GRAVITY AND GRAVITATION

CONCEPT

Gravity is, quite simply, the force that holds together the universe. People are accustomed to thinking of it purely in terms of the gravitational pull Earth exerts on smaller bodies—a stone, a human being, even the Moon—or perhaps in terms of the Sun's gravitational pull on Earth. In fact, everything exerts a gravitational attraction toward everything else, an attraction commensurate with the two body's relative mass, and inversely related to the distance between them. The earliest awareness of gravity emerged in response to a simple question: why do objects fall when released from any restraining force? The answers, which began taking shape in the sixteenth century, were far from obvious. In modern times, understanding of gravitational force has expanded manyfold: gravity is clearly a law throughout the universe—yet some of the more complicated questions regarding gravitational force are far from settled.

HOW IT WORKS

Aristotle's Model

Greek philosophers of the period from the sixth to the fourth century b.c. grappled with a variety of questions concerning the fundamental nature of physical reality, and the forces that bind that reality into a whole. Among the most advanced thinkers of that period was Democritus (c. 460-370 b.c.), who put forth a hypothesis many thousands of years ahead of its time: that all of matter interacts at the atomic level.

Aristotle (384-322 b.c.), however, rejected the explanation offered by Democritus, an unfortunate circumstance given the fact that the great philosopher exerted an incalculable influence on the development of scientific thought. Aristotle's contributions to the advancement of the sciences were many and varied, yet his influence in physics was at least as harmful as it was beneficial. Furthermore, the fact that intellectual progress began slowing after several fruitful centuries of development in Greece only compounded the error. By the time civilization had reached the Middle Ages (c. 500 a.d.) the Aristotelian model of physical reality had been firmly established, and an entire millennium passed before it was successfully challenged.

Wrong though it was in virtually all particulars, the Aristotelian system offered a comforting symmetry amid the troubled centuries of the early medieval period. It must have been reassuring indeed to believe that the physical universe was as simple as the world of human affairs was complex. According to this neat model, all materials on Earth consisted of four elements: earth, water, air, and fire.

Each element had its natural place. Hence, earth was always the lowest, and in some places, earth was covered by water. Water must then be higher, but clearly air was higher still, since it covered earth and water. Highest of all was fire, whose natural place was in the skies above the air. Reflecting these concentric circles were the orbits of the Sun, the Moon, and the five known planets. Their orbital paths, in the Aristotelian model of the universe—a model developed to a great degree by the astronomer Ptolemy (c. 100-170)—were actually spheres that revolved around Earth with clockwork precision.

On Earth, according to the Aristotelian model, objects tended to fall toward the ground in accordance with the admixtures of differing elements they contained. A rock, for instance, was mostly earth, and hence it sought its own level, the lowest of all four elements. For the same reason, a burning fire rose, seeking the heights that were fire's natural domain. It followed from this that an object falls faster or slower, depending on the relative mixtures of elements in it: or, to use more modern terms, the heavier the object, the faster it falls.

Galileo Takes Up the Copernican Challenge

Over the centuries, a small but significant body of scientists and philosophers—each working independent from the other but building on the ideas of his predecessors—slowly began chipping away at the Aristotelian framework. The pivotal challenge came in the early part of the century, and the thinker who put it forward was not a physicist but an astronomer: Nicolaus Copernicus (1473-1543.)

Based on his study of the planets, Copernicus offered an entirely new model of the universe, one that placed the Sun and not Earth at its center. He was not the first to offer such an idea: half a century after Aristotle's death, Aristarchus (fl. 270 b.c.) had a similar idea, but Ptolemy rejected his heliocentric (Sun-centered) model in favor of the geocentric or Earth-centered one. In subsequent centuries, no less a political authority than the Catholic Church gave its approval to the Ptolemaic system. This system seemed to fit well with a literal interpretation of biblical passages concerning God's relationship with man, and man's relationship to the cosmos; hence, the heliocentric model of Copernicus constituted an offense to morality.

For this reason, Copernicus was hesitant to defend his ideas publicly, yet these concepts found their way into the consciousness of European thinkers, causing a paradigm shift so fundamental that it has been dubbed "the Copernican Revolution." Still, Copernicus offered no explanation as to why the planets behaved as they did: hence, the true leader of the Copernican Revolution was not Copernicus himself but Galileo Galilei (1564-1642.)

Initially, Galileo set out to study and defend the ideas of Copernicus through astronomy, but soon the Church forced him to recant. It is said that after issuing a statement in which he refuted the proposition that Earth moves—a direct attack on the static harmony of the Aristotelian/Ptolemaic model—he protested in private:"E pur si muove!" (But it does move!) Placed under house arrest by authorities from Rome, he turned his attention to an effort that, ironically, struck the fatal blow against the old model of the cosmos: a proof of the Copernican system according to the laws of physics.

GRAVITATIONAL ACCELERATION.

In the process of defending Copernicus, Galileo actually inaugurated the modern history of physics as a science (as opposed to what it had been during the Middle Ages: a nest of suppositions masquerading as knowledge). Specifically, Galileo set out to test the hypothesis that objects fall as they do, not because of their weight, but as a consequence of gravitational force. If this were so, the acceleration of falling bodies would have to be the same, regardless of weight. Of course, it was clear that a stone fell faster than a feather, but Galileo reasoned that this was a result of factors other than weight, and later investigations confirmed that air resistance and friction, not weight, are responsible for this difference.

On the other hand, if one drops two objects that have similar air resistance but differing weight—say, a large stone and a smaller one—they fall at almost exactly the same rate. To test this directly, however, would have been difficult for Galileo: stones fall so fast that, even if dropped from a great height, they would hit the ground too soon for their rate of fall to be tested with the instruments then available.

Instead, Galileo used the motion of a pendulum, and the behavior of objects rolling or sliding down inclined planes, as his models. On the basis of his observations, he concluded that all bodies are subject to a uniform rate of gravitational acceleration, later calibrated at 32 ft (9.8 m) per second. What this means is that for every 32 ft an object falls, it is accelerating at a rate of 32 ft per second as well; hence, after 2 seconds, it falls at the rate of 64 ft (19.6 m) per second; after 3 seconds, at 96 ft (29.4 m) per second, and so on.

Newton Discovers the Principle of Gravity

Building on the work of his distinguished fore-bear, Sir Isaac Newton (1642-1727)—who, incidentally, was born the same year Galileo died—developed a paradigm for gravitation that, even today, explains the behavior of objects in virtually all situations throughout the universe. Indeed, the Newtonian model reigned until the early twentieth century, when Albert Einstein (1879-1955) challenged it on certain specifics.

Even so, Einstein's relativity did not disprove the Newtonian system as Copernicus and Galileo disproved Aristotle's and Ptolemy's theories; rather, it showed the limitations of Newtonian mechanics for describing the behavior of certain objects and phenomena. However, in the ordinary world of day-to-day experience—the world in which stones drop and heavy objects are hard to lift—the Newtonian system still offers the key to how and why things work as they do. This is particularly the case with regard to gravity and gravitation.

Like Galileo, Newton began in part with the aim of testing hypotheses put forth by an astronomer—in this case Johannes Kepler (1571-1630). In the early years of the seventeenth century, Kepler published his three laws of planetary motion, which together identified the elliptical (oval-shaped) path of the planets around the Sun. Kepler had discovered a mathematical relationship that connected the distances of the planets from the Sun to the period of their revolution around it. Like Galileo with Copernicus, Newton sought to generalize these principles to explain, not only how the planets moved, but also why they did.

Almost everyone has heard the story of Newton and the apple—specifically, that while he was sitting under an apple tree, a falling apple struck him on the head, spurring in him a great intuitive leap that led him to form his theory of gravitation. One contemporary biographer, William Stukely, wrote that he and Newton were sitting in a garden under some apple trees when Newton told him that "…he was just in the same situation, as when formerly, the notion of gravitation came into his mind. It was occasion'd by the fall of an apple, as he sat in a contemplative mood. Why should that apple always descend perpendicularly to the ground, he thought to himself. Why should it not go sideways or upwards, but constantly to the earth's centre?"

The tale of Newton and the apple has become a celebrated myth, rather like that of George Washington and the cherry tree. It is an embellishment of actual events: Newton never said that an apple hit him on the head, just that he was thinking about the way that apples fell. Yet the story has become symbolic of the creative intellectual process that occurs when a thinker makes a vast intuitive leap in a matter of moments. Of course, Newton had spent many years contemplating these ideas, and their development required great effort. What is important is that he brought together the best work of his predecessors, yet transcended all that had gone before—and in the process, forged a model that explained a great deal about how the universe functions.

The result was his Philosophiae Naturalis Principia Mathematica, or "Mathematical Principles of Natural Philosophy." Published in 1687, the book—usually referred to simply as the Principia —was one of the most influential works ever written. In it, Newton presented his three laws of motion, as well as his law of universal gravitation.

The latter stated that every object in the universe attracts every other one with a force proportional to the masses of each, and inversely proportional to the square of the distance between them. This statement requires some clarification with regard to its particulars, after which it will be reintroduced as a mathematical formula.

