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.