BLATHER REVIEW

Spork's Gravity, part 2 (foundation)

One fine day in school my sixth grade teacher taught the class that two objects of different weights fell at exactly the same rate of acceleration. I clearly remember Mr. Weatherby standing atop his desk and simultaneously dropping a text book and a #2 pencil, and we all watched very closely to make absolutely certain that they hit the floor at the same time. I couldn't be sure that they had, although they did seem to. But, with the assurance from Mr. Weatherby, I accepted that they had.
He then explained that this enchanted reality was known as Galileo's Law of Falling Bodies. Maybe it struck me at the time that if it was good enough for Galileo then it was good enough for me.

But I've always felt a nagging uneasiness whenever I thought about it. As I grew older and more interested in the finer points of physics, my curiosity about this basic law increased.
"This can not be!" I kept muttering to myself when I was sure I was alone. "More mass means more gravity. And more gravity means more acceleration, dag nab it!"

F = G(M+m) / r^2
One mass does not unilaterally attract another, the other mass is also attracting the first, even if it's one particle at a time. (Specifically; two bodies are attracted to their common center of mass.)

The Earth attracts a 1kg body with the same gravity effect as it attracts a .25kg body. But the 1kg and .25kg bodies each attract the Earth proportional to their mass. They have different masses and must attract the Earth to different degree as surely as the Moon would attract the bodies to a differing degree.
(I keep using the word "attract". It's completely inaccurate, but it's an old habit.)

If Newton's Principle of Equivalence is correct then it's a body's resistance to the applied force of gravity that counter-balances it's gravitational attraction, and each body's mass -- no matter what it is -- counters in the same way, and so they fall at the same rate of acceleration.
Assuming that's true, I wondered, then why wouldn't g on a planet of similar size but twice the mass (we'll call it Planet X) be identical to Earth's? On Planet X, a bowling ball would fall twice as fast, right?
Let's crunch some numbers, not quite planetary-sized, but varied enough to vividly make the point.

UPDATE: Daniel has pointed out to me that the following example does not make the point I was trying to make. F2 and F3 are equal, but it's because the mass and acceleration are just reversed in the two instances. That's what I get for posting an idea that struck me as I stared at the monitor, and before I thought it through.
I'll try and come up with an appropriate example (if it's even neccessary).

In this example M1=1,000,000; m1=1; M2=2,000,000; and m2=2.

There are four combinations:
F1 = G(M1m1) / r^2......F1 = G(1,000,000) / r^2
F2 = G(M1m2) / r^2......F2 = G(2,000,000) / r^2
F3 = G(M2m1) / r^2......F3 = G(2,000,000) / r^2
F4 = G(M2m2) / r^2......F4 = G(4,000,000) / r^2

F2 and F3 resolve identically, even though the two massive bodies, M1 and M2, should give very different results for the lesser bodies, m1 and m2. Substituting Earth-mass and Planet X-mass for the M1 and M2, two rocks (one twice as heavy as the other) for m1 and m2, would give the same result: F2 = F3, even though observation tells us they don't.

But, I can hear you saying "But, Spork, the Principle of Equivalence takes care of this. It's M1 and M2's resistance to moving toward m1 and m2. The inertia is almost entirely put on the lesser masses."
Okay. Then the next question is: What evidence is there that a body in free-fall in a gravitational field is, in fact, resisting?

EINSTEIN'S PRINCIPLE OF EQUIVALENCE
Albert Einstein, while developing his own theories of gravity, had tried many ideas before focusing on Newton's old Principle of Equivalence. He did not accept, at the time, that gravity could propogate across the void of space without any real, and hopefully discoverable communication between masses.
The "action-at-distance" that Newton had resigned himself to was illogical and inadequate. He sought for years to expand Relativity into a working theory of gravity.

Then Einstein was intriqued by a newspaper item about a man who had fallen some distance and survived to tell about it. The man, describing his descent to the reporter, mentioned that while he was falling he could feel no effects of gravity. He described what it was like to be weightless.
Einstein made the connection that a free-fall in gravitational field and weightlessness are indistinguishable without a frame of reference.

That Newton's Principle of Equivalence was the way to explain the Law of Falling Bodies was accepted for hundreds of years before Einstein, but it offered no real understanding of why this was so. Using his General Theory of Relativity, Einstein tried to settle the matter.

THE ELEVATOR EXPERIMENT
The elevator experiment is the most famous of Einstein's "thought experiments" that he used to explain Relativity, as well as explore it's implications.
It's important to note that it is not merely an analogy or metaphor. It's a real working example of how the world works.

