University of Virginia Library

RELATIVITY

I

Essentially, Einstein's theory of relativity has its roots
in the questions: Where are we? How are we moving?
In connection with our present purposes, these ques-
tions posed no profound problems so long as men
believed that the earth is the fixed center of the uni-
verse. With the astronomical hypothesis of a moving
earth, however, the questions began to become dis-
turbing, not only theologically but also scientifically.
This article is concerned with the scientific aspects of
the problem.

In the seventeenth-century concepts of Galileo, and
more sharply in those of Newton, one already finds
a “principle of relativity,” though the phrase itself did
not come into being until late in the nineteenth cen-
tury. We can say that the principle has to do with the
impossibility of detecting absolute motion. But to this
statement we have to attach changing caveats whose
nature will not become apparent until we have dis-
cussed the matter in detail.

In the nineteenth century, optical and electromag-
netic theory had seemed to invalidate the principle.
Reaffirming it and later generalizing it were thus revo-
lutionary acts. Their drastic scientific consequences,
affecting the basic concepts of time and space, were
worked out in the twentieth century, principally by
Einstein.

II

The considerable success of the Ptolemaic system
had brought high respectability to the intuitive concept
of a fixed earth, a doctrine strongly reinforced by the
influence of Aristotle, the dogma of the Church, and
the vanity of man.

Copernicus found the idea of a moving earth in the
writings of the ancients. In the dedication of his book,
De revolutionibus to Pope Paul III, he said that at
first he had thought it absurd. His greatness does not
reside primarily in his daring to take it seriously but
in his constructing a mathematically detailed system
capable of challenging the formidable geocentric sys-
tem expounded by Ptolemy. Like Ptolemy, he used
epicycles (Figure 1), and his system was far from sim-
ple. And though he held that the sun was fixed at the
center of the Universe, the pivotal point of the plane-
tary motions in his system was not the center of the
sun but an empty place that we may conveniently call
the center of the earth's orbit. In this sense the earth,
though relegated to the role of a planet, retained a
certain supremacy.

Dethronement of the earth was Kepler's doing, and
with it came beauty. The planets now moved in ellip-
tical orbits about a fixed sun at a common focus, their
speeds varying in orbit in such a way that the line
from the sun to a planet traces out equal areas in equal
times (Figure 2). No longer was there need for in-
tricate epicycles either for shape of orbit or speed in
orbit. Simplicity had taken their place.

In the drudgery of his lifelong search for laws of
planetary motion Kepler was sustained by a deep,
religious belief in the underlying harmony and beauty
of the heavens. Let us not forget, though, that a seeking
after beauty had motivated Copernicus, as it had the
founders of the Ptolemaic system, who believed, with
Plato, that uniform circular motion was the only one
worthy of the perfection of the heavens. These aes-
thetic aspects of their work and the work of Kepler
need to be stressed, for just such seemingly nonscientific
considerations will be playing a crucial role in the
developments waiting to be told, and we shall see that
science in its highest manifestations is more akin to
art than to the popular misconceptions concerning its
nature.

III

In the seventeenth century Galileo and Descartes
adumbrated the law of inertia, which later became
Newton's first law of motion: every body continues in
its state of rest, or of uniform motion in a straight line,
unless it is compelled to change that state by forces
impressed upon it. That this had not been formulated
millennia earlier should not surprise us, for terrestrial
experience strongly suggests that bodies left to them-
selves come to rest, and that force is needed to maintain
them in motion. True, the celestial motions seemed to
continue indefinitely, but these motions were for the
most part circular, and it was natural for the Greeks
to believe that the heavens were subject to laws far
different from those that held sway on the earth.

It is hard to overestimate the importance of the first
law. Uniform motion in a straight line was now the
natural motion, needing no external cause. Bodies,
being possessed of an innate inertia, resisted change
of motion; and only change of motion demanded the
presence of external force. Because of this new view-
point Newton was able to create a conceptual system
that brought together the dynamics of the heavens and
the earth in a mighty synthesis built seemingly on just
his three laws of motion and his law of universal gravi-
tation.

But only seemingly. By themselves, Newton's laws
made no sense. Take the first law, for example. What
does the phrase “uniform motion in a straight line”
mean? Imagine a bead on a straight wire marked off
in inches. If the bead traverses equal distances along
the wire in equal times we can certainly claim that
it is moving uniformly in a straight line. But our claim
will be superficial and ill-founded. What, for instance,
if the clock with which we timed the bead had been
unreliable? Or the wire had been whirling and reeling
—say with the Keplerian earth?

Newton was acutely aware of such problems. In his
Principia, before stating his laws of motion, he carefully
prepared a conceptual setting in which they could take
on meaning. Saying disarmingly “I do not define time,
space, place, and motion [since they are] well known
to all,” he nevertheless proceeded to define their abso-
lute as distinguished from their relative aspects:

“Absolute, true, and mathematical time, of itself, and
from its own nature, flows equably without relation
to anything external....

“Absolute space, in its own nature, without relation
to anything external, remains always similar and im-
movable....”

These are basic assertions, not operational defini-
tions. For example, they provide no method of deciding
which of our clocks comes closest to ticking uniformly.

Spurred by penetrating criticisms by Berkeley and
Leibniz, Newton added a famous Scholium in a later
edition of the Principia. Here is a short excerpt: “[God]
is not eternity and infinity, but eternal and infinite;
he is not duration or space, but he... endures forever
and is everywhere present; and by existing always and
everywhere he constitutes duration and space.”

For Newton, absolute time and absolute space were
vividly present. Without them, as we have seen, his
laws would be meaningless. With them he could form
cosmic concepts of absolute rest, of absolute uniform
motion in a straight line, and of absolute deviations
from such motion.

By noting the centrifugal effects of rotation, among
them the concavity of the surface of rotating water
in his famous bucket experiment, Newton had con-
vinced himself that rotation is absolute, in powerful
agreement with his concept of absolute space. How-
ever, his laws of motion did not faithfully mirror the
absoluteness of their setting. To appreciate this, let us
begin with everyday experience. In a vehicle, we feel
no motion when the velocity is steady. We feel the
changes in motion—the accelerations or decelerations
—when the vehicle speeds up, or swerves, or jerks, or
slows down. If we look out of the window we can learn
of our relative motion, but when a sudden acceleration

throws us off balance we need no view of the scenery to
convince us that the ride has been unsteady.

