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We show that finiteness of the Lorentzian distance is equivalent to theexistence of generalised time functions with gradient uniformly bounded awayfrom light cones. To derive this result we introduce new techniques toconstruct and manipulate achronal sets. As a consequence of these techniques weobtain a functional description of the Lorentzian distance extending the workof Franco and Moretti.

Generalised time functions and finiteness of the Lorentzian distance