Relativity

I INTRODUCTION

Relativity, theory, developed in the early 20th century, which originally attempted to account for certain anomalies in the concept of relative motion, but which in its ramifications has developed into one of the most important basic concepts in physical science (see Physics). The theory of relativity, developed primarily by German American physicist Albert Einstein, is the basis for later demonstration by physicists of the essential unity of matter and energy, of space and time, and of the forces of gravity and acceleration (see Acceleration; Energy; Gravitation).

II CLASSICAL PHYSICS

Physical laws generally accepted by scientists before the development of the theory of relativity, now called classical laws, were based on the principles of mechanics enunciated late in the 17th century by the English mathematician and physicist Isaac Newton. Newtonian mechanics and relativistic mechanics differ in fundamental assumptions and mathematical development, but in most cases do not differ appreciably in net results; the behavior of a billiard ball when struck by another billiard ball, for example, may be predicted by mathematical calculations based on either type of mechanics and produce approximately identical results. Inasmuch as the classical mathematics is enormously simpler than the relativistic, the former is the preferred basis for such a calculation. In cases of high speeds, however, assuming that one of the billiard balls was moving at a speed approaching that of light, the two theories would predict entirely different types of behavior, and scientists today are quite certain that the relativistic predictions would be verified and the classical predictions would be proved incorrect.

In general, the difference between two predictions on the behavior of any moving object involves a factor discovered by the Dutch physicist Hendrik Antoon Lorentz, and the Irish physicist George Francis FitzGerald late in the 19th century. This factor is generally represented by the Greek letter  (beta) and is determined by the velocity of the object in accordance with the following equation:

in which v is the velocity of the object and c is the velocity of light (see Light). The beta factor does not differ essentially from unity for any velocity that is ordinarily encountered; the highest velocity encountered in ordinary ballistics, for example, is about 1.6 km/sec (about 1 mi/sec), the highest velocity obtainable by a rocket propelled by ordinary chemicals is a few times that, and the velocity of the earth as it moves around the sun is about 29 km/sec (about 18 mi/sec); at the last-named speed, the value of beta differs from unity by only five billionths. Thus, for ordinary terrestrial phenomena, the relativistic corrections are of little importance. When velocities are very large, however, as is sometimes the case in astronomical phenomena, relativistic corrections become significant. Similarly, relativity is important in calculating very large distances or very large aggregations of matter. As the quantum theory applies to the very small, so the relativity theory applies to the very large.

Until 1887 no flaw had appeared in the rapidly developing body of classical physics. In that year, the Michelson-Morley experiment, named after the American physicist Albert Michelson and the American chemist Edward Williams Morley, was performed. It was an attempt to determine the rate of the motion of the earth through the ether, a hypothetical substance that was thought to transmit electromagnetic radiation, including light, and was assumed to permeate all space. If the sun is at absolute rest in space, then the earth must have a constant velocity of 29 km/sec (18 mi/sec), caused by its revolution about the sun; if the sun and the entire solar system are moving through space, however, the constantly changing direction of the earth's orbital velocity will cause this value of the earth's motion to be added to the velocity of the sun at certain times of the year and subtracted from it at others. The result of the experiment was entirely unexpected and inexplicable; the apparent velocity of the earth through this hypothetical ether was zero at all times of the year.

What the Michelson-Morley experiment actually measured was the velocity of light through space in two different directions. If a ray of light is moving through space at 300,000 km/sec (186,000 mi/sec), and an observer is moving in the same direction at 29 km/sec (18 mi/sec), then the light should move past the observer at the rate of 299,971 km/sec (185,982 mi/sec); if the observer is moving in the opposite direction, the light should move past the observer at 300,029 km/sec (186,018 mi/sec). It was this difference that the Michelson-Morley experiment failed to detect. This failure could not be explained on the hypothesis that the passage of light is not affected by the motion of the earth, because such an effect had been observed in the phenomenon of the aberration of light; see Interference; Interferometer; Wave Motion.

In the 1890s FitzGerald and Lorentz advanced the hypothesis that when any object moves through space, its length in the direction of its motion is altered by the factor beta. The negative result of the Michelson-Morley experiment was explained by the assumption that the light actually traversed a shorter distance in the same time (that is, moved more slowly), but that this effect was masked because the distance was measured of necessity by some mechanical device which also underwent the same shortening, just as when an object 2 m long is measured with a 3-m tape measure which has shrunk to 2 m, the object will appear to be 3 m in length. Thus, in the Michelson-Morley experiment, the distance which light traveled in 1 sec appeared to be 300,000 km (186,000 mi) regardless of how fast the light actually traveled. The Lorentz-FitzGerald contraction was considered by scientists to be an unsatisfactory hypothesis because it could not be applied to any problem in which measurements of absolute motion could be made.

III SPECIAL THEORY OF RELATIVITY

In 1905, Einstein published the first of two important papers on the theory of relativity, in which he dismissed the problem of absolute motion by denying its existence. According to Einstein, no particular object in the universe is suitable as an absolute frame of reference that is at rest with respect to space. Any object (such as the center of the solar system) is a suitable frame of reference, and the motion of any object can be referred to that frame. Thus, it is equally correct to say that a train moves past the station, or that the station moves past the train. This example is not as unreasonable as it seems at first sight, for the station is also moving, due to the motion of the earth on its axis and its revolution around the sun. All motion is relative, according to Einstein. None of Einstein's basic assumptions was revolutionary; Newton had previously stated “absolute rest cannot be determined from the position of bodies in our regions.” Einstein stated the relative rate of motion between any observer and any ray of light is always the same, 300,000 km/sec (186,000 mi/sec), and thus two observers, moving relative to one another even at a speed of 160,000 km/sec (100,000 mi/sec), each measuring the velocity of the same ray of light, would both find it to be moving at 300,000 km/sec (186,000 mi/sec), and this apparently anomalous result was proved by the Michelson-Morley experiment. According to classical physics, one of the two observers was at rest, and the other made an error in measurement because of the Lorentz-FitzGerald contraction of his apparatus; according to Einstein, both observers had an equal right to consider themselves at rest, and neither had made any error in measurement. Each observer used a system of coordinates as the frame of reference for measurements, and these coordinates could be transformed one into the other by a mathematical manipulation. The equations for this transformation, known as the Lorentz transformation equations, were adopted by Einstein, but he gave them an entirely new interpretation. The speed of light is invariant in any such transformation.

According to the relativistic transformation, not only would lengths in the line of a moving object be altered but also time and mass. A clock in motion relative to an observer would seem to be slowed down, and any material object would seem to increase in mass, both by the beta factor. The electron, which had just been discovered, provided a means of testing the last assumption. Electrons emitted from radioactive substances have speeds close to the speed of light, so that the value of beta, for example, might be as large as 0.5, and the mass of the electron doubled. The mass of a rapidly moving electron could be easily determined by measuring the curvature produced in its path by a magnetic field; the heavier the electron, the greater its inertia and the less the curvature produced by a given strength of field (see Magnetism). Experimentation dramatically confirmed Einstein's prediction; the electron increased in mass by exactly the amount he predicted. Thus, the kinetic energy of the accelerated electron had been converted into mass in accordance with the formula E=mc² (see Atom; Nuclear Energy). Einstein's theory was also verified by experiments on the velocity of light in moving water and on magnetic forces in moving substances.

The fundamental hypothesis on which Einstein's theory was based was the nonexistence of absolute rest in the universe. Einstein postulated that two observers moving relative to one another at a constant velocity would observe identically the phenomena of nature. One of these observers, however, might record two events on distant stars as having occurred simultaneously, while the other observer would find that one had occurred before the other; this disparity is not a real objection to the theory of relativity, because according to that theory simultaneity does not exist for distant events. In other words, it is not possible to specify uniquely the time when an event happens without reference to the place where it happens. Every particle or object in the universe is described by a so-called world line that describes its position in time and space. If two or more world lines intersect, an event or occurrence takes place; if the world line of a particle does not intersect any other world line, nothing has happened to it, and it is neither important nor meaningful to determine the location of the particle at any given instant. The “distance” or “interval” between any two events can be accurately described by means of a combination of space and time, but not by either of these separately. The space-time of four dimensions (three for space and one for time) in which all events in the universe occur is called the space-time continuum.

All of the above statements are consequences of special relativity, the name given to the theory developed by Einstein in 1905 as a result of his consideration of objects moving relative to one another with constant velocity.

IV GENERAL THEORY OF RELATIVITY

In 1915 Einstein developed the general theory of relativity in which he considered objects accelerated with respect to one another. He developed this theory to explain apparent conflicts between the laws of relativity and the law of gravity. To resolve these conflicts he developed an entirely new approach to the concept of gravity, based on the principle of equivalence.

