Kerr black holes without primary hairs

Technical Summary: Kerr Black Holes without Primary Hairs

Problem Statement
While the Kerr solution is the standard model for rotating black holes (BHs), characterized solely by mass ($M$) and spin ($a$), the detailed analytical description of its formation via gravitational collapse remains an open problem. Unlike the Schwarzschild case, where the Oppenheimer-Snyder (OS) model describes collapse to a point singularity, no simple analytical model exists for the emergence of the Kerr ring singularity from an initially regular astrophysical configuration. Recent work on nonsingular Schwarzschild configurations [27] demonstrated that singularities inevitably arise even in models devoid of "primary hairs" (new charges), suggesting the Schwarzschild solution cannot be the continuous end-state of collapse. This paper seeks to extend these findings to the axisymmetric case, asking whether a regular Kerr BH with a rich internal structure can be constructed without introducing new charges, thereby serving as a potential analog to the OS model for rotating collapse.

Methodology
The author constructs a class of stationary, axisymmetric, nonsingular black hole geometries using the following framework:

1. Kerr-Schild Subclass: The spacetime is defined within the Kerr-Schild subclass throughout the region $r > 0$, utilizing the Gurses-Gursey metric in Boyer-Lindquist coordinates.
2. Boundary Conditions: The solution recovers the standard Kerr vacuum for $r \geq h$ (where $h$ is the event horizon radius). Inside the horizon ($0 < r \leq h$), the mass function $m(r)$ is generalized from the constant $M$.
3. No Primary Hair: The configuration is constrained to be determined solely by the global parameters $\{M, a\}$ (total mass and angular momentum). No additional charges or parameters are introduced to characterize the interior.
4. Differentiability and Regularity: To avoid the trivial solution ($m(r)=M$ everywhere) while maintaining regularity at the origin, the mass function is constructed as a superposition of $N$ terms embedded in a de Sitter background. The coefficients are determined by imposing continuity conditions at the horizon ($m(h)=M, m'(h)=0$) and requiring the mass function to be of differentiability class $C^N$ ($N > 1$).
5. Mass Function Formulation: The interior mass function $m(r)$ is derived as a specific polynomial superposition (Eq. 13) dependent on a set of integer exponents $\{n_1, n_2, \dots, n_N\}$ where $n_i > 2$.

Key Contributions and Results

• Infinite Class of Regular Solutions: The paper presents an infinite family of rotating black hole solutions characterized exclusively by $\{M, a\}$. These solutions possess a regular event horizon coincident with the Kerr horizon but feature a non-trivial, regular interior structure.
• Quasi-Extremal Configurations: A significant finding is the existence of "quasi-extremal" configurations where the inner (Cauchy) horizon $h_c$ approaches the event horizon $h$ ($h_c \sim h$) without requiring the extremal condition $a \approx M$. This occurs in the limit where the parameters $n_i \to \infty$. In these cases, the surface gravity remains finite and Kerr-like, determined solely by $h$, despite the complex interior.
• Interpolation to Singularities: By extending the parameter space of the exponents $n_i$ to include values $n_i \leq 2$, the framework interpolates between regular black holes and singular configurations:
  • Regular Black Holes (RBH): Occur when all $n_i > 2$. The curvature scalars remain finite, and the ring singularity ($\rho=0$) is avoided.
  • Integrable Singularities (IS): Occur when some $n_i \in [-2, 2]$. Here, curvature scalars diverge but remain integrable (scaling as $\sim r^{-2}$).
  • Recovery of Kerr: The standard Kerr solution is recovered specifically when an index $n_i = -1$.
  • Minkowski Breaking (MB): If $n_i = -2$, the spacetime fails to approach Minkowski space at the core ($r \to 0$), preventing the definition of a smooth local Lorentz frame.
  • Strong Singularities: If $n_i < -2$, non-integrable strong singularities develop.
• Thermodynamics: The surface gravity $\kappa$ at the event horizon is continuous and independent of the interior parameters $n_i$, remaining purely Kerr-like. However, the inner horizon location $h_c$ is sensitive to the interior structure.

Significance and Claims
The paper claims to provide a "first step toward an axisymmetric analog of the OS model." By demonstrating that a regular Kerr-like geometry can exist without primary hairs, the work suggests a pathway to analytically describe the formation of a Kerr black hole from an initially regular configuration.

The author highlights that while a fully time-dependent analytical description of Kerr formation remains an "outstanding challenge" due to the lack of a rotational Birkhoff's theorem (which prevents matching a dynamic interior to a stationary Kerr exterior), the stationary solutions presented here offer a "natural setting" to investigate this process. The existence of these regular configurations implies that singularities are not inherent to the initial state but may arise inevitably during the collapse process, consistent with the strong cosmic censorship conjecture. The work serves as a theoretical foundation for future dynamical treatments seeking to model the transition from regular matter to a Kerr singularity.