MASS AND FORCE.

The three laws of motion are a subject unto themselves, covered elsewhere in this volume. However, in order to understand gravitation, it is necessary to understand at least a few rudimentary concepts relating to them. The first law identifies inertia as the tendency of an object in motion to remain in motion, and of an object at rest to remain at rest. Inertia is measured by mass, which—as the second law states—is a component of force.

Specifically, the second law of motion states that force is equal to mass multiplied by acceleration. This means that there is an inverse relationship between mass and acceleration: if force remains constant and one of these factors increases, the other must decrease—a situation that will be discussed in some depth below.

Also, as a result of Newton's second law, it is possible to define weight scientifically. People typically assume that mass and weight are the same, and indeed they are on Earth—or at least, they are close enough to be treated as comparable factors. Thus, tables of weights and measures show that a kilogram, the metric unit of mass, is equal to 2.2 pounds, the latter being the principal unit of weight in the British system.

In fact, this is—if not a case of comparing to apples to oranges—certainly an instance of comparing apples to apple pies. In this instance, the kilogram is the "apple" (a fitting Newtonian metaphor!) and the pound the "apple pie." Just as an apple pie contains apples, but other things as well, the pound as a unit of force contains an additional factor, acceleration, not included in the kilo.

BRITISH VS. SI UNITS.

Physicists universally prefer the metric system, which is known in the scientific community as SI (an abbreviation of the French Système International d'Unités —that is, "International System of Units"). Not only is SI much more convenient to use, due to the fact that it is based on units of 10; but in discussing gravitation, the unequal relationship between kilograms and pounds makes conversion to British units a tedious and ultimately useless task.

Though Americans prefer the British system to SI, and are much more familiar with pounds than with kilos, the British unit of mass—called the slug—is hardly a household word. By contrast, scientists make regular use of the SI unit of force—named, appropriately enough, the newton. In the metric system, a newton (N) is the amount of force required to accelerate 1 kilogram of mass by 1 meter per second squared (m/s²) Due to the simplicity of using SI over the British system, certain aspects of the discussion below will be presented purely in terms of SI. Where appropriate, however, conversion to British units will be offered.

CALCULATING GRAVITATIONAL FORCE.

The law of universal gravitation can be stated as a formula for calculating the gravitational attraction between two objects of a certain mass, m ₁ AND M ₂: F _grav = G · (m ₁M ₂)/R². F _grav is gravitational force, and r ² the square of the distance between m ₁ and m ₂.

As for G, in Newton's time the value of this number was unknown. Newton was aware simply that it represented a very small quantity: without it, (m ₁m ₂)/r ² could be quite sizeable for objects of relatively great mass separated by a relatively small distance. When multiplied by this very small number, however, the gravitational attraction would be revealed to be very small as well. Only in 1798, more than a century after Newton's writing, did English physicist Henry Cavendish (1731-1810) calculate the value of G.

As to how Cavendish derived the figure, that is an exceedingly complex subject far beyond the scope of the present discussion. Even to identify G as a number is a challenging task. First of all, it is a unit of force multiplied by squared area, then divided by squared mass: in other words, it is expressed in terms of (N · m²)/kg², where N stands for newtons, m for meters, and kg for kilograms. Nor is the coefficient, or numerical value, of G a whole number such as 1. A figure as large as 1, in fact, is astronomically huge compared to G, whose coefficient is 6.67 · 10⁻¹¹—in other words, 0.0000000000667.

REAL-LIFE APPLICATIONS

Weight vs. Mass

Before discussing the significance of the gravitational constant, however, at this point it is appropriate to address a few issues that were raised earlier—issues involving mass and weight. In many ways, understanding these properties from the framework of physics requires setting aside everyday notions.

First of all, why the distinction between weight and mass? People are so accustomed to converting pounds to kilos on Earth that the difference is difficult to comprehend, but if one considers the relation of mass and weight in outer space, the distinction becomes much clearer. Mass is the same throughout the universe, making it a much more fundamental characteristic—and hence, physicists typically speak in terms of mass rather than weight.

Weight, on the other hand, differs according to the gravitational pull of the nearest large body. On Earth, a person weighs a certain amount, but on the Moon, this weight is much less, because the Moon possesses less mass than Earth. Therefore, in accordance with Newton's formula for universal gravitation, it exerts less gravitational pull. By contrast, if one were on Jupiter, it would be almost impossible even to stand up, because the pull of gravity on that planet—with its greater mass—would be vastly greater than on Earth.

It should be noted that mass is not at all a function of size: Jupiter does have a greater mass than Earth, but not because it is bigger. Mass, as noted earlier, is purely a measure of inertia: the more resistant an object is to a change in its velocity, the greater its mass. This in itself yields some results that seem difficult to understand as long as one remains wedded to the concept—true enough on Earth—that weight and mass are identical.

A person might weigh less on the Moon, but it would be just as difficult to move that person from a resting position as it would be to do so on Earth. This is because the person's mass, and hence his or her resistance to inertia, has not changed. Again, this is a mentally challenging concept: is not lifting a person, which implies upward acceleration, not an attempt to counteract their inertia when standing still? Does it not follow that their mass has changed? Understanding the distinction requires a greater clarification of the relationship between mass, gravity, and weight.

F = ma.

Newton's second law of motion, stated earlier, shows that force is equal to mass multiplied by acceleration, or in shorthand form, F = ma. To reiterate a point already made, if one assumes that force is constant, then mass and acceleration must have an inverse relationship. This can be illustrated by performing a simple experiment.

Suppose one were to apply a certain amount of force to an empty shopping cart. Assuming the floor had just enough friction to allow movement, it would be easy for almost anyone to accelerate the shopping cart. Now assume that the shopping cart were filled with heavy lead balls, so that it weighed, say, 1,102 lb (500 kg). If one applied the same force, it would not move.

What has changed, clearly, is the mass of the shopping cart. Because force remained constant, the rate of acceleration would become very small—in this case, almost infinitesimal. In the first case, with an empty shopping cart, the mass was relatively small, so acceleration was relatively high.

Now to return to the subject of lifting someone on the Moon. It is true that in order to lift that person, one would have to overcome inertia, and, in that sense, it would be as difficult as it is on Earth. But the other component of force, acceleration, has diminished greatly.

Weight is, again, a unit of force, but in calculating weight it is useful to make a slight change to the formula F = ma. By definition, the acceleration factor in weight is the downward acceleration due to gravity, usually rendered as g. So one's weight is equal to mg —but on the Moon, g is much smaller than it is on Earth, and hence, the same amount of force yields much greater results.

These facts shed new light on a question that bedeviled physicists at least from the time of Aristotle, until Galileo began clarifying the issue some 2,000 years later: why shouldn't an object of greater mass fall at a different rate than one of smaller mass? There are two answers to that question, one general and one specific. The general answer—that Earth exerts more gravitational pull on an object of greater mass—requires a deeper examination of Newton's gravitational formula. But the more specific answer, relating purely to conditions on Earth, is easily addressed by considering the effect of air resistance.

Gravity and Air Resistance

One of Galileo's many achievements lay in using an idealized model of reality, one that does not take into account the many complex factors that affect the behavior of objects in the real world.

This permitted physicists to study processes that apparently defy common sense. For instance, in the real world, an apple does drop at a greater rate of speed than does a feather. However, in a vacuum, they will drop at the same rate. Since Galileo's time, it has become commonplace for physicists to discuss specific processes such as gravity with the assumption that all non-pertinent factors (in this case, air resistance or friction) are nonexistent or irrelevant. This greatly simplified the means of testing hypotheses.

Idealization of reality makes it possible to set aside the things people think they know about the real world, where events are complicated due to friction. The latter may be defined as a force that resists motion when the surface of one object comes into contact with the surface of another. If two balls are released in an environment free from friction—one of them simply dropped while the other is rolled down a curved surface or inclined plane—they will reach the bottom at the same time. This seems to go against everything that is known, but that is only because what people "know" is complicated by variables that have nothing to do with gravity.

The same is true for the behavior of falling objects with regard to air resistance. If air resistance were not a factor, one could fire a cannonball over horizontal space and then, when the ball reached the highest point in its trajectory, release another ball from the same height—and again, they would hit the ground at the same time. This is the case, even though the cannonball that was fired from the cannon has to cover a great deal of horizontal space, whereas the dropped ball does not. The fact is that the rate of acceleration due to gravity will be identical for the two balls, and the fact that the ball fired from a cannon also covers a horizontal distance during that same period is irrelevant.

TERMINAL VELOCITY.

In the real world, air resistance creates a powerful drag force on falling objects. The faster the rate of fall, the greater the drag force, until the air resistance forces a leveling in the rate of fall. At this point, the object is said to have reached terminal velocity, meaning that its rate of fall will not increase thereafter. Galileo's idealized model, on the other hand, treated objects as though they were falling in a vacuum—space entirely devoid of matter, including air. In such a situation, the rate of acceleration would continue to grow indefinitely.

