The impact of the eccentricity on the collapse of an ellipsoid into a black hole

TL;DR

The paper investigates whether a homogeneous, non-rotating, pressureless ellipsoid can collapse to a black hole and how the initial shape, encoded by the eccentricity $e$, governs the outcome. Using a Newtonian framework, it reduces the problem to oblate and prolate spheroids with a single $e$, derives the gravitational potential coefficients $\alpha(e)\u0003$ and ${\beta}(e)$, and evolves compression ratios $R(t)$ and $Z(t)$ to determine a black-hole-formation threshold. It finds a quasi-universal scaling where the maximal compression follows $\chi_{fin} \propto e_0^{15/8}$ with exponent $ u \approx 1.88$, nearly independent of spheroid type, and introduces a refined 'with flyby' estimator to improve accuracy. Applying this to early-Universe scenarios, the work yields a relation between initial eccentricity, mass, and redshift that informs primordial black hole formation from nonspherical collapse, offering a compact predictive framework for BH formation from dust-like ellipsoids.

Abstract

We consider the gravitational collapse of a homogeneous pressureless ellipsoid. We have shown that the minimal size $r$ that the ellipsoid can reach during collapse depends on its initial eccentricity $e_0$ as $r\propto e_0^ν$, where $ν\approx 15/8$, and this dependence is very universal. We have estimated the parameters (in particular, the initial eccentricity) of a homogeneous pressureless ellipsoid, whereat it collapses directly into a black hole.

Figures (2)

• Figure 1: The maximal compression ratio ($R_{fin}$ or $Z_{fin}$), as a function of the initial eccentricity $e_0$ of an oblate spheroid, if we use the simple criterion (\ref{['28b3']}) (blue line) and the criterion with flyby (red line).
• Figure 2: The maximal compression ratio ($R_{fin}$ or $Z_{fin}$), as a function of the initial eccentricity $e_0$ of a prolate spheroid, if we use the simple criterion (\ref{['28b3']}) (blue line) and the criterion with flyby (red line).