[raattgift]

Lorentzian manifolds come with a definition of causality. The spacetime of Special Relativity is a Lorentzian manifold. An additional condition of time-orientability obliterates the free choice of an observer on a "comes-before"/"comes-after" relation between observable events.

A further condition of global hyperbolicity also determines the "comes-before"/"comes-after" relation between unobservable events. This condition can be fixed by (i) a non-Minkowski metric, or (ii) by constraints on the pattern of events in the gravitation-free metric of Special Relativity (an example of such constraints is thermodynamics). Sloganizing this: "states of matter tells you what configuration came before/came after" in the second case, and also in the first case if the non-Minkowski metric's source is only matter; otherwise you need to do a causality analysis, e.g. by fixing causal cones on curves (paths, trajectories - they don't have to be geodesics) of interest or solving the relevant wave equations).

https://en.wikipedia.org/wiki/Causal_structure

https://en.wikipedia.org/wiki/Causality_conditions

As a practical matter, the initial value formulation of General Relativity <https://en.wikipedia.org/wiki/Initial_value_formulation_(gen...> (and numerical relativity built on that) is popular and of practical use because so far there is no reason to describe a natural system (where gravity isn't just ignored) in a way that breaks global hyperbolicity.