Stationary Einstein-vector-Gauss-Bonnet black holes

Abstract: We study spontaneously vectorized black holes in Einstein-vector-Gauss-Bonnet theory with a quadratic coupling function. Besides the static, spherically symmetric black holes carrying an electric charge, there are uncharged static, axially symmetric black holes that possess a magnetic dipole moment. Both types possess radial excitations. The magnetic black holes are prolate. They are hotter than the Schwarzschild black holes and possess lower free energy. The domain of existence of the rotating vectorized black holes is bounded by the Kerr black holes, the spherically and axially symmetric static black holes, and the critical solutions.

• The paper demonstrates that spontaneous vectorization in EvGB gravity gives rise to nontrivial electric and magnetic black hole solutions.
• It applies a quadratic coupling to the Gauss-Bonnet term to derive detailed thermodynamic and geometric properties of both static and stationary configurations.
• The study extends to rotating scenarios, revealing a connected existence domain with a thermodynamic preference over classical GR black holes.

Stationary Black Holes in Einstein-Vector-Gauss-Bonnet Theory

Introduction and Motivation

Einstein-vector-Gauss-Bonnet (EvGB) gravity extends general relativity (GR) via an effective field theory approach, introducing higher-curvature corrections through the Gauss-Bonnet (GB) term and coupling it to a real vector field. This framework is motivated by the search for viable modifications of gravity that remain consistent with cosmology and astrophysics, potentially addressing phenomena such as dark energy and providing new dynamics for compact objects. The investigation of spontaneous scalarization and vectorization in extended gravity theories has revealed nontrivial black hole (BH) solutions with primary scalar or vector hair, challenging the classical GR no-hair theorems.

This paper provides a systematic analysis of spontaneously vectorized BH solutions in EvGB gravity with a quadratic coupling function. It uncovers both static spherically symmetric (electric) solutions and axially symmetric (magnetic) solutions, as well as their stationary, rotating generalizations. The study details key physical and thermodynamic properties, bifurcation structure, and domain of existence as a function of the GB coupling and BH spin.

Theoretical Framework

The EvGB action studied consists of the Einstein-Hilbert term, the GB invariant, and a massless (generalized Proca) vector field coupled quadratically to the GB term. The field equations are second order due to the particular structure of the GB term and coupling, ensuring theoretical consistency. The vector field Ansatz is general enough to support both electric and magnetic-type configurations, with axial symmetry introduced for the latter. Importantly, the coupling function breaks the electromagnetic U(1) gauge invariance, so the “electric” and “magnetic” labels refer to the vector field charges and moments, distinct from Maxwell charges.

Boundary conditions enforce asymptotic flatness, regularity at the event horizon, elementary flatness on the axis, and reflection symmetry across the equator. Thermodynamic and geometric quantities—mass, angular momentum, electric charge, magnetic dipole, horizon area, entropy (including the GB contribution), Hawking temperature, and horizon geometry—are extracted from asymptotics and horizon embedding.

Static Vectorized Black Holes

Spherically Symmetric (Electric) Solutions

Recovering previous results, the paper confirms that nontrivial vectorized spherical BHs exist above a threshold value $\lambda_{\mathrm{bif}}/M^2 = 4.682$ for the dimensionless GB coupling. These BHs bifurcate from Schwarzschild at this point, coinciding with the onset of a tachyonic instability of Schwarzschild under vector perturbations. The electric charge increases monotonically with the coupling.

Entropy, horizon area, and Hawking temperature all decrease with increasing coupling compared to Schwarzschild. Notably, these BHs are thermodynamically disfavored for most of the parameter space: their free energy is always higher than that of Schwarzschild BHs with the same mass.

Axially Symmetric (Magnetic) Solutions

The study presents the first explicit construction of static, axially symmetric, uncharged (magnetic) vectorized BHs in EvGB gravity. These solutions possess a nonzero magnetic dipole moment and prolate horizon geometry. They exist within a finite interval of the coupling parameter, bifurcating from Schwarzschild at a lower threshold $\lambda_{\mathrm{bif}}/M^2 = 3.176$ and terminating at a critical value $\lambda_{\mathrm{cr}}/M^2 = 2.912$.

