Gravitation

I INTRODUCTION

Gravitation, the force of attraction between all objects that tends to pull them toward one another. It is a universal force, affecting the largest and smallest objects, all forms of matter, and energy. Gravitation governs the motion of astronomical bodies. It keeps the moon in orbit around the earth and keeps the earth and the other planets of the solar system in orbit around the sun. On a larger scale, it governs the motion of stars and slows the outward expansion of the entire universe because of the inward attraction of galaxies to other galaxies. Typically the term gravitation refers to the force in general, and the term gravity refers to the earth's gravitational pull.

Gravitation is one of the four fundamental forces of nature, along with electromagnetism and the weak and strong nuclear forces, which hold together the particles that make up atoms. Gravitation is by far the weakest of these forces and, as a result, is not important in the interactions of atoms and nuclear particles or even of moderate-sized objects, such as people or cars. Gravitation is important only when very large objects, such as planets, are involved. This is true for several reasons. First, the force of gravitation reaches great distances, while nuclear forces operate only over extremely short distances and decrease in strength very rapidly as distance increases. Second, gravitation is always attractive. In contrast, electromagnetic forces between particles can be repulsive or attractive depending on whether the particles both have a positive or negative electrical charge, or they have opposite electrical charges. These attractive and repulsive forces tend to cancel each other out, leaving only a weak net force. Gravitation has no repulsive force and, therefore, no such cancellation or weakening.

The gravitational attraction of objects for one another is the easiest fundamental force to observe and was the first fundamental force to be described with a complete mathematical theory by the English physicist and mathematician Sir Isaac Newton. A more accurate theory called general relativity was formulated early in the 20th century by the German-born American physicist Albert Einstein. Scientists recognize that even this theory is not correct for describing how gravitation works in certain circumstances, and they continue to search for an improved theory.

II EARTH'S GRAVITATION

Gravitation plays a crucial role in most processes on the earth. The ocean tides are caused by the gravitational attraction of the moon and the sun on the earth and its oceans. Gravitation drives weather patterns by making cold air sink and displace less dense warm air, forcing the warm air to rise. The gravitational pull of the earth on all objects holds the objects to the surface of the earth. Without it, the spin of the earth would send them floating off into space.

The gravitational attraction of every bit of matter in the earth for every other bit of matter amounts to an inward pull that holds the earth together against the pressure forces tending to push it outward. Similarly, the inward pull of gravitation holds stars together. When a star's fuel nears depletion, the processes producing the outward pressure weaken and the inward pull of gravitation eventually compresses the star to a very compact size.

II.A         Acceleration

 

Freefall

Falling objects accelerate in response to the force exerted on them by Earth’s gravity. Different objects accelerate at the same rate, regardless of their mass. This illustration shows the speed at which a ball and a cat would be moving and the distance each would have fallen at intervals of a tenth of a second during a short fall.

If an object held near the surface of the earth is released, it will fall and accelerate, or pick up speed, as it descends. This acceleration is caused by gravity, the force of attraction between the object and the earth. The force of gravity on an object is also called the object's weight. This force depends on the object's mass, or the amount of matter in the object. The weight of an object is equal to the mass of the object multiplied by the acceleration due to gravity.

A bowling ball that weighs 16 lb is actually being pulled toward the earth with a force of 16 lb. In the metric system, the bowling ball is pulled toward the earth with a force of 71 newtons (a metric unit of force abbreviated N). The bowling ball also pulls on the earth with a force of 16 lb (71 N), but the earth is so massive that it does not move appreciably. In order to hold the bowling ball up and keep it from falling, a person must exert an upward force of 16 lb (71 N) on the ball. This upward force acts to oppose the 16 lb (71 N) downward weight force, leaving a total force of zero. The total force on an object determines the object's acceleration.

