The gravitational collapse occurs when τ τ reaches τ τ 0 ( R ) with τ τ 0 ′ > 0 . The moment τ τ = τ τ 0 ( R ) corresponds to the arrival of matter denoted by its Lagrangian coordinate R to the center. In all three cases, as τ τ → → τ τ 0 ( R ) , the asymptotic behaviors are given by