Remarkably, these magnetic BHs display a Hawking temperature higher than the Schwarzschild solution for the same mass, and their entropy is nearly degenerate with the Schwarzschild value. The reduction in horizon area due to the deformation is almost exactly offset by the extra entropy from the GB term. Importantly, the free energy of magnetic solutions is always lower than that of their Schwarzschild counterparts, indicating a preferred thermodynamic branch at fixed mass.

Perturbative analysis near the bifurcation points determines the existence lines for each sector, confirming the full nonlinear numerical calculations and providing insight into the excitation spectrum of radial modes.

Rotating Vectorized Black Holes

The stationary sector is constructed by introducing rotation on both electric and magnetic static parents. The parameter space of rotating vectorized BHs reveals rich structure:

• The domain of existence in the $(J/M^2, \lambda/M^2)$ plane shows that both electric and magnetic families merge at sufficient rotation, forming a connected region with nontrivial boundaries corresponding to GR solutions, the onset of vectorization, and critical solutions.
• The maximal dimensionless angular momentum supported by vectorized BHs is $J/M^2 \approx 0.544$, well below the extremal Kerr limit, reflecting a significant restriction compared to GR.
• Rotating magnetic (static parent) BHs gain a small electric charge, and rotating electric (static parent) BHs acquire a magnetic dipole; the vector "hair" becomes mixed under rotation.
• Horizon geometry shifts from prolate to oblate as rotation increases for previously prolate magnetic BHs, while slowly rotating spherical BHs can paradoxically display prolate deformation.
• Entropy, as in the static case, never exceeds the value in Kerr, and the free energy remains lower than Kerr for the same angular momentum, emphasizing thermodynamic preference in this regime. The Hawking temperature remains relatively large, with no access to extremal states.

Implications and Outlook

Physical and Thermodynamic Implications

The explicit construction of static and stationary vectorized BHs in EvGB gravity both supports and extends the landscape of hairy BHs in gravity theories with higher-order curvature corrections. The existence of non-spherical, uncharged, thermodynamically favored BHs is particularly noteworthy, as it evades classic theorems such as Israel’s uniqueness result. The lower free energy suggests that, for relevant GB couplings, vectorized BHs could be dynamically preferred endpoints in realistic gravitational collapse or mergers if vector field perturbations are present.

The restriction on the maximal angular momentum excludes extremal Kerr analogs, shaping the expected astrophysical signature of highly spinning BHs within the theory. Furthermore, the near-degeneracy of entropy and the role of the GB contribution accentuate the subtleties in the thermodynamics of higher-curvature BHs beyond GR.

Theoretical Perspectives

The results confirm that spontaneous vectorization, akin to scalarization, offers a rich structure of BH solutions, topologically and dynamically distinct from GR. The emergence of magnetic, axially deformed BHs prompts further examination of their dynamical stability, gravitational wave signatures, and their gravitational lensing properties.

The observed thermodynamic preference and absence of extremal vectorized BHs imply that, if EvGB gravity is realized in nature, observations of highly spinning BHs near extremality would strongly constrain the model.

Future theoretical developments must include a rigorous stability analysis via quasinormal modes (as in [62-66]) and exploration of the nonlinear dynamical formation of vectorized states in realistic gravitational collapse or compact binary mergers.

Conclusion

This work systematically delineates the solution space of stationary black holes in Einstein-vector-Gauss-Bonnet gravity with a quadratic vector coupling. It constructs and characterizes both static and rotating vectorized BHs with electric and magnetic hair, identifying the thermodynamically preferred solutions and map of the existence domain. The findings provide strong evidence for novel, axially symmetric, magnetic BHs with prolate horizons and lower free energy than Schwarzschild—an explicit violation of uniqueness and a substantial modification of classic BH phenomenology in modified gravity. Theoretical extensions, astrophysical implications, and dynamical stability analyses remain essential directions for future research, particularly in the context of gravitational wave astronomy and strong-field tests of gravity.