If the pull of gravity is the only force acting on an object, then all objects, regardless of their weight, size, or shape, will accelerate in the same manner. At the same place on the earth, the 16 lb (71 N) bowling ball and a 500 lb (2200 N) boulder will fall with the same rate of acceleration. As each second passes, each object will increase its downward speed by about 9.8 m/sec (32 ft/sec), resulting in an acceleration of 9.8 m/sec/sec (32 ft/sec/sec). In principle, a rock and a feather both would fall with this acceleration if there were no other forces acting. In practice, however, air friction exerts a greater upward force on the falling feather than on the rock and makes the feather fall more slowly than the rock.

The mass of an object does not change as it is moved from place to place, but the acceleration due to gravity, and therefore the object's weight, will change because the strength of the earth's gravitational pull is not the same everywhere. The earth's pull and the acceleration due to gravity decrease as an object moves farther away from the center of the earth. At an altitude of 4000 miles (6400 km) above the earth's surface, for instance, the bowling ball that weighed 16 lb (71 N) at the surface would weigh only about 4 lb (18 N). Because of the reduced weight force, the rate of acceleration of the bowling ball at that altitude would be only one quarter of the acceleration rate at the surface of the earth. The pull of gravity on an object also changes slightly with latitude. Because the earth is not perfectly spherical, but bulges at the equator, the pull of gravity is about 0.5 percent stronger at the earth's poles than at the equator.

III EARLY IDEAS ABOUT GRAVITATION

The ancient Greek philosophers developed several theories about the force that caused objects to fall toward the earth. In the 4th century bc, the Greek philosopher Aristotle proposed that all things were made from some combination of the four elements, earth, air, fire, and water. Objects that were similar in nature attracted one another, and as a result, objects with more earth in them were attracted to the earth. Fire, by contrast, was dissimilar and therefore tended to rise from the earth. Aristotle also developed a cosmology, that is, a theory describing the universe, that was geocentric, or earth-centered, with the moon, sun, planets, and stars moving around the earth on spheres. The Greek philosophers, however, did not propose a connection between the force behind planetary motion and the force that made objects fall toward the earth.

At the beginning of the 17th century, the Italian physicist and astronomer Galileo discovered that all objects fall toward the earth with the same acceleration, regardless of their weight, size, or shape, when gravity is the only force acting on them. Galileo also had a theory about the universe, which he based on the ideas of the Polish astronomer Nicolaus Copernicus. In the mid-16th century, Copernicus had proposed a heliocentric, or sun-centered system, in which the planets moved in circles around the sun, and Galileo agreed with this cosmology. However, Galileo believed that the planets moved in circles because this motion was the natural path of a body with no forces acting on it. Like the Greek philosophers, he saw no connection between the force behind planetary motion and gravitation on earth.

In the late 16th and early 17th centuries the heliocentric model of the universe gained support from observations by the Danish astronomer Tycho Brahe, and his student, the German astronomer Johannes Kepler. These observations, made without telescopes, were accurate enough to determine that the planets did not move in circles, as Copernicus had suggested. Kepler calculated that the orbits had to be ellipses (slightly elongated circles). The invention of the telescope made even more precise observations possible, and Galileo was one of the first to use a telescope to study astronomy. In 1609 Galileo observed that moons orbited the planet Jupiter, a fact that could not reasonably fit into an earth-centered model of the heavens.

The new heliocentric theory changed scientists' views about the earth's place in the universe and opened the way for new ideas about the forces behind planetary motion. However, it was not until the late 17th century that Isaac Newton developed a theory of gravitation that encompassed both the attraction of objects on the earth and planetary motion.

IV NEWTON'S THEORY OF GRAVITATION

 

Gravitational Forces

Because the Moon has significantly less mass than Earth, the weight of an object on the Moon’s surface is only one-sixth the object’s weight on Earth’s surface. This graph shows how much an object that weighs w on Earth would weigh at different points between the Earth and Moon. Since the Earth and Moon pull in opposite directions, there is a point, about 346,000 km (215,000 mi) from Earth, where the opposite gravitational forces would cancel, and the object's weight would be zero.

To develop his theory of gravitation, Newton first had to develop the science of forces and motion called mechanics. Newton proposed that the natural motion of an object is motion at a constant speed on a straight line, and that it takes a force to slow down, speed up, or change the path of an object. Newton also invented calculus, a new branch of mathematics that became an important tool in the calculations of his theory of gravitation.

