Space-time Distortion By Gravity

Type of physical science: Relativity

Field of study: General relativity

Space and time are linked together, and the fabric of space-time can be distorted or warped by the presence of matter. Albert Einstein's general theory of relativity provides explanations of many diverse phenomena based on the concept that gravity is the distortion of space-time.

Overview

In the English language, "space" and "time" are two overworked and ambiguous words.

"Space" can connote emptiness, volume, or a place to stow objects. When matter is present, the space is filled. The human experience of time, unlike space, is perceived to possess structure at some fundamental level. To the layperson, space is empty but time is full of activity. Time is perceived as a flow, carrying consciousness from one present moment to the next. To a physicist, however, the terms "space" and "time" denote quite different concepts: Space possesses physical properties and many levels of structure, and the flow of time is an illusion. Indeed, some theories hold that matter, rather than being located in space and time, is nothing more than disturbances in the fabric of space and time.

Although space and time can be perceived in radically different, separate ways, they are very similar mathematically. In fact, they are linked together by motion; that is, average velocity is defined as distance divided by time. By studying the motion of material objects and light signals, one learns that space and time are actually two aspects of a single, unified reality called space-time. To locate an object in space-time, one must specify the object's three spatial coordinates and the time at which it will be there. The path followed by an object moving through space-time because of motion through space, time, or both is known as its world-line.

Objects at rest or moving with a constant velocity follow straight world-lines, while the world-lines of accelerated objects are curved.

Among other properties, space has a geometric structure. After a shortest path is constructed between any two points in space, distances and angles may be defined and a self-consistent geometry may be devised. In high-school geometry, one learns that the sum of the interior angles of every triangle is 180 degrees, a theorem proved by Euclid around 300 B.C. For more than two millennia, Euclidean geometry was assumed to be the only geometry possible.

During the late nineteenth century, however, several prominent mathematicians derived several other self-consistent geometries. In these new geometries, the sum of the angles of a triangle may be more or less than 180 degrees. A physical space described by a non-Euclidean geometry is called a curved space, as opposed to "flat" Euclidean space.

Before Albert Einstein (1879-1955), non-Euclidean geometries were mere mathematical curiosities that had nothing to do with the known universe. This situation changed drastically in 1915 when Einstein proposed his general theory of relativity. This theory proposed that the four-dimensional space-time, first introduced by Einstein in his 1905 special theory of relativity, was curved by the presence of matter. Gravity appears to exist only because mass warps space-time in its vicinity. Since space is no longer "flat," Euclidean geometry fails to describe the universe accurately.

The fundamental postulate at the heart of general relativity is the principle of equivalence. It has long been known that unrestrained objects near the earth's surface will be accelerated by the force of gravity. From the time of Galileo Galilei (1564-1642) it has been known that, ignoring air resistance, all objects dropped near the earth's surface accelerate at the same constant rate regardless of their mass. This constant acceleration is termed the "acceleration of gravity." Einstein wondered why all masses should accelerate at the same rate; that is, why should acceleration and gravity be related? Einstein's insight, like many profound ideas, is remarkably simple. He proposed that acceleration and gravity appear to be related because they are, in fact, identical (the principle of equivalence).

In a closed rocket accelerating through space at 9.8 meters per second per second, there is no experiment that an astronaut can perform which distinguishes this situation from a situation in which the rocket is at rest on the earth's surface (where the acceleration of gravity is 9.8 meters per second per second). Einstein realized that, if he could express this equivalence in a mathematical form, then he could relate both gravitation and acceleration to the curvature of space-time, thereby providing an explanation of gravity. Although special relativity theory predicts that moving nonaccelerated objects will change in length and time relative to a rest frame, the object's total extent in space-time is constant. In other words, as the three-dimensional space projection becomes shorter, the time projection lengthens such that the overall extent in four-dimensional space-time remains unchanged. The changes predicted by general relativity are more profound. To an accelerated observer, the very geometry of space-time is distorted. Thus, the presence of mass will distort the fabric of space-time, as accelerations and gravitational fields are equivalent. What is experienced as gravitation is the warping of space-time because of mass.

Like all proper physical theories, general relativity does not merely predict that space-time can be distorted, it provides equations, based on the non-Euclidean geometries, which specify exactly how mass warps space-time. These Einsteinian field equations enable one to compute, in principle, the degree of curvature at any point in space around a given mass distribution. Naturally, the bending of space-time profoundly affects the world-lines of objects moving with constant velocity. As space-time warps, the world-lines warp as well, since they are constrained to move in the straightest possible paths through a curved space. As a result, when an object moves on a world-line which is bent, one says that the object has been accelerated; that is, its uniform motion has been altered by a change of either speed or direction.

