The Modification of the Oppenheimer and Snyder Collapsing Dust Ball to a Static Ball in Discrete Space-time

TL;DR

This paper modifies the classical Oppenheimer-Snyder collapsing dust ball model by incorporating a discrete space-time framework, addressing singularity issues and internal-external symmetry problems, and proposing a quantum-based mechanism to prevent gravitational collapse.

Contribution

It introduces a finite precision function theory in discrete space-time to reformulate Einstein's equations, resulting in a static dust ball solution that overcomes key limitations of the classical model.

Findings

Discretization of space-time resolves singularity issues.

The modified model achieves internal symmetry and external connection.

Proposes a quantum-scale mechanism to resist gravitational collapse.

Abstract

Besides the singularity problem, the famous Oppenheimer and Snyder solution is discovered to be of deficiency in two aspects: the internal Friedmann space-time does not have the inherent symmetry and cannot connect to the external Schwarzschild space-time. So the process of gravitational collapse described by this solution is doubtful. The deficiency, together with the singularity problem, result from the imperfection of the field theory in continuous space-time, which is expressed by the infinite precision function theory. The space-time structure of the Oppenheimer and Snyder dust ball is founded to be discrete rather than continuous, and to describe the field theory in discrete space-time it requires a function theory with finite precision. Based on the i order real number and its equivalence class, which is defined in the real number field, the infinite precision function theory is extended to the finite precision function theory. The Einstein field equations are expressed in the form of finite precision, and then the collapsing dust ball solution in continuous space-time is modified to a static ball solution in discrete space-time. It solves all the problems of Oppenheimer and Snyder solution and shows that, with Planck length and Planck time as space-time quantum, a mechanism to resist the gravitational collapse could be obtained by the discretization of space-time.