Can one encode the causal information of a spacetime in a
metric space structure? In 2016, Sormani and Vega conjectured how this
may be done. In this talk, I will present some recent results confirming
that the answer is yes. Not only can the causal relation be encoded, but
also the condition of global hyperbolicity can be characterized via
completeness of the corresponding metric space. This is joint work with
Annegret Burtscher.