Newton proposed his law of gravitation in 1687 and stated that every particle in the universe attracts every other particle in the universe with a force that depends on the product of the two particles' masses divided by the square of the distance between them. The gravitational force between two objects can be expressed by the following equation: F= GMm/d² where F is the gravitational force, G is a constant known as the universal constant of gravitation, M and m are the masses of each object, and d is the distance between them. Newton considered a particle to be an object with a mass that was concentrated in a small point. If the mass of one or both particles increases, then the attraction between the two particles increases. For instance, if the mass of one particle is doubled, the force of attraction between the two particles is doubled. If the distance between the particles increases, then the attraction decreases as the square of the distance between them. Doubling the distance between two particles, for instance, will make the force of attraction one quarter as great as it was.

According to Newton, the force acts along a line between the two particles. In the case of two spheres, it acts along the line between their centers. The attraction between objects with irregular shapes is more complicated. Every bit of matter in the irregular object attracts every bit of matter in the other object. A simpler description is possible near the surface of the earth where the pull of gravity is approximately uniform in strength and direction. In this case there is a point in an object (even an irregular object) called the center of gravity, at which all the force of gravity can be considered to be acting.

Newton's law affects all objects in the universe, from raindrops in the sky to the planets in the solar system. It is therefore known as the universal law of gravitation. In order to know the strength of gravitational forces in general, however, it became necessary to find the value of G, the universal constant of gravitation. Scientists needed to perform an experiment, but gravitational forces are very weak between objects found in a common laboratory and thus hard to observe. In 1798 the English chemist and physicist Henry Cavendish finally measured G with a very sensitive experiment in which he nearly eliminated the effects of friction and other forces. The value he found was 6.754 x 10^-11 N-m²/kg²—close to the currently accepted value of 6.670 x 10^-11 N-m²/kg² (a decimal point followed by 10 zeros and then the number 6670). This value is so small that the force of gravitation between two objects with a mass of 1 metric ton each, 1 meter from each other, is about 67 millionths of a newton, or about 15 millionths of a pound.

Gravitation may also be described in a completely different way. A massive object, such as the earth, may be thought of as producing a condition in space around it called a gravitational field. This field causes objects in space to experience a force. The gravitational field around the earth, for instance, produces a downward force on objects near the earth surface. The field viewpoint is an alternative to the viewpoint that objects can affect each other across distance. This way of thinking about interactions has proved to be very important in the development of modern physics.

IV.A      Planetary Motion

Newton's law of gravitation was the first theory to accurately describe the motion of objects on the earth as well as the planetary motion that astronomers had long observed. According to Newton's theory, the gravitational attraction between the planets and the sun holds the planets in elliptical orbits around the sun. The earth's moon and moons of other planets are held in orbit by the attraction between the moons and the planets. Newton's law led to many new discoveries, the most important of which was the discovery of the planet Neptune. Scientists had noted unexplainable variations in the motion of the planet Uranus for many years. Using Newton's law of gravitation, the French astronomer Urbain Leverrier and the British astronomer John Couch each independently predicted the existence of a more distant planet that was perturbing the orbit of Uranus. Neptune was discovered in 1864, in an orbit close to its predicted position.

IV.B      Problems with Newton's Theory

 

Frames of Reference

A situation can appear different when viewed from different frames of reference. Try to imagine how an observer's perceptions could change from frame to frame in this illustration.

Scientists used Newton's theory of gravitation successfully for many years. Several problems began to arise, however, involving motion that did not follow the law of gravitation or Newtonian mechanics. One problem was the observed and unexplainable deviations in the orbit of Mercury (which could not be caused by the gravitational pull of another orbiting body).

