Wednesday, October 19, 2011

A reduction of spatial relations to an outdated physics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 (mamb/Fab)1/2, where Fab is the magnitude of the gravitational force of a on b at t, and ma and mb 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 Ta be the set of times at which a exists, and let ma(t) be the mass of a at t. For any pair of objects a and b and time t in both Ta and Tb, let Fab(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 Ta. Say that a function f from Q to S is an admissible position function provided that:

  1. If t is a member of both Ta and Tb, then Fab(t)=ma(t)mb(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))Fba(t)/(ma(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Perhaps the method generalizes to less out-of-date physics.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).

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