I) Historical Perspectives

A) Early ideas

We will first look at the historical development of our understanding of what is commonly called Gravity. In the section that follows, we will reintroduce and expand on the major conceptual underpinnings that are important in the development of our understanding.

Early Ideas
The very earliest ideas regarding gravity must have been based on every day experience. For example:

• All objects fall unless they are supported.
• "Down" is different from "across".
• Climbing a hill is harder work than walking on level ground.

B) Ancient Greeks: Aristotle

It was with the Ancient Greeks, and in particular Aristotle, that these every day experiences began to be unified into one idea. For Aristotle, physics was the investigation of "causes" in the widest possible sense. While "Gravity" was not yet a concept in itself, Aristotle realized that these ideas about objects must be related.

To Aristotle, the cause of falling was heaviness. The heavier the object, the more readily it falls: a large rock plummets to the earth, a leaf ambles along downward slowly and a dandelion fluff barely falls at all and frequently rises higher into the sky instead. But what was the connection between heavy objects and falling? To understand this, one must have an idea of the Aristotelian worldview.

To Aristotle, all Matter was made of four elements, Earth, Water, Air and Fire. Earth, the basest and least noble, was in the center. This was the ground we walk on. Next was the sphere of Water, followed by the sphere of Air and then of Fire. The eternal and perfect Celestial spheres of the stars and planets surrounded and limited the Universe. Everything under the moon was composed of some mixture of the four elements. Clouds, for example, were considered to be mostly air with a bit of water and fire.

The natural place for heavy objects (made principally of Earth) was back in the center. If one removed a heavy object, say a stone, from its natural place (e.g., by lifting it) it would tend to return to its "proper" place. In a similar way, fire tended to rise, because it was trying to return to its natural place above the sphere of air. Intermediate objects, such as the leaf or dandelion fluff, were made of less Earth and more Air (or Fire) and hence fell more slowly, or perhaps not at all.

C) Middle Ages

Medieval physics (and astronomy) was largely based on the ideas of Aristotle. Since the teachings of Aristotle had been given the seal of approval by the Church, they were taken to be the revealed truth and essentially unquestioned in the Universities. In general, this didn't work too badly: most objects work more or less in the manner described by Aristotle. This is not surprising since his work was essentially the codification of day-to-day experience and is very commonsensical. In certain cases, however, serious discrepancies might have been noted. One example of this was the medieval view of cannonball trajectories. Applying the Aristotelian view, when a cannonball's initial upward and forward impetus was exhausted, it would fall vertically to the Earth, its natural place. This indeed should be true of any object shot or thrown into the air. However, anyone who had watched a rock thrown into the air could tell this was not true. The trouble was that the people teaching the theories and the people dealing with the real world objects were different. Furthermore, until the beginning of the Renaissance, experimentation was discouraged and considered beneath the dignity of philosophers. The way to truth was considered to be pure thought and the Scriptures.

D) Renaissance: Galileo

Galileo's work represents the beginnings of a modern understanding of Gravity. Ironically, to achieve this, Galileo began by disavowing any interest in "causes". Instead of trying to answer the question "why do objects fall?" he explored "how do objects fall?" This is an extremely important step. Even today we do not fully understand the "why" of gravity although we understand the "how" very well indeed.

He arrived at these conclusions through a beautiful series of experiments. The first thing he realized was that he would need to slow down the motion of objects to be able to measure their fall. He did this by allowing the objects he studied to roll down a tilted board instead of falling straight down. He had to assume that this procedure was valid. Fortunately it was. Second, he knew that he had no accurate clocks with which to time the rolling objects. Instead he used his natural sense of rhythm (he may have inherited his sense of rhythm from his father, a well-known musician of the time). In the path of the objects he rolled down the plane, he placed little bumps. Every time the object went over a bump it made a click. By arranging the bumps so that the clicks came in a regular series he knew the time between the bumps was the same. The story of Galileo dropping cannonballs of different weights off the Leaning Tower of Pisa is probably apocryphal. If he did this, it was certainly less important to him than his controlled experiments.

Galileo's contribution to the understanding of gravity was threefold. First, he subtly changed the question being asked. Second, he based his answers on careful experimentation and measurement. Third, he gave a mathematical quantitative description of his results and gave the limits within which he had verified this description.

E) Enlightenment: Newton

Newton, born in the year Galileo died, developed the modern concept of gravity. Instead of simply exploring how objects fall, he posited a force of gravity that was responsible for a variety of effects. In this sense, he was getting back to causes: gravity was the cause of falling. But Newton did not know what caused gravity.

