I have previously speculated that the concept of spatial distance might be closely to connected to the difficulty of causally affecting. Roughly speaking, the further apart two things are, the harder it is for one to affect the other. This morning I was thinking about what happens if you bring time into this. Consider events a and b in spacetime, with a earlier than b. Then, keeping spatial distance constant, the greater the temporal distance, intuitively the easier it is for a to affect b. The greater the temporal distance, the greater the number of slow-moving influences from x to y that are available.
So we can think of the difficulty of causally affecting (DCA) as increasing with spatial distance and decreasing with temporal distance. And it turns out that this is pretty much what the Minkowskian relativistic metric describes: ds2=dx2+dy2+dz2−dt2 (in c=1 units).
So if we think of distance as closely connected to dca, then it is very natural to think of distance as not just a spatial but a spatiotemporal phenomenon. And without any deep considerations of physics, just using everyday observations about dca, a relativistic metric looks roughly right.
We might now have a rough functional characterization of distance: distance is the sufficiently natural relational quantity which roughly corresponds to dca. In our world it seems there is such a very natural quantity: geodesic distance in a four-dimensional spacetime. In other worlds there may not be such a quantity. Those worlds which have a distance have space or spacetime or time—which it is will depend on the mathematical structure of distance in those worlds and/or on the structure of dca.
This is, of course, vague (I said: "sufficiently natural ... which roughly corresponds"). And so it should be. Compare: Mammals have hair. That's clear. But we should not expect there to be a precise characterization of what kinds of flexible filaments in other species—especially species completely different from ours (think of aliens!)—count as hair. We can give a rough functional characterization of which biological characteristic is hair, but it's going to be very rough, and it may be vague whether some species swimming seas of liquid ammonia is hairy, and that's how it should be. Likewise, if I am right, whether there is time in a world may be quite vague and not a substantive question in Sider's sense.