It's been demonstrated since the 1500s that, when falling toward a certain body, objects fall at the same rate. Everyone from Galileo in Pisa to David Scott on the moon demonstrated that. But what if they're wrong? The Nordtvedt effect posits exactly that.
One of the most famous science legends has Galileo dropping two different-sized cannonballs off the leaning tower of Pisa demonstrating that objects of different mass fall at the same rate. Actually, he rolled two balls down ramps to demonstrate the effect and the tower had nothing to do with it. At the end of the Apollo 15 mission, commander David Scott dropped a hammer and a feather on the moon - where there is no air resistance - to prove the same thing. The mass of the object falling doesn't matter. What matters is the mass of the thing making the object fall. Whether a planet is grabbing a cannonball or a feather, the object falls at the same rate. This is called the equivalence principle, and has been held as scientific truth for about four hundred years.
Just to show that no physics principle is sacred, Professor Kenneth Nordtvedt of Montana State University proposed the idea that objects fall at different rates due to their mass. Or actually, he outlined exactly what we'd see happening between the moon, the Earth, and the sun if an object's mass was taken into account in the gravitational pull between it and the body it was orbiting. The Earth, being the more massive orbiting body, would fall towards the sun at a faster rate than the moon.
The effect justifies this by playing with three different concepts. The first concept is gravitational self-energy. This is roughly the idea that all the little pieces of an object have an effect on each other. A solid ball, for example, could be carved up into little shells inside each other like Russian nesting dolls, all of which are pulling at each other. This energy would be larger for big objects than for small objects.
The second two concepts are two different views of mass. There's inertial mass. Imagine an object is on perfectly greased wheels on a perfectly smooth, level floor. If you were to reach out and push it, you would have to exert enough force to move its inertial mass. Then there's gravitational mass. Imagine you now have to pick up that object against the pull of gravity. Outwardly, this seems like a harder task. I could easily push a large friend along on, for example, a wheeled office chair. I'd have a hard time picking them (and the chair) up off the ground. I'd have to use more force. But the difference in perceived mass is just because, when you lift, you're working against a force. Earth's gravity is pulling down. We know the force of Earth's gravity, and we know the mass of the Earth - take those away and I'd be using the same force pulling the person up into my arms as I would pushing them across a level floor. And if I'm using the same force - I must be moving the same mass. In other words, the inertial mass (mass of an object floating in space) and the gravitational mass (mass of an object sitting on the Earth), are the same. It's just the gravity of the Earth that's making the difference.
Every experiment has found these two masses to be identical, but Nordtvedt's idea of gravitational self-energy might change that. He posited that all those little pieces pulling at each other with their gravity might contribute to an object's gravitational mass, and not its inertial mass. Since a larger object has more mass, it would have more gravitational mass. So now, even accounting for the pull of gravity, more force is being exerted to lift a mass than to push it. And the bigger the mass is, the bigger the gap between the force required to lift it and the force required to shove it is. So the Earth is exerting more force on bigger objects than smaller ones, and bigger objects fall faster.
At least that's the idea. The Nordtvedt effect has been tested, and so far no evidence has been found that the more massive Earth is falling towards the sun faster than the moon. If there is an effect, it's very slight. But if it's there, everything we know about motion, and even relativity, changes. Wouldn't that be cool?