Simplicity and Newton's inverse square law

When I give talks about the way modern science is based on beauty, I give the example of how everyone will think Newton’s Law of Gravitation

1. F = Gm₁m₂/r²

is more plausible than what one might call “Pruss’s Law of Gravitation”

2. F = Gm₁m₂/r^{2.00000000000000000000000001}

even if they fit the observation data equally, and even if (2) fits the data slightly better.

I like the example, but I’ve been pressed on this example at least once, because I think people find the exponent 2 especially plausible in light of the idea of gravity “spreading out” from a source in concentric shells whose surface areas are proportional to r². Hence, it seems that we have an explanation of the superiority of (1) to (2) in physical terms, rather than in terms of beauty.

But I now think I’ve come to realize why this is not a good response to my example. I am talking of Newtonian gravity here. The “spreading out” intuition is based on the idea of a field of force as something energetic coming out of a source and spreading out into space around it. But that picture makes little sense in the Newtonian context where the theory says we have instantaneous action at a distance. The “spreading out” intuition makes sense when the field of force is emanating at a uniform rate from the source. But there is no sense to the idea of emanation at a uniform rate when we have instantaneous action at a distance.

The instantaneous action at a distance is just that: action at a distance—one thing attracting another at a distance. And the force law can then have any exponent we like.

With General Relativity, we’ve gotten rid of the instantaneous action at a distance of Newton’s theory. But my point is that in the Newtonian context, (1) is very much to be preferred to (2).

Heavier objects fall sooner

We like to say that Galileo was right that more massive objects don’t fall any faster than lighter ones, at least if we abstract away from friction.

But it occurred to me that there is a sense in which this is false. Suppose I drop an object from a meter above the moon, and measure the time until impact. If the object is more massive, the time to impact is lower. Why? Because there are two relevant gravitational accelerations that affect the time of impact: the moon pulls the object down, but simultaneously additionally the object pulls the moon up. The impact time is affected by both accelerations, and the more massive the object, the greater the upward acceleration of the moon, even though the object's acceleration is unaffected by its mass.

Of course, if we are dropping a one kilogram ball, the gravitational acceleration it induces on the moon is about 1/10²³ of the gravitational acceleration the moon induces on it. It’s negligible. But it’s still not zero. :-) A heavier object of the same size will impact sooner.

If all this is unclear, think about the extreme case where we are “dropping” a black hole on the moon.

Simplicity and gravity

I like to illustrate the evidential force of simplicity by noting that for about two hundred years people justifiably believed that the force of gravity was Gm₁m₂/r² even though Gm₁m₂/r^2 + ϵ fit the observational data better if a small enough but non-zero ϵ. A minor point about this struck me yesterday. There is doubtless some p ≠ 2 such that Gm₁m₂/r^p would have fit the observational data better. For in general when you make sufficiently high precision measurements, you never find exactly the correct value. So if someone bothered to collate all the observational data and figure out exactly which p is the best fit (e.g., which one is exactly in the middle of the normal distribution that best fits all the observations), the chance that that number would be 2 up to the requisite number of significant figures would be vanishingly small, even if in fact the true value is p = 2. So simplicity is not merely a tie-breaker.

Note that our preference for simplicity here is actually infinite. For if we were to collate the data, there would not just be one real number that fits the data better than 2 does, but a range J of real numbers that fits the data better than 2. And J contains uncountably many real numbers. Yet we rightly think that 2 is more likely than the claim that the true exponent is in J, so 2 must be infinitely more likely than most of the numbers in J.

A tweak to regularity

Let G_p be the law of gravitation that states that F = Gm₁m₂/r^p, for some real number p. There was a time when it was rational to believe G₂. But here is a problem. When 0 < |p − 2|<10⁻¹⁰⁰ (say), G_p is practically empirically indistinguishable from G₂, in the sense that within the accuracy of our instruments it predicts exactly the same observations. Moreover, there are uncountably many values of p such that 0 < |p − 2|<10⁻¹⁰⁰. This means that the prior probability for most (i.e., all but at most countably many) such values of p must have been 0. On the other hand, if the prior probability for G₂ had been 0, then the posterior probability would have always stayed at 0 in our Bayesian updates (because the probability of our measurements conditionally on the denial of G₂ never was 0, which it would have to have been to budge us from a zero prior).