Because of this, we sense that acceleration differs
significantly from velocity and from rest. But we have
been speaking in terrestrial terms. Newton's laws were
set in absolute space and absolute time, which cosmically
implied absolute rest, absolute velocity, and absolute
acceleration. Yet the laws, while making acceleration
(which term includes rotation) absolute, provided no
way of detecting absolute rest or absolute velocity. Ac-
cording to the laws, although acceleration was absolute,
both rest and velocity were, dynamically, always rela-
tive. Newton presented this as an almost immediate
consequence of his laws. His Corollary V reads “The
motions of bodies included in a given space [i.e. refer-
ence system] are the same among themselves whether
that space is at rest, or moves uniformly forward in a
straight line without any circular motion.” We shall
refer to this as the Newtonian principle of relativity,
though a better phrase might well be the Newtonian
dilemma. It troubled Newton.

Since his laws did not provide absolute location,
absolute rest, or absolute velocity, he introduced an
extraneous “Hypothesis I: That the center of the system
of the world is immovable.” This unmoving center
could not be the center of the sun, since the sun, pulled
by the planets this way and that, would be intricately
accelerated. A fortiori, no point primarily related to
the earth could fill the role of the fixed center of the
world.

The solar system did, however, have a theoretical
sort of balance point that we would now call its center
of mass; and Newton argued that according to his laws
the center of mass of the solar system would be unac-
celerated. It would thus be either at rest or in uniform
motion in a straight line—the laws could not say which.
Transcending his laws, Newton now declared that this
center of mass of the solar system, this abstract disem-
bodied point never far from the sun, was the center
of the world, and ipso facto immovable.

With the solar system pinned like a collector's but-
terfly to the immovable center of the world, absolute
location, rest, and velocity acquired human vividness.
Yet they did so only through Newton's ad hoc inter-
vention. Had Newton allowed the center of mass of
the solar system to move uniformly in a straight
line—as it had every right to do under the laws—there
would have been no dynamical effect of this motion.

A word of caution, though. In the above we have
followed Newton in ignoring the possible dynamical
effects of the distant stars.

IV

Having reached this stage, we may profitably regress
awhile. With a fixed earth the problem of relativity
could hardly arise. But in retrospect, once we accept
the idea of a moving earth, the very opposition to it
argues strongly in favor of a dynamical principle of
relativity. For if one could vividly feel the earth's
motion or intuitively recognize dynamical effects of
the motion, would men have been likely to have re-
garded the earth as fixed?

Evidently the earth's velocity has no noticeable
dynamical effect, and this is implicit in the Newtonian
principle of relativity. As for the earth's acceleration,
we realize in the light of Newton's theory that it does
have dynamical effects; but in everyday life these are
either too small to be noticed or else do not present
themselves to common sense as manifestations of the
acceleration. The path to the concept of a moving
earth had not been an easy one. Following Aristotle,
Ptolemy had argued powerfully against it, saying, for
example, that objects thrown in the air would be left
behind by a moving earth. He also argued that a rota-
ting earth would fly apart, to which argument Coper-
nicus retorted that Ptolemy should have worried rather
about the survival of the far larger sphere of the stars
if that sphere and not the earth were rotating once
a day.

Among the dynamical “proofs” advanced against the
hypothesis of a moving earth was that heavy bodies
when dropped ought to fall obliquely. By way of illus-
tration it was said that if one dropped a stone from
the top of the mast of a ship at rest it would land
at the foot of the mast, but if the ship was in rapid
motion the stone would “obviously” land closer to the
stern. Against this Galileo argued that the stone would
share the impetus of the moving ship and thus
(neglecting air resistance) would land at the foot of
the mast after all. In his Dialogues on the Two World
Systems he presents the point vividly in these words
of Salviati (emphasis added):

Galileo then has Sagredo drive the point home by
remarking:

The extent to which this is an anticipation of the
Newtonian principle of relativity needs clarification.
It can be interpreted in terms of the idea of inertia,
but on this Galileo was somewhat confused, being
unable wholly to emancipate himself from the Platonic
belief in circular inertia as the basic law. Certainly
the ship argument had powerful consequences. For
example, the parabolic motion of projectiles, a major
discovery of Galileo's, could have been deduced from
it right away. For if, relative to the moving ship, the
stone fell vertically with uniform acceleration, then as
viewed from the shore the path would indeed be
parabolic, being compounded of a vertical fall and a
uniform horizontal motion.

Some twenty years before the Principia appeared,
Huygens had used this principle of relativity brilliantly
in deducing laws of perfectly elastic impact by consid-
ering simple collisions taking place on shore and asking
how they would appear when viewed from a uniformly
moving boat. Indeed, as Huygens realized, the first law
of motion could have been deduced directly from the
Newtonian principle of relativity had that principle
been taken as basic. For a free body at rest in one
frame of reference would be moving uniformly as
viewed from a frame in uniform motion relative to
the first.

But Newton relegated this principle of relativity to
the minor role of a Corollary, and did his best to thwart
it, as we have seen. His intellectual and emotional need
for absolute space was overwhelming. How else could
he have had absolute acceleration? Besides, the
Galilean argument of the ships was not wholly satis-
factory. It compared phenomena on a stationary ship
with those on a ship in uniform motion, though such
ships could hardly exist on a spinning earth in orbit
around the sun.

As we have seen, Newton had avoided this sort of
difficulty by setting his laws in absolute space and time.
It is strange, therefore, that in commenting on his
Corollary V he himself used the illustration of station-
ary and uniformly moving ships. And this becomes even
more surprising when one notes that only a few pages
earlier, in defining absolute space, he had specifically
discussed how the motion of the earth is involved in
the absolute motion of a ship.

The Galilean argument of the ships can be defended.
Newton's laws imply that the uniform part of the
motion in absolute space of a ship or other reference
frame is not detectable within the reference frame.
Therefore, whatever the effects of the nonuniform part
of the absolute motion of the “stationary” ship, they
would be duplicated in the ship moving uniformly
relative to it. Thus in retrospect we may say that
Galileo, and later Huygens, did indeed have the New-
tonian principle of relativity, though they could not
have realized its Newtonian subtleties at the time.

V

Though Newton regarded action at a distance as
absurd, he was unable to find a satisfactory physical
model that would lead to his inverse square law of
gravitation. According to that law, every particle in
the Universe attracts every other particle with a gravi-
tational force that is utterly unaffected by intervening
matter. Or, to put it succinctly, gravitation does not
cast shadows.

Light does cast shadows, however, and this indicates
that it is something propagated. That it has finite speed
is by no means obvious. Important men like Kepler
and Descartes believed its speed infinite. Galileo's
pioneering experiment to measure its speed was incon-
clusive, and the first evidence that its speed was finite
came in 1676, when Roemer, to account for annual
variations observed in the rhythm of the eclipses of
the innermost moon of Jupiter, proposed that light is
propagated “gradually.” Since astronomical data
available at the time implied, if Roemer was correct,
a speed of some 130,000 miles per second, the word
“gradually” may sound like an understatement. Rela-
tivity will reveal in it an unexpected irony.

