Q & A: gravity and masses

Though the mass and gravity of Earth are constant, if you dropped two small objects of different masses from the same distance from Earth, would the gravity, though slight, of the small object of greater mass accelerate the object faster towards Earth than the small object of less mass and thus less of its own gravity? Basically, since all three objects (Earth, small object of greater mass, and small object of less mass) have differnt mass, should they not also have different gravitational force strengths of their own?
- Nathan (age 18)
USA

Since you're interested in the fundamental properties of gravity let's ignore air friction, which could mess up the real experiment.

The answer is- it depends on just what you mean by 'accelerate faster'. First, lets look In a standard Newtonian picture, where you pretend that space and time have simple properties. The balls accelerate at exactly the same rate when they're at the same distance from the Earth, according to a principle discovered by Galileo.

Let's say you drop the balls one at a time and measure the time before they hit. The heavier ball would cause the Earth to accelerate toward it more than the light one would. That means that the distance between the ball and the earth would shrink a tiny bit faster for the heavy ball. Then it would be a little closer to the Earth, and so its own acceleration would be a tiny bit greater. It and the Earth would collide a little quicker.

Now if you drop both balls at the same time, they're BOTH pulling the Earth up, and they still accelerate exactly the same as each other at some height so they both hit the Earth at the same time.

If you try to use more general coordinate frames of General Relativity, incorporating gravity as part of the properties of space-time rather than as a 'force', the choice of how much you say something accelerates becomes arbitrary. However, the basic fact remains true that if the balls are dropped together, they hit at the same time. If they are dropped separately, the heavy one hits a tiny bit more rapidly, as measured by beats of some standard clock.

I'm not quite sure what your last question means. However, it helps to remember Newton's third law. Whatever force the ball exerts on the Earth (we're back in Newtonian descriptions) has exactly the same strength as the force the Earth exerts on the ball, but pointed the opposite way.

Mike W.

(published on 10/22/2007)

Could this mean "equivalence principle" is nothing but an approximation of how two objects(3 objects? including either the earth or an accelerating reference frame?) move or attract each other? and, you can actually differentiate the effect of gravitational field and an accelerated reference frame?? You said, "Now if you drop both balls at the same time, they're BOTH pulling the Earth up, and they still accelerate exactly the same as each other at some height so they both hit the Earth at the same time" ->but if two balls are separated by some (a bit of) distance, the heavier one is pulling up the earth slightly stronger toward itself and reach the earth a tiny bit quicker because the heavier ball can distort the space-time tiny bit more than the lighter one? On the other hand, if you observe a heavy and a light object from an accelerating reference frame, they both seem to move at the same acceleration and reach the floor etc. (of a space craft etc.) at the same time. "equivalence principle" seems correct because we cannot "yet" detect the extremely tiny difference of two falling objects with different mass by gravity? Is this the case or not? and if we accept a tiny bit of break in "equivalence principle" does any theoretical or observable inconsistency arise? Or, could the break explain/predict broader phenomena more comprehensively?
- Anonymous

You're right that even a little break in the equivalence principle would wreak havoc on general relativity. We don't thing there's any such break, at least on distances longer than the Planck scale, below which we don't know what goes on.

The particular example you give- of the earth getting pulled more toward a big ball than toward a little one- involves differences in the gravitational field from place to place, so called tidal effects. The EP only says that acceleration and gravity are equivalent for local, not tidal, effects. So this sort of example fits perfectly in the general relativistic framework, with no break.

Mike W.

(published on 06/07/12)