Relativity

I  INTRODUCTION

Albert Einstein

In 1905 German-born American physicist Albert Einstein published his first paper outlining the theory of relativity. It was ignored by most of the scientific community. In 1916 he published his second major paper on relativity, which altered mankind’s fundamental concepts of space and time.

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. 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.

II  CLASSICAL PHYSICS

 

Michelson-Morley Apparatus

In 1887 Albert Michelson and Edward Morley attempted to measure Earth’s speed with respect to the ether, a hypothetical substance thought to fill empty space. Their apparatus split a beam of light so that half went straight ahead and half went sideways. If the ether existed, they theorized, it should exert more of a drag on one of the beams than on the other as Earth moves through the ether along its orbit. They found no evidence of a difference in speed, however, which led to the demise of the ether theory.

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 β (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. 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.

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

Frames of Reference

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

 

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. 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 . 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.

V  CONFIRMATION AND MODIFICATION

 

Dragging Space and Time

The results of two studies announced in early November 1997 provide unprecedented support for “frame-dragging,” a concept predicted by physicist Albert Einstein's general theory of relativity. Frame-dragging describes how massive objects actually distort space and time around themselves as they rotate. One of the studies examined frame-dragging around black holes, an example of which is shown here in an artist's conception.

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. 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. 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. 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.

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.

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.

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