Newton started from Galileo's law of falling objects and applied it to an unlikely object: the Moon. Why, he asked, did the moon not fall to the earth? Other unsupported objects (like rocks, sticks, etc.) fall immediately to the ground. The Moon seems to flout the law of gravity. That's the trick, however. The moon only seems to be immune to gravity. Newton realized that the Moon is not immune to gravity... it is continuously falling towards the Earth, but it keeps missing it! A little explanation of this somewhat outrageous sounding statement is in order.

Imagine standing on a tower on a flat earth. Throw a rock sideways out the window. Eventually, the rock will fall to the ground. Now throw the rock harder. It will hit the ground farther from the tower. On a flat earth this can be continued, throwing the rock harder and harder with the rock's impact farther and farther away. But not so on a spherical earth. On a spherical Earth, the surface curves away under the falling rock. When the rock is thrown with only a little speed, the distance is small, and over the distance spanned, the Earth's surface is almost flat. But if the rock is thrown hard enough, the ground will drop a great deal. In fact, if the rock is thrown very, very hard indeed it will never hit the ground because the earth keeps receding beneath it! This is what is called being in orbit. The secret to flying is falling but missing the ground!

As a result of these thoughts, Newton realized that gravity was not something special to the Earth. Gravity also acts in space. This was a profound, even revolutionary idea. According to Aristotle, the laws governing the heavens were considered to be completely different from the laws of physics here on Earth. If, however, the Moon was affected by gravity, then it made sense that the rest of the Solar System should also be subject to gravity. Newton found that he could explain the entire motion of the Solar System from the planets to the moons to the comets with a single Law of Gravity:

All bodies attract all other bodies, and the strength of the attraction is proportional to the masses of the two bodies and inversely proportional to the square of the distance between the bodies.

A modern mathematical way of saying this is:

where G (Newton's Constant) is a constant value equal to 6.67x10-11 m³/s²/kg, M is the mass of one object, m is the mass of the other object, R (radius) is the distance between the objects and F is the resulting gravitational force pulling the objects together.

This is called the Universal Law of Gravitation. It is called "Universal" because it applies to all bodies in the Universe regardless of their nature. It is important to note that Gravity is not just about falling. Gravity is about attraction! As I write this, the keyboard in front of me pulls ever so slightly on the phone to my right, the penny I left at home gently tugs at the umbrella I lost in San Diego and the flight of a bird above the Adler makes me fractionally lighter! All objects pull on all other objects! What a fantastic statement! Of course, for most objects, the force of attraction is incredibly tiny and not noticeable, but it is always there.

Despite its power in explaining the orbits of the Solar System, Newton (and his critics) were unhappy with the lack of a mechanism by which gravity worked. Until then, all forces were believed to be "contact" forces. That is to say, to push an object one had to be touching it. I push a pen across the table using my hand directly. Even if I blow a piece of paper, I am really moving the air with my lungs which then moves across to the paper and pushes it along. Almost everything in our experience works this way except for gravity. The Newtonian concept of "action-at-a-distance" was profoundly disturbing to his opponents, who attacked his theory as "occult" and explaining nothing.

F) A Side Note on Cavendish

Newton proposed this principle of Universal Gravitational attraction as the basis of his Universal Law of Gravitation in 1665. However, it was not until 1798 that Henry Cavendish became the first person to determine the value of the constant. The Gravitational Constant (or sometimes referred to as the Gravitational Coupling Constant) is commonly represented by the letter G and has a value of 6.67x10^-11 m³kg^-1s^-2. In order to determine G, Cavendish conducted an experiment to directly measure the gravitational attraction between objects. In his experiment he used a barbell arrangement with 5.0 cm diameter lead ball masses at its ends (M₂) suspended and balanced in the middle by a long fine fiber. This formed a very sensitive "torsional balance" that would rotate when even very small forces were applied at the ends (see figure below). Cavendish took possession of the device from Rev. John Michell of the Royal Society, who actually designed the Torsional balance, but was unable to complete his work before his death in 1793.

