Formation of the Kerr black hole: an exact model

Technical Summary: Formation of the Kerr Black Hole: An Exact Model

Problem Statement
While the Oppenheimer-Snyder (OS) model provides an exact analytical description of spherically symmetric gravitational collapse leading to a Schwarzschild black hole, no analogous model exists for the axisymmetric collapse of a rotating stellar system into a Kerr black hole. The primary obstacle is the absence of a rotational counterpart to Birkhoff's theorem; in the axisymmetric case, the exterior geometry is not fixed by the interior mass distribution alone, as the evolution of the collapsing object inevitably modifies the exterior spacetime. Consequently, constructing an exact analytical model for the formation of a Kerr black hole has been considered intractable, with the exterior geometry belonging to an infinite family of admissible solutions.

Methodology
The author proposes a strategy that bypasses the difficulty of matching separate interior and exterior solutions. Instead, a single, time-dependent line element is postulated to describe the entire spacetime, encompassing both the region occupied by the collapsing matter and the exterior. This metric is constructed as a time-dependent extension of the Gurses-Gursey metric (a Kerr-Schild class metric) reported in previous work [29].

The model is defined by the following components:

1. Metric Structure: The line element is expressed in Eddington-Finkelstein-like coordinates $(v, r, \theta, \phi)$, incorporating an ingoing null coordinate $v$ and a shifted azimuthal coordinate.
2. Parameters: The geometry depends exclusively on the total mass $M$ and a time-dependent rotation parameter $a(v) = J(v)/M$. The model carries no "primary hair" (no additional independent parameters).
3. Mass Function: The mass profile $\tilde{m}(v, r)$ is piecewise defined. For the exterior ($r > h(v)$), $\tilde{m} = M$. For the interior ($r \leq h(v)$), the mass function $m(v, r)$ is a specific polynomial form belonging to the class $C^N$, determined by a set of integer parameters $\{n_i\}$.
4. Evolutionary Mechanism: To describe the formation of the Kerr black hole, the model couples the evolution of the rotation parameter $a(v)$ with the evolution of the interior mass parameters $n_i(v)$. Specifically, the discrete labels $n_i \in \mathbb{N}$ are extended to continuous functions $n_i(v)$ that evolve from initial values (corresponding to regular configurations) to a final critical value of $n_i = -1$, which recovers the singular Kerr geometry.

Key Contributions and Results

• Exact Analytical Model of the Final Stage: The paper presents an exact analytical solution for the final stage of axisymmetric gravitational collapse, immediately preceding the formation of a stationary Kerr black hole. The model covers both the extremal case ($a=M$) and a novel "quasi-extremal" regime where the inner and outer horizons are close ($h_c \sim h$) even when $a$ is not close to $M$. The author explicitly notes that this is not a complete description of the entire collapse process (which would include a prolonged regular stage), but rather an exact framework for the transition to the final stationary state.
• Dynamics of the Horizon: The evolving event horizon radius is given by $h(v) = M + \sqrt{M^2 - a(v)^2}$. The analysis shows that the collapse ($\dot{h} < 0$) is driven by an increase in the spin parameter ($\dot{a} > 0$). The system satisfies a relation formally identical to the first law of black hole mechanics ($\delta M = \frac{\kappa}{8\pi}\delta A + \Omega \delta J$).
• Singularity Structure:
  • Kinematic Singularities: The time dependence of $a(v)$ introduces curvature singularities. The scalar curvature $R$ contains terms proportional to $\dot{a}^2/r^2$ and $\ddot{a}/r^2$, and a problematic non-integrable term $\sim \dot{a}/r^3$.
  • Mass-Induced Singularities: As the parameters $n_i(v)$ evolve toward the critical value $-1$ (required for the Kerr limit), the mass function generates additional singularities.
  • Trapped Region: Despite these singularities, the metric function $\Delta(v, r)$ maintains two real roots ($h_c(v) < h(v)$) throughout the evolution. The author argues that this strongly suggests the curvature singularities remain enclosed within a trapped region near the evolving horizon, preventing them from becoming naked.
• Transient Exterior Curvature: The model predicts a transient anisotropic curvature in the exterior region ($r \gg a$) that decays as $1/r^2$. This curvature is induced by the kinetic contributions of the time-dependent rotation parameter.

Significance and Claims
The paper claims to provide the first exact analytical description of the final stage of rotational gravitational collapse, immediately preceding the formation of the Kerr black hole. The author explicitly states that the model is not intended to be a complete description of the entire collapse process (which would include a prolonged regular stage), but rather an exact framework for the transition to the final stationary state.

The construction establishes that:

1. The formation of a Kerr black hole can be described analytically without introducing additional gravitational hairs, relying solely on mass and angular momentum.
2. The transition to the Kerr geometry ($n_i \to -1$) necessitates a transient, non-stationary, and singular phase.
3. The model offers a potential framework for investigating observable signatures associated with the formation of rotating black holes, specifically through the transient anisotropic curvature in the exterior spacetime.

The work acknowledges that the physical source generating this geometry and the rigorous determination of the causal structure (specifically the existence of the trapped surface) remain open questions for future investigation. The model is presented as a theoretical tool to analyze the emergence of the Kerr singularity and horizon structure during the collapse.