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*=*mc ^{2}* (

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/sec^{2} (32 ft/sec^{2}) (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.

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.

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}/*c*^{2})^{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}/*c*^{2})^{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 10^{12}
miles, or 9.46 10^{12}
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:

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*.

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, *m*_{0}, by a formula in which *m*
equals *m*_{0} divided by the square root of one minus the
fraction ^{2}/*c*^{2}:

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 *m*_{0} by an amount that equals its kinetic energy
*E*, divided by *c*^{2}: *m* - *m*_{0} = *E*/*c*^{2}.
Hence the hypothesis that generally the energy is *c*^{2} times
the mass, or *E *=* mc*^{2}, 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.

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}
= *T*^{2} -* L*^{2}/*c*^{2}. This will
have the same value as *T*^{2} - *L*^{2}/*c*^{2},
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}
= *L*^{2} - *c*^{2}*T*^{2}, 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.

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.

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, *s*^{2} = *x*^{2} + *y*^{2} +
*z*^{2}, 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 *L*^{2} - *c*^{2}*T*^{2}
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 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.

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.

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.

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.

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 10^{8}, and in
the 1960s by an American physicist, Robert
Dicke, who achieved an accuracy of one part in 10^{11}.
Subsequently the Soviet physicist V.L. Braginsky
further improved the accuracy to one part in 10^{12}. 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.

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.

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.

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.

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.

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.**

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.

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.

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 (10^{8}). 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.

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 (*m*_{0})
is balanced by a corresponding change in energy (*E*),
divided by the square of the speed of light (*c*^{2}): *m*_{0}
= -*c*^{-2}*E*.
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 *c*^{2}. 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.

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 10^{5} light-years. Equivalently, if the expansion has been
occurring at a constant rate, it must have started about 2 10^{10}
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.

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 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** 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).

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).