Cavendish first applied very small "known forces" and measured how much the device rotated. The "known forces" were obtained by hanging very small, calibrated weights on pulleys applied at right angles to the barbells. A known mass exerts a specific force and would twist the apparatus a specific, finite amount. He could then determine the size of any other force acting on the balance simply by measuring how far the balance twisted. Once the device was properly calibrated, he placed the known masses, 20 cm diameter lead balls (M1), on opposite the ends of the barbell masses and recorded the amount of rotation produced. The known masses (M₁) were not connected to the barbell, but produced a twist nonetheless. The twist had to be due to the gravitational attraction between the known masses, M₁, and the barbell masses, M₂. The angle of rotation told him what the force's value was. Once he measured the distance between the masses, he was able to put all the known quantities into the Universal Law equation and calculate the Gravitational Constant. Cavendish's real goal was to determine the mass of the Earth. With an accurate value for the Gravitational Constant, this calculation can be done in a straightforward manner.

Example:
Take a 1.0 kg mass on earth. Its weight is 9.8 N (w = m*g, w = 1.0kg * 9.8 m/s²). This weight is a result of the force of gravitational attraction between the 1.0 kg mass and Earth. Therefore using the Universal Law of Gravitation stated earlier,

(where F is the weight of the 1.0 kg mass, m is the mass of the 1.0 kg mass, M is the mass of the earth and r is the distance between the 1.0 kg mass and the center of the Earth) the mass of the Earth can be calculated. If we place the known values into the equation,

and solve for the mass of the earth, we find M (MassoftheEarth) to be 6.0x10²⁴ kg.

G) Post Enlightenment - 1700s,1800s

From the period immediately following Newton's discovery of his Universal Law of Gravitation, to about the turn of the 20th century, the theory of gravitation stayed essentially unchanged. More sophisticated mathematical tools for understanding the interplay of the planets were developed, but the underlying theory remained stable. As in the earlier Aristotelian world-view, gravitation was intricately connected with the structure of the Universe. The moons revolved around the planets, the planets revolved around the Sun, the Sun floated through space passing other stars, all with clockwork precision. The Universe was orderly and controlled by gravity and the laws of motion. The excitement during this period mainly came from the systematic application of the theory of gravity to the heavens. For example:

1) Comets: an understanding of how objects orbited the Sun allowed predictions of the path of comets. The best-known case of this was Halley's prediction of the return of the comet that now bears his name.

2) Discovery of Neptune: In the time of Newton, only six of the nine planets had been discovered. While the discovery of the planet Uranus was by and large accidental, the discovery of Neptune was a triumph of the Newtonian theory of gravity. Why? After the discovery of Uranus, great attention was paid to this newest of planets. Its orbit was carefully mapped out in great detail. And something strange was found: The orbit of Uranus did not seem to follow Newton's laws precisely! The motion across the skies was just slightly different from the motion predicted on the basis of the theory of gravity. Astronomers were presented with a choice. Either Newton was wrong, or their calculations were somehow incomplete. Many influential scientists thought that perhaps the law of gravitation did not apply so far from the Sun.

After the discovery of Uranus, great attention was paid to this newest of planets. Its orbit was carefully mapped out in great detail. And something strange was found... The orbit of Uranus did not seem to follow Newton's laws precisely! The motion across the skies was just slightly different from the motion predicted on the basis of the theory of gravity. Astronomers were presented with a choice. Either Newton was wrong, or their calculations were somehow incomplete. Many influential scientists thought that perhaps the law of gravitation did not apply so far from the Sun.

In the period 1843-1846 John Adams and Urbain Leverrier independently came to the conclusion that the perturbations of Uranus's orbit were due to an eighth planet. Shortly thereafter, the new planet, now called Neptune, was discovered precisely where Adams and Leverrier had predicted. Newton was spectacularly vindicated!

3) Binary Stars: William Hershel's observations of binary stars during the early 1800s showed that the Newtonian laws of gravity also applied to the stars. They also allowed, for the first time, the calculation of the mass of stars other than our Sun.

4) Rings of Saturn: The law of gravitation also illuminated the origin and nature of the rings of Saturn. The rings could not be thin, solid sheets as previously thought. James Maxwell showed that such rings would break apart under the combined actions of their own motion and the gravity of Saturn. He suggested instead that the rings were made up of many individual particles.

We now know that Newtonian Gravity is only an approximation, but we still use it all the time. Why is this? The reason is that it is a superb approximation for almost all

uses in the Solar System. Spacecraft are sent to the Moon and back, probes are put into orbit around Jupiter, and mile-long bridges are built all using Newtonian gravity with only the smallest of errors. Newtonian gravity works unless the speeds involved are close to that of light, the masses are tremendous, or both.