Since any change in an object's motion is considered to be caused by a force, the curvature of space-time is ultimately manifested as a force. Because all objects experience the same distortion of space-time regardless of their masses, this force will indiscriminately affect all matter. The force that accelerates all masses at the same rate is gravity. In other words, gravity is more a property of the surrounding fabric of space-time than of the bodies that move through it.

Physicist John Wheeler, who contributed enormously to relativity theory, provided this description of space-time: "Space tells matter how to move and matter tells space how to curve."

From the perspective of general relativity, gravity is not to be considered as a force, but rather as geometry--a distortion of space-time.

If curved space is difficult to visualize, it would seem impossible to form any intuitive concepts of curved space-time. Fortunately, the curved space-time of general relativity can be represented by the curved space of three-dimensional, non-Euclidean geometry. In fact, the application of the equations of general relativity to the structure of the universe predicts either a negative or a positive geometry for space. The curvature of time may be considered separately.

The issue of a negative or a positive geometry becomes the pivot for debate over an "open" or a "closed" universe. The overall curvature of space is the sum of all the space warps that are caused by all the mass in the universe. If the average curvature of space is negative, then the universe is infinite, as is time. Such a universe is said to be open and unbounded. Since cosmological theory indicates that the universe is expanding, an open universe would continue to expand forever. If, on the other hand, the curvature of space is positive, then the universe is finite, or closed. This type of universe is not only bounded in space, but also has a finite existence in time. Gravitational retardation would slow the expansion of a closed universe until, ultimately, it ceased to expand. Gravity would then cause this type of universe to contract into an ever-smaller volume until the universe extinguished itself.

Einstein's equations do not reveal whether the universe is positively or negatively curved, or whether it has a finite or an infinite future. Either type of universe is possible, and only observational evidence can decide the issue. The average curvature of space depends on the total amount of matter in the universe, as the total curvature is the sum of all the warps caused by matter. If the total average mass is greater than a critical amount (about three hydrogen atoms per cubic meter), then the universe is closed. Most available evidence suggests that the universe contains less than 10 percent of this critical amount, but the issue is far from being settled. Many astronomers who feel intuitively that the universe is closed have searched for the "missing matter" necessary to verify their hypotheses.

The theory that matter distorts space-time led Einstein to make several remarkable predictions. In Euclidean geometry, the shortest distance between two points is a straight line. In a curved space, the shortest path is curved because space itself is curved. In order to gain a clearer understanding of this concept, consider that New York and Tokyo have similar latitudes but that the shortest distance between them, along the earth's surface, passes close to the North Pole. Although incomprehensible on a flat map, this fact is easily visualized on the curved surface of a globe. Similarly, a light beam traveling toward the earth from a distant star is forced to follow a curved path as it passes through the gravitational field of the sun. In 1911, Einstein predicted that starlight grazing the sun (visible only during a total eclipse) would appear to have been deflected by this warping of space-time.

The bending of light in the presence of gravitational fields is also mandated by the equivalence principle. In an accelerated reference frame, a transverse light beam would appear to deflect toward the upwardly accelerating floor. Since the equivalence principle requires that there be no distinction between acceleration and gravitational fields, light would, by comparison, have to be deflected toward a source of gravity. Although Einstein's prediction for the deviation of a light beam grazing the sun was a mere fraction of a degree, the prediction was verified by experiment in 1919.

Since space and time are intrinsically bound together, the effect of mass on the geometry of space-time means that time will be warped as well as space. Einstein predicted that time would be slowed by the warped space-time in the vicinity of matter; the greater is the concentration of mass, the greater is the effect. Thus, clocks at the earth's surface should run slightly slower than clocks at higher altitudes (by 10-12 for each vertical kilometer). This minuscule effect was measured in 1960, again verifying Einstein's predictions.

Ingenious experiments continue to test, and to support, Einstein's theory of space-time.

If the effects of matter on space-time were limited to the minute, the general theory of relativity would be no more than an intellectual curiosity. It has become apparent, however, that the universe contains fantastic objects whose forces of gravitation are so intense that they distort space-time in fascinating and bizarre ways. If a gravitational field is intense enough, then space is warped to such a degree that light passing through the region will lose all of its energy and be unable to escape. Since all light is absorbed, this region appears black to a distant observer. The slowing of time will also be dramatic, in severe cases causing time to stop completely. Objects that distort space-time to this degree are the collapsed remnants of very massive stars (greater than three solar masses). General relativity theory predicts that the formation of such objects, called black holes, is inevitable when a massive star collapses. According to present understanding, the entire mass of the star would be concentrated at the center in a single point of infinite density. Although there is no direct observational evidence for black holes, the circumstantial evidence for their existence is overwhelming. Furthermore, since Einstein's basic concepts of warped space-time have remained essentially unaltered since their inception, there is no reason to believe that the predictions of general relativity concerning phenomena observed in the vicinity of a black hole are inaccurate.

