Saving a Newtonian intuition

Here is a Newtonian intuition:

1. Space and time themselves are unaffected by the activities of spatiotemporal beings.

General Relativity seems to upend (1). If I move my hand, that changes the geometry of spacetime in the vicinity of my hand, since gravity is explained by the geometry of spacetime and my hand has gravity.

It’s occurred to me this morning that a branching spacetime framework can restore the Newtonian intuition of the invariance of space. Suppose we think of ourselves as inhabiting a branching spacetime, with the laws of nature being such as to require all the substances to travel together (cf. the traveling forms interpretation of quantum mechanics). Then we can take this branching spacetime to have a fixed geometry, but when I move my hand, I bring it about that we all (i.e., all spatiotemporal substances now existing) move up to a branch with one geometry rather than up to a branch with a different geometry.

On this picture, the branching spacetime we inhabit is largely empty, but one lonely red line is filled with substances. Instead of us shaping spacetime, we travel in it.

I don’t know if (1) is worth saving, though.

A philosophical advantage of quantum mechanics over Newtonian mechanics

We often talk as if quantum mechanics were philosophically much more puzzling than classical mechanics. But there is also a deep philosophical puzzle about Newtonian mechanics as originally formulated—the puzzle of velocities—which disappears on quantum mechanics.

The puzzle of velocities is this. To give a causal explanation of a Newtonian system’s behavior, we have to give the initial conditions for that system. These initial conditions have to include the positions and velocities (or momenta) of all the bodies in the system.

To see why this is puzzling, let’s imagine that t₀ is the first moment of the universe’s existence. Then the conditions at t₀ explain how things are at all times t > t₀. But how can there be velocities at t₀? A velocity is a rate of change of position over time. But if t₀ is the first moment of the universe’s existence, there were no earlier positions. Granted, there are later positions. But these later positions, given Newtonian dynamics, depend on the velocities at t₀ and hence cannot help determine what these velocities are.

One might try to solve this by saying that Newtonian dynamics implies that there cannot be a first moment of physical reality, that physical reality has to have always existed or that it exists on an interval of times open at the lower end. On either option, then, Newtonian dynamics would have to be committed to an infinite temporal regress, and that seems implausible.

Another solution would be to make velocities (or, more elegantly, momenta) equally primitive with positions (indeed, some mathematical formulations will do that). On this view, that the velocity is the rate of change of position would no longer be a definition but a law of nature. This increases the number of laws of nature and the fundamental properties of things. And if it is a mere law of nature that velocity is the rate of change of position, then it would be metaphysically possible, by a miracle, that an object standing perfectly still for days would nonetheless have a high velocity. If that seems wrong, we could just introduce a technical term, say “movement propensity” (that’s kind of what “momentum” is), in place of “velocity”, and it would sound better. However, anyway, while the resulting theory would be mathematically equivalent to Newton’s, and it would solve the velocity problem, it would be a metaphysically different theory, since it would have different fundamental properties.

On the other hand, the whole problem is absent in quantum mechanics. The Schroedinger equation determines the values of the wavefunction at times later than t₀ simply on the basis of the values of the wavefunction at t₀. Granted, the cost is that we have a wavefunction instead of just positions. And in a way it is really a variant of the making-momenta-primitive solution to the Newtonian problem, because the wavefunction encodes all the information on positions and momenta.

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).