An elevator carrying a few scientists is suspended at some altitude above the Earth's surface. There are no windows.
The scientists know that they are at rest in the Earth's gravitational field. They can sense up from down, and their pencils fall to the floor when dropped.

Suddenly the cables are released and they begin a free-fall toward the ground. Now they float weightlessly. One scientist takes a pencil from his pocket-protector and releases it. It floats in mid-air; an object at rest and remaining at rest. He gives it a nudge, and it moves at a constant speed -- an object in motion and remaing in motion -- until acted upon by an external force: the wall of the elevator.
The scientists conclude that they have either a) been magically transported to a zero-gravity system, obeying all of Newton's laws of motion, or b) are about to crash.

Now imagine that they have indeed been transported, still floating around, into a vast region of empty space. Suddenly the cables begin to reel in the car at a constant rate of acceleration.
Once again their feet can plant firmly on the floor, and pencil accelerate at a constant rate when dropped.
The scientists could then conclude that they are, once again, at rest in Earth's gravitational field.

The elevator thought experiment illustrates the central premise of the General Theory of Relativity; that rest within a gravitational field in indistiguishable from a constant acceleration without a frame of reference. They are relative states of inertia. This is Einstein's Principle of Equivalence.

F = G(M+m) / r^2 RETURNS
One important element of the elevator experiment is that Einstein used it to confirm the Law of Falling Bodies. When the elevator was accelerated by the cables in the void of space, all the objects met the floor with the same urgency regardless of each one's individual mass.

But, as I wrote earlier, the experiment isn't just an analogy but an example of how the world works. The gravitational effects between the walls of the elevator and the pencils is so negligible that they are, for all practical observational purposes, meaningless. (And, if the elevator were a uniform sphere, the effects would be zero.)
To say that the elevator experiment proves the Law of Falling Bodies is no different than saying that dropping two unequal masses from the Leaning Tower of Piza proves it. The difference would be too irrelevant to matter.

In an effort to test Newton's principle of equivalence (which Einstein claimed to validate) a man named Baron von Eotvos came up with an experiment in which he attempted to observe any discrepancy between the velocity of falling unequal masses. He devised an apparatus -- known as a tension fiber balance -- and an experiment whose results seemed to indicate that the Law of Falling Bodies is valid to at least 5 parts per billion.
In the mid-60's the experiment was refined by Robert Dicke by a factor of several hundred. It was regarded as rock solid evidence that Galileo's Law, and Newton's Principle of Equivalence, were scientifically canonical.

However, if we invoke the principle that all forces are additive and look at the example of two bodies -- one 1kg, the other .25kg -- falling toward the surface of the Earth (with a mass of approximately 6 x 10^24kg, we can calculate the difference between the two rates of acceleration in free-fall.

The mass of the Earth-1kg body system:
6,000,000,000,000,000,000,000,001.kg
The mass of the Earth-.25kg body system:
6,000,000,000,000,000,000,000,000.25kg

The difference, being a mere .75kg -- or, one part per 8 trillion trillion -- is a smaller difference than the Eotvos/Dicke experiments could measure by a factor of about 40 thousand trillion. No wonder they didn't notice it.

QUICK CLEAN-UP Q&A BEFORE WE MOVE ON:
Q: If F=G(Mm)/r^2 is wrong, then why is a 1kg body 4 times as heavy as a .25kg body?
A: The weight of the body is what it is because of the force needed to keep it from falling. But what's at work here is F=ma.
When you hold out a hammer and release it, it wall fall until it hits the ground. This event is not due to a force (gravity) acting on the hammer; it's due to the sudden absence of force (you, holding it) and the sudden reappearance of force (the ground) that are the only forces at work.

Einstein showed that a body in free-fall is in an inertial state of rest. It's the act of holding it that creates an acceleration.
Even though a falling body appears to be accelerating, it is actually WE, standing still on the ground, who are experiencing an acceleration.
("a" is the acceleration cause by "g", and varies at different altitudes. So the force needed to hold up a body -- while "at rest" in a gravitational field (it's "weight")-- is specifically described by F=mg.)

CLIFF-HANGER
But, I hear you thinking, this is all academic. All of the discrepancies so far are debatable, and the "evidence" is on a scale too small to allow any confirmation. Where is some real observable evidence? Surely there are macrocosmic examples that could justify a radical change in such long-held physical laws.

Yep, there are! Coming tomorrow!!
(Kepler, Cavendish, Jupiter, light spheres, Hawking, and the fact that gravity is not an applied external force. And maybe, just maybe, some of my working hypothesis of what gravity actually is!)
Hey, c'mon, I love this stuff, but I'm not about to give myself carpal-tunnel syndrome for it!!! ;)

Posted by Tuning Spork at September 24, 2003 11:06 PM