An analytic model for a total process of gravitational collapse: from star to Schwarzschild black hole

Abstract We present an exact analytical model describing a complete gravitational collapse of matter from horizonless initial conditions to black hole formation, tracing the full evolution of the horizon H ( t ) from its formation at microscopic scales to macroscopic stabilization. The solution reveals two main stages: (i) a dynamical horizon growth, where an apparent horizon H ( t ) emerges at a critical time $$t_b$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>t</mml:mi> <mml:mi>b</mml:mi> </mml:msub> </mml:math> until reaching its final size $$h=2\mathcal{M}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>h</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mi>M</mml:mi> </mml:mrow> </mml:math> , demonstrating how trapped surfaces form dynamically in finite time, and (ii) a naked-singularity resolution, where an integrable Ricci curvature singularity ( $$R^\mu {}_\nu \sim r^{-2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>μ</mml:mi> </mml:msup> <mml:mmultiscripts> <mml:mrow/> <mml:mi>ν</mml:mi> <mml:mrow/> </mml:mmultiscripts> <mml:mo>∼</mml:mo> <mml:msup> <mml:mi>r</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> ) develops at $$r=0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , but remains causally hidden by the horizon growth, preserving weak cosmic censorship without exotic matter. The model could offer a framework to study the quantum-to-classical transition ( $$H(t) \sim \ell _\textrm{Planck}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>∼</mml:mo> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mtext>Planck</mml:mtext> </mml:msub> </mml:mrow> </mml:math> ).