  Minkowski space is the local model of 3 dimensionnal flat spacetimes. Recent
progress in the description of globally hyperbolic flat spacetimes showed
strong link between Lorentzian geometry and Teichm{\"u}ller space. We notice
that Lorentzian generalisations of conical singularities are useful for the
endeavours of descripting flat spacetimes, creating stronger links with
hyperbolic geometry and compactifying spacetimes. In particular massive
particles and extreme BTZ singular lines arise naturally. This paper is
three-fold. First, prove background local properties which will be useful for
future work. Second, generalise fundamental theorems of the theory of globally
hyperbolic flat spacetimes. Third, defining BTZ-extension and proving it
preserves Cauchy-maximality and Cauchy-completeness.