The principle of equivalence holds that forces produced by gravity are in every way equivalent to forces produced by acceleration, so that it is theoretically impossible to distinguish between gravitational and accelerational forces by experiment. In the theory of special relativity, Einstein had stated that a person in a closed car rolling on an absolutely smooth railroad track could not determine by any conceivable experiment whether he was at rest or in uniform motion. In general relativity he stated that if the car were speeded up or slowed down or driven around a curve, the occupant could not tell whether the forces so produced were due to gravitation or whether they were acceleration forces brought into play by pressure on the accelerator or on the brake or by turning the car sharply to the right or left.

Acceleration is defined as the rate of change of velocity. Consider an astronaut standing in a stationary rocket. Because of gravity his or her feet are pressed against the floor of the rocket with a force equal to the person's weight, w. If the same rocket is in outer space, far from any other object and not influenced by gravity, the astronaut is again being pressed against the floor if the rocket is accelerating, and if the acceleration is 9.8 m/sec² (32 ft/sec²) (the acceleration of gravity at the surface of the earth), the force with which the astronaut is pressed against the floor is again equal to w. Without looking out of the window, the astronaut would have no way of telling whether the rocket was at rest on the earth or accelerating in outer space. The force due to acceleration is in no way distinguishable from the force due to gravity. According to Einstein's theory, Newton's law of gravitation is an unnecessary hypothesis; Einstein attributes all forces, both gravitational and those associated with acceleration, to the effects of acceleration. Thus, when the rocket is standing still on the surface of the earth, it is attracted toward the center of the earth. Einstein states that this phenomenon of attraction is attributable to an acceleration of the rocket. In three-dimensional space, the rocket is stationary and therefore is not accelerated; but in four-dimensional space-time, the rocket is in motion along its world line. According to Einstein, the world line is curved, because of the curvature of the continuum in the neighborhood of the earth.

Thus, Newton's hypothesis that every object attracts every other object in direct proportion to its mass is replaced by the relativistic hypothesis that the continuum is curved in the neighborhood of massive objects. Einstein's law of gravity states simply that the world line of every object is a geodesic in the continuum. A geodesic is the shortest distance between two points, but in curved space it is not generally a straight line. In the same way, geodesics on the surface of the earth are great circles, which are not straight lines on any ordinary map. See Geometry; Navigation.

V CONFIRMATION AND MODIFICATION

As in the cases mentioned above, classical and relativistic predictions are generally virtually identical, but relativistic mathematics is more complex. The famous apocryphal statement that only ten people in the world understood Einstein's theory referred to the complex tensor algebra and Riemannian geometry of general relativity; by comparison, special relativity can be understood by any college student who has studied elementary calculus.

General relativity theory has been confirmed in a number of ways since it was introduced. For example, it predicts that the world line of a ray of light will be curved in the immediate vicinity of a massive object such as the sun. To verify this prediction, scientists first chose to observe a star appearing very close to the edge of the sun. Such observations cannot normally be made, because the brightness of the sun obscures a nearby star. During a total eclipse, however, stars can be observed and their positions accurately measured even when they appear quite close to the edge of the sun. Expeditions were sent out to observe the eclipses of 1919 and 1922 and made such observations. The apparent positions of the stars were then compared with their apparent positions some months later, when they appeared at night far from the sun. Einstein predicted an apparent shift in position of 1.745 seconds of arc for a star at the very edge of the sun, with progressively smaller shifts for more distant stars. The expeditions that were sent to study the eclipses verified these predictions. In recent years, comparable tests were made of radio-wave deflections from distant quasars, using radio-telescope interferometers (see Radio Astronomy). The tests yielded results that agreed, to within 1 percent, with the values predicted by general relativity.

Another confirmation of general relativity involves the perihelion of the planet Mercury. For many years it had been known that the perihelion (the point at which Mercury passes closest to the sun) revolves about the sun at the rate of once in 3 million years, and that part of this perihelion motion is completely inexplicable by classical theories. The theory of relativity, however, does predict this part of the motion, and recent radar measurements of Mercury's orbit have confirmed this agreement to within about 0.5 percent.

Yet another phenomenon predicted by general relativity is the time-delay effect, in which signals sent past the sun to a planet or spacecraft on the far side of the sun experience a small delay, when relayed back, compared to the time of return as indicated by classical theory. Although the time intervals involved are very small, various tests made by means of planetary probes have provided values quite close to those predicted by general relativity (see Radar Astronomy). Numerous other tests of the theory could also be described, and thus far they have served to confirm it.

The general theory of relativity predicts that a massive rotating body will drag space and time around with it as it moves. This effect, called frame dragging, is more noticeable if the object is very massive and very dense. In 1997 a group of Italian astronomers announced that they had detected frame dragging around very dense, rapidly spinning astronomical objects called neutron stars. The astronomers found evidence of frame dragging by examining radiation emitted when the gravitational pull of a dense neutron star sucks matter onto its surface. This radiation showed slight differences from the radiation that was predicted by classical physics.

In 1998 another group of astronomers from the United States and Europe announced that the orbits of some artificial satellites around the earth showed the effects of frame dragging. The earth is much lighter and less dense than a neutron star, so the effects of the earth’s frame dragging are much more subtle than those of the neutron star’s frame dragging. The astronomers found that the orbits of two Italian satellites seem to shift about 2 m (about 7 ft) in the direction of the earth’s rotation every year. The launch of the U.S. spacecraft Gravity Probe B in 2000 should provide even more evidence of frame dragging around the earth and other bodies.

VI LATER OBSERVATIONS

Since 1915 the theory of relativity has undergone much development and expansion by Einstein and by the British astronomers James Hopwood Jeans, Arthur Stanley Eddington, and Edward Arthur Milne, the Dutch astronomer Willem de Sitter, and the German American mathematician Hermann Weyl. Much of their work has been devoted to an effort to extend the theory of relativity to include electromagnetic phenomena (see Unified Field Theory). Although some progress has been made in this area, these efforts have been marked thus far by less success. No complete development of this application of the theory has yet been generally accepted. See Elementary Particles.

The astronomers mentioned above also devoted much effort to developing the cosmological consequences of the theory of relativity. Within the framework of the axioms laid down by Einstein, many lines of development are possible. Space, for example, is curved, and its exact degree of curvature in the neighborhood of heavy bodies is known, but its curvature in empty space is not certain. Moreover, scientists disagree on whether it is a closed curve (such as a sphere) or an open curve (such as a cylinder or a bowl with sides of infinite height). The theory of relativity leads to the possibility that the universe is expanding; this is the most likely theoretical explanation of the experimentally observed fact that the spectral lines of all distant nebulae are shifted to the red; on the other hand the expanding-universe theory also supplies other possible explanations. The latter theory makes it reasonable to assume that the past history of the universe is finite, but it also leads to alternative possibilities. See Cosmology.

Much of the later work on relativity was devoted to creating a workable relativistic quantum mechanics. A relativistic electron theory was developed in 1928 by the British mathematician and physicist Paul Dirac, and subsequently a satisfactory quantized field theory, called quantum electrodynamics, was evolved, unifying the concepts of relativity and quantum theory in relation of the interaction between electrons, positrons, and electromagnetic radiation. In recent years, the work of the British physicist Stephen Hawking has been devoted to an attempted full integration of quantum mechanics with relativity theory.

Contributed By: Lawrence A. Bornstein

Microsoft ® Encarta ® Reference Library 2003. © 1993-2002 Microsoft Corporation. All rights reserved.

relativity

in physics, the problem of how physical laws and measurements change when considered by observers in various states of motion. Thus, relativity is concerned with measurements made by different observers moving relative to one another. In classical physics it was assumed that all observers anywhere in the universe, whether moving or not, obtained identical measurements of space and time intervals. According to relativity theory, this is not so, but their results depend on their relative motions.

There are actually two distinct theories of relativity known in physics, one called the special theory of relativity, the other the general theory of relativity. Albert Einstein proposed the first in 1905, the second in 1916. Whereas the special theory of relativity is concerned primarily with electric and magnetic phenomena and with their propagation in space and time, the general theory of relativity was developed primarily in order to deal with gravitation. Both theories centre on new approaches to space and time, approaches that differ profoundly from those useful in everyday life; but relativistic notions of space and time are inextricably woven into any contemporary interpretation of physical phenomena ranging from the atom to the universe as a whole.

This article will set forth the principal ideas comprising both special and general relativity. It will also deal with some implications and applications of these theories. For treatment of the motion of relativistic bodies, see the article relativistic mechanics.

The special theory of relativity

Historical background

Classical physics owes its definitive formulation to the British scientist Sir Isaac Newton. According to Newton, when one physical body influences another body, this influence results in a change of that body's state of motion, its velocity; that is to say, the force exerted by one particle on another results in the latter's changing the direction of its motion, the magnitude of its speed, or both. Conversely, in the absence of such external influences, a particle will continue to move in one unchanging direction and at a constant rate of speed. This statement, Newton's first law of motion, is known as the law of inertia.

