2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/205962Significant advances in numerical simulations of black-hole binaries have recently been achieved using the puncture method. We examine how and why this method works by evolving a single black hole. The coordinate singularity and hence the geometry at the puncture are found to change during evolution, from representing an asymptotically flat end to being a cylinder. We construct an analytic solution for the stationary state of a black hole in spherical symmetry that matches the numerical result and demonstrates that the evolution is not dominated by artefacts at the puncture but indeed finds the analytical result.4 pages, 2 figures. Replaced with version that matches the one published in PRL: one extra figure, and modified abstract and introductionGeneral Relativity and Quantum Cosmologytext