For the most part, Roemer's idea met with little
favor, though Huygens and Newton were among those
who took it seriously. Not till 1728 was independent
corroboration found of the finite yet stupendous speed
of light. In that year Bradley deduced from the aberra-
tion of light (tiny annual elliptic apparent motions of
the stars) a speed close to the currently accepted value
of some 186,300 miles per second. Since aberration has
an important role to play, we briefly describe its es-
sence. If we stand still, vertically falling rain falls on
our hat. If, remaining upright, we run forward, it
strikes our face. If we ran in a circle, the rain would
seem to come from an ever-changing direction always
somewhat ahead of us. Analogously, because of the
orbital motion of the earth, light from a star seems
to come to us from a position always somewhat ahead
of where we would see it if we were not orbiting. The
stars thus seem to move in tiny ellipses once a year,
and from the size of the effect Bradley calculated the
ratio of the orbital speed of the earth to the speed
of light.

The discovery that light has a finite speed was to
prove of world-shaking importance. It lies at the heart
of the modern theory of relativity, with all its conse-
quences. That the discovery came from astronomers,

as did the basis of Newton's law of gravitation, under-
lines the enormous practical consequences of the
astronomers' seemingly ivory-tower pursuits.

Ingenious laboratory methods have been devised for
measuring the speed of light with extraordinary preci-
sion, but the details need not concern us. It suffices
to know that the speed is finite and can be measured
in the laboratory.

VI

In optics Newton developed a powerful particle-
and-wave theory of light that, if misread, can seem
an extraordinary foreshadowing of the modern quan-
tum theory. His rejection of the pure wave theory
propounded by his contemporary Huygens and others,
and the superiority of his own theory in accounting
for the optical phenomena known at the time were
major reasons for the neglect of the wave theory during
the eighteenth century.

In the early nineteenth century, however, Young and
Fresnel brilliantly revived the wave theory and brought
it to victory over the prevalent particle theory. The
rise of the wave theory, with its ubiquitous aether as
the bearer of the waves, brought a threat to the New-
tonian principle of relativity, and one that Newton
would probably have welcomed. For aberration im-
plied an aether essentially undisturbed by the passage
of matter through it. The aether could thus be consid-
ered stationary, so that though mechanical experiments
were powerless, optical experiments had a chance to
succeed in detecting absolute rest and absolute velocity
—meaning now rest and velocity relative to the sta-
tionary, all-pervading aether.

The aether was not what one might reasonably con-
sider a credible concept. Because of the phenomenon
of polarization, light waves were taken to be trans-
verse, and the aether to be an elastic solid. Yet it had
to offer no perceptible impediment to the motions of
the planets, for these motions were in excellent accord
with Newton's system of mechanics. Nevertheless, since
the wave theory of light, developed in detail by
Fresnel, was as successful in encompassing the intricate
phenomena of optics as Newton's laws were in encom-
passing the intricate phenomena of celestial and ter-
restrial mechanics, the aether could hardly be ignored,
for all its conflicting properties.

If v is the speed of the earth through the aether
and c the speed of light, an experiment to detect the
quantity v/c is said to be of the first order, as distin-
guished from a second-order experiment designed to
detect v2/c2. A first-order experiment was soon per-
formed, but it failed to detect v/c. To account for the
failure, Fresnel proposed that matter carries aether
wholly entrapped within it yet allows aether to pass
freely through it. Moreover the amount of aether en-
trapped in, say, glass had to depend on the wavelength
of the light passing through it, so that if various wave-
lengths were present, as they certainly were, the
amount of entrapped aether was given by a self-
contradictory formula. Fresnel's extraordinary hypoth-
esis, which goes by the misleading name partial aether
drag, proved highly successful. Without hurting the
theory of aberration, it implied that every feasible
first-order experiment to detect the earth's motion
through the aether would fail. And since, over the
years, all such experiments did fail, Fresnel's hypothesis
had to be taken seriously. Indeed, it was confirmed by
difficult laboratory experiments on the speed of light
in streaming water.

VII

The experimenter Faraday, being unskilled in math-
ematics, created simple pictorial concepts to help him
interpret his pioneering researches in electromagnet-
ism. The theorists had been content to find mathe-
matical, action-at-a-distance formulas for the forces
exerted by magnets and electric charges. But Faraday
created a revolution in physics by consistently envi-
sioning a magnet or charge as surrounded by a “field”
of tentacle-like lines of force reaching through space,
so that all space became the domain of the important
aspects of electromagnetic phenomena.

Building on Faraday's work, Maxwell imagined an
electromagnetic aether with a pseudo-mechanical
structure so bizarre that he himself did not take it
seriously. Nevertheless he took it just seriously enough
to extract from it electromagnetic field equations that
play a key role in the development of the theory of
relativity. Since Maxwell required an electric current
—his crucial “displacement current”—in free space,
where there was no electric charge, his theory hardly
seemed credible to physicists. Yet the displacement
current gave an elegant mathematical symmetry to
Maxwell's equations and because of it his theory pre-
dicted the existence of transverse electromagnetic
waves moving with the speed of light; and when in
1888, nine years after his death, these waves were
detected by Hertz, Maxwell's theory could no longer
be easily resisted. It yielded a superb unification of the
hitherto disparate disciplines of optics and electro-
magnetism, with visible light occupying a narrow band
of wavelengths in a broad spectrum of electromagnetic
radiation. It also dispensed with electromagnetic action
at a distance by having electromagnetic effects trans-
mitted by the aether acting as intermediary. As for
the all-important aether, Maxwell's equations deline-
ated for it an inner structure that could not be en-
visaged in credible Newtonian mechanical terms.

Gradually physicists, becoming accustomed to its
mathematical properties, learned to live with it, and
an era of mechanistic physics faded.

VIII

By analogy with water waves and sound waves, and
more specifically because of Maxwell's equations, we
can expect light waves to travel through free aether
with a fixed speed. If, in our laboratory, we find that
light waves have different speeds in different directions,
we can conclude that we are moving through the
aether. Suppose, for example, that we find that their
greatest speed is 186,600 miles per second in this di-
rection → and their least speed 186,000 miles per
second in this direction ←. Then we can say that our
laboratory is moving through the aether in this direc-
tion → at 300 miles per second (half the difference
of the speeds), and that the light waves are travelling
through the aether at 186,300 miles per second (half
the sum). Thus we shall have discovered our absolute
velocity, and this despite Fresnel. But in speaking of
Fresnel's so-called aether drag, we said it implied that
every feasible first order experiment would fail. The
above is not feasible. The direct laboratory methods
of measuring the speed of light have involved not
one-way but round-trip speeds.

