Proca-Maxwell System in an Infinite Tower of Higher-Derivative Gravity

Abstract: We numerically construct a five-dimensional Proca-Maxwell system coupled to an infinite tower of higher-derivative gravity, parameterized by the correction order and coupling constant. While the first-order correction case recovers standard Einstein gravity results, and the second-order correction (Gauss-Bonnet) case fails to resolve the central singularity in the vanishing frequency limit, we demonstrate that higher-order corrections effectively regularize the spacetime, yielding globally regular solutions. A key finding is the emergence of a frozen state'' in the supercritical regime: as the field frequency approaches zero, matter concentrates entirely within a critical radius, creating a regular core that externally mimics an extremal black hole. We further reveal that introducing the electric charge fundamentally alters this behavior; the electrostatic repulsion counteracts the gravitational collapse, effectivelyunfreezing'' the system and preventing the formation of the critical core. Significantly, unlike models relying on exotic matter, our solutions satisfy all standard energy conditions across the entire parameter space, establishing a physically viable pathway for constructing regular black hole mimickers.

• The paper demonstrates that infinite-tower higher-derivative corrections enable the construction of globally regular charged Proca stars, resolving singularities without exotic matter.
• It employs numerical solutions of coupled nonlinear ODEs to explore the transition from conventional boson star behavior to a novel frozen state mimicking an extremal black hole.
• The study reveals that electromagnetic charge alters stability, with increased charge unfreezing the system and preventing quasi-horizon formation.

Proca-Maxwell Systems in Infinite-Tower Higher-Derivative Gravity

Introduction

This work constructs and analyzes charged Proca (massive vector) stars in five-dimensional quasi-topological gravities comprising an infinite tower of higher-derivative curvature terms. The context is the persistent challenge of resolving spacetime singularities predicted by General Relativity (GR) during gravitational collapse, which is usually addressed by quantum gravity or, more speculatively, by modifications of GR involving higher-curvature interactions. While traditional regular black hole (BH) models typically require the ad hoc introduction of exotic matter, the recent surge in quasi-topological, non-polynomial, or infinite-derivative gravitational actions opens a purely gravitational avenue to singularity resolution. Against this backdrop, the present paper systematically investigates the equilibrium and phenomenology of spherically symmetric charged Proca stars (vector boson stars with $U(1)$ charge), in particular focusing on the novel "frozen state" that emerges from non-perturbative higher-order gravity corrections, and the interplay between electromagnetic and higher-order gravitational repulsions.

Theoretical Construction

The model considers the action of a minimally coupled Proca-Maxwell system,

$$S = \int d^5x\, \sqrt{-g} \left[ \frac{R}{16\pi G} + \sum_{n=2}^{n_{\max}} \frac{\alpha_n \mathcal{Z}_n}{16\pi G} + \mathcal{L}_\mathrm{Proca} + \mathcal{L}_\mathrm{Maxwell} \right],$$

where the $\mathcal{Z}_n$ are higher-derivative quasi-topological invariants, systematically added to all orders ($n \to \infty$) with couplings $\alpha_n = \alpha^{n-1}$. The matter sector comprises a massive complex vector field (Proca) with electric charge $q$ under an additional Maxwell field.

A spherically symmetric metric, static Proca ansatz, and electrostatic gauge field yield a system of coupled nonlinear ODEs for the radial profiles of the metric functions and matter fields. The coupling hierarchy in the higher-derivative terms translates to physically distinct regimes: the $n=1$ case reduces to the classical Einstein-Proca-Maxwell system, $n=2$ reproduces Einstein-Gauss-Bonnet gravity, and $n \geq 3$, as well as the $n=\infty$ infinite tower, access the non-perturbative regime responsible for singularity regularization.

Global solutions are constructed numerically with requisite regularity at the origin and asymptotic flatness. Key physical observables such as ADM mass, Proca particle number, energy density, principal pressures, and binding energy are computed to characterize stability and compactness.

