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This licentiate thesis consists of two papers dealing with two central problems in mathematical general relativity: the question of positivity of mass and the problem of establishing a comprehensive metric theory for spacetimes.

In the first paper we rigorously analyze the generalized Jang equation in the asymptotically anti-de Sitter setting modelled on constant time slices of anti-de Sitter spacetimes. We provide a novel construction of barriers in this setting which allows us to handle the most general asymptotics. By virtue of methods from geometric measure theory, our analysis applies in dimensions greater than 2 and less than 8. The obtained results have applications in the context of spacetime positive mass theorems for asymptotically anti-de Sitter initial data sets. More specifically, we show that the positivity of mass follows provided that a certain geometrically motivated system involving the generalized Jang equation has a solution.

In the second paper we show that any point in a smooth spacetime admits a so-called uniform Temple chart. The key property of this chart is that we can completely recover its causal structure using a time function satisfying very general assumptions and the associated null distance. We also show that these uniform Temple charts are bi-Lipschitz charts that can be used for converting spacetimes to integral current spaces of Sormani and Wenger. Furthermore we strengthen the previous result of Sakovich and Sormani and prove a theorem showing that the Lorentzian metric is uniquely determined by a Lipschitz time function with norm of the gradient equal to 1 almost everywhere and the associated null distance, without the assumption that the causality is globally encoded by the time function and the associated null distance. This is joint work with Anna Sakovich and Christina Sormani.