Applications

Since space and time are like two sides of the same coin, the existence of warped space-time, as predicted by Einstein's theory, has led to the possibility of several intriguing, yet theoretically possible, future applications. Two of these are "wormholes" through space and time machines.

The wormhole, like its cousin the black hole, is one of general relativity's progeny.

Both are solutions of Einstein's field equations, which occur when space-time is severely warped by a dense, massive object. Unlike a black hole, however, a wormhole has no infinite concentration of mass at its center. Rather, a wormhole has two "mouths" (openings) connected by a "throat" (passageway). Even if the passageway is short, the openings may be very far apart in space (or time). Thus, in theory, a wormhole could serve (like the shortcut a worm takes through an apple) as a cosmic shortcut, enabling the vast distances of space to be traveled rapidly without exceeding the speed of light. Unlike the black hole, however, no one has suggested how a macroscopic wormhole might form or how its passageway might be kept open. Nevertheless, physicist Kip Thorne and his associates at the California Institute of Technology performed a thought experiment which indicated that nothing in the laws of physics prohibits a traversable wormhole. In addition, they devised two possible methods for keeping a wormhole open. Thus, future engineers may be able to realize what is routine for science-fiction characters: traveling to galactic outposts by means of shortcuts through hyperspace.

Even more intriguing is the use of space-time distortions to travel backward in time. In 1974, mathematician Frank J. Tipler predicted that under certain conditions and according to equations of general relativity, space-time may be warped in such a manner that a hypothetical voyager in this region can return to his or her starting point at an earlier time. Furthermore, he showed that the necessary conditions for such a space-time warp could be created artificially.

Tipler's mathematical proof indicated that, in the vicinity of an infinitely long, massive cylinder whose surfaces rotate at one-half the speed of light, space-time is warped to such a degree that the space and time coordinates are interchanged and their roles reversed. By moving through the region surrounding this cylinder, an astronaut would move through time (as viewed from a region outside the influence of the rotating cylinder). Through a suitable choice of path, the traveler could voyage into his or her own past or future. These travels are, in theory, limited only by the lifetime of the rotating mass. Before or after its existence, there is nothing to warp space-time and, hence, no time machine.

There is no guarantee, however, that because such a region of warped space-time can exist it will exist. The engineering obstacles to constructing such a time machine would be formidable. For example, it is questionable whether the matter composing such a device could remain stable if a large object rotated in it at these incredible speeds. Even more problematic, however, is an important theoretical difficulty: Tipler's calculation was made for an infinitely long cylinder. Assuming an infinite length for a physical object is a mathematical abstraction often employed to simplify difficult calculations. Tipler speculated that the time-travel effect might also be observed around a very long but finite cylinder, although he made no calculations concerning such a cylinder.

Many scientists are convinced that time travel would not be possible near a finite rotating cylinder simply because it would be impossible to construct such a device. An infinite cylinder has no ends and thus no force is required to prevent its collapse. A finite cylinder, however, requires that the material from which it is constructed exert sufficient pressure to support the ends in order to prevent its collapse. Yet, there is an absolute maximum amount of pressure which ordinary matter can exert. Any finite cylinder existing under Tipler's required conditions would need ends constructed of gravitationally repulsive matter having negative mass in order to support it. Such a substance, if it could exist, might violate the known laws of physics.

Thus, even if travel to an earlier time is a theoretical possibility, a device capable of achieving this goal may remain forever beyond technological capabilities.

Context

In 1827, the German mathematician Carl Friedrich Gauss published a paper in which he recorded his measurements of the interior angles of a large triangle formed by three mountain peaks. His measurement was an attempt to ascertain whether space was Euclidean. The experiment was inconclusive: He obtained 180 degrees within the experimental accuracy of his measuring devices. Gauss's experiment may have been the first attempt to test the long-established assumption that the universe is best described by Euclidean geometry. By the end of the nineteenth century, two other self-consistent geometries had been devised: the geometry of negatively curved space by the Russian mathematician Nikolay Ivanovich Lobachevsky (1792-1856) and the geometry of positively curved space by the German mathematician Georg Friedrich Riemann (1826-1866). Until 1915, however, when Einstein published his general theory of relativity, most scientists assumed that non-Euclidean geometries were mathematical curiosities. In Einstein's theory, space-time is curved by the presence of mass and gravity exists only because mass gives space a non-Euclidean character.