By means of a graph, one can compare the behavior of an object falling through air with that of an object falling in a vacuum. If the x axis measures time and the y axis downward speed, the rate of an object falling in a vacuum describes a 60°-angle. In other words, the speed of its descent is increasing at a much faster rate than is the rate of time of its descent—as indeed should be the case, in accordance with gravitational acceleration. The behavior of an object falling through air, on the other hand, describes a curve. Up to a point, the object falls at the same rate as it would in a vacuum, but soon velocity begins to increase at a much slower rate than time. Eventually, the curve levels off at the point where the object experiences terminal velocity.

Air resistance and friction have been mentioned separately as though they were two different forces, but in fact air resistance is simply a prominent form of friction. Hence air resistance exerts an upward force to counter the downward force of mass multiplied by gravity—that is, weight. Since g is a constant (32 ft or 9.8 m/sec²), the greater the weight of the falling object, the longer it takes for air resistance to bring it to terminal velocity.

A feather quickly reaches terminal velocity, whereas it takes much longer for a cannonball to do the same. As a result, a heavier object does take less time to fall, even from a great height, than does a light one—but this is only because of friction, and not because of "elements" seeking their "natural level." Incidentally, if raindrops (which of course fall from a very great height) did not reach terminal velocity, they would cause serious injury by the time they hit the ground.

Applying the Gravitational Formula

Using Newton's gravitational formula, it is relatively easy to calculate the pull of gravity between two objects. It is also easy to see why the attraction is insignificant unless at least one of the objects has enormous mass. In addition, application of the formula makes it clear why G (the gravitational constant, as opposed to g, the rate of acceleration due to gravity) is such a tiny number.

If two people each have a mass of 45.5 kg (100 lb) and stand 1 m (3.28 ft) apart, m ₁m ₂ is equal to 2,070 kg (4,555 lb) and r ² is equal to 1 m² . Applied to the gravitational formula, this figure is rendered as 2,070 kg²/1 m². This number is then multiplied by gravitational constant, which again is equal to 6.67 · 10⁻¹¹ (N · m²)/kg². The result is a net gravitational force of 0.000000138 N (0.00000003 lb)—about the weight of a single-cell organism!

EARTH, GRAVITY, AND WEIGHT.

Though it is certainly interesting to calculate the gravitational force between any two people, computations of gravity are only significant for objects of truly great mass. For instance, there is the Earth, which has a mass of 5.98 · 10²⁴ kg—that is, 5.98 septillion (1 followed by 24 zeroes) kilograms. And, of course, Earth's mass is relatively minor compared to that of several planets, not to mention the Sun. Yet Earth exerts enough gravitational pull to keep everything on it—living creatures, manmade structures, mountains and other natural features—stable and in place.

One can calculate Earth's gravitational force on any one person—if one wants to take the time to do so using Newton's formula. In fact, it is much simpler than that: gravitational force is equal to weight, or m · g. Thus if a woman weighs 100 lb (445 N), this amount is also equal to the gravitational force exerted on her. By dividing 445 N by the acceleration of gravity—9.8 m/sec²—it is easy to obtain her mass: 45.4 kg.

The use of the mg formula for gravitation helps, once again, to explain why heavier objects do not fall faster than lighter ones. The figure for g is a constant, but for the sake of argument, let us assume that it actually becomes larger for objects with a greater mass. This in turn would mean that the gravitational force, or weight, would be bigger than it is—thus creating an irreconcilable logic loop.

Furthermore, one can compare results of two gravitation equations, one measuring the gravitational force between Earth and a large stone, the other measuring the force between Earth and a small stone. (The distance between Earth and each stone is assumed to be the same.) The result will yield a higher quantity for the force exerted on the larger stone—but only because its mass is greater. Clearly, then, the increase of force results only from an increase in mass, not acceleration.

Gravity and Curved Space

As should be clear from Newton's gravitational formula, the force of gravity works both ways: not only does a stone fall toward Earth, but Earth actually falls toward it. The mass of Earth is so great compared to that of the stone that the movement of Earth is imperceptible—but it does happen. Furthermore, because Earth is round, when one hurls a projectile at a great distance, Earth curves away from the projectile; but eventually gravity itself forces the projectile to the ground.

However, if one were to fire a rocket at 17,700 MPH (28,500 km/h), at every instant of time the projectile is falling toward Earth with the force of gravity—but the curved Earth would be falling away from it at the same moment as well. Hence, the projectile would remain in constant motion around the planet—that is, it would be in orbit.

The same is true of an artificial satellite's orbit around Earth: even as the satellite falls toward Earth, Earth falls away from it. This same relationship exists between Earth and its great natural satellite, the Moon. Likewise, with the Sun and its many satellites, including Earth: Earth plunges toward the Sun with every instant of its movement, but at every instant, the Sun falls away.

WHY IS EARTH ROUND?

Note that in the above discussion, it was assumed that Earth and the Sun are round. Everyone knows that to be the case, but why? The answer is "Because they have to be"—that is, gravity will not allow them to be otherwise. In fact, the larger the mass of an object, the greater its tendency toward roundness: specifically, the gravitational pull of its interior forces the surface to assume a relatively uniform shape. There is a relatively small vertical differential for Earth's surface: between the lowest point and the highest point is just 12.28 mi (19.6 km)—not a great distance, considering that Earth's radius is about 4,000 mi (6,400 km).

It is true that Earth bulges near the equator, but this is only because it is spinning rapidly on its axis, and thus responding to the centripetal force of its motion, which produces a centrifugal component. If Earth were standing still, it would be much nearer to the shape of a sphere. On the other hand, an object of less mass is more likely to retain a shape that is far less than spherical. This can be shown by reference to the Martian moons Phobos and Deimos, both of which are oblong—and both of which are tiny, in terms of size and mass, compared to Earth's Moon.

Mars itself has a radius half that of Earth, yet its mass is only about 10% of Earth's. In light of what has been said about mass, shape, and gravity, it should not surprising to learn that Mars is also home to the tallest mountain in the solar system. Standing 15 mi (24 km) high, the volcano Olympus Mons is not only much taller than Earth's tallest peak, Mount Everest (29,028 ft [8,848 m]); it is 22% taller than the distance from the top of Mount Everest to the lowest spot on Earth, the Mariana Trench in the Pacific Ocean (−35,797 ft [−10,911 m])

A spherical object behaves with regard to gravitation as though its mass were concentrated near its center. And indeed, 33% of Earth's mass is at is core (as opposed to the crust or mantle), even though the core accounts for only about 20% of the planet's volume. Geologists believe that the composition of Earth's core must be molten iron, which creates the planet's vast electromagnetic field.

THE FRONTIERS OF GRAVITY.

The subject of curvature with regard to gravity can be both a threshold or—as it is here—a point of closure. Investigating questions over perceived anomalies in Newton's description of the behavior of large objects in space led Einstein to his General Theory of Relativity, which posited a curved four-dimensional space-time. This led to entirely new notions concerning gravity, mass, and light. But relativity, as well as its relation to gravity, is another subject entirely. Einstein offered a new understanding of gravity, and indeed of physics itself, that has changed the way thinkers both inside and outside the sciences perceive the universe. Here on Earth, however, gravity behaves much as Newton described it more than three centuries ago.

Meanwhile, research in gravity continues to expand, as a visit to the Web site <www.Gravity.org> reveals. Spurred by studies in relativity, a branch of science called relativistic astrophysics has developed as a synthesis of astronomy and physics that incorporates ideas put forth by Einstein and others. The <www.Gravity.org> site presents studies—most of them too abstruse for a reader who is not a professional scientist—across a broad spectrum of disciplines. Among these is bioscience, a realm in which researchers are investigating the biological effects—such as mineral loss and motion sickness—of exposure to low gravity. The results of such studies will ultimately protect the health of the astronauts who participate in future missions to outer space.

WHERE TO LEARN MORE

Ardley, Neil. The Science Book of Gravity. San Diego, CA: Harcourt Brace Jovanovich, 1992.

Beiser, Arthur. Physics, 5th ed. Reading, MA: Addison-Wesley, 1991.

Bendick, Jeanne. Motion and Gravity. New York: F. Watts, 1972.

Dalton, Cindy Devine. Gravity. Vero Beach, FL: Rourke, 2001.

David, Leonard. "Artificial Gravity and Space Travel." Bio-Science, March 1992, pp. 155-159.

Exploring Gravity—Curtin University, Australia (Web site). <http://www.curtin.edu.au/curtin/dept/physsci/gravity/> (March 18, 2001).

The Gravity Society (Web site). <http://www.gravity.org>(March 18, 2001).

Nardo, Don. Gravity: The Universal Force. San Diego, CA: Lucent Books, 1990.

Rutherford, F. James; Gerald Holton; and Fletcher G. Watson. Project Physics. New York: Holt, Rinehart, and Winston, 1981.

Stringer, John. The Science of Gravity. Austin, TX: Raintree Steck-Vaughn, 2000.

KEY TERMS

FORCE:

The product of mass multiplied by acceleration.

