In parts one and two of this series, I discussed some common misconceptions about the way that projectiles fall to earth, lamenting that gravity is still being taught using ideas from Galileo that could - and perhaps ought to be - replaced by the more precise Newtonian model of gravity. Certainly, Einstein's formulation of gravity will remain beyond the grasp of most high school students for the foreseeable future, but if we wish to exploit Einstein's legacy to the fullest, we would do well to ensure that common education at least rises to the level of his great predecessor.

Einstein himself failed to convey some simple and still-important Newtonian concepts in the presentation of his theory of gravity, even dismissing them in a way that may continue to cause some confusion for serious students, as Einstein's approach to explaining general relativity persists beyond his time. And so I introduce the third and final subject in this series.

Einstein’s general theory of relativity is often regarded as the crowning achievement of the greatest scientific mind in history. His name is synonymous with genius. It would seem arrogant for me as an amateur to criticize his writings. Nevertheless, after sifting through many textbooks and lectures, I have found some credible voices who have expressed criticisms similar to mine. Ironically, this is more agreement than I have yet found regarding the points I made in the first and second parts in this series.

After publishing in 1905 his groundbreaking analysis of uniform motion at extremely high speeds, Einstein struggled for ten years to generalize his theory of relativity to accelerated systems. His breakthrough, he said, came as "the happiest thought of his life:" that if a person were to fall from a rooftop, he would feel weightless for the duration of his fall. He began thinking about the similarities between falling freely in the presence of gravity and being in a place where there was no gravity at all; and this thought process led to the completion of his general theory of relativity in 1915. When the theory was proven correct by astronomical observation in 1919, Einstein became an instant celebrity.

Einstein’s " equivalence principle " ‐ the idea that all forms of acceleration are functionally equal, whether they are due to gravity or some other force ‐ is still taught in most introductory texts on Einstein’s general theory. The idea is that if you wake up in an elevator or some other enclosed space and have no way of looking outside, you will have no way of knowing for certain whether the floor is pushing on you due to Earth’s gravity or whether the elevator is accelerating upward through the shaft of a rocket ship blasting its way through space. Likewise, if you wake up floating in the middle of the elevator, you will have no way of knowing if the cable has been cut or if you are adrift in space.

What many science authors fail to mention is* *that there *is* a way to detect the presence of gravity in a space, provided that gravity is strong enough, observations are long enough, measurement is precise enough, and/or the space is large enough. To accomplish this, one releases a number of objects from rest at various points in the enclosed space and observes how they fall. It is not enough to measure *whether* the objects fall, for as we have discussed, this fall can be caused by forces other than gravity; one must measure *differences* between how the objects move. If gravity is present, objects on opposite sides of the space will drift apart; these sides will then be identifiable as the " top " and " bottom " sides of the space. If we draw an imaginary line between top and bottom, we will also observe that all of the objects in the space will contract toward that line. Why is this so?

Though Newton’s law of gravity has been superseded by relativity, both theories agree that the strength of gravity diminishes with distance from the mass causing the attraction, and that the direction of gravity’s pull varies from place to place. And it is these two principles that are too often glossed over when discussing equivalence.

The answer to the previous question ‐ why objects in a space which is affected by gravity drift apart in one direction and converge in another direction ‐ is that gravity is non-uniform in that space. Gravity is stronger at the bottom of the space than at the top, which is further away from the source of gravity; the objects at the bottom are therefore pulled down faster than the ones at the top, and the objects drift apart from top to bottom. Gravity pulls objects toward a single point (the center of gravity) rather than pulling all of them in strictly parallel directions; as the objects approach the center of gravity, objects at equal distances from the center will converge. These two related phenomena which make gravity measurable regardless of one’s frame of reference are known as *tidal forces*, and as you may guess, they are the reason the moon is able to cause tides on the Earth.

The tidal forces surrounding a black hole are predicted to be so strong as to be deadly. An astronaut falling into a black hole feet first would eventually find that the pull of gravity on his feet would be so much stronger than the pull on his head that he would be stretched and torn. Among scientists, the short-hand term for this gruesome fate is " spaghettification. "

There is another " equivalence principle " often discussed in introductory texts in relativity, though it has been known since Galileo’s time. This other equivalence principle is the equivalence of gravitational mass to inertial mass, which means simply that the amount of force required to accelerate an object is in direct proportion to its weight, and vice versa. Newton gave us two separate formulas for calculating mechanical and gravitational force, each of them having a term for mass, and it need not be taken for granted that the " mass " in one formula meant the same thing as the " mass " in the other. The equivalence of gravitational and inertial mass is not the equivalence principle which I intend to criticize. Rather, I wish to conclude this series by pointing out some of the various ways in which physics educators support or contradict Einstein’s idea of the equivalence of gravity and uniform acceleration by some other force.