As motion of a particle can be described only in relation to some agreed frame of reference, Newton's law of inertia may also be stated as the assertion that there exist frames of reference (so-called inertial frames of reference) with respect to which particles not subject to external forces move at constant speed in an unvarying direction. Ordinarily, all laws of classical mechanics are understood to hold with respect to such inertial frames of reference. Each frame of reference may be thought of as realized by a grid of surveyor's rods permitting the spatial fixation of any event, along with a clock describing the time of its occurrence.

According to Newton, any two inertial frames of reference are related to each other in that the two respective grids of rods move relative to each other only linearly and uniformly (with constant direction and speed) and without rotation, whereas the respective clocks differ from each other at most by a constant amount (as do the clocks adjusted to two different time zones on Earth) but go at the same rate. Except for the arbitrary choice of such a constant time difference, the time appropriate to various inertial frames of reference then is the same: If a certain physical process takes, say, one hour as determined in one inertial frame of reference, it will take precisely one hour with respect to any other inertial frame; and if two events are observed to take place simultaneously by an observer attached to one inertial frame, they will appear simultaneous to all other inertial observers. This universality of time and time determinations is usually referred to as the absolute character of time. The idea that a universal time can be used indiscriminately by all, irrespective of their varying states of motion--that is, by a person at rest at his home, by the driver of an automobile, and by the passenger aboard an airplane--is so deeply ingrained in most people that they do not even conceive of alternatives. It was only at the turn of the 20th century that the absolute character of time was called into question as the result of a number of ingenious experiments described below.

As long as the building blocks of the physical universe were thought to be particles and systems of particles that interacted with each other across empty space in accordance with the principles enunciated by Newton, there was no reason to doubt the validity of the space-time notions just sketched. This view of nature was first placed in doubt in the 19th century by the discoveries of a Danish physicist, Hans Christian Ørsted, the English scientist Michael Faraday, and the theoretical work of the Scottish-born physicist James Clerk Maxwell, all concerned with electric and magnetic phenomena. Electrically charged bodies and magnets do not affect each other directly over large distances, but they do affect one another by way of the so-called electromagnetic field, a state of tension spreading throughout space at a high but finite rate, which amounts to a speed of propagation of approximately 186,000 miles (300,000 kilometres) per second. As this value is the same as the known speed of light in empty space, Maxwell hypothesized that light itself is a species of electromagnetic disturbance; his guess has been confirmed experimentally, first by the production of lightlike waves by entirely electric and magnetic means in the laboratory by a German physicist, Heinrich Hertz, in the late 19th century.

Both Maxwell and Hertz were puzzled and profoundly disturbed by the question of what might be the carrier of the electric and magnetic fields in regions free of any known matter. Up to their time, the only fields and waves known to spread at a finite rate had been elastic waves, which appear to the senses as sound and which occur at low frequencies as the shocks of earthquakes, and surface waves, such as water waves on lakes and seas. Maxwell called the mysterious carrier of electromagnetic waves the ether, thereby reviving notions going back to antiquity. He attempted to endow his ether with properties that would account for the known properties of electromagnetic waves, but he was never entirely successful. The ether hypothesis, however, led two U.S. scientists, Albert Abraham Michelson and Edward Williams Morley, to conceive of an experiment (1887) intended to measure the motion of the ether on the surface of the Earth in their laboratory. On the reasonable hypothesis that the Earth is not the pivot of the whole universe, they argued that the motion of the Earth relative to the ether should result in slight variations in the observed speed of light (relative to the Earth and to the instruments of a laboratory) travelling in different directions. The measurement of the speed of light requires but one clock, if, by use of a mirror, a pencil of light is made to travel back and forth so that its speed is measured by clocking the total time elapsed in a round trip at one site; such an arrangement obviates the need for synchronizing two clocks at the ends of a one-way trip. Finally, if one is concerned with variations in the speed of light, rather than with an absolute determination of that speed itself, then it suffices to compare with each other round-trip-travel times along two tracks at right angles to each other, and that is essentially what Michelson and Morley did. To avoid the use of a clock altogether, they compared travel times in terms of the numbers of wavelengths travelled, by making the beams travelling on the two distinct tracks interfere optically with each other. (If the waves meet at a point when both are in the same phase--e.g., both at their peak--the result is visible as the sum of the two in amplitude; if the peak of one coincides with the trough of the other, they cancel each other and no light is visible. Since the wavelengths are known, the relative positions of the peaks give an exact measure of how far one wave has advanced with respect to the other.) This highly precise experiment, repeated many times with ever-improved instrumental techniques, has consistently led to the result that the speed of light relative to the laboratory is the same in all directions, regardless of the time of the day, the time of the year, and the elevation of the laboratory above sea level.

The special theory of relativity resulted from the acceptance of this experimental finding. If an Earth-bound observer could not detect the motion of the Earth through the ether, then, it was felt, probably any observer, regardless of his state of motion, would find the speed of light the same in all directions.

Relativity of space and time

An Irish and a Dutch physicist, George Francis FitzGerald and Hendrik Antoon Lorentz, independently showed that the negative outcome of Michelson's and Morley's experiment could be reconciled with the notion that the Earth is travelling through the ether, if one hypothesizes that any body travelling through the ether is foreshortened in the direction of travel (though its dimensions at right angles to the motion remain undisturbed) by a ratio that increases with increasing speed. If denotes the speed of the body relative to the ether, and c is the speed of light, that ratio equals the quantity (1 - ²/c²)^1/2. At ordinary speeds, c is so much greater than that the fraction, practically speaking, is zero, and the ratio becomes 1, which is 1; i.e., the foreshortening is nil; as approaches c, however, the fraction becomes significant. The travelling body would be flattened completely if its speed through the ether should ever reach that of light.

Suppose, now, that the variations in the speed of light were to be determined not by interference but by means of an exceedingly accurate clock and assume further that in such a modified experiment (whose actual performance is precluded at present, because even the best atomic clocks available do not possess the requisite accuracy) the motion through the ether were still imperceptible, then, Lorentz showed, one would have to conclude that all clocks moving through the ether are slowed down compared to clocks at rest in the ether, again by the factor (1 - ²/c²)^1/2. Thus, all rods and all clocks would be modified systematically, regardless of materials and construction design, whenever they were moving relative to the ether. Accordingly, for theoretical analysis, one would have to distinguish between "apparent" and "true" space and time measurements, with the further proviso that "true" dimensions and "true" times could never be determined by any experimental procedure.

Conceptually, this was an unsatisfactory situation, which was resolved by Albert Einstein in 1905. Einstein realized that the key concept, on which all comparisons between differently moving observers and frames of reference depended, is the notion of universal, or absolute, simultaneity; that is to say, the proposition that two events that appear simultaneous to any one observer will also be judged to take place at the same time by all other observers. This appears to be a straightforward proposition, provided that knowledge of distant events can be obtained practically instantaneously. Actually, however, there is no known method of signalling faster than by means of light or radio waves or any other electromagnetic radiation, all of which travel at the same rate, c.

Suppose, now, that someone on Earth observes two events, say two supernovae (suddenly erupting very bright stars) appearing in different parts of the sky. Nothing can be said about whether these two supernovae emerged simultaneously or not from merely noting their appearance in the sky; it is necessary to know also their respective distances from the observer, which typically may amount to several hundred or several thousand light-years (one light-year, the distance light moves in one year, equals approximately 5.88 10¹² miles, or 9.46 10¹² kilometres). By the time one sees the eruption of a supernova, it has in actuality faded back into invisibility hundreds of years ago. Applying this simple idea to the observations and measurements made by different observers of the same events, Einstein demonstrated that if each observer applied the same method of analysis to his own data, then events that appeared simultaneous to one would appear to have taken place at different times to observers in different states of motion. Thus, it is necessary to speak of relativity of simultaneity.

Once this theoretical deduction is accepted, the findings of FitzGerald and Lorentz lend themselves to a new interpretation. Whenever two observers are associated with two distinct inertial frames of inference in relative motion to each other, their determinations of time intervals and of distances between events will disagree systematically, without one being "right" and the other "wrong." Nor can it be established that one of them is at rest relative to the ether, the other in motion. In fact, if they compare their respective clocks, each will find that his own clock will be faster than the other; if they compare their respective measuring rods (in the direction of mutual motion), each will find the other's rod foreshortened. The speed of light will be found to equal the same value, c = 186,000 miles per second, relative to every inertial frame of reference and in all directions. The status of Maxwell's ether is thereby cast in doubt, as its state of motion cannot be ascertained by any conceivable experiment. Consequently, the whole notion of an ether as the carrier of electromagnetic phenomena has been eliminated in contemporary physics.

The mathematical equations that relate space and time measurements of one observer to those of another, moving observer are known as Lorentz transformations. If the relative motion is measured along the x-axis and if its magnitude is , these expressions are:

Consequences

The limiting character of the speed of light

As the speed of one inertial frame of reference relative to another is increased, its rods appear increasingly foreshortened and its clocks more and more slowed down. As this relative speed approaches c, both of these effects increase indefinitely. The relative speed of the two frames cannot exceed c if light and other electromagnetic phenomena are to travel at the speed c in all directions when viewed from either frame of reference. Hence the special theory of relativity forecloses relative speeds of frames of reference greater than c. As an inertial frame of reference can be associated with any material object in uniform nonrotational motion, it follows that no material object can travel at a rate of speed exceeding c.

