Stochastic control of Tolman-Oppenheimer collapse of zero-pressure fluid stars: rigorous criteria for density boundedness and singularity smoothing

Abstract

The Tolman-Oppenheimer 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.