Daniel has a critique of the gravity posts so far here, and I responded in the comments. I want to explain more precisely what I'm trying to prove, as it might be a bit unclear so far.
I'm attempting to show that Isaac Newton's equation
F = G(Mm) / r^2
should be replaced with
a = G(M+m) / r^2.
Also, since Newton's Principle of Equivalence was first theorized only because Galileo's Law of Falling Bodies seemed irrefutable (and I refute it!), the Principle of Equivalence is unneccessary.
But let me explain exactly what I mean by a=G(M+m)/r^2.
(And I like to use visual examples rather than just deal with cold abstract point particles. I'm like that.)
Imagine that we are in a space ship that is at rest relative to a distant planet called M1. We have a very powerful spy-telescope than not only see the planet clearly, but also any incoming meteors that are only the size of bowling ball.
The planet is just sitting out there in the void of space and doesn't even seem to be moving against the background stars.
We then notice a small meteor heading straight for M1 on a collision course. We know how far away these bodies are from us, and from each other. Observing m2's motion against the background stars, we quickly calculate that the meteor is "falling" toward M1 acording to F=G(M1)/r^2.
We don't notice M1 moving at all.
We call our fellow astronaut, who's camping on M1's surface, and tell him to look up through his telescope at the incoming meteor. "Yep," he says, "Here it comes, and with the exact same rate of acceleration we expect all falling bodies to have here.
Eventually the meteor hits the planet, but misses the astronaut, and M1 seems to still be in the exact same place it was before the meteor ever showed up.
The next day we look out again from our spaceship and... holy cow, look what's coming in now! It's another planet with exactly the same mass as M1 (we're physic, too btw) heading on the same path that the meteor followed! We call this planet: M2.
Observing M2's motion against the background stars we calculate that M2 is falling with an acceleration toward M1 according to F=G(M1)/r^2, as expected.
Now we turn our telescope to M1 an notice that it's moving across the background stars, in a straight line toward M2, with an acceleration in accordance with F=G(M2)/r^2.
We call our astronaut who's camping on M1 and tell him to look up. "What the..." he says. That thing is heading toward me with an acceleration rate that's TWICE what it should be! I mean, all bodies are supposed to fall at the same rate!"
"Uh, Dave?" we say, "M1 -- where you are -- is accelerating toward M2 at the same rate that M2 is accelerating toward M1."
"But, I don't feel any differant. How can you tell me I'm accelerating?"
"Because we can see it happening!"
To Dave the Astronaut, in his frame of reference he is still where we was along along, in a state of rest; and the incoming bodies were all in a state of acceleration. He has no idea why that falling planet is coming in at twice the rate of acceleration that's twice the rate with which he's seen every other falling body fall.
The two forces at work are defined by F=G(M1)/r^2 and F=G(M2)/r^2. The total acceleration due to those "forces" involved in the M1-M2 system is a=G(M1+M2)/r^2. The accelerations due to the masses of the two components of the system are additive; and the relative acceleration of M2 toward M1 is twice the rate than that of the insignificant meteor toward M1.
Newton's F=G(Mm)/r^2 (along with the corresponding Principle of Equivalence that negates the accelerating effects resulting from the large product: F) assumes that two bodies are "action-reaction pairs," and that the force exerted on M1 by a meteor is "equal and opposite" to that exerted on the meteor by M1.
This comes from Newton's hypothesis that gravity is essentially an "action-at-a-distance" where the bodies somehow detect each other's mass, and then act accordingly.
All of this is based on the idea that gravity is an applied external force.
Einstein demonstrated via the elevator experiment that, without a frame of reference, a body falling in a gravitational field is indistiguishable from a body at rest outside of a gravitational field. Therefore; bodies fall (accelerate toward other bodies) not because they are being acted upon by an external force, but because they are at rest in a gravitational field.
Stephen Hawking, in his book A Brief History of Time, attempted to explain the Principle of Equivalence (between Gravitational and Inertial mass) with an analogy invoking the different sized engines required to move a light car versus a heavy car; that the inertia mass of the heavier car needs a stronger engine to move it.
Hawking wrote, as Newton had, that greater energy is needed to move the heavier car and thus, it followed that the greater gravitational mass of a heavier body is needed to overcome the body's greater resistance to a change in it's inertia.
Why the analogy fails -- I contend -- is that the car needs a force applied to it in order to accelerate it (F=ma), but a body in free-fall does not; it is in an inertial state of rest even as it appears to be accelerating.
As I wrote before: Even though a falling body appears to be accelerating, it is not. It is WE, standing on the ground in the gravitational field, that are undergoing an acceleration.
Just to belabor the point one more time (egad!): The presumption that gravity is an applied external force is the entire premise behind Newton's "action-reaction pairs" idea that led to F=G(Mm)/r^2 and the Principle of Equivalence.
Next post will be in a little while! Yay!