H) Twentieth Century: Einstein

The twentieth century was a time of tremendous progress in physical science. For the understanding of gravity, the century began with two puzzles.

The first of these puzzles concerned the orbit of the planet Mercury. In the Newtonian theory of gravity, the orbit of a single planet around the Sun should be a perfect ellipse. In the real world, however, the planet is subject to the gravitational forces from the other planets in the Solar System, and hence, does not move in a perfect ellipse (This is how Neptune was discovered). In the case of Mercury, the motion was expected to look almost like an ellipse, but the point of closest approach to the Sun (perihelion) was expected to slowly revolve around the Sun. This is called the "perihelion advance of Mercury". Astronomers carefully measuring the position of Mercury over a period of time came to a startling conclusion: The perihelion advance was there, but it was occurring too quickly (see illustration below).

At first, astronomers assumed this was due to the influence of another undiscovered planet. After all, unexplained perturbations in planetary orbits was how Neptune was discovered. The proposed planet, tentatively called Vulcan, was expected to orbit closer to the Sun than Mercury. But after extensive searches, no new planet was found. What was going on?

The second puzzle was related to a series of experiments performed by the Hungarian physicist Roland Eotvos at the end of the 19th century. Eotvos was intrigued by a curious fact about Newton's laws of gravity and motion.

Newton's Law of Gravitation says that the gravitational force felt by an object is proportional to its "mass". Newton's Law of Motion also involves a "mass". But why should both laws involve the same quantity? After all, motion and gravity seem to be two very different things. Why should they both depend on the same property of an object? One might imagine a world where the force of gravity depended on how green an object was, or perhaps some other property. Scientists call the mass in the Law of Gravitation "gravitational" mass and the mass involved in motion "inertial" mass. Amazingly, Eotvos' experiments showed that the gravitational mass was the same as the inertial mass to at least a few parts in a hundred million. One consequence of this is that all objects fall towards the Earth at the same rate.

A larger mass is pulled with a larger force, but a larger mass also needs a larger force to get it moving. If one calculates the acceleration of an object, the mass cancels out entirely. Nobody had any idea why this should be the case.

Both of these puzzles were solved by the epochal work of Albert Einstein. Earlier, in the first years of the 20th century, Einstein had proposed his Special Theory of Relativity. This is the theory that sets the maximum speed as that of light and gives the famous relationship between energy and matter (E=mc²). Einstein then turned his thoughts towards gravity. His greatest insight can be illustrated by a very simple "thought experiment". Imagine that you are sitting inside a room on a comfortable chair with a desk full of equipment in front of you. You are then told that one of two situations is true:

a) the room is sitting on the Earth.
b) the room is in space (far from the Sun) being accelerated by a powerful rocket.

You are told to figure out which is true without leaving the room or obtaining information from outside. Einstein's realization was that your task is impossible. There is no observable difference between acceleration and gravity. The reason that the inertial mass and the gravitational mass are identical is that acceleration (motion) and gravity are really, on some deep level, the same thing. This can be turned around. Imagine one is in the same room, but now the room is weightless. Are you in space far from a source of gravity, or are you in an elevator whose cable has been cut (freely falling)? No experiment that you do can help you decide. What is going on, and why is this helpful in understanding gravity?

Einstein realized that one doesn't really need to deal with gravity directly - one can always cancel it out by moving in the right way. If one moves in the right way (falling), then one doesn't feel any gravity. In fact, one is weightless. This is called being in an inertial frame. But the direction you have to move is different when moving/falling from different locations (see following image). It's not possible to cancel out gravity everywhere by one single motion, only locally. Einstein's great achievement was to show how to connect, to patch together, the inertial frames in different locations.

In doing so, Einstein showed that space itself is bent by the presence of matter. Objects don't feel a force of gravity, they simply move in straight lines. But the space they move through is bent, and so it appears that they move in arcs. An example may help to clarify things. Imagine an ant walking on the surface of a table. Being lazy, it wants to walk straight ahead. If the surface is perfectly flat, then its path will be a line. But what if the table has an upside-down bowl in the middle? The ant walking along will reach the bowl and then continue walking along its side. But the side of the bowl is curved so when it reaches the other edge it may not be going in the same direction as before. It looks like a force acted on the ant to change its direction; but, in reality the ant continued to walk as straight as it could, and it was the geometry of the table that changed the direction the ant walked. So in the Einsteinian world picture, matter affects space and space affects matter. They are inextricably tied together.