Shortly before his death, Maxwell outlined a way
to measure the earth's velocity through the aether by
comparing not one-way but round-trip speeds of light
in various directions in the laboratory. But since there
would be only a residual effect of the second order—if
v is the earth's orbital speed and the sun is at rest v2/c2
is about 10-8—he dismissed the effect as “far too small
to be observed.”

In 1881, however, Michelson succeeded in perform-
ing the experiment with borderline accuracy for de-
tecting the orbital speed. And in 1887, with Morley,
he repeated the experiment, this time with ample
accuracy. It gave a null result, and thereby precipitated
a crisis. For it suggested, and this was indeed Michel-
son's own interpretation, that the earth carries the
nearby aether along with it. But aberration implied
that the earth does not.

To resolve the conflict, FitzGerald, and later Lorentz
independently, proposed that objects moving through
the aether contract by an amount of the second order
in the direction of their motion.

Lorentz, assuming a fixed aether, untrapped and
undragged, had nevertheless obtained an electromag-
netic derivation of Fresnel's formula far more con-
vincing than that given by Fresnel. Thus Lorentz could
account for the null results of the feasible first-order
experiments to detect absolute motion. His task was
to express Maxwell's equations in a reference frame
moving uniformly through the aether with velocity v,
and to do so in such a way that, to the first order,
the v did not show up. But the Maxwell equations were
far from being pliable. In the moving frame they more
or less forced Lorentz to replace the t representing
the time by a new mathematical quantity that he called
“local time” because it was not the same everywhere.

By incorporating the contraction of lengths, he was
able to account for the null result of the Michelson-
Morley experiment without spoiling the theory of ab-
erration. But again the Maxwell equations forced his
hand, causing him to introduce with the contraction
a corresponding dilatation, or slowing down, of the
local time. Specifically, he found in 1904 what we now
call the Lorentz transformation, a name given it by
Poincaré in 1905. Consider two reference frames simi-
larly oriented, one at rest in the aether and the other
moving with uniform speed v in the common x-direc-
tion. Ordinarily one would have related the coordinates
(x, y, z) of the former to the coordinates (x′, y′, z′) of
the latter by what P. Frank named the Galilean trans-
formation:

. But the Lorentz transformation relates (x, y, z) and the
true time t to (x′, y′, z′) and the local time t′ of the
moving frame as follows:


By means of these equations, Lorentz succeeded, ex-
cept for a small blemish removed by Poincaré in 1905,
in transferring the Maxwell equations to the moving
reference frame in such a way that they remained
unchanged in form. Since no trace of the v survived,
neither the Michelson-Morley nor any other electro-
magnetic experiment could now be expected to yield
a value for v.

It is of interest that the Lorentz transformation, (2),
had already been obtained on electromagnetic grounds
by Larmor in 1898, and its essentials by Voigt on the
basis of wave propagation as early as 1887, the very
year of the Michelson-Morley experiment.

IX

The background has now been presented for Ein-
stein's accomplishments of 1905, which we shall con-
sider in conjunction with the accomplishments of
Poincaré. Along with the later fame of Einstein there
grew a popular mythology correctly attributing the
theory of relativity to him, but seriously slighting the
work of Poincaré. A considerable controversy was
created when Whittaker claimed that the 1905 theory

of relativity was due to Poincaré and Lorentz, with
Einstein playing a negligible role. Whittaker was justi-
fied in seeking to bring the situation into better per-
spective, but in his zeal he went too far, forsaking his
usually impeccable scholarship. This led to a counter-
reaction that has also sometimes gone too far. And
meanwhile the work of Larmor has received less rec-
ognition than it merits.

Maxwell led Larmor, Lorentz, and Poincaré to
mathematical equations identical with equations be-
longing to the theory of relativity. Poincaré had so
many of the crucial ideas that, in retrospect, it seems
amazing that he did not put them together to create
the theory of relativity. He raised aesthetic objection
to the piecemeal, ad hoc patching up of theory to meet
emergencies—Fresnel's entrapped aether to account
for the null results of first order experiments, and the
contraction to account for the second order experiment
of Michelson and Morley—and as early as 1895
Poincaré adumbrated a principle of relativity that
denied the possibility of detecting uniform motion
through the aether. His were the aesthetic strictures
that led Lorentz to seek a transformation to a moving
frame that would leave Maxwell's equations invariant
in form. Since, for example, the Lorentz contraction
factor √1 - v2/c2 reduces lengths to zero when
v = c, Lorentz had limited the application of his 1904
theory to systems moving through the aether with
speeds less than c; it was Poincaré who suggested in
1904 the need for a new dynamics in which speeds
exceeding c would be impossible. And in 1905 he wrote
a major article, sent in almost simultaneously with that
of Einstein, in which extraordinary amounts of the
mathematics of relativity are explicitly developed.

Einstein, in his epoch-making paper of 1905 “On
the Electrodynamics of Moving Bodies,” introduced a
new viewpoint. He began by discussing an aesthetic
blemish in electromagnetic theory as then conceived.
When a magnet and a wire loop are in relative motion,
there is an induced electric current in the wire. But
the explanation differed according as the magnet or
the wire was at rest. A moving magnet was accompa-
nied by an electric field that was not present when
the magnet was at rest and the wire moving. Thus what
was essentially one phenomenon had physically differ-
ent explanations within the same theory.

Because the phenomenon depended on the relative
motion of magnet and wire and not on any absolute
motion through the aether, and because experiments
to detect motion through the aether had given null
results, Einstein postulated as a basic principle that
there is no way of determining absolute rest or uniform
motion—he worded it more technically—and he called
it the principle of relativity, as Poincaré had done.

The phrase was not wholly new with Poincaré. In
1877 Maxwell, in his little book Matter and Motion,
had spoken of “the doctrine of relativity of all physical
phenomena,” which he proceeded to explain in these
eloquent words (emphasis added): “There are no land-
marks in space; one portion of space is exactly like
every other portion, so that we cannot tell where we
are. We are, as it were, on an unruffled sea, without
stars, compass, soundings, wind or tide, and we cannot
tell in what direction we are going. We have no log
which we can cast out to take a dead reckoning by;
we may compute our rate of motion with respect to
the neighboring bodies but we do not know how these
bodies may be moving in space.”