FRICTION:

The force that resists motion when the surface of one object comes into contact with the surface of another.

INERTIA:

The tendency of an object in motion to remain in motion, and of an object at rest to remain at rest.

INVERSE RELATIONSHIP:

A situation involving two variables, in which one of the two increases in direct proportion to the decrease in the other.

LAW OF UNIVERSAL GRAVITATION:

A principle, put forth by Sir Isaac Newton (1642-1727), which states that every object in the universe attracts every other one with a force proportional to the masses of each, and inversely proportional to the square of the distance between them.

MASS:

A measure of inertia, indicating the resistance of an object to a change in itsmotion.

TERMINAL VELOCITY:

A term describing the rate of fall for an object experiencing the drag force of air resistance. In a vacuum, the object would continue to accelerate with the force of gravity, but in most real-world situations, air resistance creates a powerful drag force that causes a leveling in the object's rate off all.

VACUUM:

Space entirely devoid of matter, including air.

WEIGHT:

A measure of the gravitational force on an object; the product of mass multiplied by the acceleration due to gravity.

Gravity and Gravitation

The history of gravity

Newtonian gravity

General relativity

Resources

Gravity is a force of attraction that exists between every pair of objects in the universe. This force is proportional to the mass of each object in each pair, and inversely proportional to the square of the distance between the two. Thus, F = Gm₁m²/r2, where m₁ is the mass of the first object, m₂ is the mass of the second object, r is the distance between their centers, and G is a fixed number termed the gravitational constant. (If m₁ and m₂ are given in kilograms and r in meters, then G = 6.673 x 10^-1N m²/kg².)

The history of gravity

The Greek philosopher Aristotle (384–322 BC) posed, following earlier traditions, that the material world consisted of four elements: earth, water, air, and fire. For example, a rock was mostly earth with a little water, air, and fire, a cloud was mostly air and water with a little earth and fire. Each element had a natural or proper place in the universe to which it spontaneously inclined; earth belonged at the very center, water in a layer covering the earth, air above the water, and fire above the air. Each element had a natural tendency to return to its proper place, so that, for example, rocks fell toward the center and fire rose above the air. This was one of the earliest explanations of gravity: that it was the natural tendency for the heavier elements, earth and water, to return to their proper positions near the center of the universe.

Aristotle’s theory was for centuries taken as implying that objects with different weights should fall at different speeds; that is, a heavier object should fall faster because it contains more of the center-trending elements, earth and water. However, this is not correct. Objects with different weights fall, in fact, at the same rate. (This statement still only an approximation, however, for it assumes that the Earth is perfectly stationary, which it is not. When an object is dropped the Earth accelerates “upward” under the

influence of their mutual gravitation, just as the object “falls,” and they meet somewhere in the middle. For a heavier object, this meeting does take place slightly sooner than for a light object, and thus, heavier objects actually do fall slightly faster than light ones. In practice, however, the Earth’s movement is not measurable for “dropped” objects of less than planetary size, and so it is accurate to state that all small objects fall at the same rate, regardless of their mass.)

Aristotle’s model of the universe also included the moon, sun, the visible planets, and the fixed stars. Aristotle assumed that these were outside the layer of fire and were made of a fifth element, the ether or quintessence (the term is derived from the Latin expression quinta essentia, or fifth essence, used by Aristotle’s medieval translators). The celestial bodies circled Earth attached to nested ethereal spheres centered on Earth. No forces were required to maintain these motions, since everything was considered perfect and unchanging, having been set in motion by a Prime Mover—God.

Aristotle’s ideas were accepted in Europe and the Near East for centuries, until the Polish astronomer Nicolaus Copernicus (1473–1543) developed a heliocentric (sun-centered) model to replace the geocentric (Earth-centered) one that had been the dominant cos-mological concept ever since Aristotle’s time. (Non-European astronomers unfamiliar with Aristotle, such as the Chinese and Aztecs, had developed geocentric models of their own; no heliocentric model existed prior to Copernicus.) Copernicus’s model placed the sun in the center of the universe, with all of the planets orbiting the sun in perfect circles. This development was such a dramatic change from the previous model that it is now called the Copernican Revolution. It was an ingenious intellectual construct, but it still did not explain why the planets circled the sun, in the sense of what caused them to do so.

While many scientists were trying to explain these celestial motions, others were trying to understand terrestrial mechanics. It seemed to be a commonsense fact that heavier objects fall faster than light ones: drop a feather and a rock and see which hits the ground first. The fault in this experiment is that air resistance affects the rate at which objects fall. What about another experiment, one in which air resistance plays a smaller role: observing the difference between dropping a large rock and a small rock? This is an easy experiment to perform, and the results have profound implications. As early as the sixth century AD, Johannes Philiponos (c. 490–566) claimed that the difference in landing times was small for objects of different weight but similar shape. Galileo’s friend, Italian physicist Giambattista Benedetti (1530–1590), in 1553, and Dutch physicist Simon Stevin (1548–1620), in 1586, also considered the falling-rock problem and concluded that rate of fall was independent of weight. However, the individual most closely associated with the falling-body problem is Italian physicist Galileo Galilei (1564–1642), who systematically observed the motions of falling bodies. (It is unlikely that he actually dropped weights off the Leaning Tower of Pisa, but he did write that such an experiment might be performed.)

Because objects speed up (accelerate) quickly while falling, and Galileo was restricted to naked-eye observation by the technology of his day, he studied the slower motions of pendulums and of bodies rolling and sliding down incline. From his results, Galileo formulated his law of falling bodies. This states that, disregarding air resistance, bodies in free fall speed up with a constant acceleration (rate of change of velocity) that is independent of their weight or composition. The acceleration due to gravity near Earth’s surface is given the symbol g and has a value of about 32 ft per second per second (9.8 m/s²) This means that 1 second after a release a falling object is moving at about 10 m/s; after 2 seconds, 20 m/s; after 10 seconds, 100 m/s. That is, after falling for 10 seconds, it is dropping fast enough to cross the length of a football field in less than one second. Writing v for the velocity of the falling body and t for the time since commencement of free fall, we have v = gt.

Galileo also determined a formula to describe the distance d that a body falls in a given time: d = 1/2gt²

That is, if one drops an object, after 1 second it has fallen approximately 5 m; after 2 seconds, 20 m; and after 10 seconds, 500 meters.

Galileo did an excellent job of describing the effect of gravity on objects on Earth, but it wasn’t until English physicist Isaac Newton (1642–1727) studied the problem that it was understood just how universal gravity is. An old story says that Newton suddenly understood gravity when an apple fell out of a tree and hit him on the head; this story may not be exactly true, but Newton did say that a falling apple helped him develop his theory of gravity.

Newtonian gravity

Newton’s universal law of gravitation states that all objects in the universe attract all other objects. Thus the sun attracts Earth, Earth attracts the sun, Earth attracts a book, a book attracts Earth, the book attracts the desk, and so on. The gravitational pull between small objects, such as molecules and books, is generally negligible; the gravitational pull exerted by larger objects, such as stars and planets, organizes the universe. It is gravity that keeps us on the Earth, the moon in orbit around the Earth, and the Earth in orbit around the sun.

Newton’s law of gravitation also states that the strength of the force of attraction depends on the masses of the two objects. The mass of an object is a measure of how much material it has, but it is not the same as its weight, which is a measure of how much force a given mass experiences in a given gravitational field; a given rock, say, will have the same mass anywhere in the universe but will weight more on Earth than on the moon.

We do not feel the gravitational forces from objects other than the Earth because they are weak. For example, the gravitational force of attraction between two friends weighing 100 lb (45.5 kg) standing 3 ft (1 m) apart is only about 3 x 10~8 N = 0.00000003 lb, which is about the weight of a bacterium. (Note: the pound is a measure of weight—the gravitational force experienced by an object—while the kilogram is a measure of mass. Strictly speaking, then, pounds and kilograms cannot be substituted for each other as in the previous sentence. However, near Earth’s surface weight and mass can be approximately equated because Earth’s gravitational field is approximately constant; treating pounds and kilograms as proportional units is therefore standard practice under this condition.)

The gravitational force between two objects becomes weaker if the two objects are moved apart and stronger if they are brought closer together; that is, the force depends on the distance between the objects. If we take two objects and double the distance between them, the force of attraction decreases to one fourth of its former value. If we triple the distance, the force decreases to one ninth of its former value. The force depends on the inverse square of the distance.

All these statements are derived from one simple equation: for two objects having masses m₁ and m₁ respectively, the magnitude of the force of gravity acting on each object is given by: F= Gm₁m₂/r², where r is the distance between the objects’ centers and G is the gravitational constant (6.673 x 10^-11N m²/kg².) Note that the gravitational constant is an extremely small number; this explains why we only feel gravity when we are near a large mass (e.g., the Earth).

