  Linear temporal logic was introduced in order to reason about reactive
systems. It is often considered with respect to infinite words, to specify the
behaviour of long-running systems. One can consider more general models for
linear time, using words indexed by arbitrary linear orderings. We investigate
the connections between temporal logic and automata on linear orderings, as
introduced by Bruy\`ere and Carton. We provide a doubly exponential procedure
to compute from any LTL formula with Until, Since, and the Stavi connectives an
automaton that decides whether that formula holds on the input word. In
particular, since the emptiness problem for these automata is decidable, this
transformation gives a decision procedure for the satisfiability of the logic.