It is surprising that these words should have come
from Maxwell. Not only did he build his electromag-
netic field theory on the concept of an aether but, in
later propounding the idea that led to the Michelson-
Morley experiment, he was envisaging the light waves
that ruffle the ethereal sea as a means for determining
our motion through the aether. It is not clear precisely
what Maxwell had in mind when speaking of the rela-
tivity of all physical phenomena. There is in the phrase
an echo of the views of Berkeley, of which more later.
Perhaps Maxwell was not here regarding the aether
as kinematically synonymous with absolute space. But
later in the book, citing Newton's bucket experiment
and the Foucault pendulum, he specifically contradicts
the “all” by affirming the absoluteness of rotation.

Poincaré's concept of the principle of relativity, like
Einstein's, went beyond what, for convenience, we
have been referring to as the Newtonian principle of
relativity. That principle referred to the impossibility
of detecting one's absolute uniform motion by dynam-
ical means. The new principle, while retaining the
restriction to uniform motion, extended the impossi-
bility to include the use of all physical means, particu-
larly the optical. Yet it is fair to say that in Newton's
time, in the absence of a generally accepted wave
theory of light, the Newtonian principle of relativity
could have been thought of as implying the impotence
of all physical phenomena to detect one's absolute
uniform motion. If so, the Newtonian principle, after
a period of grave doubt as to its validity, was now
being reaffirmed. But, as will appear, its reaffirmation
in the Maxwellian context played havoc with funda-
mental tenets of Newtonian mechanics.

In speaking of the principle of relativity, Poincaré
had an aether in mind. But Einstein declared that in
his theory the introduction of an aether would be
“superfluous” since he would not need an “absolute
stationary space.” Moreover, unlike Poincaré, Einstein
audaciously treated the principle of relativity as a
fundamental axiom suggested by the experimental hints

already available but not in itself subject to question.

Einstein next introduced a second postulate: that the
speed of light in vacuo is constant and independent
of the motion of its source—again he expressed it more
technically. In terms of an aether, this postulate seems
almost a truism. For a wave, once generated, is on its
own. It has severed its connection with the sources that
gave it birth and moves according to the dictates of
the medium through which it travels.

On these two principles Einstein built his theory.
Each by itself seemed reasonable and innocent. But,
as Einstein well realized, they formed an explosive
compound. This is easy to see, especially if, for con-
venience, we begin by talking in terms of an aether.
Imagine two unaccelerated spaceships, S and S′, far
from earth and in uniform relative motion (Figure 3).
In S and S′ are lamps L, L′ and experimenters E, E′
as indicated. Assume that S happens to be at rest in
the aether. E measures the speed with which the light
waves from L pass him and obtains the value c. In
S′ a similar measurement is performed by E′ using light
waves from L′. What value does E′ find? Since the
speed of the light waves is independent of the motion
of their sources, the waves from L and L′ keep pace
with one another. And since S′ is moving towards the
waves with speed v we expect him to find that they
pass him with speed c + v. But the principle of rela-
tivity forbids this. For if E′ found the value c + v, he
could place another lamp at the opposite end of S′
and measure the speed of the light waves in the oppo-
site direction, obtaining the value c - v. By taking half
the difference of these values he could find his speed
through the aether, in violation of the principle of
relativity. Therefore he must obtain not c + v, nor
c - v, but simply c, no matter how great his speed
v relative to S, or indeed relative to any source of light
towards which, or away from which, he is moving.

Viewing this without reference to the aether, we see
from Einstein's two postulates that no matter how fast
we travel towards or away from a source of light, the
light waves will pass us with the same speed c. Clearly
this is impossible within the context of Newtonian
physics. Either we must give up the first postulate or
else give up the second. But Einstein retained both,
and found a way to keep them in harmony by giving
up instead one of our most cherished beliefs about the
nature of time.

X

Einstein reexamined the concept of simultaneity.
Accepting it as intuitively clear for events occurring
at the same place, he asked what meaning could be
given to it when the events were at different places.
Realizing that this must be a matter of convention,
he proposed a definition that we shall now illustrate.

Imagine the spaceships S, S′ equipped with identi-
cally constructed clocks fore and aft, as shown (Figure
4). Pretend that c is small, or that the spaceships are
of enormous length, so that we can use convenient
numbers in what follows. When clock C1 reads noon,
E sends a light signal from C1 to C2 where it is reflected
back to C1. Suppose that the light reached clock C2
when C2 read 1 second after noon, and returned to
C1 when C1 read 3 seconds after noon. Then Einstein
would have E say that his clocks C1 and C2 were not
synchronized. To synchronize the clocks Einstein
would have E advance C2 by half a second so that
according to the readings of C1 and C2 the light would
take equal times for the outward and return journeys.
With C1 and C2 thus synchronized, if events occurred
at C1 and C2 when these clocks read the same time,
the events would be deemed simultaneous.

In 1898 Poincaré had already questioned the concept
of simultaneity at different locations, and in 1900 he

had considered the use of light signals for adjusting
clocks in a manner strikingly similar to that used by
Einstein in 1905. However, Poincaré was concerned
with adjusting rather than synchronizing clocks. More-
over, he did not build on two kinematical postulates
but worked in terms of the Maxwell equations; nor
did he take the following step, and it is things such
as these that set Einstein's work sharply apart.

Consider E and E′ synchronizing the clocks in their
respective spaceships S and S′. E arranges C1 and C2
so that they indicate equal durations for the forward
and return light journeys, and declares them synchro-
nized. But E′, watching him from S′, sees S moving
backwards with speed v relative to him. Therefore,
according to S′, the light signals sent by E did not travel
equal distances there and back (Figure 5), but unequal
distances (Figure 6). And so, according to E′, the very
fact that C1 and C2 indicated equal durations for the
forward and backward light journeys showed that C1
and C2 were not synchronized.

However, E′ has synchronized his own clocks, C′1,
C′2, according to Einstein's recipe, and E says they are
not synchronized, since relative to S the light signals
used by E′ travel unequal distances there and back.

With E and E′ disagreeing about synchronization
we naturally ask which of them is correct. But the
principle of relativity, as Einstein rather than Poincaré
viewed it, forbids our favoring one over the other. E
and E′ are on an equal footing, and we have to regard
both as correct. Since events simultaneous according
to E are not simultaneous according to E′, and vice
versa, Einstein concluded that simultaneity is relative,
being dependent on the reference frame. By so doing
he gave up Newtonian universal absolute time.