Results: Phenomenology and Structure

Einstein and Low-Order Quasi-Topological Gravity ($$S = \int d^5x\, \sqrt{-g} \left[ \frac{R}{16\pi G} + \sum_{n=2}^{n_{\max}} \frac{\alpha_n \mathcal{Z}_n}{16\pi G} + \mathcal{L}_\mathrm{Proca} + \mathcal{L}_\mathrm{Maxwell} \right],$$0)

In the $$S = \int d^5x\, \sqrt{-g} \left[ \frac{R}{16\pi G} + \sum_{n=2}^{n_{\max}} \frac{\alpha_n \mathcal{Z}_n}{16\pi G} + \mathcal{L}_\mathrm{Proca} + \mathcal{L}_\mathrm{Maxwell} \right],$$1 (Einstein) and $$S = \int d^5x\, \sqrt{-g} \left[ \frac{R}{16\pi G} + \sum_{n=2}^{n_{\max}} \frac{\alpha_n \mathcal{Z}_n}{16\pi G} + \mathcal{L}_\mathrm{Proca} + \mathcal{L}_\mathrm{Maxwell} \right],$$2 (Gauss-Bonnet) sectors, the system exhibits standard compact star phenomenology. At weak coupling, ADM mass and Proca number follow spiral-like behavior in parameter space (as a function of field frequency $$S = \int d^5x\, \sqrt{-g} \left[ \frac{R}{16\pi G} + \sum_{n=2}^{n_{\max}} \frac{\alpha_n \mathcal{Z}_n}{16\pi G} + \mathcal{L}_\mathrm{Proca} + \mathcal{L}_\mathrm{Maxwell} \right],$$3), typical of boson stars, but the binding energy is always negative, suggesting gravitationally unbound, unstable configurations. Increasing $$S = \int d^5x\, \sqrt{-g} \left[ \frac{R}{16\pi G} + \sum_{n=2}^{n_{\max}} \frac{\alpha_n \mathcal{Z}_n}{16\pi G} + \mathcal{L}_\mathrm{Proca} + \mathcal{L}_\mathrm{Maxwell} \right],$$4 shifts the spiral but does not produce physically viable, stable stars. Introduction of electric charge $$S = \int d^5x\, \sqrt{-g} \left[ \frac{R}{16\pi G} + \sum_{n=2}^{n_{\max}} \frac{\alpha_n \mathcal{Z}_n}{16\pi G} + \mathcal{L}_\mathrm{Proca} + \mathcal{L}_\mathrm{Maxwell} \right],$$5 further compresses the existence window in frequency space due to growing electromagnetic repulsion.

For sufficiently large curvature coupling ($$S = \int d^5x\, \sqrt{-g} \left[ \frac{R}{16\pi G} + \sum_{n=2}^{n_{\max}} \frac{\alpha_n \mathcal{Z}_n}{16\pi G} + \mathcal{L}_\mathrm{Proca} + \mathcal{L}_\mathrm{Maxwell} \right],$$6 for $$S = \int d^5x\, \sqrt{-g} \left[ \frac{R}{16\pi G} + \sum_{n=2}^{n_{\max}} \frac{\alpha_n \mathcal{Z}_n}{16\pi G} + \mathcal{L}_\mathrm{Proca} + \mathcal{L}_\mathrm{Maxwell} \right],$$7), the multi-branch spiral structure disappears in favor of a monotonic sequence as $$S = \int d^5x\, \sqrt{-g} \left[ \frac{R}{16\pi G} + \sum_{n=2}^{n_{\max}} \frac{\alpha_n \mathcal{Z}_n}{16\pi G} + \mathcal{L}_\mathrm{Proca} + \mathcal{L}_\mathrm{Maxwell} \right],$$8 decreases. However, as $$S = \int d^5x\, \sqrt{-g} \left[ \frac{R}{16\pi G} + \sum_{n=2}^{n_{\max}} \frac{\alpha_n \mathcal{Z}_n}{16\pi G} + \mathcal{L}_\mathrm{Proca} + \mathcal{L}_\mathrm{Maxwell} \right],$$9, the Proca field collapses toward the origin, producing a divergent Kretschmann scalar and hence a singular core in the Gauss-Bonnet case—higher-order corrections are necessary for true regularity.

