Closure of Traceful Radiation-Gauge Reconstruction with Matter: General Kerr Conditions, a Thin-Shell Benchmark, and a Conserved Torus

Abstract

We formulate the closure problem for first-order, traceful radiation-gauge metric reconstruction on a Kerr background with matter. The reconstruction is decomposed into Weyl-curvature, Ricci-trace, and mass--angular-momentum completion sectors. For compatible initial, gauge, and completion data, we establish local existence and uniqueness of the resulting triangular transport system on a regular Kerr domain, and identify the additional conditions needed for a closed global formulation. This is not a global existence theorem. We test the framework using two conserved matter sources. For a thin shell around a Schwarzschild black hole, Israel matching, regularity, asymptotic flatness, a barotropic equation of state, and fixed rest mass determine the static, axial, and leading rotational sectors. The ADM-mass shift is $\sqrt{1-2M/R}$ times the shell rest mass, while the axial sector determines the angular momentum, moment of inertia, and interior frame dragging. We also construct a stationary constant-angular-momentum fluid torus on Kerr and compute its Newman--Penrose Ricci projections, showing that nonspherical sources generally excite coupled Ricci sectors. The shell provides a closed benchmark, while the torus illustrates the additional structure required for general Kerr sources.