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We prove that the topology, smooth structure, and metric of a compactLorentzian manifold with boundary is uniquely determined by data at theboundary. The data consists of the lengths and directions of future-directedonce-broken geodesics connecting points on the boundary, which are firsttimelike and then lightlike. This requires the strong causality condition and aweak convexity assumption, but it holds without any assumptions about conjugatepoints. With an additional convexity assumption we prove the analogousstatement for future-directed once-broken timelike geodesics.  If there are no conjugate points and lightlike geodesics never refocus, theanalogous data using lightlike geodesics and once-broken lightlike geodesicsmay be used to reconstruct the manifold up to a conformal factor. This is acorollary of a result which shows that the conformal class is determined by thecollection of sets of future-directed lightlike vectors at the boundary whichgive geodesics which all intersect in a single point.

Broken causal lens rigidity and sky shadow rigidity of Lorentzian manifolds