Explore the uniqueness problem for spacetime extensions in this 47-minute talk by Jan Sbierski, presented at the Workshop on "Non-regular Spacetime Geometry" at the Erwin Schrödinger International Institute for Mathematics and Physics (ESI). Delve into the conditions under which two extensions of a Lorentzian manifold must be identical at the boundary. Examine a classical example demonstrating that uniqueness at the boundary can fail even with analytic extensions, contrasting sharply with function extensions in Euclidean space. Learn about a recent result providing a necessary condition for two extensions with Lipschitz continuous metrics to agree at the boundary. Investigate the relationship to Chruściel's previous work and discover a new non-uniqueness mechanism for extensions below Lipschitz regularity.