That gravitation is a central force of magnitude inversely proportional to the square of the distance between objects is a physical fact, at least so far as we've been able to measure. While there is solid math here, and connections are made between empirical facts and assumptions we often take for granted, the concept of why is not really meaningful within the purview of the natural sciences (which are entirely concerned with what and how), relegating this to the large gray area between solid and well-tested things like the theory of relativity and the pure myth of Just So Stories. The argument builds on a few intuitive assumptions and is not especially rigorous about which properties of space it introduces and when. Nevertheless, it gave me a taste of nothing short of religious ecstasy when I first came up with it, and it's true as far as it goes, so here you go:

Consider the gravitational field emanating from a single point mass in isolation. We don't know anything about the magnitude or direction of the force a priori, but we can start with the basic assumption of both Einstein's and Newton's relativity: that space itself has no intrinsic sense of location or direction (which is to say, it is homogeneous and isotropic), so no particular point is distinguishable from any other except through relation to actual things in space. Right away this tells us that

There is only one distinguishable direction for a force to point in: either toward or away from the particle. Thus gravity must be a central force—and thus it is conservative, with a curl of 0.

Next we introduce a related intuition: the only source or sink (in the strict mathematical sense of a point where divergence is nonzero) for gravity can be the particle itself. This makes an intuitive sort of sense when you consider that a source or a sink is a "special" point, and we still have no way of differentiating any point other than the particle's location as more or less "special" than any other. (Perhaps, in fact, it is possible to prove this rigorously from symmetry, but I can't currently see how.)

Since the divergence of the force is zero almost everywhere, the integral of the divergence over a submanifold of our space can be nonzero only if it contains the point mass, and the integrals of the divergence over any two submanifolds that contain the point mass will be equal. (We can talk only about the integral as its own thing, not the divergence per se, since we haven't yet introduced the bits of measure theory that we would need for that. But that's OK, we only need the integral anyway.)

Having said something about the integral of the divergence of the force over a submanifold, we can now introduce Stokes' Theorem for differential manifolds of three dimensions:

δM g = M dg = M ∇·g = c

(where c is some constant). If M is a solid sphere of radius r centered on the point mass, then we can use the fact that the magnitude of g is a constant |g|(r) (since g is a central force) to say that

δM g = 4πr2 |g|(r) = c

or

|g|(r) = k/r2

thus proving that gravitation is a central inverse square force (QED). And I think that's pretty cool.


Precisely the same argument, of course, applies to the electric force and any other force that we claim is symmetric and has a point source. Sometimes I fantasize about beating the current (rather low, by modern standards) precision in measuring gravity by taking measurements of its symmetry and some measure of point-sourced-ness and forcing the error down obliquely.


As I said to unperson: "Sure it's a proof—it just doesn't prove anything directly about the real world. It proves that IF gravitation is a symmetric, point-source force on a 3-fold, THEN it is a central inverse-square force. I should be more explicit about that." But just coming out and saying that sounds so dry...

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