Understanding how curvature of space can appear to be a force is conceptually very difficult. The following analogy gives a feeling for how this might be so:

Bob and Janet start out at the equator, 1 km apart. Both begin to walk due north. They are moving along parallel tracks. After a couple thousand miles Janet notices a very strange thing: they are closer togther! "Bob," she says, "why didn't you walk directly north as we agreed?" "But I did", Bob replies... A couple thousand miles later, they are only 1/2 a kilometer apart. Finally, after a bit more than 6000 miles they meet! What is going on?

From our perspective it is obvious, they both walked north and eventually met at the North Pole! The curvature of the Earth meant that even though they started out parallel, their paths must eventually cross. Janet and Bob, however, can't see the curvature of the Earth. To them, it seems like somehow, despite their best efforts to walk in a straight line, a force is pushing them together.

General Relativity requires us to change how we think of gravity. Gravity is not really a force, but rather a distortion of space. Since all objects travel through the same space, it is no wonder that all objects move through this space or "fall" in the same way. It is a property of the space not of the falling object. When the distortion of space is small, the effect on moving objects is almost exactly the same as would be predicted from Newton's Theory. But, only "almost exactly"! There are very small differences between the predications of the two theories. One of these differences shows up in the prediction of Mercury's orbit. The observed perihelion advance of Mercury (see p. 92) is exactly that predicted by General Relativity.

Extraordinary claims require extraordinary proof, and General Relativity is no exception. General Relativity states that space itself is dynamic and that time and distance are dependent on matter. This is a very strong claim. If we only had the evidence from Mercury's orbit, acceptance of General Relativity would be doubtful. Luckily, the support for Einstein's claims is extremely strong. Evidence comes in many forms, such as from measurements of the bending of starlight around the Sun or around massive objects, such as black holes. It is also seen in the slowing down of clocks in airplanes and the indirect detection of gravitational waves in a binary pulsar system. In fact, the Global Positioning System requires corrections based on GR to achieve its highest accuracy!

I) Twenty First Century: Beyond Relativity

General Relativity is perhaps the most beautiful physical theory yet created. It is powerful, pleasing to the aesthetic sense and well-tested. It is one of the crowning glories of modern physics. At about the same time General Relativity was born, another theory was being created. This was Quantum Mechanics. If General Relativity deals with very massive objects, then Quantum Mechanics deals with the interactions of very small objects, such as electrons and protons. Quantum Mechanics, in the form of Quantum Field Theory, (QFT) has been verified to a stunning degree of accuracy. It is perhaps the most successful theory in all of physics.

So what would happen if one had a very massive, but small, object? Both GR and QFT would apply. This seems reasonable... until one tries to do the math! It turns out that the two theories are incompatible. Not that they predict different results (that would be straightforward to test), but rather we don't even know how to express a theory that combines both GR and QFT! The usual method for obtaining a quantum theory of a physical process is to take the classical theory and to "quantize" it. But if one does this to General Relativity, the answers to all calculations become infinite! Nothing makes sense anymore. One of the hurdles in unifying these two fields can be seen in the ways they treat mass. In GR, mass is a property of an object. But in QFT, all particles are actually massless. The appearance of mass is due to interactions with what is called the Higgs Field.

Think of motion in a vat of molasses. Even light objects tend to be hard to move! The Higgs field is a bit like the vat of molasses, and the difficulty in moving an object is like the inertial mass. You can see that the relationship between the two kinds of mass is made even less clear in QFT: why should there be a connection between how strongly a particle interacts with the Higgs field and how much it bends space? Most physicists believe that a true combination of GR and QFT is possible, but it won't be found as merely an extension of GR. The search for a theory that combines GR and QFT is called the search for the Theory of Everything (TOE).

Recently, a theory known as "string theory" has gained a lot of support as a candidate TOE. What is different about string theory? Normal Quantum Mechanics treats all particles as points of zero size. This leads to a lot of problems when distances get small or energies get large. String theory says that particles are not points after all, but instead small little loops. The size of these loops is about 10-34 cm. Very small indeed, but not zero! Most of the problems reconciling GR and QM go away when one uses this theory. The full consequences of string theory have not been worked out yet (the mathematics is incredibly complex). But so far it seems very promising. Although, the final theory of gravity may be something else entirely. Whatever it is, however, we can be certain that the attempts to understand it will have profound consequences for our understanding of the Universe.