A simple causal theory of the arrow of time

Typical causal theories of time say that the order of time is determined by the order of causal relationships between events in time. This tends to be a difficult theory to develop, if only because of the possibilities of simultaneous causation and time travel. But suppose with substantivalists that there really are moments (or intervals) of time. Then it is possible to have a very simple and elegant theory of the order of time:

1. Time u is prior to time v if and only if time u causes time v.

This theory is, I suppose, inspired by Tooley's idea that earlier points of spacetime cause later ones. It has no worries about the possibility of simultaneous causation. For even if there is simultaneous causation, there is no simultaneous causation between times, since distinct times can never be simultaneous (if u and v are times and they are simultaneous, they are the same time--and of course nothing can cause itself). Likewise, while there is some cost in denying time travel and backwards causation, one can accept time travel and backwards causation while denying that a later time can cause an earlier one.

There is also an interesting and slightly more complex variant:

2. Time u is prior to time v if and only if some event at u at least partially causes v.

This variant, too, is compatible with simultaneous and backwards causation between events--it just rules out simultaneous or backwards causation between an event and a time. It has the advantage that perhaps the dynamic evolution of times isn't just causally driven by earlier times, but by the events at those earlier times. For instance, if a time is a maximal spacelike hypersurface, it is very natural to think on the grounds of General Relativity that the distribution of matter at earlier times causes that hypersurface. We can, further, deem each time's existence to be an event that happens at that time. If we do that, then (2) becomes a generalization of (1).