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 $[0,t_{\ast }]$, 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 $t=t_{\ast }=\pi /2(8\pi G/3{\rho }_o{)}^{1/2}$. The blowup interval $[0,t_{\ast }]$ is partitioned into domains $[0,t_{\epsilon }]\bigcup [t_{\epsilon },t_{\ast }]$,with $t_{\ast }=t_{\epsilon }+|\epsilon |$ and $|\epsilon |\ll 1$, so that $t_{\epsilon }$ can be infinitesimally close to $t_{\ast }$. 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.

91 pages, no figures. New title, some minor errors and typos corrected, some notation changes

References in corpus (4)

• Black holes in an ultraviolet complete quantum gravity
• Master equations and the theory of stochastic path integrals
• Planck star phenomenology
• Singularities and Quantum Gravity