Stochastic Control of Tolman-Oppenheimer-Snyder Collapse of Zero-Pressure Stars to Black Holes: Rigorous Criteria for Density Bounds and Singularity Smoothing

Abstract

The Tolman-Oppenheimer-Snyder description gives exact analytical solutions for an Einstein-matter system describing total gravitational collapse of a zero-pressure perfect-fluid sphere, representing a massive star which has exhausted its nuclear fuel. The star collapses to a point of infinite density within a finite comoving proper time interval $[0,t_{*}]$, and the exterior metric matches the Schwarzchild black hole metric. The description is re-expressed in terms of a 'density function' $u(t)=(\rho(t)/\rho_{o}))^{1/3}=R^{-1}(t)$ for initial density $u_{0}=R^{-1}(0)=1$ and radius $R(0)$, whereby the general-relativistic formulation reduces to an autonomous nonlinear ODE for $u(t)$. The solution blows up or is singular at $t=t_{*}=\pi/2(8\pi G/3\rho_{o})^{1/2}$. The blowup interval $[0,t_{*}]$ is partitioned into domains $[0,t_{\epsilon}]\bigcup[t_{\epsilon},t_{*}]$,with $t_{*}=t_{\epsilon}+|\epsilon|$ and $|\epsilon|\ll 1$, so that $t_{\epsilon}$ can be infinitesimally close to $t_{*}$. Randomness or 'stochastic control' is introduced via the 'switching on' of specific (white-noise) perturbations at $t=t_{\epsilon}$. Hybrid nonlinear ODES-SDES are then 'engineered' over the partition. Within the Ito interpretation, the resulting density function diffusion $\overline{u(t)}$ is proved to be a martingale whose supremum, volatility and higher-order moments are finite, bounded and singularity free for all finite $t>t_{\epsilon}$. The collapse is (comovingly) eternal but never becomes singular. Extensive and rigorous boundedness and no-blowup criteria are established via various methods, and blowup probability is always zero. The density singularity is therefore smoothed or 'noise-suppressed'. Within the Stratanovitch interpretation, the singularity formation probability is unity; however, null recurrence ensures the expected comoving time for this to occur is now infinite.