[evanb]

They do fall at the same rate, even with Newtonian gravity. For,

    F = m a
    F_gravity = GMm/r^2

so that

    ma = GMm/r^2.

Now cancel m from both sides and get

    a = GM/r^2

If you plug in G=6.674e-11 m^3 kg^-1 s^-2, M = M_Earth = 5.972e+24 kg and r = R_earth = 6.378e+6 m you get

    a = 9.79... m/s^2

which ought to be familiar.

[splitwheel]

The force is on both objects at the same time. The force in F = ma is a function of the mass of both and their distance. If the mass is different in the two scenarios, then the force is different. On earth with small weights, they seem the same because of the precision of the measurement.

This is why you _weigh_ less on the moon.

[evanb]

Is what you're getting at the fact that the distance between the earth and the other object changes from two effects (the first being the ball falling towards the Earth and the second being the Earth falling towards the ball)? That's right, of course. But that distance's second derivative is not the acceleration a in F=ma. Indeed, in both Galilean and Einsteinian relativity acceleration is detectable locally without a needed reference to another object.

[splitwheel]

Yes - I was making a mistake. I was trying to describe the effect of both masses. When one is much smaller than the other, then the movement is mostly in one direction. When they are closer in mass or even equal, they move toward each other. For example, if you have a 1 liter water bottle filled with a material that gives it the same mass as the earth, then the two bodies will move toward each other, and the water bottle will seem to move toward the earth much faster that the 1 filled with water (1kg). If it is filled with a material, that gives it much grater mass than the earth, the earth will move toward it.