This conclusion is self-consistent only because under the Lorentz transformations the velocity of a body with respect to one inertial frame of reference is related to its velocity with respect to another frame not by the Newtonian rule that the difference in velocities equals the relative velocity between the two frames but by a more involved formula, which takes into account the changes in scale length, in clock time, and in simultaneity. If all velocities involved have the same direction, then the velocity (see Figure 1

Figure 1: Velocities of the same body in two frames of reference (see text).

) in one frame, u, is related to the velocity in the other frame, u', by the expression stating that u' equals the sum of u and divided by 1 plus the product of u and divided by the square of c:

As long as neither u nor exceeds the speed of light, c, u' also will be less than c.

Variable mass

The mass of a material body is a measure of its resistance to a change in its state of motion caused by a given force. The larger the mass, the smaller the acceleration. If a material body is already moving at a speed approaching the speed of light, it must offer increasing resistance to any further acceleration so as not to cross the threshold of c. Hence the special theory of relativity leads to the conclusion that the mass of a moving body m is related to the mass that it would have if at rest, m₀, by a formula in which m equals m₀ divided by the square root of one minus the fraction ²/c²:

This changing value of the mass of the moving body, m, is called the relativistic mass. As approaches c, the figure within the parentheses approaches zero and the mass m becomes infinitely large.

The relativistic mass formula may be interpreted as indicating that the relativistic mass of a body exceeds its rest mass m₀ by an amount that equals its kinetic energy E, divided by c²: m - m₀ = E/c². Hence the hypothesis that generally the energy is c² times the mass, or E = mc², and that energy and mass are, in fact, equivalent physical concepts, differing only by the choice of their units. This hypothesis has been verified experimentally, in that all massive particles have been converted into forms of energy (for instance, gamma radiation) and conversely have been created out of pure energy. It was in part the recognition of this relationship that led to research out of which grew the technology of nuclear fission and fusion.

Invariant intervals

Data on pure time intervals obtained with respect to two relatively moving inertial frames of reference will differ and so will data on spatial distances. It is possible, however, to form from time intervals plus distances a single expression that will have the same value with respect to all inertial frames of reference. If the time interval between two distant events be denoted by T and their distance from each other by L, an expression involving a quantity symbolized by can be derived in which squared equals the square of the time interval minus the fraction of distance squared over speed of light squared: ² = T² - L²/c². This will have the same value as T² - L²/c², with T and L having been obtained in another inertial frame of reference. If ² is positive, then is called the invariant (timelike) interval between the two events. If ² is negative, then the expression , derived from the above as ² = L² - c²T², will be called the invariant (spacelike) interval.

The invariant interval between two instants in the history of one physical system equals the ordinary time lapse T measured by means of a clock at rest relative to that physical system, because, in such a comoving frame of reference, L vanishes. That is why such an invariant (timelike) interval is also referred to as the "proper time" elapsed between the two instants. Any clock will read its own proper time.

The "twin paradox"

Given an inertial frame of reference and two similar material systems ("twins")--for instance, two atomic clocks of identical design--suppose that one of these clocks remains permanently at rest in the given frame, whereas the other clock is moved at a high speed first in one direction away from the first clock and subsequently in the opposite direction until the two clocks are again close to each other. According to the Lorentz transformation, the second clock has been slower than the first throughout its journey, and hence it shows a smaller lapse of time than the clock that has remained at rest. By reading the clocks, one can then tell which clock has remained at rest, which one has moved. This difference in behaviour of the two clocks has been called the clock paradox or the twin paradox.

The "paradox" supposedly consists of a violation of the principle of relativity, according to which no asymmetric distinctions exist between different inertial frames of reference. The fallacy of this argument lies in the fact that no inertial frame of reference is associated with the second clock, as it cannot have moved free of acceleration throughout its journey: at least once its velocity (i.e., the direction of its motion) must have been changed drastically, so as to enable it ever to return to its mate. Hence no violation of the principle of relativity; no paradox is involved. Various experiments on moving particles and atoms have indeed confirmed the predictions of the theory.

Four-dimensional space-time

The German mathematical physicist Hermann Minkowski pointed out that the invariant interval between two events has some of the properties of the distance in Euclidean geometry. Based on Euclidean geometry, the Cartesian coordinate system is designed to identify any point (event) in space by its reference to three mutually perpendicular lines or axes meeting at an arbitrary point of origin. The distance s between two events, in accordance with Pythagoras' theorem, in any Cartesian (rectilinear) coordinate system is obtained by taking the square root of the sum of the squares of coordinate distances, s² = x² + y² + z², and its value is independent of the choice of coordinate system, though the values of x, y, and z are not. The invariant interval, similarly, is the square root of a sum and difference of squares of intervals of both space and time. Accordingly, Minkowski suggested that space and time should be thought of as comprising a single four-dimensional continuum, space-time, often also referred to as the Minkowski universe. Events, localized as regards both space and time, are the natural analogues of points in ordinary three-dimensional geometry; in the history of one particle, its proper time resembles the arc length of a curve in three-space.

In Minkowski's space-time the invariant interval may be either timelike or spacelike. If L² - c²T² for two events happens to be zero, the invariant interval is neither, but null, or lightlike, as a light signal emanating from the earlier of the two events may just pass the second as the latter occurs. By contrast, in ordinary geometry the distance between two points, s, vanishes only if the two points coincide. To this extent the analogy between space-time and ordinary space is imperfect.

Minkowski's four-dimensional, geometric approach to relativity appears to add to the original physical concepts of relativity mostly a new terminology but not much else. Nevertheless, for the further conceptual development of relativity Minkowski's contribution has been of inestimable value.

The general theory of relativity

Physical origins

The general theory of relativity derives its origin from the need to extend the new space and time concepts of the special theory of relativity from the domain of electric and magnetic phenomena to all of physics and, particularly, to the theory of gravitation. As space and time relations underlie all physical phenomena, it is conceptually intolerable to have to use mutually contradictory notions of space and time in dealing with different kinds of interactions, particularly in view of the fact that the same particles may interact with each other in several different ways--electromagnetically, gravitationally, and by way of so-called nuclear forces.

Newton's explanation of gravitational interactions must be considered one of the most successful physical theories of all time. It accounts for the motions of all the constituents of the solar system with uncanny accuracy, permitting, for instance, the prediction of eclipses hundreds of years ahead. But Newton's theory visualizes the gravitational pull that the Sun exerts on the planets and the pull that the planets in turn exert on their moons and on each other as taking place instantaneously over the vast distances of interplanetary space, whereas according to relativistic notions of space and time any and all interactions cannot spread faster than the speed of light. The difference may be unimportant, for practical reasons, as all of the members of the solar system move at relative speeds far less than ¹/_1,000 of the speed of light; nevertheless, relativistic space-time and Newton's instantaneous action at a distance are fundamentally incompatible. Hence Einstein set out to develop a theory of gravitation that would be consistent with relativity.

Proceeding on the basis of the experience gained from Maxwell's theory of the electric field, Einstein postulated the existence of a gravitational field that propagates at the speed of light, c, and that will mediate an attraction as closely as possible equal to the attraction obtained from Newton's theory. From the outset it was clear that mathematically a field theory of gravitation would be more involved than that of electricity and magnetism. Whereas the sources of the electric field, the electric charges of particles, have values independent of the state of motion of the instruments by which these charges are measured, the source of the gravitational field, the mass of a particle, varies with the speed of the particle relative to the frame of reference in which it is determined and hence will have different values in different frames of reference. This complicating factor introduces into the task of constructing a relativistic theory of the gravitational field a measure of ambiguity, which Einstein resolved eventually by invoking the principle of equivalence.

The principle of equivalence

Everyday experience indicates that in a given field of gravity, such as the field caused by the Earth, the greater the mass of a body the greater the force acting on it. That is to say, the more massive a body the more effectively will it tend to fall toward the Earth; in fact, in order to determine the mass of a body one weighs it--that is to say, one really measures the force by which it is attracted to the Earth, whereas the mass is properly defined as the body's resistance to acceleration. Newton noted that the ratio of the attractive force to a body's mass in a given field is the same for all bodies, irrespective of their chemical constitution and other characteristics, and that they all undergo the same acceleration in free fall; this common rate of acceleration on the surface of the Earth amounts to an increase in speed by approximately 32 feet (about 9.8 metres) per second every second.

