Global Continuation Across the Kerr Ring: Completion Data, Junction Obstructions, and Disk-Crossing Geodesics

Abstract

With the stationary Killing field continued from $r>0$, the fixed-$M$ Boyer--Lindquist continuation of Kerr has a second asymptotically flat end with ADM mass $-M$. We separate this metric fact from the sign of Komar functionals and formulate the data required to compare other continuations across the regular ring-bounded disk. The resulting tuple is a bookkeeping device; four cases are treated only as representative prescriptions. Re-deriving the classical Darmois--Israel data for the timelike disk world tube, we prove that no real smooth tangential identification preserving the first fundamental form can join two identical positive-$M$ Kerr blocks by a normal-reflection collar as a $C^1$ vacuum spacetime. For the identity and simultaneous $(t,\phi)$ reversal maps, the jump is twice the one-sided second fundamental form and produces the familiar disk layer. We also record the standard Carter crossing conditions $R(0)=-a^2Q$ and polar admissibility. Their role here is diagnostic: because an open set of timelike and null geodesics reaches the regular disk transversely in finite affine parameter, a one-ended model must supply a genuine continuation or identification there. No global completion, time orientation, chronology, or stability result is selected.