Newton also explained how bodies respond to forces (including gravitational forces) that act on them. His second law of motion states that a net force (i.e., force not canceled by a contrary force) causes a body to accelerate. The amount of this acceleration is inversely proportional to the mass of the object. This means that under the influence of a given force, more massive objects accelerate more slowly than less massive objects. Alternatively, to experience the same acceleration, more massive objects require more force. Consider the gravitational force exerted by the Earth on two rocks, the first with a mass of 2 lb (1 kg) and a second with a mass of 22 lb (10 kg). Since the mass of the second is 10 times the mass of the first, the gravitational force on the second will be 10 times the force on the first. But a 22-lb (10-kg) mass requires 10 times more force to accelerate it, so both masses accelerate Earthward at the same rate. Ignoring the Earth’s own acceleration toward the rocks (which is extremely small), it follows that equal falling rates for small objects are a natural consequence of Newton’s law of gravity and second law of motion.

What if one throws a ball horizontally? If one throws it slowly, it will hit the ground a short distance away. If one throws it faster, it will land farther away. Since the Earth is round, the Earth will curve slightly away from the ball before it lands; the farther the throw, the greater the amount of curve. If one could throw or launch the ball at 18,000 mi/h (28,800 km/h), the Earth would curve away from the ball by the same amount that the ball falls. The ball would never get any closer to the ground, and would be in orbit around the Earth. Gravity still accelerates the ball at 9.8 m/s² toward the Earth’s center, but the ball never approaches the ground. (This is exactly what the moon is doing.) In addition, the orbits of the Earth and other planets around the sun and all the motions of the stars and galaxies follow Newton’s laws. This is why Newton’s law of gravitation is termed “universal;” it describes the effect of gravity on all objects in the universe.

Newton published his laws of motion and gravity in 1687, in his seminal Philosophiae Naturalis Principia Mathematica (Latin for Mathematical Principles of Natural Philosophy, or Principia for short). When we need to solve problems relating to gravity, Newton’s laws usually suffice. There are, however, some phenomena that they cannot describe. For example, the motions of the planet Mercury are not exactly described by Newton’s laws. Newton’s theory of gravity, therefore, needed modifications that would require another genius, Albert Einstein, and his theory of general relativity.

General relativity

German physicist Albert Einstein (1879–1955) realized that Newton’s theory of gravity had problems. He knew, for example, that Mercury’s orbit showed unexplained deviations from that predicted by Newton’s laws. However, he was worried about a much more serious problem. As the force between two objects depends on the distance between them, if one object moves closer, the other object will feel a change in the gravitational force. According to Newton, this change would be immediate, or instantaneous, even if the objects were millions of miles apart. Einstein saw this as a serious flaw in Newtonian gravity. Einstein assumed that nothing could travel instantaneously, not even a change in force. Specifically, nothing can travel faster than light in a vacuum, which has a speed of approximately 186,000 mi/s (300,000 km/s). In order to fix this problem, Einstein had not only to revise Newtonian gravity, but to change the way we think about space, time, and the structure of the universe. He stated this new way of thinking mathematically in his general theory of relativity.

Einstein said that a mass bends space, like a heavy ball making a dent on a rubber sheet. Further, Einstein contended that space and time are intimately related to each other, and that we do not live in three spatial dimensions and time (all four quite independent of each other), but rather in a four-dimensional space-time continuum, a seamless blending of the four. It is thus not “space,” naively conceived, but space-time that warps in reaction to a mass. This, in turn, explains why objects attract each other. Consider the sun sitting in space-time, imagined as a ball sitting on a rubber sheet. It curves the space-time around it into a bowl shape. The planets orbit around the sun because they are rolling across through this distorted space-time, which curves their motions like those of a ball rolling around inside a shallow bowl. (These images are intended as analogies, not as precise explanations.) Gravity, from this point of view, is the way objects affect the motions of other objects by affecting the shape of space-time.

Einstein’s general relativity makes predictions that Newton’s theory of gravitation does not. Since particles of light (photons) have no mass, Newtonian theory predicts that they will not be affected by gravity. However, if gravity is due to the curvature of space-time, then light should be affected in the same way as matter. This proposition was tested as follows: During the day, the sun is too bright to see any stars. However, during a total solar eclipse the sun’s disk is blocked by the moon, and it is possible to see stars that appear in the sky near to the sun. During the total solar eclipse of 1919, astronomers measured the positions of several stars that were close to the sun in the sky. It was determined that the measured positions were altered as predicted by general relativity; the sun’s gravity bent the starlight so that the stars appeared to shift their locations when they were near the sun in the sky. The detection of the bending of starlight by the sun was one of the great early experimental verifications of general relativity; many others have been conducted since.

Another surprising prediction made by general relativity is that waves can travel in gravitational forces just as waves travel through air or other media. These gravitational waves are formed when masses move back and forth in space-time, much as sound waves are created by the oscillations of a speaker cone. In 1974, two stars were discovered orbiting around each other, and scientists found out that the stars were losing energy at the exact rate required to generate the predicted gravity waves; that is, they were steadily radiating energy away in the form gravitational waves. So far, gravitational waves have not been detected directly, but new detectors will be completed in the United States, Japan, and Europe in 2003

KEY TERMS

Acceleration— The rate at which the velocity of an object changes over time.

Force— Influence exerted on an object by an outside agent which produces an acceleration changing the object’s state of motion.

General relativity— Einstein’s theory of space and time, which explains gravity and the shape of space.

Mass— A measure of the amount of material in an object.

Velocity— The speed and direction of a moving object.

Weight— The gravitational force pulling an object toward a large body, e.g., the Earth, that depends both on the mass of the object and its distance from the center of the larger body.

and it is expected that these devices will detect gravitational waves produced by violent cosmic events such as supernovae. Scientists have already verified that changes in gravitation do propagate at the speed of light, as predicted by Einstein’s theory but not by Newton’s.

Of all the predictions of general relativity, the strangest is the existence of black holes. When a very massive star runs out of fuel, the gravitational self-attraction of the star makes it shrink. If the star is massive enough, it will collapse it to a point having finite mass but infinite density. Space-time will be so distorted in the vicinity of this “singularity”, as it is termed, that not even light will be able to escape; hence the term “black hole.” Astronomers have been searching for objects in the sky that might be black holes, but since they do not give off light directly, they must be detected indirectly. When material falls into a black hole, it must heat up so much that it glows in x rays. Astronomers look for strong x-ray sources in the sky because these sources may be likely candidates to be black holes. Numerous black holes have been detected by these means, and it is now believed that many or most galaxies contain a supermassive black hole at their center, having a mass millions or billions of times greater than that of the sun.

The greatest remaining challenge for gravity theory is unification with quantum mechanics. Quantum theory describes the physics of phenomena at the atomic and subatomic scale, but does not account for gravitation. General relativity, which employs continuous variables, does not describe the behavior of objects at the quantum scale. Physicists therefore seek a theory of “quantum gravity,” a unified set of equations that will describe the whole range of known phenomena. Some theorists believe that the highly mathematical field known as string theory will be able to reconcile gravity and quantum mechanics, but as of 2006 no experimental test of string theory had yet been devised.

See also Geocentric theory; Heliocentric theory; X-ray astronomy; Relativity, general; Relativity, special.

Resources

BOOKS

Cheng, Ta-Pei. Relativity, Gravitation, and Cosmology: A Basic Introduction. New York: Oxford University Press, USA, 2005.

Hartle, James B. Gravity: An Introduction to Einstein’s General Relativity. Boston: Addsion-Wesley, 2002.

Hawkings, Stephen and Mlodinow, Leonard. A Briefer History of Time. New York: Bantam, 2005.

PERIODICALS

“Einstein Was Right on Gravity’s Velocity.” New York Times. (January 8, 2003).

Jim Guinn

Gravity and gravitation

Gravity is a force of attraction that exists between every pair of objects in the Universe. This force is proportional to the mass of each object in each pair, and inversely proportional to the square of the distance between the two; thus,

where m1 is the mass of the first object, m2 is the mass of the second object, r is the distance between their centers, and G is a fixed number termed the gravitational constant. (If m1 and m2 are given in kilograms and r in meters, then G = 6.673 × 10-11N m2/kg2.)

The history of gravity

The Greek philosopher Aristotle (384–322 b.c.) posed, following earlier traditions, that the material world consisted of four elements: earth , water , air, and fire. For example, a rock was mostly earth with a little water, air, and fire, a cloud was mostly air and water with a little earth and fire. Each element had a natural or proper place in the Universe to which it spontaneously inclined; earth belonged at the very center, water in a layer covering the earth, air above the water, and fire above the air. Each element had a natural tendency to return to its proper place, so that, for example, rocks fell toward the center and fire rose above the air. This was one of the earliest explanations of gravity: that it was the natural tendency for the heavier elements, earth and water, to return to their proper positions near the center of the Universe. Aristotle's theory was for centuries taken as implying that objects with different weights should fall at different speeds; that is, a heavier object should fall faster because it contains more of the centertrending elements, earth and water. However, this is not correct. Objects with different weights fall, in fact, at the same rate . (This statement still only an approximation , however, for it assumes that the Earth is perfectly stationary, which it is not. When an object is dropped the Earth accelerates "upward" under the influence of their mutual gravitation, just as the object "falls," and they meet somewhere in the middle. For a heavier object, this meeting does take place slightly sooner than for a light object, and thus, heavier objects actually do fall slightly faster than light ones. In practice, however, the Earth's movement is not measurable for "dropped" objects of less than planetary size, and so it is accurate to state that all small objects fall at the same rate, regardless of their mass.)

