Re: Knowing what you know not about what should be known

[John Denker <jsd@AV8N.COM>]

Bob LaMontagne wrote:
> My explanation for including both masses has relied on an
> argument based on Newton's 3rd law. If the gravitational
> forces on the two objects are to be equal and opposite, and
> if the mass of one of the objects is to appear in the
> formula, then the mass of the other must appear also - and
> as a product so the forces will double if either of the
> masses doubles.

That is an elegant and powerful argument.

It can be refined by clarifying some unstated assumptions.
Most notably it seems to implicitly assume the equivalence
principle and the non-relativistic limit.

Consider a force law of the form
   F = G h(m1, m2) / r^2               [1]
where h() is some function.  As long as h() is symmetric
w.r.t interchange of its two arguments, the 3rd law is
upheld.  You can come up with lots of perverse h()
functions.

Now to enforce the equivalence principle, in the non-
relativistic limit, we require h() to be proportional
to m1 and proportional to m2, which pretty much narrows
things down to the expected result: h(m1, m2) = m1 m2.

========

Note that one should not be too glib about "proving"
equation [1], because it isn't valid in the relativitic
regime.  (Kinetic energy enters the picture in nontrivial
ways.)

It's bad luck to prove things that aren't true.