This common rate of gravitationally caused acceleration is illustrated dramatically in space travel during periods of coasting. The vehicle, the astronauts, and all other objects within the space capsule undergo the same acceleration, hence no acceleration relative to each other. The result is apparent weightlessness: no force holds the astronaut to the floor of his cabin or a liquid in an open container. To this extent, the behaviour of objects within the freely coasting space capsule is indistinguishable from the condition that would be encountered if the space capsule were outside all gravitational fields in interstellar space and moved in accordance with the law of inertia. Conversely, if a space capsule were to be accelerated upward by its rocket engines in the absence of gravitation, all objects inside would behave exactly as if the capsule were at rest but in a gravitational field. The principle of equivalence states formally the equivalence, in terms of local experiments, of gravitational forces and reactions to an accelerated noninertial frame of reference (e.g., the capsule while the rockets are being fired) and the equivalence between inertial frames of reference and local freely falling frames of reference. Of course, the principle of equivalence refers strictly to local effects: looking out of his window and performing navigational observations, the astronaut can tell how he is moving relative to the planets and moons of the solar system.

Einstein argued, however, that in the presence of gravitational fields there is no unambiguous way to separate gravitational pull from the effects occasioned by the noninertial character of one's chosen frame of reference; hence one cannot identify an inertial frame of reference with complete precision. Thus the principle of equivalence renders the gravitational field fundamentally different from all other force fields encountered in nature. The new theory of gravitation, the general theory of relativity, adopts this characteristic of the gravitational field as its foundation.

Curved space-time

The principles

In terms of Minkowski's space-time, inertial frames of reference are the analogues of rectilinear (straight-line) Cartesian coordinate systems in Euclidean geometry. In a plane these coordinate systems always exist, but they do not exist on the surface of a sphere: any attempt to cover a spherical surface with a grid of squares breaks down when the grid is extended over a significant fraction of the spherical surface. Thus a plane is a flat surface, whereas the surface of a sphere is curved. This distinction, based entirely on internal properties of the surface itself, classifies the surface of a cylinder as flat, as it can be rolled off on a plane and thus is capable of being covered by a grid of squares.

Einstein conjectured that the presence of a gravitational field causes space-time to be curved (whereas in the absence of gravitation it is flat), and that this is the reason that inertial frames cannot be constructed. The curved trajectory of a particle in space and time resulting from the effects of gravitation would then represent not a straight line (which exists only in flat spaces and space-times) but the straightest curve possible in a curved space-time, a geodesic. Geodesics on a sphere (such as the surface of the Earth) are the great circles. (The plane of any great circle goes through the centre of the Earth.) They are the least curved lines one can construct on the surface of a sphere, and they are the shortest curves connecting any two points. The geodesics of space-time connect two events (or two instants in the history of one particle) with the greatest lapse of proper time, as was indicated in the earlier discussion of the twin paradox.

If the presence of a gravitational field amounts to a curvature of space-time, then the description of the gravitational field in turn hinges on a mathematical elucidation of the curvature of four-dimensional space-time. Before Einstein, the German mathematician Bernhard Riemann (1826-66) had developed methods related directly to the failure of any attempt to construct square grids. If one were to construct within any small piece of (two-dimensional) surface a quadrilateral whose sides are geodesics, if the surface were flat, the sum of the angles at the four corners would be 360. If the surface is not flat, the sum of the angles will not be 360. The deviation of the actual sum of the angles from 360 will be proportional to the area of the quadrilateral; the amount of deviation per unit of surface will be a measure of the curvature of that surface. If the surface is imbedded in a higher dimensional continuum, then one can consider similarly unavoidable angles between vectors constructed as parallel as possible to each other at the four corners of the quadrilateral, and thus associate several distinct components of curvature with one surface. And, of course, there are several independent possible orientations of two-dimensional surfaces--for instance, six in a four-dimensional continuum, such as space-time. Altogether there are 20 distinct and independent components of curvature defined at each point of space-time; in mathematics these are referred to as the 20 components of Riemann's curvature tensor.

The mathematical expression

Einstein discovered that he could relate 10 of these components in a natural way to the sources of the gravitational field, mass (or energy), density, momentum density, and stress, if he were to duplicate approximately Newton's equations of the gravitational field and, at the same time, formulate laws that would take the same form regardless of the choice of frame of reference. The remaining 10 components may be chosen arbitrarily at any one point but are related to each other by partial differential equations at neighbouring points. Einstein derived a field equation that, along with the rule that a freely falling body moves along a geodesic, forms the comprehensive treatment of gravitation known as the general theory of relativity.

Confirmation of the theory

The general theory of relativity is constructed so that its results are approximately the same as those of Newton's theories as long as the velocities of all bodies interacting with each other gravitationally are small compared with the speed of light--i.e., as long as the gravitational fields involved are weak. The latter requirement may be stated roughly in terms of the escape velocity. The escape velocity is defined as the minimal speed with which a projectile must be endowed at any given location to enable it to fly off to infinitely removed regions of the universe without the application of further force. On the surface of the Earth the escape velocity is approximately 11.2 kilometres (6.95 miles) per second. A gravitational field is considered strong if the escape velocity approaches the speed of light, weak if it is much smaller. All gravitational fields encountered in the solar system are weak in this sense.

The success of Newton's theory, incidentally, must be considered a confirmation of the general theory of relativity to the extent that that application of the theory remains confined to situations involving small relative speeds and weak fields. Obviously, any superiority of the new theory over the old one may be inferred only if their predictions disagree and if those of the general theory of relativity are confirmed by experiment and observation.

As the principle of equivalence forms the cornerstone of general relativity, its verification is crucial. Highly precise experiments with this objective were performed between 1888 and 1922 by a Hungarian physicist, Roland, Baron von Eötvös, and his collaborators, who confirmed the principle to an accuracy of one part in 10⁸, and in the 1960s by an American physicist, Robert Dicke, who achieved an accuracy of one part in 10¹¹. Subsequently the Soviet physicist V.L. Braginsky further improved the accuracy to one part in 10¹². Through this work the principle of equivalence has become one of the most precisely confirmed general principles of contemporary physics.

Some other new predictions of general relativity are explained below.

Advance of Mercury's perihelion

The major axes of the elliptical trajectories of the planets about the Sun turn slowly within their planes because of the interactions of the planets with each other, but it was discovered in the 19th century that interplanetary perturbations could not account fully for the turning rate of Mercury's orbit, leaving unexplained about 43 of arc per century. The general theory of relativity, however, accounts exactly for this discrepancy. In 1967 Dicke--and more recently Henry Allen Hill, also of the United States--suggested that a small part of Mercury's perihelion advance may be caused by the slight flattening of the Sun at its poles, thus opening the way for possible modification of general relativity. On the other hand, support for Einstein's original version of the theory has come from a comprehensive evaluation of solar system data by the American investigator Ronald W. Hellings and from investigations of the binary pulsar system PSR 1913+16 by the American astronomer Joseph H. Taylor.

Gravitational redshift

General relativity predicts that the wavelength of light emanating from sources within a gravitational field will increase (shift toward the red end of the spectrum) by an amount proportional to the gravitational potential at the site of the source. This effect was found first in astronomical objects, particularly in stars called white dwarfs, on whose surfaces the gravitational potential is large. The best quantitative confirmation of gravitational redshift was obtained in laboratory experiments in Great Britain and the United States in the 1960s; an accuracy of one part in 100 was achieved in measuring the minute difference in gravitational potential between two sites differing in altitude by a few metres.

Optical effects of gravitation

General relativity predicts that the curvature of space-time results in the apparent bending of light rays passing through gravitational fields and in an apparent reduction of their speeds of propagation. The bending was first observed, within a couple of years of Einstein's publication of the new theory, during a total eclipse, when stellar images near the occulted disk of the Sun appeared displaced by fractions of 1 of arc from their usual locations in the sky. The associated delay in travel time was observed in the late 1960s, when ultraintense radar pulses were reflected off Mercury and Venus just as these planets were passing behind the Sun. These experiments are difficult to perform and their accuracy is difficult to evaluate, but it seems conservative to conclude that they confirm the relativistic effect within a few parts in 100. Finally, extended massive objects such as galaxies may act as "gravitational lenses," providing more than one optical path for light emanating from a source far behind the lens and thus producing multiple images. Such multiple images, typically of quasars, had been discovered by the early 1980s.

Gravitational waves

General relativity predicts the occurrence of gravitational waves, whose properties should resemble in some respects those of electromagnetic waves: they should travel at the same speed, c, and they should be polarized. Joseph Weber, an American physicist, announced in 1969 that he had detected events that might be caused by incoming gravitational waves--namely, vibrations occurring simultaneously in pairs of large aluminum cylinders, about 1,000 kilometres apart and each weighing several tons. Although these detectors had been insulated with great care from all other potential sources of such vibrations, the separation of gravitational signals from ordinary thermal noise (Brownian motion) presents delicate problems of instrumentation and interpretation, which proved difficult to resolve to the satisfaction of other experimenters attempting to repeat Weber's observations.

