By "symmetry of space", I actually mean the spatial symmetry of physical laws: that is, Newtonian relativity. The basic idea here is that there is no absolute space: the position, direction, and so on of a particle must be described in terms of other particles.
Now consider the gravitational force, whatever that is, between two particles, Alice and Bob, who are alone in the Newtonian universe at a particular instant. We can determine the distance between them, and whether a force is drawing them together or apart, but nothing else: we can't say "this way is up" or "Bob is to Alice's right" or whatever. So the force vector has to be radial, and its magnitude can depend only on the distance between Alice and Bob. So far, so good.
Now, since the universe (and hence our force) is homogeneous, the force field F coming from Alice must have no more than one point source or sink, which must reside at Alice itself. In other words, ∇·F = 0 everywhere except at Alice. That means the integral of the divergence over any region of space that doesn't include Alice is zero, and the integral over any region of space that does include Alice is some constant, k.
Now consider a sphere S of radius r centered on Alice in light of the above and the divergence theorem. If ∂S is the boundary of S (a spherical shell), we have both
![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
![[identity profile]](https://www.dreamwidth.org/img/silk/identity/openid.png){kind=small}
no subject
Date: 2003-07-29 02:20 pm (UTC)Now consider the gravitational force, whatever that is, between two particles, Alice and Bob, who are alone in the Newtonian universe at a particular instant. We can determine the distance between them, and whether a force is drawing them together or apart, but nothing else: we can't say "this way is up" or "Bob is to Alice's right" or whatever. So the force vector has to be radial, and its magnitude can depend only on the distance between Alice and Bob. So far, so good.
Now, since the universe (and hence our force) is homogeneous, the force field F coming from Alice must have no more than one point source or sink, which must reside at Alice itself. In other words, ∇·F = 0 everywhere except at Alice. That means the integral of the divergence over any region of space that doesn't include Alice is zero, and the integral over any region of space that does include Alice is some constant, k.
Now consider a sphere S of radius r centered on Alice in light of the above and the divergence theorem. If ∂S is the boundary of S (a spherical shell), we have both
∫S ∇·F dV = k
and
∫S ∇·F dV = ∫∂S F·dA = |F|(r) 4πr2
or, in other words,
|F|(r) = k/(4πr2),
an inverse-square force.
(For extra points, figure out where I cheated.)