infty$): For sufficiently large $\alpha$ in $n \geq 3$ models, solutions persist down to $\omega \to 0$, but the divergence is eliminated. Fields and densities remain finite everywhere—a crucial qualitative change indicating geometric resolution of singularities.

The behavior is summarized in the table:

Truncation	ω→0 Limit Exists?	Central Divergence?	Regular "Frozen" State?
n = 1	No	—	No
n = 2	Yes (α>α_c)	Yes	No
n ≥ 3	Yes (α>α_c)	No	Yes

4. Frozen Star States: Definition and Properties

In the $\omega \to 0$ limit for $n \geq 3$, five-dimensional Proca stars approach a "frozen star" configuration—an object featuring a regular, horizonless, and nonsingular geometry that achieves maximal possible compactness for a given mass. In this state:

• The metric function $N(r)$ develops a "double-zero" at a finite radius $r_c$: $N(r_c) = 0$ and $N'(r_c) = 0$. This critical radius acts as an effective, infinitely redshifted shell.
• Metric functions $-g_{tt}$ and $1/g_{rr}$ both approach zero at $r_c$.
• For $r > r_c$, the solution for $N(r)$ coincides with the extremal black hole exterior:

$$N_\infty(r) = 1 - \frac{8M r^2}{3\pi r^4 + 8\alpha M}$$

The extremality condition yields $r_c^2 = 4M/(3\pi)$ and compactness $C_{5D} = 3\pi/4$, exactly matching the extremal five-dimensional black hole value.

• All curvature invariants, including the Kretschmann scalar, remain finite throughout the spacetime, including at $r=0$ and across $r_c$. There is no event horizon but an infinite-redshift surface.

5. Physical Quantities and Exterior–Interior Structure

The mass $M(\omega)$ and charge $Q(\omega)$ exhibit the typical spiral form for $n=1$ and evolve monotonically towards finite, nonzero values at $\omega \to 0$ for $n \geq 3$. The relation between compactness and frequency elucidates that as $\omega$ decreases, compactness increases, saturating at the extremal value in the frozen branch, where the critical radius $r_c$ replaces the usual effective radius.

In the exterior ($r > r_c$), the metric is indistinguishable from that of the extremal higher-derivative black hole: classical probes such as light deflection, shadow size, or quasinormal modes cannot distinguish a frozen star from a genuine extremal black hole. Contrastingly, within $r < r_c$, the geometry is nonsingular—a supported Proca field yields a regular interior in place of a black hole's null singularity.

6. Theoretical and Physical Significance

Five-dimensional Proca stars in higher-derivative gravity, particularly in the presence of an infinite tower of quasi-topological corrections, provide explicit, geometric, and nonsingular resolutions to the pathologies afflicting the traditional $\omega \to 0$ branch. The emergent frozen stars act as counterexamples to the widely held association between maximal compactness, event horizons, and curvature singularities. Their existence demonstrates that, for suitable curvature corrections, it is possible to achieve ultracompact, horizonless, and everywhere regular objects that mimic black hole exteriors without exotic matter. This advances the study of singularity resolution and highlights the physical viability of nonsingular, ultracompact configurations in higher-derivative gravity (Chen et al., 31 Dec 2025).

7. Relationship to Related Models and Future Directions

Frozen states of scalar boson stars in similar settings have also been analyzed, notably in Ma et al. (Eur. Phys. J. C 85 (2025) 542), confirming that the geometric regularization mechanism extends to other matter content. The results parallel findings in five-dimensional higher-derivative theories for scalar fields (Ma et al., 2024). The persistence of regular, ultracompact, horizonless solutions in higher-curvature regimes suggests a general framework for singularity resolution, meriting further exploration in various spacetime dimensions, matter sectors, and under dynamical perturbations. A plausible implication is that such "frozen" configurations could offer novel astrophysical signatures in higher-dimensional scenarios and sharpen the theoretical boundary between black holes and horizonless compact objects.