Weber's approach has been refined by the choice of different materials for the vibrating masses, by cryogenic techniques reducing the level of thermal noise, and by other improvements. A fundamentally different technique, replacing Weber's stationary cylinders by independently moving masses whose distances from each other would be measured by interferometric means, also has been investigated. While these efforts at direct detection of gravitational waves were under way, observations of the binary pulsar PSR 1913+16 indicated that this double star system is losing energy at precisely the rate that corresponds to the emission of gravitational radiation according to the theory of general relativity.

The discovery of gravitational waves would represent an important confirmation of the validity of the theory. Also, such waves might become the basis of an entirely new technology of astronomical observation, as they are believed to be the most penetrating kind of radiation imaginable.

Future astrophysical tests

The properties of certain astronomical objects, such as quasars (see below Relativistic cosmology ), pulsars (extremely dense stars that emit electromagnetic pulses with great regularity), very bright galaxies at the cores of which extraordinary amounts of energy are being emitted, and jets of matter moving at relativistic speeds, imply that there are processes involving gravitational fields so strong that general relativity is needed to interpret the observations, which in turn will provide new tests of that theory.

Copyright © 1994-2000 Encyclopædia Britannica, Inc.

Conceptual implications of general relativity

The general theory of relativity represents a further modification of classical concepts of space and time that goes far beyond those implicit in the special theory. The special theory does away with the absolute character of time and with the absolute distance between two objects that are at rest relative to each other. The geometric concepts appropriate to the special theory are the four-dimensional space-time continuum, in which events that are fixed in space and in time are represented by points, often referred to as world points (to distinguish them from the points of ordinary three-dimensional space), and the histories of particles moving through space in the course of time by curves (world curves); the representations of particles that are not accelerated by forces are straight lines.

Minkowski's space-time is a rigidly flat continuum, as is the three-dimensional space of Euclid's geometry. Distances between world points are measured by the invariant intervals, whose magnitudes do not depend on the particular coordinate system, or frame of reference, used. The Minkowski universe is homogeneous; that is to say, geometric figures constructed at any site may be transferred to another site without distortion. Finally, among all the possible frames of reference there is a special set, the inertial frames of reference, just as in ordinary space the rectilinear coordinate systems are distinguished by their simplicity among all conceivable coordinate systems. Space-time serves as the immutable backdrop of all physical processes, without being affected by them.

In general relativity, space-time also is a four-dimensional continuum, with invariant intervals being defined at least locally between events taking place close to each other. But only small regions of space-time resemble the continuum envisaged by Minkowski, just as small bits of a spherical surface appear nearly planar. In the broad sense, according to general relativity, space-time is curved, and this curvature is equivalent to the presence of a gravitational field. Far from being rigid and homogeneous, the general-relativistic space-time continuum has geometric properties that vary from point to point and that are affected by local physical processes. Space-time ceases to be a stage, or scaffolding, for the dynamics of nature; it becomes an integral part of the dynamic process. General relativity, it has been said, makes physics part of geometry. It may also be claimed that general relativity makes geometry part of physics, that is to say, of a natural science. Not only are the properties of space and time subject to scientific investigation, to a study by means of experiments, but specific properties, such as the amount of curvature in a particular location at a specified time, are to be measured with the help of physical instruments.

Though the general theory of relativity is universally accepted as the most satisfactory basis of the gravitational force now known, it has not been completely fused with quantum mechanics, of which the central concept is that energy and angular momentum exist only in finite and discrete lumps, called quanta. Since the 1920s quantum mechanics has been the sole rationale of the forces that act between subatomic particles; gravitation doubtless is one of these forces, but its effects are unobservably small in comparison to electromagnetic and nuclear forces. Relativistic phenomena in the subatomic realm have been adequately dealt with by merging quantum mechanics with the special, not the general, theory.

Many physicists, foremost among them Einstein himself, tried during the first half of the 20th century to enrich the geometric structure of space-time so as to encompass all known physical interactions. Their goal, a unified field theory, remained elusive but was brought nearer during the late 1960s by the successful unification of the electromagnetic and the so-called weak nuclear force.

Schwarzschild's solution of the field equations

Immediately on publication of Einstein's paper on general relativity, the German astronomer Karl Schwarzschild found a mathematical solution to the new field equations, which corresponds to the gravitational field of a compact massive body, such as a star or planet, and which is now referred to as Schwarzschild's field. If the mass that serves as the source of the field is fairly diffuse, so that the gravitational field on the surface of the astronomical body is fairly weak, Schwarzschild's field will exhibit physical properties similar to those described by Newton. Gross deviations will be found if the mass is so highly concentrated that the field on the surface is strong. At the time of Schwarzschild's work, in 1916, this appeared to be a theoretical speculation; but with the discovery of pulsars and their interpretation as probable neutron stars composed of matter that has the same density as atomic nuclei (so-called nuclear matter), the possibility exists that strong fields may soon be accessible to astronomical observation.

The most conspicuous feature of the Schwarzschild field is that if the total mass is thought of as concentrated at the very centre, a point called a singularity, then at a finite distance from that centre, the Schwarzschild radius, the geometry of space-time changes drastically from that to which we are accustomed. Particles and even light rays cannot penetrate from inside the Schwarzschild radius to the outside and be detected. Conversely, to an outside observer any objects approaching the Schwarzschild radius appear to take an infinite time to penetrate toward the inside. There cannot be any effective communication between the inside and the outside, and the boundary between them is called an event horizon.

The exterior and the interior of the Schwarzschild radius are not cut off from each other entirely, however. Suppose an observer were to attach himself to a particle that is falling freely straight toward the centre and that this observer is equipped with a clock that reads its own proper time. This observer would penetrate the Schwarzschild radius within a finite proper time; moreover, he would find no abnormalities in his environment as he did so. The reason is that his clock would deviate from one permanently kept outside and at a constant distance from the centre, so grossly that the same event that seen from the outside takes forever occurs within a finite time to the free-falling observer.

These peculiarities of the Schwarzschild field may well have practical applications in astronomy. In 1931 the Indian-born U.S. astrophysicist Subrahmanyan Chandrasekhar, and in 1939 the U.S. physicist J. Robert Oppenheimer, established that a star whose mass exceeds the mass of the Sun by an appreciable factor is bound to contract and, eventually, to collapse under the influence of its own gravity, no matter how resistant its constituent matter. As many stars are believed to have such large masses, it is likely that there already exist some collapsed stars, so-called black holes. Though continuing to make its presence known by the gravitational attraction it exerts on other stars, a black hole would not emit light, and thus be invisible, hence its name.

Applications of relativistic principles

Particle accelerators

Modern particle accelerators raise particles to speeds very near that of light. At these energies and speeds the differences in behaviour predicted by classical physics and by the special theory of relativity are huge; the machines must be designed in accordance with relativistic principles, or they will not operate.

Electron synchrotrons operate at energies of several thousand million electron volts, which means that the relativistic mass of an electron orbiting at maximum energy is roughly 10,000 times its rest mass. Accordingly, the magnetic field required to maintain the electrons in orbit is 10,000 times as powerful as it would have to be if nonrelativistic physics held, at the same speed. On the other hand, at that given energy the speed of the electrons is in fact very nearly equal to the speed of light, the difference amounting to no more than one part in 100,000,000 (10⁸). At the same energy, but by nonrelativistic mechanics, the speed of the electrons would be about 100 times the speed of light. This difference has a very practical consequence: in those particle accelerators designed for highly relativistic energies, the synchrotrons, particles are injected into a circular orbit already near the speed of light, and their velocities change only slightly as their energies are brought up to the highest design value. If the orbit diameter is kept nearly constant, particles at all energies will circulate at the same frequency, and only the magnetic field that keeps them in orbit needs to be increased to keep pace with the increasing mass. The accelerating voltage is applied at the constant frequency required so that the particles will always be accelerated forward.

Relativistic particle physics

The physics of subatomic particles depends on the principles of the special theory of relativity. These principles have their most direct application when particles are created, annihilated, or converted into different particles. In most particle transformations, large amounts of energy are involved; the total (rest) masses of the particles involved in the transformations will change, and this change will be related to the amounts of energy expended or gained by the rule that the change in mass (m₀) is balanced by a corresponding change in energy (E), divided by the square of the speed of light (c²): m₀ = -c^-2E. This rule has been confirmed universally and, by now, is being taken for granted.

The units, or quanta, of electromagnetic energy, called photons, have long been regarded as a species of particle in which are combined the properties of zero rest mass with nonvanishing relativistic mass, because they travel at the speed of light. The relativistic mass equals its total energy E divided by c². The energy of a photon also is equal to the product of its frequency and Planck's constant h. The relativistic mass of a photon can be checked experimentally if the photon is absorbed or deflected in its interactions with particles, when the change in its linear momentum (product of velocity and relativistic mass) results in a recoil by the other particles. If a high-frequency photon, a gamma photon, collides with a free electron, the result is called the Compton effect, which involves both an observable recoil on the part of the electron and an altered frequency of the deflected photon. Again, relativity is confirmed by experiment.

