Singularities, initial and boundary problems for the Tolman-Bondi model

TL;DR

This work addresses the boundary-value formulation of the Tolman-Bondi model and its relation to the initial-value formulation. It generalizes Olson and Silk by treating the bang-time function $t_0(M_i)$ as an extra condition within the Cauchy framework of Gromov, revealing a dual description via initial and boundary problems. The authors establish a one-to-one correspondence between two density singularities (at $R=0$ and $\partial_{M_i}R=0$) and the two bang-time functions $t_1(M_i)$ and $t_2(M_i)$, and provide explicit relations that connect boundary data to initial data, supported by the Korn–Korn transform. They also discuss the role of spherical symmetry in constraining singularities and how small perturbations could split a single singularity into two, highlighting the physical significance for inhomogeneous cosmologies and collapse scenarios.

Abstract

Boundary problem for Tolman-Bondi model is formulated. One-to-one correspondence between singularities hypersurfaces and initial conditions of the Tolman-Bondi model is constructed.