Aristotle's model of the Universe also included the Moon , Sun , the visible planets, and the fixed stars. Aristotle assumed that these were outside the layer of fire and were made of a fifth element, the ether or quintessence (the term is derived from the Latin expression quinta essentia, or fifth essence, used by Aristotle's medieval translators). The celestial bodies circled the Earth attached to nested ethereal spheres centered on Earth. No forces were required to maintain these motions, since everything was considered perfect and unchanging, having been set in motion by a Prime Mover—God.

Aristotle's ideas were accepted in Europe and the Near East for centuries, until the Polish astronomer Nicolaus Copernicus (1473–1543) developed a heliocentric (Sun-centered) model to replace the geocentric (Earth-centered) one that had been the dominant cosmological concept ever since Aristotle's time. (Non-European astronomers unfamiliar with Aristotle, such as the Chinese and Aztecs, had developed geocentric models of their own; no heliocentric model existed prior to Copernicus.) Copernicus's model placed the Sun in the center of the Universe, with all of the planets orbiting the Sun in perfect circles. This development was such a dramatic change from the previous model that it is now called the Copernican Revolution. It was an ingenious intellectual construct, but it still did not explain why the planets circled the Sun, in the sense of what caused them to do so.

While many scientists were trying to explain these celestial motions, others were trying to understand terrestrial mechanics. It seemed to be the common-sense fact that heavier objects fall faster than light ones of the same mass: drop a feather and a pebble of equal mass and see which hits the ground first. The fault in this experiment is that air resistance affects the rate at which objects fall. What about another experiment, one in which air resistance plays a smaller role: observing the difference between dropping a large rock and a medium rock? This is an easy experiment to perform, and the results have profound implications. As early as the sixth century a.d. Johannes Philiponos (c. 490–566) claimed that the difference in landing times was small for objects of different weight but similar shape. Galileo's friend, Italian physicist Giambattista Benedetti (1530–1590), in 1553, and Dutch physicist Simon Stevin (1548–1620), in 1586, also considered the falling-rock problem and concluded that rate of fall was independent of weight. However, the individual most closely associated with the falling-body problem is Italian physicist Galileo Galilei (1564–1642), who systematically observed the motions of falling bodies. (It is unlikely that he actually dropped weights off the Leaning Tower of Pisa, but he did write that such an experiment might be performed.)

Because objects speed up (accelerate) quickly while falling, and Galileo was restricted to naked-eye observation by the technology of his day, he studied the slower motions of pendulums and of bodies rolling and sliding down incline. From his results, Galileo formulated his Law of Falling Bodies. This states that, disregarding air resistance, bodies in free fall speed up with a constant acceleration (rate of change of velocity ) that is independent of their weight or composition. The acceleration due to gravity near Earth's surface is given the symbol g and has a value of about 32 feet per second per second (9.8 m/s2) This means that 1 second after a release a falling object is moving at about 10 m/s; after 2 seconds, 20 m/s; after 10 seconds, 100 m/s. That is, after falling for 10 seconds, it is dropping fast enough to cross the length of a football field in less than one second. Writing v for the velocity of the falling body and t for the time since commencement of free fall, we have v = gt.

Galileo also determined a formula to describe the distance d that a body falls in a given time:

That is, if one drops an object, after 1 second it has fallen approximately 5m; after 2 seconds, 20m; and after 10 seconds, 500 meters.

Galileo did an excellent job of describing the effect of gravity on objects on Earth, but it wasn't until English physicist Isaac Newton (1642–1727) studied the problem that it was understood just how universal gravity is. An old story says that Newton suddenly understood gravity when an apple fell out of a tree and hit him on the head; this story may not be exactly true, but Newton did say that a falling apple helped him develop his theory of gravity.

Newtonian gravity

Newton's universal law of gravitation states that all objects in the Universe attract all other objects. Thus the Sun attracts Earth, Earth attracts the Sun, Earth attracts a book, a book attracts Earth, the book attracts the desk, and so on. The gravitational pull between small objects, such as molecules and books, is generally negligible; the gravitational pull exerted by larger objects, such as stars and planets, organizes the Universe. It is gravity that keeps us on the Earth, the Moon in orbit around the Earth, and the Earth in orbit around the Sun.

Newton's law of gravitation also states that the strength of the force of attraction depends on the masses of the two objects. The mass of an object is a measure of how much material it has, but it is not the same as its weight, which is a measure of how much force a given mass experiences in a given gravitational field; a given rock, say, will have the same mass anywhere in the universe but will weight more on the Earth than on the Moon.

We do not feel the gravitational forces from objects other than the Earth because they are weak. For example,
the gravitational force of attraction between two friends weighing 100 lb (45.5 kg) standing 3 ft (1 m) apart is only about 3 = 10-8 N = 0.00000003 lb, which is about the weight of a bacterium. (Note: the pound is a measure of weight—the gravitational force experienced by an object—while the kilogram is a measure of mass. Strictly speaking, then, pounds and kilograms cannot be substituted for each other as in the previous sentence. However, near Earth's surface weight and mass can be approximately equated because Earth's gravitational field is approximately constant; treating pounds and kilograms as proportional units is therefore standard practice under this condition.)

The gravitational force between two objects becomes weaker if the two objects are moved apart and stronger if they are brought closer together; that is, the force depends on the distance between the objects. If we take two objects and double the distance between them, the force of attraction decreases to one fourth of its former value. If we triple the distance, the force decreases to one ninth of its former value. The force depends on the inverse square of the distance.

All these statements are derived from one simple equation: for two objects having masses m1 and m1 respectively, the magnitude of the force of gravity acting on each object is given by:

where r is the distance between the objects' centers and G is the gravitational constant (6.673 × 10-11N m2/kg2.) Note that the gravitational constant is an extremely small number; this explains why we only feel gravity when we are near a large mass (e.g., the Earth).

Newton also explained how bodies respond to forces (including gravitational forces) that act on them. His Second Law of Motion states that a net force (i.e., force not canceled by a contrary force) causes a body to accelerate. The amount of this acceleration is inversely proportional to the mass of the object. This means that under the influence of a given force, more massive objects accelerate more slowly than less massive objects. Alternatively, to experience the same acceleration, more massive objects require more force. Consider the gravitational force exerted by the Earth on two rocks, the first with a mass of 2 lb (1 kg) and a second with a mass of 22 lb (10 kg). Since the mass of the second is 10 times the mass of the first, the gravitational force on the second will be 10 times the force on the first. But a 22-lb (10-kg) mass requires 10 times more force to accelerate it, so both masses accelerate Earthward at the same rate. Ignoring the Earth's acceleration toward the rocks (which is extremely small), it follows that equal falling rates for small objects are a natural consequence of Newton's law of gravity and second law of motion.

What if one throws a ball horizontally? If one throws it slowly, it will hit the ground a short distance away. If one throw sit faster, it will land farther. Since the Earth is round, the Earth will curve slightly away from the ball before it lands; the farther the throw, the greater the amount of curve. If one could throw or launch the ball at 18,000 mi/h (28,800 km/h), the Earth would curve away from the ball by the same amount that the ball falls. The ball would never get any closer to the ground, and would be in orbit around the Earth. Gravity still accelerates the ball at 9.8 m/s2 toward the Earth's center, but the ball never approaches the ground. (This is exactly what the Moon is doing.) In addition, the orbits of the Earth and other planets around the Sun and all the motions of the stars and galaxies follow Newton's laws. This is why Newton's law of gravitation is termed "universal;" it describes the effect of gravity on all objects in the Universe.

Newton published his laws of motion and gravity in 1687, in his seminal Philosophiae Naturalis Principia Mathematica (Latin for Mathematical Principles of Natural Philosophy, or Principia for short). When we need to solve problems relating to gravity, Newton's laws usually suffice. There are, however, some phenomena that they cannot describe. For example, the motions of the planet Mercury are not exactly described by Newton's laws. Newton's theory of gravity, therefore, needed modifications that would require another genius, Albert Einstein, and his Theory of General Relativity.

General relativity

German physicist Albert Einstein (1879–1955) realized that Newton's theory of gravity had problems. He knew, for example, that Mercury's orbit showed unexplained deviations from that predicted by Newton's laws. However, he was worried about a much more serious problem. As the force between two objects depends on the distance between them, if one object moves closer, the other object will feel a change in the gravitational force. According to Newton, this change would be immediate, or instantaneous, even if the objects were millions of miles apart. Einstein saw this as a serious flaw in Newtonian gravity. Einstein assumed that nothing could travel instantaneously, not even a change in force. Specifically, nothing can travel faster than light in a vacuum , which has a speed of approximately 186,000 mi/s (300,000 km/s). In order to fix this problem, Einstein had not only to revise Newtonian gravity, but to change the way we think about space , time, and the structure of the Universe. He stated this new way of thinking mathematically in his general theory of relativity.