It has been conjectured that gravitational waves, also, are composed of zero-rest-mass quanta travelling at the speed of light (gravitons). As the quantum theory of the gravitational field has not been definitely established and as the detection of individual gravitons may remain beyond experimental capabilities for years to come, the existence of gravitons cannot be considered assured.

There is another species of zero-rest-mass particles, produced in radioactive decay involving the ejection of electrons or positrons from atomic nuclei (so-called beta decay). These particles, known as neutrinos, have no electric charge and travel at the speed of light. Several distinct species of neutrinos are now known, each produced in a different kind of beta decay. Neutrinos interact with other particles extremely weakly. As a result, they can traverse large amounts of matter with little chance of being deflected or absorbed. Though their existence has been confirmed beyond a doubt, their detection and detailed examination remain challenging problems.

Relativistic cosmology

Theories concerning the structure and history of the whole universe have assumed an increasingly empirical aspect in the 20th century. Beginning in the 1960s, particularly, a combination of new observation techniques, new discoveries, and applications of special and general relativity has resulted in considerable progress. The most important techniques added to those of observations by means of visible light were radio astronomy; infrared, ultraviolet, X-ray, and gamma-ray astronomy from extraterrestrial platforms; cosmic-ray investigations; neutrino astronomy; and examination of the Moon and other astronomical bodies by unmanned and manned space exploration.

Edwin Powell Hubble, a U.S. astronomer, had discovered that the more distant astronomical objects exhibited a shift of spectral lines toward the red (long wavelength) end of the spectrum, the extent of the shift increasing the greater their distance from Earth. This cosmological red shift has been generally interpreted as evidence of rapid recession of these distant objects in an expanding universe. The present rate of expansion is expressed as the amount of recession per unit distance and is known as the Hubble constant. It amounts to about a mile per second recessional velocity for a distance of 10⁵ light-years. Equivalently, if the expansion has been occurring at a constant rate, it must have started about 2 10¹⁰ years ago.

Quasars, also called quasi-stellar objects (QSO's), appear to be structures that combine extreme luminosity (100 times that of a bright galaxy) with great compactness, taking up less space than the distance between the Sun and its nearest neighbour star. Wherever a spectral analysis of a quasar's emitted light has been possible, the spectral lines have been found considerably red shifted. If these red shifts are cosmological (an interpretation now accepted by most astronomers), some quasars are more distant from the Galaxy than any other known objects. As such they may provide indications of the large-scale structure of the universe, which could not be obtained from investigations confined to "close" surroundings. The term close is to be understood in relation to distance in which Hubble's red shift becomes large ("cosmological distances"), distances amounting to thousands of millions of light-years.

Finally, the term primeval fireball refers to the discovery of an all-pervasive background of radiation whose frequencies lie in the border region between microwave radio frequencies and infrared, corresponding to wavelengths of the order of millimetres and centimetres. In the early 1970s this radiation was interpreted as a remnant of the original intensive radiation that must have been associated with the early history of the universe, when matter was both extremely dense and extremely hot; hence its name. Its spectral composition, which has been the object of intensive investigation, might provide some clues to the early history of the universe.

General relativity contributes to a theoretical discussion of cosmology the idea that the universe as a whole need not be flat even on the average and that it probably is not. Even if one were to assume that on a very large (cosmological) scale the universe is homogeneous and isotropic (i.e., having the same properties in all directions), which appears a reasonable working hypothesis in the absence of any evidence to the contrary, there are a number of different possibilities. The universe might be spatially open (as a flat one surely is), or it might be closed, somewhat as the surface of a sphere is closed, without boundaries. Likewise, in the time direction the universe might be either open or closed; it is a little difficult to visualize a time-wise closed universe, which appears to be inconsistent with ordinary notions of cause and effect. But because these notions are distilled out of normal experience, they might be inapplicable on the scale of billions of years. In brief, many different cosmological models have been proposed and investigated theoretically, but observational information does not seem to favour one particular type. The information appears to favour types that expand from an early stage involving fireball conditions.

Modifications of general relativity

An outgrowth of a unified field theory of the early 1920s has been the development of a class of theories based on the hypothesis that underlying the four-dimensional space-time of our experience is a manifold having a higher dimensionality, whose geometric structure can accommodate all known force fields, including those associated with stable and unstable subatomic particles. Though these concepts remained highly speculative, they offered much promise and occupied many investigators.

Apart from the attempts to devise unified field theories, several modifications of general relativity have been proposed during the late 20th century. One of these was presented by the British scientist Fred Hoyle, whose results, together with the proposals of the astronomers Hermann Bondi and Thomas Gold, became the basis of the so-called steady-state cosmological theory. Bondi, Gold, and Hoyle opposed the "big-bang" theory of the origin of the universe, arguing instead that matter is being created continuously at a very low rate, just sufficient to maintain the constant average density of the universe in spite of the observed expansion. Though the steady-state hypothesis evoked much interest for some years, the existence of the cosmic background radiation (established in the 1960s) has been generally accepted as proof that the universe has in fact passed through a highly dense stage.

Relativity, Philosophical Consequences of

RELATIVITY: PHILOSOPHICAL CONSEQUENCES. Of the consequences in philosophy which may be supposed to follow from the theory of relativity, some are fairly certain, while others are open to question. There has been a tendency, not uncommon in the case of a new scientific theory, for every philosopher to interpret the work of Einstein in accordance with his own metaphysical system, and to suggest that the outcome is a great accession of strength to the views which the philosopher in question previously held. This cannot be true in all cases; and it may be hoped that it is true in none. It would be disappointing if so fundamental a change as Einstein has introduced involved no philosophical novelty. (See SPACE-TIME.)

Space-Time.--For philosophy, the most important novelty was present already in the special theory of relativity; that is, the substitution of space-time for space and time. In Newtonian dynamics, two events were separated by two kinds of interval, one being distance in space, the other lapse of time. As soon as it was realised that all motion is relative (which happened long before Einstein), distance in space became ambiguous except in the case of simultaneous events, but it was still thought that there was no ambiguity about simultaneity in different places. The special theory of relativity showed, by experimental arguments which were new, and by logical arguments which could have been discovered any time after it became known that light travels with a finite velocity, that simultaneity is only definite when it applies to events in the same place, and becomes more and more ambiguous as the events are more widely removed from each other in space.

This statement is not quite correct, since it still uses the notion of "space." The correct statement is this: Events have a four-dimensional order, by means of which we can say that an event A is nearer to an event B than to an event C; this is a purely ordinal matter, not involving anything quantitative. But, in addition, there is between neighbouring events a quantitative relation called "interval," which fulfils the functions both of distance in space and of lapse of time in the traditional dynamics, but fulfils them with a difference. If a body can move so as to be present at both events, the interval is time-like. If a ray of light can move so as to be present at both events, the interval is zero. If neither can happen, the interval is space-like. When we speak of a body being present "at" an event, we mean that the event occurs in the same place in space-time as one of the events which make up the history of the body; and when we say that two events occur at the same place in space-time, we mean that there is no event between them in the four-dimensional space-time order. All the events which happen to a man at a given moment (in his own time) are, in this sense, in one place; for example, if we hear a noise and see a colour simultaneously, our two perceptions are both in one place in space-time.

When one body can be present at two events which are not in one place in space-time, the time-order of the two events is not ambiguous, though the magnitude of the time-interval will be different in different systems of measurement. But whenever the interval between two events is space-like, their time-order will be different in different equally legitimate systems of measurement; in this case, therefore, the time-order does not represent a physical fact. It follows that, when two bodies are in relative motion, like the sun and a planet, there is no such physical fact as "the distance between the bodies at a given time"; this alone shows that Newton's law of gravitation is logically faulty. Fortunately, Einstein has not only pointed out the defect, but remedied it. His arguments against Newton, however, would have remained valid even if his own law of gravitation had not proved right.

Time not a Single Cosmic Order.--The fact that time is private to each body, not a single cosmic order, involves changes in the notions of substance and cause, and suggests the substitution of a series of events for a substance with changing states. The controversy about the aether thus becomes rather unreal. Undoubtedly, when light-waves travel, events occur, and it used to be thought that these events must be "in" something; the something in which they were was called the aether. But there seems no reason except a logical prejudice to suppose that the events are "in" anything. Matter, also, may be reduced to a law according to which events succeed each other and spread out from centres; but here we enter upon more speculative considerations.

Physical Laws.--Prof. Eddington has emphasised an aspect of relativity theory which is of great philosophical importance, but difficult to make clear without somewhat abstruse mathematics. The aspect in question is the reduction of what used to be regarded as physical laws to the status of truisms or definitions. Prof. Eddington, in a profoundly interesting essay on "The Domain of Physical Science," [Footnote 1] states the matter as follows:--

In the present stage of science the laws of physics appear to be divisible into three classes--the identical, the statistical and the transcendental. The "identical laws" include the great field-laws which are commonly quoted as typical instances of natural law--the law of gravitation, the law of conservation of mass and energy, the laws of electric and magnetic force and the conservation of electric charge. These are seen to be identities, when we refer to the cycle so as to understand the constitution of the entities obeying them; and unless we have misunderstood this constitution, violation of these laws is inconceivable. They do not in any way limit the actual basal structure of the world, and are not laws of governance (op. cit., pp. 214-5).

