From rotating attractors to extremal black holes with axionic hair

Technical Summary: From Rotating Attractors to Extremal Black Holes with Axionic Hair

Problem Statement
This work investigates the existence and properties of extremal, rotating black holes in four-dimensional Einstein-Maxwell-axion (EMA) theory. The study addresses a critical gap in understanding the interplay between the near-horizon extremal geometry (NHEG) and the global spacetime structure for black holes carrying axionic hair. While the attractor mechanism posits that horizon data are fixed by conserved charges and decoupled from asymptotic moduli, recent analyses suggest that smooth extremal horizons are not guaranteed in general. The paper specifically targets the constraints imposed by the axion field on rotating attractors and the extent to which these near-horizon solutions extend to regular, asymptotically flat global black holes.

Methodology
The research employs a dual approach combining near-horizon analysis via the entropy-function formalism and a global bulk analysis using numerical relativity techniques.

1. Near-Horizon Analysis (NHEG):

   • The authors utilize the entropy-function formalism to derive the equations of motion for the near-horizon geometry, which possesses an $SO(2,1) \times U(1)$ isometry.
   • Perturbative Analysis: Solutions are constructed perturbatively around two analytic backgrounds: the Near-Horizon Extremal Kerr-Newman (NHEKN) and the Near-Horizon Extremal Kerr (NHEK). The axion-photon coupling $g_{\psi\gamma\gamma}$ is treated as a small expansion parameter.
   • Non-Perturbative Analysis: An integral constraint is derived directly from the axion equation of motion within the NHEG framework to establish necessary conditions for regularity without relying on perturbation theory.

2. Global Bulk Analysis:

   • The authors construct asymptotically flat, stationary, and axisymmetric solutions to the full EMA field equations.
   • A numerical scheme based on a sigma-model reduction is employed, solving the coupled partial differential equations (PDEs) for the metric, gauge, and axion fields.
   • Boundary conditions are imposed at the horizon (using a compactified radial coordinate to handle extremality) and at spatial infinity. The numerical solver (CADSOL) is used to generate families of solutions interpolating between the NHEG and flat infinity.

Key Contributions and Results

• Exclusion of Regular Dyonic Attractors:
  The central theoretical finding is that regular rotating attractors with axionic hair exist only in the purely electric or purely magnetic sectors. Regular rotating dyonic (both electric and magnetic) attractors are strictly excluded.

  • Perturbative Evidence: In the expansion around NHEKN, the first-order axion correction contains logarithmic terms ($\log(1-u)$ and $\log(1+u)$) that diverge at the poles ($u=\pm 1$) unless the electric and magnetic charges satisfy $Q_e Q_m = 0$.
  • Non-Perturbative Proof: By integrating the axion equation, the authors derive a boundary condition requiring $b(1) = \pm b(-1)$. This condition forces either the magnetic charge ($Q_m$) or the electric charge ($Q_e$) to vanish for the solution to remain smooth ($C^1$) at the poles.

• Construction of Purely Electric Attractors:
  Focusing on the allowed purely electric sector, the authors construct families of rotating EMA attractors for representative values of the axion-photon coupling ($g_{\psi\gamma\gamma} = 1$ and $g_{\psi\gamma\gamma} = \sqrt{3}/2$). The horizon data (entropy, angular momentum, charge) are shown to be fixed by the extremization of the entropy function, confirming the attractor mechanism's validity in this sector.

• Global Solutions and the Critical Point (P):
  The study constructs global, asymptotically flat extremal black holes that interpolate with the electric NHEG.

  • Agreement: Along a specific branch of solutions, the horizon data extracted from the global PDE solutions match the predictions from the near-horizon entropy function extremization. This confirms that the attractor mechanism controls the horizon data and is decoupled from asymptotic moduli.
  • Breakdown of Smoothness: As the electric charge increases and angular momentum decreases (approaching the static limit), the solutions encounter a critical configuration denoted as point (P). Beyond this point, while extremal (zero temperature) solutions can be constructed numerically, they lose smoothness. Specifically, the first derivative of the axion field becomes discontinuous at the horizon.
  • Divergence of Descriptions: Past point (P), the near-horizon description (NHEG) and the global bulk description "peel apart." The NHEG constructed from the entropy function no longer describes the horizon of the global solution, indicating that the existence of an NHEG does not guarantee the existence of a corresponding smooth global black hole.

• Curvature and Singularity Structure:
  Numerical results indicate that within the smooth branch, curvature invariants (Ricci and Kretschmann scalars) remain finite at the horizon, consistent with standard extremal black holes. The singularity at $r=0$ remains hidden behind the horizon.

Significance and Claims
The paper claims to provide a concrete characterization of the extremal sector in EMA theory, highlighting the limitations of the near-horizon analysis when applied to global spacetimes.

1. Refinement of the Attractor Mechanism: The work demonstrates that while the attractor mechanism successfully fixes horizon data for smooth rotating attractors in the purely electric sector, this mechanism is not universally sufficient to guarantee the existence of a smooth global solution.
2. Diagnostic Value of NHEG: The authors emphasize that the absence of a smooth NHEG is a diagnostic for non-smooth extremal limits. Conversely, the existence of an NHEG does not imply the existence (or uniqueness) of a corresponding global black hole solution.
3. Axionic Constraints: The exclusion of dyonic rotating attractors is presented as an intrinsic feature of the axion equation of motion combined with regularity requirements, rather than an artifact of perturbation theory.
4. Comparison with EMDA: The paper briefly notes that in the more general Einstein-Maxwell-dilaton-axion (EMDA) theory, regular rotating solutions were not found for generic couplings, appearing only at the special coupling values that reproduce the Kerr-Sen solution. This suggests a selectivity in the allowed electromagnetic sectors for smooth extremal black holes in theories with scalar fields.

The study concludes that the landscape of rotating extremal black holes in EMA theory is organized into smooth branches where the attractor mechanism holds, and non-smooth branches where extremal solutions exist but lack a smooth near-horizon limit, with the transition marked by a critical point where the axion field loses differentiability.