Einstein said that a mass bends space, like a heavy ball making a dent on a rubber sheet. Further, Einstein contended that space and time are intimately related to each other, and that we do not live in three spatial dimensions and time (all four quite independent of each other), but rather in a four-dimensional space-time continuum, a seamless blending of the four. It is thus not "space," naively conceived, but space-time that warps in reaction to a mass. This, in turn, explains why objects attract each other. Consider the Sun sitting in space-time, imagined as a ball sitting on a rubber sheet. It curves the spacetime around it into a bowl shape. The planets orbit around the Sun because they are rolling across through this distorted space-time, which curves their motions like those of a ball rolling around inside a shallow bowl. (These images are intended as analogies, not as precise explanations.) Gravity, from this point of view, is the way objects affect the motions of other objects by affecting the shape of space-time.

Einstein's general relativity makes predictions that Newton's theory of gravitation does not. Since particles of light (photons) have no mass, Newtonian theory predicts that they will not be affected by gravity. However, if gravity is due to the curvature of space-time, then light should be affected in the same way as matter . This proposition was tested as follows: During the day, the Sun is too bright to see any stars. However, during a total solar eclipse the Sun's disk is blocked by the Moon, and it is possible to see stars that appear in the sky near to the Sun. During the total solar eclipse of 1919, astronomers measured the positions of several stars that were close to the Sun in the sky. It was determined that the measured positions were altered as predicted by general relativity; the Sun's gravity bent the starlight so that the stars appeared to shift their locations when they were near the Sun in the sky. The detection of the bending of starlight by the Sun was one of the great early experimental verifications of general relativity; many others have been conducted since.

Another surprising prediction made by general relativity is that waves can travel in gravitational forces just as waves travel through air or other media. These gravitational waves are formed when masses move back and forth in space-time, much as sound waves are created by the oscillations of a speaker cone. In 1974, two stars were discovered orbiting around each other, and scientists found out that the stars were losing energy at the exact rate required to generate the predicted gravity waves; that is, they were steadily radiating energy away in the form gravitational waves. So far, gravitational waves have not been detected directly, but new detectors will be completed in the U.S., Japan, and Europe in 2003 and it is expected that these devices will detect gravitational waves produced by violent cosmic events such as supernovae. Scientists have already verified that changes in gravitation do propagate at the speed of light, as predicted by Einstein's theory but not by Newton's.

Of all the predictions of general relativity, the strangest is the existence of black holes. When a very massive star runs out of fuel, the gravitational self-attraction of the star makes it shrink. If the star is massive enough, it will collapse it to a point having finite mass but infinite density . Space-time will be so distorted in the vicinity of this "singularity," as it is termed, that not even light will be able to escape; hence the term "black hole." Astronomers have been searching for objects in the sky that might be black holes, but since they do not give off light directly, they must be detected indirectly. When material falls into a black hole , it must heat up so much that it glows in x rays . Astronomers look for strong x-ray sources in the sky because these sources may be likely candidates to be black holes. Numerous black holes have been detected by these means, and it is now believed that many or most galaxies contain a supermassive black hole at their center, having a mass millions or billions of times greater than that of the Sun.

The greatest remaining challenge for gravity theory is unification with quantum mechanics . Quantum theory describes the physics of phenomena at the atomic and subatomic scale, but does not account for gravitation. General relativity, which employs continuous variables, does not describe the behavior of objects at the quantum scale. Physicists therefore seek a theory of "quantum gravity," a unified set of equations that will describe the whole range of known phenomena.

See also Geocentric theory; Heliocentric theory; X-ray astronomy; Relativity, general; Relativity, special.

Resources

books

Hartle, James B. Gravity: An Introduction to Einstein's General Relativity Boston: Addsion-Wesley, 2002.

Hawking, Stephen W. A Brief History of Time: From the Big Bang to Black Holes. New York: Bantam Books, 1988.

Thorne, Kip S. Black Holes and Time Warps: Einstein's Outrageous Legacy. New York: W. W. Norton, 1994.

periodicals

"Einstein Was Right on Gravity's Velocity." New York Times. (January 8, 2003).

Jim Guinn

KEY TERMS

Acceleration

—The rate at which the velocity of an object changes over time.

Force

—Influence exerted on an object by an outside agent which produces an acceleration changing the object's state of motion.

General relativity

—Einstein's theory of space and time, which explains gravity and the shape of space.

Mass

—A measure of the amount of material in an object.

Velocity

—The speed and direction of a moving object.

Weight

—The gravitational force pulling an object toward a large body, e.g., the Earth, that depends both on the mass of the object and its distance from the center of the larger body.

Gravity and gravitation

Gravity is the force of attraction between any two objects in the universe. That force depends on two factors: the mass of each object and the distance between them.

Historical background

The story behind English physicist Isaac Newton's (1642–1727) discovery of the gravitational force is one of the most fascinating in all of science. It begins in ancient Greece in the period from the sixth to the third century b.c. During that time, a number of Greek philosophers attempted to explain common observations from the natural world—such as the fact that most objects fall to the ground if they are not held up in some way.

Aristotle. Among the explanations developed for this tendency was one offered by Greek philosopher Aristotle (384–322 b.c.). Aristotle developed a grand scheme of natural philosophy stating that all objects "belonged" in one place or another. Heat belonged in the atmosphere because it originally came from the Sun (as Aristotle taught). For that reason, heat rises. Objects fall toward Earth's surface, Aristotle said, because that's where "earthy" objects belong. Aristotle's philosophy was an attempt to explain why objects fall.

Galileo and Newton. Aristotle's philosophy dominated the thinking of European scholars for nearly 2,000 years. Then, in the sixteenth century, Italian physicist Galileo Galilei (1564–1642) suggested another way of answering questions in science. Scientists should not trouble themselves trying to understand why things happen in the natural world, Galileo said. Instead, they should focus simply on describing how things occur. Galileo also taught that the way to find out about the natural world is not just to think logically about it but to perform experiments that produce measurable results.

One of the most famous experiments attributed to Galileo was the one he conducted at the Leaning Tower of Pisa. He is said to have dropped two balls from the top of the tower and discovered that they both took the same time to reach the ground. Galileo's greatest achievements were not in defining the true nature of gravity, then, but in setting the stage for the work of Isaac Newton, who was born the year Galileo died.

Newton's accomplishments in the field of gravity also are associated with a famous story. Legend has it that Newton was hit on the head by an apple falling from a tree. That event got him wondering about the force between two objects on Earth (the apple and the ground) and the force between two objects in the universe (the force between a planet and the Sun).

Gravity on Earth and in the heavens. The connection between gravitational forces on Earth and in the heavens is a very important one. Measuring the force of gravity on Earth is very difficult for one simple reason. Suppose we want to measure what happens when an object falls on Earth. In terms of gravity, what actually happens is that the object and the planet Earth are attracted toward each other. The object moves downward toward Earth, and Earth moves upward toward the object. The problem is that Earth is so much larger than the object that it's impossible to see any movement on the part of the planet.

The situation is quite different in the heavens. The reason planets travel in an orbit around the Sun, Newton said, is that they are responding to two forces. One force is caused simply by their motion through the skies. Just imagine that at some time in the past, someone grabbed hold of Mars and threw it past the Sun. Mars would be traveling through space, then, because of the initial velocity that was given to it.

But Mars does not travel in a straight line. It moves in a circle (or nearly a circle) around the Sun. What changes Mars's motion from a straight line to a curve, Newton asked? The answer he proposed was gravity. The gravitational force between the Sun and Mars causes the planet to move out of a straight line and towards the Sun. The combination of the straight line motion and the gravitational force, then, accounts for the shape of Mars's orbit.

Words to Know

Mass: A measure of the amount of matter in a body.

Orbit: The path followed by a body (such as a planet) in its travel around another body (such as the Sun).

Proportionality constant: A number inserted into an equation to make both sides equal.

Weight: The gravitational attraction of Earth on an object.

But a huge point in Newton's favor was that he already knew all the main points about Mars and its orbit around the Sun. He had a good idea as to how fast the planet was traveling, its mass, the mass of the Sun, and the size of its orbit. Furthermore, the difference in size between Mars and the Sun was great—but not nearly as great as the difference between an apple and Earth.

So Newton derived his idea of the gravitational force by studying the orbit of the planets. He applied that idea to what he knew about the planets and found that he was able to predict almost perfectly the orbits followed by the planets.

Cavendish's findings. Proving the gravitational law on Earth was somewhat more difficult. Probably the most important experiment conducted for this purpose was one carried out by English chemist and physicist Henry Cavendish (1731–1810) in 1798. Cavendish suspended a light rod horizontally from a silk thread. At each end of the rod he hung a lead ball. Then he brought a third lead ball close to one of the two lead balls suspended from the rod. He was able to notice that the two lead balls attracted each other. As they did so, they caused the metal rod to pivot slightly on its silk thread. The amount by which the rod pivoted, Cavendish found out, depended on how closely the lead balls were brought next to each other and how much the two balls weighed (what their masses were). Cavendish's results turned out to confirm Newton's predictions exactly.