Another problem with Newton's theory involved reference frames, that is, the conditions under which an observer measures the motion of an object. According to Newtonian mechanics, two observers making measurements of the speed of an object will measure different speeds if the observers are moving relative to each other. A person on the ground observing a ball that is on a train passing by will measure the speed of the ball as the same as the speed of the train. A person on the train observing the ball, however, will measure the ball's speed as zero. According to the traditional ideas about space and time, then, there could not be a constant, fundamental speed in the physical world because all speed is relative. However, near the end of the 19th century the Scottish physicist James Clerk Maxwell proposed a complete theory of electric and magnetic forces that contained just such a constant, which he called c. This constant speed was 300,000 km/sec (186,000 mi/sec) and was the speed of electromagnetic waves, including light waves. This feature of Maxwell's theory caused a crisis in physics because it indicated that speed was not always relative.

Albert Einstein resolved this crisis in 1905 with his special theory of relativity. An important feature of Einstein's new theory was that no particle, and even no information, could travel faster than the fundamental speed c. In Newton's gravitation theory, however, information about gravitation moved at infinite speed. If a star exploded into two parts, for example, the change in gravitational pull would be felt immediately by a planet in a distant orbit around the exploded star. According to Einstein's theory, such forces were not possible.

Though Newton's theory contained several flaws, it is still very practical for use in everyday life. Even today, it is sufficiently accurate for dealing with earth-based gravitational effects such as in geology (the study of the formation of the earth and the processes acting on it), and for most scientific work in astronomy. Only when examining exotic phenomena such as black holes (points in space with a gravitational force so strong that not even light can escape them) or in explaining the big bang (the origin of the universe) is Newton's theory inaccurate or inapplicable.

V EINSTEIN'S THEORY OF RELATIVITY

In 1915 Einstein formulated a new theory of gravitation that reconciled the force of gravitation with the requirements of his theory of special relativity. He proposed that gravitational effects move at the speed of c. He called this theory general relativity to distinguish it from special relativity, which only holds when there is no force of gravitation. General relativity produces predictions very close to those of Newton's theory in most familiar situations, such as the moon orbiting the earth. Einstein's theory differed from Newton's theory, however, in that it described gravitation as a curvature of space and time.

In Einstein's general theory of relativity, he proposed that space and time may be united into a single, four-dimensional geometry consisting of 3 space dimensions and 1 time dimension. In this geometry, called spacetime, the motions of particles from point to point as time progresses are represented by curves called world lines. If there is no gravity acting, the most natural lines in this geometry are straight lines, and they represent particles that are moving always in the same direction with the same speed—that is, particles that have no force acting on them. If a particle is acted on by a force, then its world line will not be straight. Einstein also proposed that the effect of gravitation should not be represented as the deviation of a world line from straightness, as it would be for an electrical force. If gravitation is present, it should not be considered a force. Rather, gravitation changes the most natural world lines and thereby curves the geometry of spacetime. In a curved geometry, such as the two-dimensional surface of the earth, there are no straight lines. Instead, there are special curves called geodesics, an example of which are great circles around the earth. These special curves are at each point as straight as possible, and they are the most natural lines in a curved geometry. The effect of gravitation would be to influence the geodesics in spacetime. Near sources of gravitation the space is strongly curved and the geodesics behave less and less like those in flat, uncurved spacetime. In the solar system, for example, the effect of the sun and the earth is to cause the moon to move on a geodesic that winds around the geodesic of the earth 12 times a year.

V.A         Testing Einstein's Theory

Einstein's theory required verification, but the level of precision in measurement needed to distinguish between Einstein's theory and Newton's theory was difficult to achieve in the early 20th century and remains so today. There were two predictions, however, that could be tested. One involved deviations in the orbit of Mercury. Astronomers had observed that the ellipse of Mercury's orbit itself rotated—that is, the point nearest the sun, called the perihelion, slowly advanced around the sun. The rate of advancement predicted by Newton's theory differed slightly from what astronomers had measured, but Einstein's theory predicted the correct rate.