At one end of the spectrum is Chad Orzel’s argument that " The key to understanding Einstein’s happiest thought is Galileo’s result . . . If there were any difference between the effects of gravity and any other force, some objects would fall faster than others and allow the falling observer to distinguish [between the two] " (*How to Teach Relativity To Your Dog,* p. 203).

James B. Hartle, author of *Gravity: An Introduction to Einstein’s General Relativity*, establishes " the equivalence of uniform acceleration and a uniform gravitation field, " (p. 112) omitting any mention that gravitational fields are inherently *non*-uniform. Robert Resnick and David Halliday do likewise in *Basic Concepts in Relativity and Early Quantum Theory* (p. 292)

N. David Mermin, in *It’s About Time*, provides the disclaimer which Hartle, *et al*. do not: " The direction in which things accelerate under the Earth’s gravity is not fixed; it is directed toward the center . . . And the magnitude of that acceleration depends on the distance [from the Earth] . . . But if one is only interested in a region that is small on the scale of the whole planet . . . then the magnitude and direction . . . hardly vary at all. One *says* that the gravitation field is uniform. " (p. 172, emphasis mine)

It is valid to assert that Einstein’s equivalence principle is understood as being in the context of small extents of space and short durations of time. In such cases, tidal forces would not be evident. But this point of view could be parodied thus: " Gravity creates conditions just like uniform acceleration, at least in cases where it is so weak that most of its effects are too small to be measured. " It seems a Zeno-like argument, examining the flight of an arrow at instants in time and questioning whether the arrow is really moving at all.

Bernard Schutz, in *Gravity From the Ground Up*, writes: " If gravity were everywhere uniform we could not distinguish it from acceleration. This is the sense of the word *equivalence* in *equivalence principle*. Therefore, the changes in gravity over distances tell us that we are really dealing with gravity and not simply a uniform acceleration. We will elaborate on this subject in Chapter 5, " (p. 21) Schutz’s Chapter Five is entitled " Tides and tidal forces: the real signature of gravity. "

At the other end of the spectrum from Orzel’s statements above are arguments that the ability to detect gravity’s tidal forces falsifies the equivalence principle to some extent. In the audio lecture series *Einstein’s Theory and the Quantum Revolution*, Richard Wolfson argues that tidal forces are the only real aspect of gravity and that all others can be made to disappear by making a change in one’s frame of reference (lecture 13). On that same note, Hermann Bondi said, " A gravitational field is a relative acceleration of neighboring particles. "

The equivalence principle is discussed as one of *Einstein’s Mistakes* by Hans Ohanian:

It should have given Einstein pause that his 1911 calculation of the bending of rays of light, which was based on the Principle of Equivalence, yielded a result half as large as the new calculation based on the theory of gravitation. . . . For the physicist enclosed in a box, the implication is clear: the bending of a ray of light in an accelerated box is half as large as the bending in a box at rest in a gravitational field. This means that acceleration and gravitation are not interchangeable ‐ and the Principle of Equivalence fails! (p. 226)

He goes on to say, " Although Einstein was an avid sailor and sailboats were his favorite hobby, he . . . failed to recognize that the tides undermine the Principle of Equivalence that he adopted as the cornerstone of his theory of general relativity. " (p. 228-29)

A common problem in all three of the misconceptions I have discussed in this series is the problem of measurement. In the laboratory, we find it a challenge to measure the miniscule difference between the parabolic trajectories predicted by Galileo and the elliptical ones predicted by Newton, or to measure the tidal forces which distinguish gravity from uniform acceleration. We are unable to measure any difference in the rate of fall of a heavy object as opposed to a lighter one.

These challenges all arise from the weakness of Earth’s gravity and indeed of gravity in general. Gravity does not seem weak when it causes serious injury, but when one pits gravity against electricity or magnetism, it pales in comparison. A little bit of electric charge on a balloon can give that balloon the ability to defy the gravitational attraction of our entire planet.

This relative weakness has the further consequence that we do not know to a high degree of precision what the strength of gravity is. The gravitational constant used to relate mass and distance to force in Newton’s theory of gravity is only known to a few decimal places. An 18^{th}-century experiment to determine the force between two masses in the laboratory gave results that were less than 1% different from the results of the best modern experiments. In other words, over two hundred years of improvements in theory and technology can only compensate so much in the measurement of an extremely weak force. Walter Lewin, at the end of his first lecture on classical mechanics, says that a Nobel Prize would be in order for any scientist who can experimentally demonstrate that the time for an object to fall is dependent on its mass.