Higher-Dimensional Boson Star Solutions

• The paper develops a numerical framework using Chebyshev expansions and Newton–Raphson relaxation to construct rotating boson star solutions in asymptotically AdS spacetimes.
• Key findings show that mass, angular momentum, and frequency exhibit damped oscillatory behavior with increasing central energy density, indicating a critical transition to black hole formation.
• Implications include new insights into gravitational symmetry breaking, the influence of higher-curvature corrections, and potential holographic applications in high-dimensional gravitational physics.

Higher-dimensional boson star solutions are self-gravitating, horizonless configurations of complex scalar fields in spacetimes with five or more dimensions, extending both the conceptual framework and phenomenology of four-dimensional boson stars. These solutions are of significant interest in the context of classical and semiclassical gravitation, string theory, higher-curvature gravity, and holographic dualities. They exhibit novel features, stability properties, and critical behaviors not seen in the lower-dimensional case, and their construction reveals both the intricate mathematical structure and the rich dynamical landscape of scalar solitons in higher-dimensional gravity.

1. Mathematical Framework and Construction

The canonical action for a higher-dimensional, asymptotically locally AdS boson star comprises the Einstein–Hilbert term (and often higher-curvature corrections), a negative cosmological constant (for AdS backgrounds), and a multiplet of complex scalar fields, typically arranged as a minimally coupled $\frac{D-1}{2}$-tuplet for odd $D$: $$S = \int d^D x\, \sqrt{-g} \left[ \frac{1}{16\pi G} \left( R - 2\Lambda \right) - \nabla_{\mu} \overline{\Pi}_i \nabla^{\mu} \Pi_i \right].$$ A crucial ansatz for constructing rotating solutions is

$$\Pi_i(r, t, \Omega) = \Pi(r) e^{-i\omega t} z_i(\Omega),$$

where $z_i$ is a complex vector chosen such that the stress-energy tensor inherits the symmetries of the background metric, ensuring the reduction to ordinary differential equations in the radial coordinate.

To enable robust numerical solution, the infinite radial domain is compactified, for example by $y = r^2/(r^2+\ell^2)\in[0,1]$, and the unknown functions are regularized via appropriate factorization of their asymptotic behavior. All dynamical fields are then represented as Chebyshev expansions in $y$; a relaxation (Newton–Raphson) method is deployed to solve the resulting nonlinear algebraic system to precision better than $10^{-8}$.

A variety of potentials, self-interaction terms, and higher-curvature modifications (Gauss–Bonnet and higher) have been included to paper both physically motivated scenarios and the impact of field-theoretical generalizations.

2. Symmetry Structure and Killing Vectors

A distinctive feature of higher-dimensional boson stars is the interplay between the symmetries of the metric and those of the scalar field. While the metric in $D$-dimensional rotating AdS spacetime may possess multiple commuting Killing vector fields (such as temporal translations and multiple rotational symmetries), the multiplet scalar field ansatz is constructed to be invariant only under a single "helical" Killing vector: $K = \partial_t + \omega \partial_\chi,$ where $\chi$ denotes the coordinate of rotation associated with $z_i$. This symmetry breaking is nontrivial: the full configuration (metric plus scalar) retains only this helical symmetry, even though the metric itself generally admits a larger isometry group. The solutions thus exemplify "spontaneous symmetry breaking" in strong gravity, with nontrivial implications for gravitational hair and holographic applications.

3. Parametrization and Physical Properties

The space of higher-dimensional boson star solutions forms a one-parameter family that can be cleanly labeled by the central value of the energy density, typically encoded in the derivative of the scalar field at the origin: $q_0 = \ell \,\Pi'(0).$ Low-$q_0$ solutions are accurately described by linear perturbations around AdS, with additional physics emerging as $q_0$ increases.

Global charges, including ADM mass $M$, total angular momentum $J$, and angular frequency $\omega$, are extracted from the asymptotic behavior of metric functions. For example, in $D=n+2$ dimensions,

$$M = \frac{(n+1)\pi^{\frac{n-1}{2}} \ell^{n-1}}{16[(n+1)/2]!}\left[(n+1)C_h - nC_f\right],$$

$$J = \frac{(n+1)^2\pi^{\frac{n-1}{2}} \ell^n}{16[(n+1)/2]!}C_\Omega,$$

where $C_h, C_f, C_\Omega$ are asymptotic coefficients of the fields. Plots of $(M,J,\omega)$ versus $q_0$ reveal damped oscillatory (spiral) behavior around finite limiting values.

For $D>5$ (e.g., $D=7,9,11$), as $q_0\to\infty$, these quantities exhibit smaller amplitude spirals, which become tighter as dimension increases. The amplitude of oscillations in $M$, $J$, and $\omega$ all decrease with higher $D$—a signature of the modified gravitational binding in large dimensions.

