Static multipolar Einstein-vector-Gauss-Bonnet black holes

Technical Summary: Static Multipolar Einstein-Vector-Gauss-Bonnet Black Holes

Problem Statement
This paper addresses the existence and classification of static, asymptotically flat, axisymmetric black hole solutions in Einstein-Vector-Gauss-Bonnet (EvGB) theory. While General Relativity (GR) forbids static, nonspherical vacuum black holes (Israel's theorem), extensions of GR involving additional fields coupled to curvature invariants can evade this restriction. Specifically, the authors investigate the "vectorization" mechanism in EvGB theory, where a vector field $A_\mu$ couples quadratically to the Gauss-Bonnet (GB) invariant ($R^2_{GB}$) via a coupling function $\lambda A_\mu A^\mu$. The study focuses on constructing static black holes that bifurcate from the Schwarzschild solution, carrying nontrivial vector hair characterized by angular multipole structures, distinct from the previously studied spherically symmetric ($\ell=0$) or rotating solutions.

Methodology
The authors employ a numerical approach to solve the coupled system of partial differential equations (PDEs) derived from the EvGB action:
$$S = \frac{1}{16\pi} \int \left[ R - F_{\mu\nu}F^{\mu\nu} + \lambda A_\mu A^\mu R^2_{GB} \right] \sqrt{-g} \, d^4x$$
where $F_{\mu\nu}$ is the vector field strength.

1. Ansatz and Coordinates: The spacetime is assumed to be static and axially symmetric, utilizing isotropic coordinates. The vector field is treated in two distinct sectors:
   • Electric Sector: $A_\mu dx^\mu = A_t dt$ (purely electric).
   • Magnetic Sector: $A_\mu dx^\mu = A_\phi d\phi$ (purely magnetic).
     Simultaneous electric and magnetic components are excluded in the strictly static ansatz to avoid angular momentum sources.
2. Perturbative Analysis: To identify the onset of vectorization, the authors perform a linear perturbation analysis on the Schwarzschild background. This reduces the problem to an eigenvalue problem for the radial function $\gamma_\ell(r)$ and discrete coupling values $\lambda_\ell$. The solutions are labeled by the angular multipole number $\ell$ and the radial excitation number $n$. The study restricts itself to the fundamental radial branch ($n=0$).
3. Numerical Construction: The nonlinear field equations are solved numerically on a compactified domain ($0 \le x \le 1$, where $x=0$ is the horizon and $x=1$ is spatial infinity). The coupling parameter $\lambda$ is varied along solution branches to map the global structure of the solutions.
4. Boundary Conditions: Regularity is imposed at the horizon and spatial infinity (asymptotic flatness), along with specific symmetries on the axis and equatorial plane.

Key Contributions and Results

• Multipolar Bifurcation: The authors demonstrate that vectorized black holes bifurcate from the Schwarzschild solution at discrete values of the coupling parameter $\lambda_\ell/M^2$. These bifurcation points are determined by the angular multipole number $\ell$ (calculated for $\ell = 0, 1, 2, 3$).
• Electric Branches:
  • The electric sector contains the known spherically symmetric branch ($\ell=0$), which corresponds to an electric vector monopole, and new axisymmetric branches for $\ell > 0$.
  • These branches extend to arbitrarily large values of the coupling parameter $\lambda/M^2$; no upper critical endpoint was found.
  • Nonlinearities generate higher-order multipoles of the same parity (even $\ell$ generates even moments, odd $\ell$ generates odd moments), but the dominant moment remains the one associated with the bifurcation mode.
• Magnetic Branches:
  • The magnetic sector features a sequence of axisymmetric branches, including the previously known magnetic dipole branch at $\ell=0$.
  • Unlike the electric sector, magnetic branches exist only on finite intervals of the coupling parameter: $\lambda_{cr, \ell} \le \lambda \le \lambda_\ell$.
  • These branches terminate at critical solutions ($\lambda_{cr, \ell}$) where the nonlinear solver ceases to converge to regular black hole solutions.
  • The magnetic bifurcation values $\lambda_\ell$ are systematically smaller than their electric counterparts for the same $\ell$.
• Thermodynamics and Geometry: The paper provides detailed profiles for the ADM mass, horizon area, entropy (calculated via the Wald-Iyer Noether-charge expression), Hawking temperature, and horizon deformation. The results show that as the coupling moves away from the bifurcation point, the horizon deformation and thermodynamic quantities deviate significantly from the Schwarzschild values.

Significance and Claims
The paper claims to provide the first systematic construction and classification of static, multipolar vectorized black holes in EvGB theory with a quadratic coupling function. The primary significance lies in:

1. Evasion of Israel's Theorem: Demonstrating that static, nonspherical vacuum black holes can exist in modified gravity theories due to the nontrivial coupling of the vector field to the GB invariant.
2. Sectoral Differences: Highlighting a fundamental structural difference between electric and magnetic vectorization. While both exhibit multipolar bifurcations, their nonlinear continuations differ drastically: electric branches are unbounded in coupling space, whereas magnetic branches are bounded and terminate at critical solutions.
3. Parity Selection: Establishing that equatorial parity acts as a selection rule for nonlinear multipoles, keeping even and odd moments separated even in the fully nonlinear regime.

Limitations and Outlook
The authors explicitly state that the work is restricted to the construction and classification of stationary solutions. They do not address the dynamical formation or stability of these solutions. The paper notes that in related vector-tensor theories, tachyonic vectorization mechanisms have been associated with ghost-like degrees of freedom or gradient instabilities. Consequently, the dynamical viability of the specific multipolar branches found here remains an open question requiring dedicated time-dependent perturbation analysis. The solutions serve as a reference for future studies into the broader landscape of vector-tensor theories and the physical relevance of spontaneous vectorization.