The second test involved measuring the bending of light as it passed around the sun. Both Newton's and Einstein's theories predicted that light would be deflected by gravitation. But the amount of deflection predicted by the two theories differed. The light to be measured in such a test originates in distant stars. However, under ordinary conditions the sun's brightness prevents scientists from observing the light from these stars. This problem disappears during an eclipse, when the moon blocks the sun's light. In 1919 a special British expedition took photographs during an eclipse. Scientists measured the deflection of starlight as it passed by the sun and arrived at numbers that agreed with Einstein's prediction. Subsequent eclipse observations also have confirmed Einstein's theory.

Other physicists have proposed relativistic theories of gravitation to compete with Einstein's. In these competing theories, almost all of which are geometrical like Einstein's, gravitational effects move at the speed c. They differ mostly in the mathematical details. Even using the technology of the late 20th century, scientists still find it very difficult to test these theories with experiments and observations. But Einstein's theory has passed all tests that have been made so far.

V.B         Applications of Einstein's Theory

Einstein's general relativity theory predicts special gravitational conditions. The Big Bang theory, which describes the origin and early expansion of the universe, is one conclusion based on Einstein's theory that has been verified in several independent ways.

Another conclusion suggested by general relativity, as well as other relativistic theories of gravitation, is that gravitational effects move in waves. Astronomers have observed a loss of energy in a pair of neutron stars (stars composed of densely packed neutrons) that are orbiting each other. The astronomers theorize that energy-carrying gravitational waves are radiating from the pair, depleting the stars of their energy. Very violent astrophysical events, such as the explosion of stars or the collision of neutron stars, can produce gravitational waves strong enough that they may eventually be directly detectable with extremely precise instruments. Astrophysicists are designing such instruments with the hope that they will be able to detect gravitational waves by the beginning of the 21st century.

Another gravitational effect predicted by general relativity is the existence of black holes. The idea of a star with a gravitational force so strong that light cannot escape from its surface can be traced to Newtonian theory. Einstein modified this idea in his general theory of relativity. Because light cannot escape from a black hole, for any object—a particle, spacecraft, or wave—to escape, it would have to move past light. But light moves outward at the speed c. According to relativity, c is the highest attainable speed, so nothing can pass it. The black holes that Einstein envisioned, then, allow no escape whatsoever. An extension of this argument shows that when gravitation is this strong, nothing can even stay in the same place, but must move inward. Even the surface of a star must move inward, and must continue the collapse that created the strong gravitational force. What remains then is not a star, but a region of space from which emerges a tremendous gravitational force.

VI OTHER MODERN THEORIES

Einstein's theory of gravitation revolutionized 20th-century physics. Another important revolution that took place was quantum theory. Quantum theory states that physical interactions, or the exchange of energy, cannot be made arbitrarily small. There is a minimal interaction that comes in a packet called the quantum of an interaction. For electromagnetism the quantum is called the photon. Like the other interactions, gravitation also must be quantized. Physicists call a quantum of gravitational energy a graviton. In principle, gravitational waves arriving at the earth would consist of gravitons. In practice, gravitational waves would consist of apparently continuous streams of gravitons, and individual gravitons could not be detected.

Einstein's theory did not include quantum effects. For most of the 20th century, theoretical physicists have been unsuccessful in their attempts to formulate a theory that resembles Einstein's theory but also includes gravitons. Despite the lack of a complete quantum theory, it is possible to make some partial predictions about quantized gravitation. In the 1970s, British physicist Stephen Hawking showed that quantum mechanical processes in the strong gravitational pull just outside of black holes would create particles and quanta that move away from the black hole, thereby robbing it of energy.

VI.A      Theory of Everything

An important trend in modern theoretical physics is to find a theory of everything (TOE), in which all four of the fundamental forces are seen to be really different aspects of the same single universal force. Physicists already have unified electromagnetism and the weak nuclear force and have made progress in joining these two forces with the strong nuclear force. However, relativistic gravitation, with its geometric and mathematically complex character, poses the most difficult challenge. Einstein spent most of his later years searching for an all-encompassing theory by trying to make electromagnetism geometrical like gravitation. The modern approach is to try to make gravitation fit the pattern of the other fundamental forces. Much of this work involves looking for mathematical patterns. A TOE is difficult to test using experiments because the effects of a TOE would be important only in the rarest circumstances.

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