2019/09/09 by Steven D Miller, Miller, Steven D
Mathematics · Physics and Astronomy · #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Mathematical Physics (math-ph) #gr-qc #math-ph #math.MP
paper · pdf · doi:10.48550/arxiv.1909.04183
91 pages, no figures. New title, some minor errors and typos corrected, some notation changes
arxiv created 2021/05/15 · arxiv updated 2021/05/18
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)=(ρ(t)/ρo))1/3=R-1(t) for initial density u0=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*=π/2(8πG/3ρo)1/2. The blowup interval [0,t*] is partitioned into domains [0,tε]\bigcup[tε,t*],with t*=tε+|ε| and |ε|≪ 1, so that tε can be infinitesimally close to t*. Randomness or 'stochastic control' is introduced via the 'switching on' of specific (white-noise) perturbations at t=tε. Hybrid nonlinear ODES-SDES are then 'engineered' over the partition. Within the Ito interpretation, the resulting density function diffusion 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>tε. 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.