Previously we spoke of measurements of the one-way
speed of light as not being feasible, deliberately sug-
gesting by the wording that this might be because of
practical difficulties. The lack of feasibility can now
be seen to have a deeper significance. To measure the
one-way speed of light over a path AB in a given
reference frame we need synchronized clocks at A and
B by which to time the journey of the light. With the
synchronization itself performed by means of light, the
measurement of the one-way speed of light becomes,
in principle, a tautology. The mode of synchronization
is a convention permitting the convenient spreading
of a time coordinate over the reference frame one uses.
What can be said to transcend convention (with apolo-
gies to conventionalists) is the rejection of Newtonian
absolute time, with its absolute simultaneity.

Once the concept of time is changed, havoc spreads
throughout science and philosophy. Speed, for exam-
ple, is altered, and acceleration too, and with it force,
and work, and energy, and mass, so that we wonder
if anything can remain unaffected.

Not even distance remains unscathed, as is easily
seen. Imagine the spaceships S and S′ marked off in
yard lengths by E and E′ respectively. E measures a
yard length of S′ by noting where the ends of the yard
are at some particular time. Since E and E′ disagree
about simultaneity, E′ accuses E of noting the positions
of the yard marks non-simultaneously, thus obtaining
an incorrect value for the length. When the roles are
reversed, E similarly accuses E′. Because of the princi-
ple of relativity, both are adjudged correct. Thus once

simultaneity is relative, so too is length. Indeed, the
disagreement as to lengths corresponds in magnitude
to the FitzGerald-Lorentz contraction, but here it is
a purely kinematical effect of relative motion and not
an absolute effect arising from motion relative to a fixed
aether. While E says that the yards of E′ are con-
tracted, E′ says the same about those of E.

Let x, y, z, and t denote the coordinates and syn-
chronized clock times used by E in his spaceship refer-
ence frame S; and let x′, y′, z′, and t′ denote the corre-
sponding quantities used by E′ in S′. Einstein derived
directly from his two postulates a mathematical rela-
tion between these quantities, and it turned out to be
the Lorentz transformation (2). The interpretation,
though, was different, because E and E′ were now on
an equal footing: t′, for example, was just as good a
time as t.

Because of the principle of relativity, the rela-
tionships between E and E′ are reciprocal. We have
already discussed this in connection with lengths. It
is instructive to consider it in relation to time: when
E and E′ compare clock rates each says the other's
clocks go the more slowly. This, like the reciprocal
contractions of lengths, is immediately derivable from
the Lorentz transformation in its new interpretation.
It can be understood more vividly by giving E and
E′ “clocks,” each of which consists of a framework
holding facing parallel mirrors with light reflected
tick-tock between them. Each experimenter regards his
own clocks as using light paths as indicated (Figure 8).
But because of the relative motion of E and E′, we
have the situation shown (Figure 9). Since these longer
light paths are also traversed with speed c, each ex-
perimenter finds the other's clocks ticking more slowly
than his own. Indeed, a simple, yet subtle, application
of Pythagoras' theorem to the above diagrams yields
the mutual time dilatation factor √ 1 - v /c2. More-
over, since E′ and E, in addition to agreeing to dis-
agree, agree about the speed of light, each says that
the other's relative lengths are contracted in the
direction of the relative motion by this same factor
√1 - v/c2, for otherwise the ratio distance/time for
light would not be c for both.

A further kinematical consequence is easily deduced
directly from Einstein's postulates. We begin by noting
that no matter how fast E′ moves relative to E, light
waves recede from him with speed c: he cannot over-
take them. But these light waves also move with speed
c relative to E. Therefore E′ cannot move relative to
E with a speed greater than c. Nor can any material
object relative to any other: c is the speed limit. (It
has been proposed that particles exist that move faster
than light. They have been named tachyons. According
to the Lorentz transformation, tachyons can never
move slower than light; their speed exceeds c in all
reference frames.)

XI

An immediate victim of relativity was Newton's law
of gravitation with its instantaneous action at a dis-
tance; for with simultaneity relative, one could no
longer accept a force acting with absolute simultaneity
on separated bodies. We can safely ignore the routine
modifications of Newton's law that were proposed to
let it fit into the relativistic framework; Einstein's by
no means routine theory of gravitation will be de-
scribed later.

That the startling relativistic kinematics, of which
we have just seen samples, did not also play havoc with
Maxwell's equations need not surprise us. Larmor,
Lorentz, and Poincaré had shown the intimate rela-
tionship between Maxwell's equations and the Lorentz
transformation. We can now appreciate the achieve-
ments of Fresnel and Maxwell: Fresnel's self-contra-
dictory “aether drag” was a relativistic effect, as too
was Maxwell's displacement current. Little wonder
that these concepts had seemed incredible. They were
valiant attempts to fit relativistic effects into the
kinematics of Newtonian absolute space and absolute
time. In retrospect the work of Fresnel and Maxwell
takes on that aspect of inspired madness that is the
highest form of the art we call science. So too does

the work of Einstein, for his theory of 1905 was itself
built on a contradiction: its basic principles assumed
reference frames made out of rigid rods while denying
their possibility. For a rigid rod would transmit impacts
instantly and could be used to synchronize clocks in
a manner conflicting with that proposed by Einstein.

XII

Relativistic kinematics required a relativistic dy-
namics for which Newtonian concepts were not well
suited. But the success of Newtonian dynamics for
speeds small compared with c had created habits of
thought that could not be easily broken. Accordingly,
Einstein and others sought to distort Newtonian con-
cepts to fit relativistic kinematics. Mass became rela-
tive, increasing in value with increasing relative speed.
Thus the greater the relative speed of an object, the
greater its inertial resistance to change in its speed.
If the object could attain speed c its mass would be-
come infinite, and no increase in speed would be possi-
ble. While this sounds like dynamics and uses Newto-
nian concepts, it is basically a reflection of the existence
of the speed limit c, which, as we saw, is an immediate
kinematic consequence of Einstein's two postulates.
(For ordinary matter and radiation, c is the upper speed
limit. For tachyons, if they exist, c is the lower speed
limit. In either case, c is a speed limit.)

In a second paper on relativity in 1905 Einstein
made a daring extrapolation. He began by showing
mathematically that if a body gives off an amount of
energy L in the form of electromagnetic radiation, its
mass decreases by L/c2. Now came these momentous
words: “The fact that the energy... [is] energy of
radiation evidently makes no difference.” Therefore,
Einstein concluded, all energy, of whatever sort, has
mass. And herein lay the germ of the famous equation
E = mc2.

In 1907 Einstein completed the derivation by a
further daring step. Arguing that a body of mass m
has the same inertia as an amount of energy mc2, and
that one should not make a distinction between “real”
and “apparent” mass, he concluded that all mass should
be regarded as a reservoir of energy. At the time, and
for many years after, there was not the slightest direct
experimental evidence for this, yet Einstein not only
asserted the equivalence of mass and energy, but rec-
ognized it in 1907 as a result of extraordinary theoret-
ical importance.