Several months after Einstein's theory was published, the German astronomer Karl Schwarzschild found rigorously exact solutions to Einstein's field equations for two different cases: an ideal point mass and a finite spherical mass. The first case predicted that, at a certain radius from the mass point, some of the mathematical terms become infinite. This prediction indicates such an intense warping of space-time that any signal within this boundary would be unable to escape. This radius, now called the Schwarzschild radius, defines a surface (called the event horizon) such that any mass or energy within this surface is forever trapped. Most physicists of the time, Einstein included, believed that it would be impossible for any real massive object to contract to such a small volume that its mass would be contained within this surface, despite a 1939 theorem which demonstrated that this is precisely what would happen to a massive collapsing star. Thus, for the four decades between 1920 and 1960, general relativity lay dormant. Although considered to be a remarkable creation of the human intellect and a theory of fundamental importance, it was assumed to have little connection to the rest of physics and astronomy.

In the early 1960's, two events brought about a renaissance of general relativity theory.

First was the discovery of quasars, incredibly energetic radio sources which are so small that they appear to be starlike. The enormous amounts of energy pouring out of these concentrated objects forced theorists to work with the relativistic equations for collapsing superdense objects, previously solved in 1939. The second event was the 1963 discovery by Roy Kerr of a new exact solution to Einstein's equations. Kerr's solution turned out to be the unique solution for a rotating black hole, and the Schwarzschild solution for a mass point was seen to be a special case of Kerr's solution. Since matter and energy could cross the event horizon in the inward direction but nothing could escape from inside the event horizon, the term "black hole" was coined.

In addition to predicting the existence of black holes, the renaissance of general relativity caused a revolution in the use of astronomical techniques, both theoretical and observational, for understanding the origin and nature of the universe. General relativity became a central tool in the astronomical enterprise. The applications to black holes, gravity waves, and cosmology imbued the subject with a new vitality, despite the fact that its basic structure had remained unaltered since 1915. It is truly amazing that an abstract theory concerning the warping of space-time, invented without experimental guidance as a product of Einstein's intellect, turned out to be so correct and so useful.

Principal terms

ACCELERATION: the change in the velocity of an object divided by the time required for the change to occur; commonly expressed in units of meters per second per second

ACCELERATION OF GRAVITY: the average acceleration of an object which is in free-fall near the earth's surface, ignoring air resistance; this approximately constant acceleration has a value of 9.8 meters per second per second

CURVED SPACE: a space which does not obey the rules of Euclidean geometry; may be positively curved or negatively curved space

FLAT SPACE: a space of any number of dimensions which obeys the rules of Euclidean geometry

SPACE-TIME: a four-dimensional "space" consisting of the three spatial dimensions and time as a fourth dimension

WORLD-LINE: the graph of the motion of an object through space-time; objects moving with constant velocities have straight world-lines, while accelerated motion is represented by curved world-lines

Bibliography

Davies, P. C. W. SPACE AND TIME IN THE MODERN UNIVERSE. New York: Cambridge University Press, 1977. Written in an authoritative yet lucid style, this book explores the changing ideas of space and time and their applications in astronomical and cosmological scenarios.

Gribbin, John. SPACEWARPS. New York: Delacorte Press, 1983. Describes the universal implications of Albert Einstein's general theory of relativity in physics and astronomy.

Gribbin, John. TIMEWARPS. New York: Delacorte Press, 1979. A clear and imaginative examination of some of the dramatic questions raised by the new concepts of time.

Pagels, Heinz. THE COSMIC CODE. New York: Simon & Schuster, 1982. Although only one chapter is devoted to general relativity, it is a particularly lucid summary of the main principles and how Albert Einstein arrived at these ideas.

Rucker, Rudolf. GEOMETRY, RELATIVITY, AND THE FOURTH DIMENSION. New York: Dover, 1977. A highly readable and amusing exposition of four-dimensional space-time and the structure of the universe.

Will, Clifford. WAS EINSTEIN RIGHT? PUTTING GENERAL RELATIVITY TO THE TEST. New York: Basic Books, 1986. The renaissance of relativity is described with splendid clarity by one of the professional participants. Observations and theories that test the experimental basis for general relativity are presented without mathematics.

The Evolution of the Universe