## Tuesday, March 14, 2017

### Instantaneous Newtonian gravitational causation at a distance?

It’s widely thought that Newtonian gravity, when causally interpreted, involves instantaneous causation at a distance. But I think this is technically not right.

Suppose we have two masses m1 and m2 with distance r apart at time t1. The location of m2 at t1 causes m1 to accelerate at t1 towards m2 of magnitude Gm2/r2. And this sure looks like instantaneous causation at a distance.

But this isn’t an instance of instantaneous causation. For facts about what m1’s acceleration is at t1 are not facts about how the mass is instantaneously at t1, but facts about how the mass is at t1 and at times shortly before and after t1: acceleration is the rate of change of velocity over time. Suppose that a poison ingested at t1 caused Smith to be dead at all subsequent times. That wouldn’t be a case of instantaneous causation, even though we could say: “The poison caused t1 to be the last moment of Smith’s life.” For the statement that t1 is the last moment of Smith’s life isn’t a statement about what the world is instantaneously like at t1, but is a conjunctive statement that at t1 he’s alive (that part isn’t caused by the poison) and that at times after t1 he’s dead (that part is caused by the poison, but not instantaneously). Similarly, m1’s velocity (and position) at times after t1 is caused by m2’s location at t1, but m1’s velocity (or position) at t1 itself is inot.

Let’s call cases where a cause at t1 causes an effect at interval of times starting at, but not including, t1 a case of almost instantaneous causation. In the gravitational case, what I have described so far is only almost instantaneous causation. Of course, people balking at instantaneous action at a distance are apt to balk at almost instantaneous action at a distance, but the two are different.

The above is pretty much the whole story about instantaneous Newtonian causation if one is not a realist about forces. But if one is a realist about forces, then things will be a bit more complicated. For m2’s location at t1 causes a force on m1 at t1, which complicates the causal story. On the bare story above, we had m2’s location causing an acceleration of m1. When we add realism about forces, we have an intermediate step: m2’s location causes a force on m1, which force then causes an acceleration of m1. (There might even be further complications depending on the details of the realism about forces: we may have component forces causing a net force.) Now, when the force-at-t1 causes an acceleration-at-t1, this is, for the reasons given above, a case of almost instantaneous causation. But the causing of the force-at-t1 by the location-at-t1 of m2 is a case of genuinely instantaneous causation.

But is it a case of causation at a distance? It seems to be: after all, the best candidate for where the force on m1 is located is that it is located where m1 is, namely at distance r from m2. (There are two less plausible candidates: the force acting on m1 is located at m2, and almost instantaneously pulls on m1; or it’s bilocated between the two locations; in any case, those candidates won’t improve the case for instantaneous action at a distance.) But here is another problem. The force on m1 is not produced by m2. It is produced by m1 and m2 together. After all, the Newtonian force law is Gm1m2/r2. (It is only when we divide the force by m1 to get the acceleration that m1 disappears.) Rather than m2 pulling on m1, we have m1 and m2 pulling each other together. Thus, m2 instantaneously partially causes the force on m1 at a distance. But the full causation, where m1 and m2 cause the force on m1, is not causation at a distance, because m2 is at the location of that force.

In summary, the common thought that Newtonian gravitation involves instantaneous causation at a distance is wrong:

• If forces are admitted as genuine causal intermediates (“realism about forces”), then we have almost instantaneous causation of acceleration by force (moreover, not at a distance), and instantaneous partial causation of force at a distance.

• Absent force realism, we have almost instantaneous causation at a distance.