Towards a Probabilistic Paradigm for Gravitational Collapse, Black Hole Creation and Singularity Smearing

Abstract

A prototype probability interpretation ispresented for the Oppenheimer-Snyder model ofspherically symmetric, gravitational collapse of apressureless ensemble of n point particles. A transitionprobability P(R(t), t; R1, t1) isderived for an initial sphere or fluid star of radius Rat comoving time t, collapsing smoothly andhomogeneously to any finite radii R(t, r) t1 and R(t) = 0 at t = tf. The transitionprobability is evaluated in two cases. In the firstcase, Planck's constant is assumed zero and smoothdifferential limits exist for space and matter on alllength scales down to zero. The probability for singularityformation converges smoothly to unity as R → 0 ort → tf: the collapse is deterministic atall scales. There is also a finite, nonzero probabilityof event horizon formation at R = Rh = 2GM, but the starcontinues to collapse through this radius since there isalways a higher probability of reaching any smallerradius R < Rh. An event horizon forms sothe collapsed state is still a black hole. In the classical limit(as ℏ → 0) the singularity returns with unitprobability. Finally, we briefly discuss how the final,fuzzy, collapsed state may be related to aspects ofstring theory. The emphasis of the paper is on theconceptual ideas and general possibilities which couldarise when incorporating stochastic mechanics andanalysis into general relativistic collapse.