Ordinary-Matter Dynamics of Timelike Kantowski–Sachs–Schwarzschild Shells: Local Existence and Finite-Radius Signed-Root Exit

Abstract

For collapsing timelike material shells joining homogeneous Kantowski–Sachs (KS) dust to the retained increasing-radius Schwarzschild interior, define λ = [(4πG/c⁴)σ]/√[2(m − m_C)/B³], where m_C and m are the KS Misner–Sharp and Schwarzschild mass lengths. Within the admitted ordinary-matter domain, the positive Schwarzschild angular-root sector is exactly 0 < λ < 1, and its zero-root boundary is λ = 1. We prove local existence on the invariant open set 0 < B < 2m_C < 2m for every continuously differentiable surface law p_s = P(σ) satisfying 0 < P(σ) < σ; bulk dust and shell matter then obey the null, weak, dominant, and strong energy conditions strictly. Codazzi evolution drives λ upward at least as B^(−1/2). Consequently the positive-root sector cannot persist to B = 0. If the KS dust background remains regular, λ = 1 is reached at a finite radius satisfying B₀λ₀² < B_* < B₀; the only earlier obstruction admitted by the bulk data is a longitudinal dust caustic, A → 0, with ρ and |H_A| divergent. At the zero-root endpoint the surface density is finite, not divergent, the rapidity equation has a finite one-sided limit, and all shell energy conditions remain strict. The exact B_* is trajectory-dependent. By contrast, every compatible finite-rapidity junction on the retained increasing-radius side has negative surface density for F ≥ 0.