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.
Interesting post! A few thoughts: shouldn't there also be a requirement that a relation, in order to count as a distance relation, must also meet certain structural requirements, e.g. (at least roughly) satisfy the definition of a metric function? But of course, the Minkowski relativistic "metric" doesn't. (It fails triangle inequality, and it also permits distinct points to have a "distance" of 0.)
ReplyDeleteAlso, if I'm not mistaken, a consequence of collapsing DCA and the relativistic metric is that, for any event x, every event along x's forward light cone is easier for x to affect than every event *within* x's forward light cone. That's a bit counterintuitive, I think. Much more counterintuitive is this: every event along x's *backward* lightcone would turn out to be easier for x to affect than any event within x's forward light cone.
Brian:
ReplyDeleteAgreed about the first part: the relation needs to have some kind of formal properties, akin to those of a metric. Again, it's going to be vague which properties it needs. As you point out, the relativistic "metric" isn't a metric in the "metric space" sense.
If b is on the boundary of a's forward light cone, ds^2 will be 0, and if it's within, ds^2 will be negative (at least the way I defined it). You're right, however, that I do get the wrong result in the backwards light cone. That's embarrassing.
The post is much more of a manifesto than a worked out theory, and I am grateful to you for pointing out some of the deficiencies.
Regarding the forwards and backwards light cones:
ReplyDeleteCausality has a vector direction, but it looks to me that the DCA measure AX is the same as the DCA measure XA for points A and X.
So should there be an additional equation or fact in the model to indicate whether A causes X or X causes A?
Yeah. Oops.
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ReplyDeletePerhaps we should instead work in terms of ds^2 = dt^2-dx^2-dy^2-dz^2, and consider ds to be a metric of "ease of causally affecting". We can then stipulate that when ds is real, it gets the same sign as dt (there are two square roots to choose from, after all).
ReplyDeleteBut ds is a very funny sort of metric, since it can take on imaginary values, namely when the two points are spacelike related (each outside the other's light cone). We then say: When ds is negative or not real, there is no possibility of causal affecting.
Or maybe we should just say: Where a can affect b, ds^2 is a measure of how easy or hard it is for it to do that?
Or maybe the line of thought is to be abandoned.
I still like it. Maybe you just have to add a sign to the DCA which, for example, for XA is positive if X can affect A but A cannot affect X?
ReplyDeleteKind of like a generalization of choosing the arrow of time for the light cones.