XIII

In 1907 Minkowski showed in detail that the natural
habitat of the equations of relativity is a four-dimen-
sional “space-time,” an idea already explicitly fore-
shadowed by Poincaré in 1905.

The Galilean transformation (1) exhibits the aloof
absoluteness of Newtonian time. Though t enters the
transformation of x (and, more generally, of y and of
z), it itself remains untouched: one does not even bother
to write t′ = t. In the Lorentz transformation (2), x
mixes with t as intimately as t does with x; and in more
general Lorentz transformations x, y, z, and t thor-
oughly intermingle.

In ordinary analytical geometry, if a point P has
coordinates (x, y, z) its distance, OP, from the origin,
O, is given by

If we rotate the reference frame about O to a different
orientation, the coordinates of P change, say to (x′, y′,
z′), but the value of the sum of their squares remains
the same:

OP2 = x′2 + y′2 + z′2 = x2 + y2 + z2.

Under the Lorentz transformation (2) there is an
analogous quantity s such that
s2 = x′2 + y′2 + z′2 - c2t′2    = x2 + y2 + z2 - c2t2.

The analogy with (4), already close, can be made even
closer by introducing τ = √ -1 ct, τ′ = √ -1 ct′,
for now

s2 = x′2 + y′2 + z′2 + τ′2    = x2 + y2 + z2 + τ2.

and the Lorentz transformation (2) can be envisaged
as a change to a new four-dimensional reference frame
obtained by rotating the first about O to a different
orientation. While (6) may give us initial confidence
that relativity pertains to a four-dimensional world in
which time is a fourth dimension, the nature of this
four-dimensional world is more vividly seen by avoid-
ing √-1 and returning to (5).

Let E in his spaceship S press button A on his instru-
ment panel, and a minute later, according to his clock,
press a neighboring button B; and let us refer to these
pressings as events A and B. According to E, the spatial
distance between events A and B is a matter of inches.
According to E′, because of the rapid relative motion
of S and S′, events A and B are separated by many
miles; also, according to E′, who says the clocks in S
go more slowly than his own, the time interval between
events A and B is very slightly longer than a minute.
The importance of (5) is that, despite these disparities,
it affords a basis of agreement between E and E′. If
each calculates for events A and B the quantity
ds2 = (spatial distance)2 - (time interval)2
he will get the same result as the other. The large

discrepancy in the spatial distances is offset by the very
small discrepancy in the time intervals, this latter being
greatly magnified by the factor c2.

Take two other events: E switching on a lamp in
S, and the light from the lamp reaching a point on
the opposite wall. Here (7) gives ds = 0 for both E
and E′, since for each of them the distance travelled
by light is the travel time multiplied by c.

The quantity ds is the relativistic analogue of dis-
tance, but the effect of the minus sign in (5) is drastic.
This is easily seen if we ignore two spatial dimensions,
use x and ct as coordinates and try to fit the resulting
two-dimensional Minkowskian geometry onto the fa-
miliar Euclidean geometry of this page. We draw a
unit “circle,” all of whose points are such that the
magnitude of ds2 equals 1. Because ds = 0 along the
lines OL, OL′ given by x = ±ct, this “circle” obviously
cannot cut these lines. It actually has the shape shown,
consisting of two hyperbolas (Figure 10). When we add
a spatial dimension the lines OL, OL′ blossom into a
cone. When we add a further spatial dimension, so that
we have the x, y, z, ct of the four-dimensional Min-
kowski world, the cone becomes a three-dimensional
conical hypersurface—do not waste time trying to
visualize it. Since it represents the progress of a wave-
front of light sent out from O, it is called the light
cone; there is one at each point of Minkowski space-
time.

Because a particle has duration, it is represented not
by a point but a line, called its world line. If it is at
rest relative to the reference frame used, its world line
is parallel to the ct axis. If it moves relative to the
frame, its world line slants away from the ct direction,
the greater the speed the greater the slant. Since the
speed cannot exceed c—we are ignoring the possi-
bility of tachyons here—the world line must remain
within all the light cones belonging to the points on it.

An event M within the light cone at O can be
reached by an influence from event O moving with
a speed less than c, and can thus be caused by the
event O. It turns out that in all reference frames, event
M is later than event O.

An event N outside the light cone at O cannot be
similarly reached: the speed would have to exceed c.
Thus O could not cause N. This is intimately related
to the theorem that in some reference frames O is
earlier than N and in others it is later. We have here
been using the word “cause” rather loosely. The con-
cept of causality poses enormous problems, but the
situation here is superficially simple: if for some exper-
imenters O is earlier than N while for others it is later,
we are not likely to regard it as a possible cause of N.

The light cone at an event O separates space-time
into three regions: the absolute future of O, the abso-
lute past of O, and a limbo that is neither the one nor
the other.

In Minkowski space-time the mutual contractions of
yardsticks and the mutual slowing of clocks become
mere perspective foreshortenings. Also, the hitherto
unrelated laws of conservation of energy and momen-
tum become welded together into a single space-time
law. As for the hard-won Maxwell equations, they take
on a special elegance. One could almost have obtained
them uniquely by writing down the simplest nontrivial
equations for a four-dimensional mathematical quantity
(called an antisymmetric tensor of the second order)
that combines electricity and magnetism into a single
Minkowskian entity. These are but samples of the
beauty of the theory in its Minkowskian setting. Space
does not permit a discussion of the many triumphs of
the theory of relativity, either by itself or when applied
to the quantum theory.

No matter in what theory, the symbol t is at best
a pale shadow of time, lacking what, for want of better
words, we may call time's nowness and flow. In treating
time as a fourth dimension, Minkowski presented the
bustling world as something static, laid out for all
eternity in frozen immobility. This geometrization of
time, however, was crucial for the development of
Einstein's general theory of relativity, of which we must
now tell.

XIV

The absence of absolute rest and of absolute uniform
motion becomes intuitively acceptable if we assume
that space is featureless. In that case, though, how
could there be absolute acceleration?