It is these identical laws that form the subject-matter of relativity theory; the other laws of physics, the statistical and transcendental, lie outside its scope. Thus the net result of relativity theory is to show that the traditional laws of physics, rightly understood, tell us almost nothing about the course of nature, being rather of the nature of logical truisms.

This surprising result is an outcome of increased mathematical skill. As the same author [Footnote 2] says elsewhere:--

In one sense deductive theory is the enemy of experimental physics. The latter is always striving to settle by crucial tests the nature of the fundamental things; the former strives to minimise the successes obtained by showing how wide a nature of things is compatible with all experimental results.

The suggestion is that, in almost any conceivable world, something will be conserved; mathematics gives us the means of constructing a variety of mathematical expressions having this property of conservation. It is natural to suppose that it is useful to have senses which notice these conserved entities; hence mass, energy, and so on seem to have a basis in our experience, but are in fact merely certain quantities which are conserved and which we are adapted for noticing. If this view is correct, physics tells us much less about the real world than was formerly supposed.

Force and Gravitation.--An important aspect of relativity is the elimination of "force." This is not new in idea; indeed, it was already accepted in rational dynamics. But there remained the outstanding difficulty of gravitation, which Einstein has overcome. The sun is, so to speak, at the summit of a hill, and the planets are on the slopes. They move as they do because of the slope where they are, not because of some mysterious influence emanating from the summit. Bodies move as they do because that is the easiest possible movement in the region of space-time in which they find themselves, not because "forces" operate upon them. The apparent need of forces to account for observed motions arises from mistaken insistence upon Euclidean geometry; when once we have overcome this prejudice, we find that observed motions, instead of showing the presence of forces, show the nature of the geometry applicable to the region concerned. Bodies thus become far more independent of each other than they were in Newtonian physics: there is an increase of individualism and a diminution of central government, if one may be permitted such metaphorical language. This may, in time, considerably modify the ordinary educated man's picture of the universe, possibly with far-reaching results.

Realism in Relativity.--It is a mistake to suppose that relativity adopts an idealistic picture of the world--using "idealism" in the technical sense, in which it implies that there can be nothing which is not experience. The "observer" who is often mentioned in expositions of relativity need not be a mind, but may be a photographic plate or any kind of recording instrument. The fundamental assumption of relativity is realistic, namely, that those respects in which all observers agree when they record a given phenomenon may be regarded as objective, and not as contributed by the observers. This assumption is made by common sense. The apparent sizes and shapes of objects differ according to the point of view, but common sense discounts these differences. Relativity theory merely extends this process. By taking into account not only human observers, who all share the motion of the earth, but also possible "observers" in very rapid motion relatively to the earth, it is found that much more depends upon the point of view of the observer than was formerly thought. But there is found to be a residue which is not so dependent; this is the part which can be expressed by the method of "tensors." The importance of this method can hardly be exaggerated; it is, however, quite impossible to explain it in non-mathematical terms.

Relativity Physics.--Relativity physics is, of course, concerned only with the quantitative aspects of the world. The picture which it suggests is somewhat as follows:--In the four-dimensional space-time frame there are events everywhere, usually many events in a single place in space-time. The abstract mathematical relations of these events proceed according to the laws of physics, but the intrinsic nature of the events is wholly and inevitably unknown except when they occur in a region where there is the sort of structure we call a brain. Then they become the familiar sights and sounds and so on of our daily life. We know what it is like to see a star, but we do not know the nature of the events which constitute the ray of light that travels from the star to our eye. And the space-time frame itself is known only in its abstract mathematical properties; there is no reason to suppose it similar in intrinsic character to the spatial and temporal relations of our perceptions as known in experience. There does not seem any possible way of overcoming this ignorance, since the very nature of physical reasoning allows only the most abstract inferences, and only the most abstract properties of our perceptions can be regarded as having objective validity. Whether any other science than physics can tell us more, does not fall within the scope of the present article.

Meanwhile, it is a curious fact that this meagre kind of knowledge is sufficient for the practical uses of physics. From a practical point of view, the physical world only matters in so far as it affects us, and the intrinsic nature of what goes on in our absence is irrelevant, provided we can predict the effects upon ourselves. This we can do, just as a person can use a telephone without understanding electricity. Only the most abstract knowledge is required for practical manipulation of matter. But there is a grave danger when this habit of manipulation based upon mathematical laws is carried over into our dealings with human beings, since they, unlike the telephone wire, are capable of happiness and misery, desire and aversion. It would therefore be unfortunate if the habits of mind which are appropriate and right in dealing with material mechanisms were allowed to dominate the administrator's attempts at social constructiveness.

Bibliography A. S. Eddington, Space, Time, and Gravitation (Cambridge, 1921); Bertrand A. W. Russell, The A. B. C. of Relativity (1925).

(B. A. W. R.)

Footnote 1: In Science, Religion and Reality, ed. by Joseph Needham (1925).

Footnote 2: A. S. Eddington, Mathematical Theory of Relativity, p. 238 (Cambridge, 1924).

Bibliography

Among expositions for general readers are Albert Einstein, Relativity: The Special and General Theory: A Popular Exposition, 17th ed. (1961; originally published in German, 1917), a popularization for the lay reader of a classic work written by one of the greatest scientists of all time; Bertrand Russell, The ABC of Relativity, 4th rev. ed. edited by Felix Pirani (1985); Albert Einstein and Leopold Infeld, The Evolution of Physics (1938, reissued 1961); Leopold Infeld, Albert Einstein: His Work and Its Influence on Our World (1950), two books that cover the whole of physics, with special emphasis on relativity (Infeld was one of Einstein's chief collaborators in the 1930s); Hermann Bondi, Relativity and Common Sense: A New Approach to Einstein (1964, reissued 1980); Robert Geroch, General Relativity from A to B (1978), a beautiful book explaining general relativity in an exciting and insightful manner to an audience of humanists; Peter G. Bergmann, The Riddle of Gravitation, rev. and updated ed. (1987, reissued 1992), a work that emphasizes the general theory of relativity and includes a discussion of research; Sam Lilley, Discovering Relativity for Yourself (1981), a work that covers both theories; George F.R. Ellis and Ruth M. Williams, Flat and Curved Space-times (1988); Eric Chaisson, Relatively Speaking: Relativity, Black Holes, and the Fate of the Universe (1988); and Clifford M. Will, Was Einstein Right?: Putting General Relativity to the Test, 2nd ed. (1993), the last two works stressing the astronomical aspect of relativity.

Presentations for readers with technical training include H.A. Lorentz et al., The Principle of Relativity (1923, reissued 1952), a collection of fundamental research papers, all in English; Albert Einstein, The Meaning of Relativity, 5th ed., trans. from German (1955, reprinted 1988), based on lectures by Einstein delivered in 1921, with two appendixes containing Einstein's views on cosmology through 1945, and his work on the "nonsymmetric" unified field theory to the time of his death in 1955; Abraham Pais, "Subtle Is the Lord--": The Science and the Life of Albert Einstein (1982), containing a wealth of material on relativity, its history, and its relationship to the whole of physics; David Bohm, The Special Theory of Relativity (1965, reprinted 1989), a thoroughgoing treatment of the special theory combined with a discussion of the philosophical foundations of physics; A.P. French, Special Relativity (1968, reissued 1984), an introduction at the undergraduate level; Hermann Bondi, Cosmology, 2nd ed. (1961), a survey of cosmology at a technical level, including observational data through the late 1950s; Peter G. Bergmann, Introduction to the Theory of Relativity (1942, reissued 1976); C. Møller, The Theory of Relativity, 2nd ed. (1972); J.L. Synge, Relativity: The Special Theory, 2nd ed. (1964, reissued 1972), and Relativity: The General Theory (1960, reissued 1971); Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler, Gravitation (1973), technical texts, on the graduate level, that represent distinct approaches to the subject by active research workers; Steven Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity (1972), by a Nobel laureate; J.L. Martin, General Relativity: A Guide to Its Consequences for Gravity and Cosmology (1988), a text on the general theory; S.W. Hawking and G.F.R. Ellis, The Large Scale Structure of Space-Time (1973), a work principally concerned with the geometric aspects of general relativity on a global scale; Robert M. Wald, General Relativity (1984), and Space, Time, and Gravity, 2nd ed. (1992), written by one of the experts and an active contributor in the field; Roberto Torretti, Relativity and Geometry (1983), an exposition of the general and special theories from a geometric perspective, for the advanced reader; and Wolfgang Rindler, Introduction to Special Relativity, 2nd ed. (1991). Two historical works are Don Howard and John Stachel (eds.), Einstein and the History of General Relativity (1989); and Jean Eisenstaedt and A.J. Kox (eds.), Studies in the History of General Relativity (1992).