Abstract: This talk discusses the problem under what conditions two extensions of a Lorentzian manifold have to be the same at the boundary. As will be explained, this question arises naturally in the analysis of Einstein?s equations of general relativity. After making precise the notion of two extensions agreeing at the boundary, we recall a classical example that shows that even under the assumption of analyticity of the extensions, uniqueness at the boundary is in general false ? in stark contrast to the extension problem for functions on Euclidean space. We proceed by presenting a recent result that gives a necessary condition for two extensions with at least Lipschitz continuous metrics to agree at the boundary. Furthermore, we discuss the relation to a previous result by Chru?ciel and demonstrate a new non-uniqueness mechanism for extensions below Lipschitz regularity.