Infinite-Tower Corrections ($\mathcal{Z}_n$0; $\mathcal{Z}_n$1 Case)

Beyond $\mathcal{Z}_n$2, the system transitions to a new regime. For $\mathcal{Z}_n$3 and the infinite tower ($\mathcal{Z}_n$4), the higher-order repulsive corrections become dominant as $\mathcal{Z}_n$5, leading to globally regular, compact equilibria with positive binding energy over broad parameter ranges, indicating potential dynamical stability. In the $\mathcal{Z}_n$6 neutral case, as the field frequency approaches zero, matter becomes completely confined within a critical radius $\mathcal{Z}_n$7, and the metric develops a "quasi-horizon" where $\mathcal{Z}_n$8 and $\mathcal{Z}_n$9 are non-vanishing but extremely small (order $n \to \infty$0). The geometry exterior to $n \to \infty$1 coincides exactly with the extremal black hole solution in the same theory. The interior, however, hosts no singularity or horizon, and the energy-momentum tensor everywhere satisfies all standard energy conditions.

This "frozen state" constitutes a regular black hole mimicker: from the exterior, the ADM mass and causal structure are indistinguishable from a classical extremal BH, yet the interior is everywhere regular and supported solely by non-exotic matter fields.

Impact of Electric Charge and the "Unfreezing" Mechanism

The inclusion of electric charge introduces a long-range Coulomb repulsion that qualitatively alters the zero-frequency "frozen" state. As $n \to \infty$2 increases, the minimum achievable frequency $n \to \infty$3 rises, and matter is no longer able to condense within the critical radius; the system becomes "unfrozen," preventing the formation of a quasi-horizon. The equilibrium is now dictated by the balance between gravitational attraction, higher-derivative curvature repulsion, and electromagnetic repulsion. For sufficiently high charge, a window of negative binding energy emerges, indicating loss of stability, and the system ceases to mimic an extremal black hole in the frozen sense. Unlike the neutral case, the exterior solution for $n \to \infty$4 is genuinely electrovacuum with two independent parameters, causing loss of the universal matching between the star and extremal black hole geometries.

Energy Conditions and Physical Viability

A crucial property of the solutions for all $n \to \infty$5, including the $n \to \infty$6 case, is that all standard energy conditions—weak, dominant, null, and strong—are obeyed everywhere. No regime of exotic or phantom matter appears. This sharply distinguishes these regular BH mimickers from other regular black hole constructions in modified gravity or non-linear electrodynamics, where energy conditions are typically violated.

Implications and Future Directions

The demonstration that infinite-order higher-curvature corrections can support regular, ultra-compact, charged and neutral Proca star configurations satisfying all energy conditions has multiple ramifications:

• Theoretical Significance: It provides an explicit realization that singularity resolution and ultra-compact object formation can be accomplished in a physically consistent manner within purely gravitational extensions of GR, without recourse to uncertain matter models.
• Astrophysical Relevance: These frozen stars and their charged analogs represent observationally distinct black hole mimickers. Their shadow structure, gravitational-wave signatures (especially light ring and ringdown differences), and possible dynamical signatures—if they form or persist in astrophysical scenarios—warrant further study. The unfreezing mechanism suggests that charge, potentially even minute, could robustly distinguish between true BHs and compact horizonless objects in the strong-gravity regime.
• Stability and Dynamics: Positive binding energy in the neutral infinite-derivative case indicates the possibility of stability, but actual dynamical evolution and nonlinear stability must be confirmed via time-dependent numerical relativity simulations. The fate of these objects under perturbation, accretion, or mergers is of central interest.

Conclusion

Charged Proca stars in infinite-tower higher-derivative gravity demonstrate that globally regular, ultra-compact, horizonless configurations—forming "frozen states"—naturally arise from the competition between gravitational attraction and high-order curvature corrections. Electrostatic charge "unfreezes" these objects, preventing the formation of a regular core and reintroducing extended matter profiles. The fact that all solutions, for realistic matter sectors, satisfy all energy conditions establishes these objects as some of the most consistent and tractable black hole mimicker and regularization mechanisms yet constructed in higher-dimensional gravity. Extensions to four dimensions, dynamical studies, phenomenology, and observational discriminants remain as critical avenues for exploration.