Einsteinian gravity. Newton's description of gravitational forces proved to be satisfactory for almost two and a half centuries. Then, observations began to appear in which his gravitational law turned out to be not exactly correct. The differences between predictions based on Newton's law and actual observations were small—too small to have been noticed for many years. But scientists eventually realized that Newton's law was not entirely and always correct.

In the early 1900s, German-born American physicist Albert Einstein (1879–1955) proposed a solution for problems with Newton's law. Interestingly enough, Einstein did not suggest modifications in Newton's law to make it more accurate. Instead, he proposed an entirely new way to think about gravity.

The way to think about gravitational forces, Einstein said, is to imagine that space has shape. Imagine, for example, a thin sheet of rubber stretched very tightly in all directions. Then imagine that the rubber sheet has indentations in it, similar to the depressions caused by pushing in on the sheet with your thumb. Finally, imagine that this dented rubber sheet represents space.

Using this model, Einstein suggested that gravity is nothing other than the effect produced when an object moving through space approaches one of these indentations. If a planet were moving through space and came close to an indentation, for example, it would tend to roll inward toward the dent. The effect to an outside observer would be exactly the same as if the planet were experiencing a gravitational force of attraction to the center of the dent.

Finally, Einstein said, these dents in space are caused by the presence of objects, such as stars and planets. The larger the object, the deeper the dent. Again, the effect observed is the same as it would be with Newtonian gravity. An object traveling through space will be pulled out of its orbit more by a deep dent (a heavy object) than it will be by a shallow dent (a lighter object).

So what's the point of thinking about gravity in Einstein's terms rather than Newton's? The answer is that the mathematics used by Einstein does everything that Newton's law of gravitation does plus it solves all of the problems that Newtonian gravity cannot explain.

Fundamental forces

Physicists now believe that all forces in the universe can be reduced to one of four fundamental forces: gravitation, electromagnetism, and the strong and weak force. The strong and weak force are forces discovered in the twentieth century; they are responsible for the way atoms and particles smaller than the atom interact with each other. Electromagnetic forces affect charged or magnetic particles. And the gravitational force affects all bodies of any size whatsoever. Of the four forces, the gravitational force is by far the weakest and probably least understood.

One of the great efforts among physicists during the twentieth century was the attempt to show how all four fundamental forces are really different symptoms of a single force. They have been successful in doing so for the electromagnetic and weak forces, which are now recognized as two forms of a single force. The attempts to unify the remaining forces, including gravitation, however, have been unsuccessful so far.

[See also Celestial mechanics ]

Gravitation

Gravitation is a universal attractive force exerted by any two physical bodies on each other, even though they may be separated by a large distance. Gravitation is responsible for making objects fall to the surface of the Earth (gravitational attraction of the object by the Earth), for the nearly circular motions of the planets around the sun (gravitational attraction of the planets by the sun), for the structure of stars and planets (gravitational attraction balanced by pressure forces of constituent particles towards each other), and for the structure of star clusters and galaxies (hundreds of millions of stars would fly apart from each other if not held together by gravity). Gravitation also controls the rate at which the universe expands, and is responsible for the growth of small inhomogeneities in the expanding universe into galaxies and clusters of galaxies.

Gravity is the weakest of the four fundamental forces known to physics, but it dominates on large scales because it is a long-distance force that is locally always attractive (in contrast to the far stronger electromagnetic force, which can both attract and repel, and cancels itself out on large scales). Thus gravitation is a dominant force in every day life, as well as in the motions of stars and planets and in the evolution of the cosmos. Indeed, it is one of the forces that makes our existence possible by enabling the formation and stability of plants like Earth that are hospitable to life. Without gravity (at approximately the strength it has on Earth) evolution of life would be difficult if not impossible. This fact can naturally lead to speculation that the existence and specific nature of gravitation could be part of a grand design allowing self-assembling structures to come into existence and lead to intelligent life. In this way, gravity can have theological significance.

Classical physics

Italian astronomer Galileo Galilei (1564–1642) first recognized in the early seventeenth century that when air resistance can be neglected, objects accelerate at the same rate towards the surface of the earth, irrespective of their physical composition. Thus a feather and a cannon ball will arrive at the same time at the earth's surface if simultaneously released from rest at the same height in a vacuum chamber. This means there is a universal rate of acceleration downwards caused by the earth's gravitational field—approximately 32 feet per second squared—irrespective of the nature of the object considered. Gravitational potential energy can be converted to kinetic energy, with total energy conserved, as for example in a roller coaster or a pole vaulter. This enables gravity to do useful work, as in a clock driven by weights or a water mill, but it also means people must work to go uphill. Gravity can also be a danger to people, who can fall or be hurt by falling objects. Despite this danger, gravity is an essential part of the stability of every day life—it is the reason that objects stay firmly rooted on the ground rather than floating into the air.

In the late seventeenth century, Isaac Newton (1642–1727) showed that the gravitational attraction of objects towards the earth and the motion of the planets around the sun could be described accurately by assuming a universal attractive force between any two bodies, proportional to each of their masses and to the inverse of the square of the distance between them. The attractive nature of gravity results because masses are always positive. On this basis he was able to explain both the universal acceleration towards the surface of the earth observed by Galileo and the laws of motion of planets around the sun that had been observationally established earlier in the century by Johannes Kepler (1571–1630). This was the first major unification of explanation attained in theoretical physics, showing that two phenomena that initially appeared completely unrelated had a unified origin. Newton's account of gravitation also explained why the direction of gravity varies at different places on the surface of the earth (always being directed towards its center), allowing "up" to be different directions at different places on the earth's surface (Australia and England, for example).

In conformity with the rest of theoretical physics, Newton's theory of gravity can be reformulated as a variational principle (Hamilton's principle or Lagrange's equations) based on minimisation of particular combinations of kinetic energy and gravitational potential energy along the trajectory followed by a particle. Gravity by itself is a conservative theory (energy is conserved), so there is no friction associated with the motion of stars and planets in the sky, and their motion is fully reversible; the past and future directions of time are indistinguishable, as far as gravity is concerned. Newton was puzzled as to how the force of gravity, as described by his equations, could succeed in acting at a distance when there was no apparent contact between the bodied concerned. Pierre Laplace (1749-1827), a French physicist and mathematician, essentially resolved this puzzle by introducing the idea of a gravitational force field that fills the empty space between massive bodies and mediates the gravitational force between them. The concept of such fields became one of the major features of classical physics, particularly in the case of electromagnetism. In quantum theory the idea gravitational fields is revised and understood as a force mediated by the interchange of force-carrying particles.

Einstein and after

In the early twentieth century, Albert Einstein (1879–1955) radically reshaped the understanding of gravity through his proposal of the general theory of relativity, based on the idea that space-time is curved, with the space-time curvature determined by the matter in it. This theory predicts the motion of planets round the sun more accurately than Newtonian theory can, and also predicts radically new phenomena, in particular, black holes and gravitational radiation. Insofar as science has been able to test these predictions, they are correct. A problem with the theory is that it predicts that under many conditions (for example, at the start of the universe and at the end of gravitational collapse to form a black hole), space-time singularities will occur. Scientists still do not properly understand this phenomenon, but presumably it means that they will have to take the effect of quantum theory on gravity into account. General Relativity does not do so; it is a purely classical theory.

Quantum gravity theories try to develop a theory of gravity that generalizes Einstein's theory and is also compatible with quantum theory. Even the way to start such a project is unclear. Approaches include twistor theory, lattice theories, noncommutative geometries, loop variable theories, and superstring theories. None has reached a satisfactorily developed state, however, much less been tested and shown to be correct. Indeed, in many ways such theories are likely to be untestable. The most ambitious are the superstring theories, now extended into a metatheory of uncertain nature known as M-theory, which promises to provide a unified theory of all fundamental forces and particles. M-theory still has far to go before making good on that promise.

Despite the lack of a definite quantum theory of gravity, various attempts have been made to develop quantum theories of cosmology. These theories also face considerable conceptual and calculational problems. The satisfactory unification of quantum theory and general relativity theory, perhaps in some unified theory of all the fundamental forces, remains one of the most significant outstanding problems of theoretical physics.

The desire to develop a practical antigravity machine remains one of humanity's outstanding wishes. No present theory offers a way to such a machine, but the negative gravitational effect of the vacuum energy will continue to inspire some to hope that one day such a machine might exist.

See also Black Holes; Cosmology, Physical Aspects; Forces of Nature; Galileo Galilei; Newton, Isaac; Physics, Quantum; Quantum Theory; Relativity, General Theory of; Singularity; String Theory; Superstrings

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d'inverno, ray. introducing einstein's relativity. oxford: oxford university press, 1996.

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george f. r. ellis