4. Limiting Behavior, Kretschmann Invariant, and Black Hole Threshold

The curvature properties of these solutions are quantified via the Kretschmann scalar,

$$K_n = \frac{2(n+1)}{\ell^4} \left[ (n+2) + \frac{3(n-1)(n+3)}{n} (q'_h(0))^2 - 4q_0^2 + \frac{4}{n}q_0^4\right],$$

where $q'_h(0)$ is a near-origin metric expansion coefficient. As $q_0$ increases, $K_n$ diverges, indicating that the central curvature grows without bound. In the asymptotic regime (large $q_0$), the $(M,J)$ curves flatten, suggesting that the system approaches a critical configuration reminiscent of black hole formation: an infinitesimal further increase in energy triggers collapse.

Notably, this behavior marks the end of regular boson star solutions and signals a dynamical transition to black hole spacetimes—a geometric realization of soliton collapse in gravitationally bounded scalar field systems.

5. Extensions: Higher-Curvature Gravity and Uniqueness Properties

Modification of the gravitational sector via the Gauss–Bonnet term and higher-order curvature corrections has dramatic impact on solution structure. In five dimensions, sufficiently strong Gauss–Bonnet coupling prevents the central scalar field from attaining large values. As a result, the spiral behavior in the parameter curves disappears for large coupling, and the $(M,R)$ relation becomes unique—eliminating the multivaluedness seen in pure Einstein gravity and yielding a pattern similar to neutron stars (Hartmann et al., 2013). In even more generalized higher-derivative gravity (including an infinite tower of curvature corrections), "frozen star" configurations emerge: the central divergence is regularized, the scalar field localizes within a critical radius $r_c$, and the exterior spacetime closely approaches that of an extreme black hole (Ma et al., 13 Jun 2024).

Additionally, the interplay with compactification and the inclusion of nontrivial topology (e.g., global monopole sectors or multi-configurational chains (Reid et al., 2015, Gervalle, 2022)) yield a rich taxonomy of stationary solitonic solutions, some of which may mimic black holes to arbitrary precision in their external observables.

6. Dynamical Stability and Radial Oscillations

The stability of higher-dimensional boson stars is deeply sensitive to both the spacetime dimension and the structure of the scalar potential. Plain "mini" boson stars (complex scalar with a mass term but no self-interaction) are radially unstable for $D > 4$—that is, all equilibrium solutions are dynamically unstable against radial perturbations (Franzin, 14 Aug 2024). This instability is confirmed by direct calculation of the spectrum of fundamental oscillation frequencies (pulsation analysis) and by binding energy arguments: no dynamically stable branch exists, in contrast to the four-dimensional case where solutions up to the Kaup mass are stable.

However, if one introduces either a quartic self-interaction (massive or solitonic potentials) or more complex potential forms, radially stable branches are possible in $D=5,6$ and likely for all $D\geq 4$ (Marks et al., 15 Oct 2025). The stability analysis rests on generalizing the radial pulsation equations to arbitrary dimension and potential, yielding eigenvalue problems whose lowest mode $\chi_0^2 > 0$ is necessary and sufficient for stability. Nonlinear dynamical evolutions in fully general-relativistic simulations confirm that configurations with positive $\chi_0^2$ are stable, supporting the robustness of the perturbative predictions.

7. Broader Physical and Theoretical Implications

Higher-dimensional boson stars have direct implications in several domains of gravitational theory:

• Holography: The existence of horizonless, "hairy" solitons in AdS spacetimes, with only a single Killing symmetry, provides controlled laboratories for studying non-thermalizing phases in dual CFTs, the spectrum of black hole microstates, and holographic turbulence (Stotyn et al., 2013, Henderson et al., 2014).
• Black hole mimickers: The unique mass-radius relations (especially under higher-curvature corrections) and the "frozen star" phenomenon point to configurations that can closely reproduce the external field of black holes while avoiding horizon/singularity problems (Hartmann et al., 2013, Ma et al., 13 Jun 2024).
• Astrophysical relevance: Rotating higher-dimensional boson stars and chains can support quantized angular momentum in multiple planes, differing substantially in topology and geodesic structure from both four-dimensional boson stars and higher-dimensional classical fluid stars (Brihaye et al., 2016, Gervalle, 2022). The presence of toroidal structures, stable circular orbits, and exotic geodesic behavior may have observable consequences in hypothetical higher-dimensional astrophysical settings.
• Critical phenomena and gravitational collapse: The transition from boson star solutions to black holes as energy density is increased provides a model setting for studying critical collapse, with the onset of instability and black hole formation sharply marked by divergent curvature invariants.

A plausible implication is that higher-dimensional solitonic gravity systems have a richer stability, criticality, and collapse structure than in $D=4$, with new solution branches parameterized by self-interaction strengths and higher-curvature effects.

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In summary, higher-dimensional boson star solutions constitute a well-defined and technically rich class of self-gravitating scalar solitons. Their construction underlies advances in understanding gravitational symmetry breaking, nonlinear stability, extremal compact objects, and the interplay between matter and higher-curvature gravity—all of which are essential themes in high-dimensional gravitational physics and related quantum gravity programs.