So, G₂ is exceptional in the sense that it has a non-zero prior probability, whereas most hypotheses G_p have zero prior probability. This embodies a radical preference for a more elegant theory.

Let N be the set of values of p such that the rational prior probability P(G_p) is non-zero. Then N contains at most countably many values of p. I conjecture that N is the set of all the real numbers that can be specifically defined in the language of mathematics (e.g., 2, 3.8, e^π and the smallest real root of z⁷ + 3z⁶ + 2z⁵ + 7πz³ − z + 18).

If this is right, then Bayesian regularity—the thesis that all contingent hypotheses should have non-zero probability—should be replaced by the weaker thesis that all contingent expressible hypotheses should have non-zero probability.

Note that all this doesn’t mean that we are a priori certain that the law of gravitation involves a mathematically definable exponent. We might well assign a non-zero probability to the disjunction of G_p over all non-definable p. We might even assign a moderately large non-zero probability to this disjunction.

A ray of Newtonian particles

Imagine a Newtonian universe consisting of an infinite number of equal masses equidistantly arranged at rest along a ray pointing to the right. Each mass other than first will experience a smaller gravitational force to the left and a greater (but still finite, as it turns out) gravitational force to the right. As a result, the whole ray of masses will shift to the right, but getting compressed as the masses further out will experience less of a disparity between the left-ward and right-ward forces. There is something intuitively bizarre about a whole collection of particles starting to move in one direction under the influence of their mutual gravitational forces. It sure looks like a violation of conservation of momentum. Not that such oddities should surprise us in infinitary Newtonian scenarios.

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.

A reduction of spatial relations to an outdated physics

Consider a Newtonian physics with gravity and point particles with non-zero mass. Take component forces and masses as primitive quantities. Then we can reduce the distance at time t between distinct particles a and b as (m_am_b/F_ab)^1/2, where F_ab is the magnitude of the gravitational force of a on b at t, and m_a and m_b are the masses at t of a and b respectively (I am taking the units to be ones where the gravitational constant is 1); we can define the distance between a and a to be zero. For every t, we may suppose that by law that the forces are such as to define a metric structure on the point particles.

If we want to extend this to a spatiotemporal structure, rather than just a momentary temporal structure, we need to stitch the metric structure into a whole. One way to do that is to abstract a little further. Let S be a three-dimensional Euclidean space. Let P be the set of all particles. Let T be the real line. For each object a in P, let T_a be the set of times at which a exists, and let m_a(t) be the mass of a at t. For any pair of objects a and b and time t in both T_a and T_b, let F_ab(t) be the magnitude of the gravitational force of a on b at t. Let Q be the set of all pairs (a,t) such that t is a member of T_a. Say that a function f from Q to S is an admissible position function provided that:

1. If t is a member of both T_a and T_b, then F_ab(t)=m_a(t)m_b(t)/|f(b,t)−f(a,t)|².
2. f''(a,t) is equal to the sum over all particles b distinct from a of (f(b,t)−f(a,t))F_ba(t)/(m_a(t)|f(b,t)−f(a,t)|).

The laws can then be taken to say that the world is such that there is an admissible position function. We can then relativize talk of location to an admissible position function, which plays the role of a reference frame: the location of a relative to f at t is just f(a,t).

The above account generalizes to allow for other forces in the equations.

So, instead of taking spatial structure to be primitive, we can derive it from component forces, masses and objects, taking the latter trio as primitive.

I don't know how to generalize this to work in terms of a spatiotemporal position function instead of just a spatial position function.

Of course, component forces are hairy.

Perhaps the method generalizes to less out-of-date physics. Perhaps not. But at least it's a nice illustration of how spatial relations might be non-fundamental, as in Leibniz (though Leibniz wouldn't like this particular proposal).