Berkeley, in Newton's day, had insisted that all mo-
tion must be relative and that absolute space was a
fiction. As for the seemingly absolute centrifugal effects

of rotation, he argued that they must indicate not
absolute rotation but rotation relative to the stars.
Towards the end of the nineteenth century, Mach
subjected the Newtonian theory to a searching epis-
temological analysis that was to have a profound effect
on Einstein. Amplifying Berkeley's kinematical views,
Mach gave them dynamical substance by proposing
that inertia—which gives rise to the seemingly absolute
effects of rotation and other types of acceleration—is
due to a physical interaction involving all matter in
the universe. In Newton's theory, acceleration was
referred to absolute space. Thus absolute space had
inertial dynamical effects on bodies, yet despite New-
ton's third law that to every action there is an equal
opposite reaction, there was no corresponding reaction
by the bodies on absolute space. This anomalous, one-
way dynamical influence of absolute space on matter
was aesthetically and epistemologically unpleasant. Yet
Einstein's theory of relativity suffered from an analo-
gous defect. It had replaced Newton's absolute space
and absolute time by a space-time in which, though
the essence of the Newtonian principle of relativity
was retained, acceleration was nevertheless absolute.

As early as 1907 Einstein was attacking the problem
of acceleration. Aesthetically, one would like to extend
the principle of relativity not just kinematically but
physically to include all motion. But despite the pro-
posals of Berkeley and Mach, experience and experi-
ment had hitherto seemed sharply against this. Follow-
ing the dictates of aesthetics, Einstein was able to show
how experiment could be made to serve the ends of
beauty. His weapon was the well-known observation,
going back to Galileo and earlier, that all dropped
bodies fall to the earth with the same acceleration g
(neglecting air resistance and assuming everyday
heights). Newton had incorporated this by giving mass
two roles to play: inertial and gravitational. The gravi-
tational pull of the earth on a body was proportional
to the mass of the body, and thus to its inertia. The
larger the mass, the larger the pull but also the larger
the inertia, with the result that the acceleration re-
mained independent of the mass.

That the gravitational mass of a body should be
proportional to its inertial mass was an extraneous
assumption having no inherent Newtonian raison
d'être. Einstein made it a cornerstone of his new the-
ory.

Starting in purely Newtonian terms, Einstein
imagined a laboratory K, far removed from external
gravitational influence, moving with uniform acceler-
ation g as indicated. He compared it with a similar
laboratory K′ at rest in a uniform gravitational field
which, for convenience, we may pretend is furnished
by the earth (Figure 11).

In K′ all free bodies fall with acceleration g. Where
K is, though, all free bodies are unaccelerated; but
because of the “upward” acceleration of K, they “fall”
relative to K with acceleration g. It is a simple exercise
in Newtonian mechanics to show that, so far as purely
mechanical experiments within K and K′ are con-
cerned, there is no way of distinguishing between K
and K′.

Now came the stroke of genius: Einstein propounded
a principle of equivalence stating that no experiment
of any sort within the laboratories could distinguish
between K and K′. At once this permitted a general
principle of relativity embracing all motion, for if an
experimenter in K, or in K′, could no longer determine
the extent to which physical effects were due to
uniform acceleration and to what extent to uniform
gravitation, acceleration need no longer be regarded
as absolute. Indeed, acceleration was now seen to be
intimately linked to gravitation. In addition, the equal-
ity of gravitational and inertial mass took on the aspect
of a truism. For, consider equal particles suspended
from equal springs in K and K′. In K, because of the
acceleration, the inertia of the particle causes the
stretching of the spring. In K′, there being no acceler-
ation, inertia does not come into play. Instead the
stretching is due to the gravitational mass of the parti-
cle. By the principle of equivalence, one cannot distin-
guish between these gravitational and inertial effects.

Suppose, further, that each particle absorbs energy,
thus gaining inertial mass. Since the spring in K is now
extended further than before, so too, according to the
principle of equivalence, must the spring in K′ be. Thus
the inertial mass of the energy must also have an
equivalent gravitational mass.

If one looks too closely at the principle of equiva-
lence as Einstein initially used it one finds inconsisten-
cies. Yet its fertility was extraordinary. Consider, as a
further example of this, a ray of light sent laterally
across K. The acceleration of K causes the path of the
light to appear to be curved “downwards” relative to
K. Therefore light rays must be correspondingly bent

by the gravitational field in K′. Moreover, just as the
bending of a light ray passing from air to glass implies
a decreased speed of propagation of the light waves,
so too does the gravitational bending of light rays imply
a slowing down by gravitation of the speed of light.
Thus the 1905 theory, now called the special theory
of relativity, could hold only approximately in the
presence of gravitation.

Again, let C1, C2, C′1, C′2, as shown in K and K′, be
“standard” clocks, by which we mean that they are
ticking at identical rates. At each tick of C1 a light
signal is sent from C1 towards C2. Because K is moving
faster and faster, each light signal has farther to travel
than its predecessor to reach the receding C2. So the
light signals reach C2 separated by greater time inter-
vals than the time intervals separating the ticks of C2.
When thus compared by means of light signals, there-
fore, clock C1, which ticks at the same rate as clock
C2, nevertheless seems to be going more slowly than
C2. The principle of equivalence now requires that the
same shall hold for C′1 and C′2 in K′, so that standard
clock C′1 seems to go more slowly than standard clock
C′2 because of gravitation. Einstein argued that the
spectral frequencies of light emitted by atoms can be
regarded as standard time-keepers, and thus as substi-
tutes for C′1 and C′2. Therefore spectral lines arriving
at C′2 from C′1 would have lower frequencies than those
in the spectra produced locally by C′2, which would
mean that they were shifted towards the red end of
the spectrum. This is the famous gravitational red shift.
But the most important lesson to be learned here is
that gravitation warps time.

XV

At this stage we must pause to consider the imposing
edifice of Euclidean geometry on which Newton and
Maxwell had based their theories. The Greeks had built
it on idealized concepts like sizeless points and
breadthless lines, and postulates concerning them. The
naturalness of these postulates so deeply impressed
Kant that he regarded Euclidean geometry as inescap-
able and existing a priori. Yet, from the start, Euclid's
fifth postulate had caused disquiet. In context it implied
that through a point P not on a line l there is one
and only one line parallel to l. Because parallelism
entered the dangerous realm of infinity, where intuition
is particularly fallible, numerous attempts were made
to avoid the fifth postulate or deduce it from the other
postulates.

In 1733 Saccheri sought a reductio ad absurdum
proof of the postulate by assuming it untrue, and
managed to convince himself that the consequences
were unacceptable. However, in the early nineteenth
century, Gauss, Lobachevsky, and Bolyai inde
pendently made a momentous discovery: that if one
denies the fifth postulate by assuming more than one
straight line through P parallel to l, a viable geometry
results. Later, Riemann found a different non-Euclid-
ean geometry in which there are no parallel lines. Thus
Euclidean geometry could no longer be logically re-
garded as God-given or existing a priori.

The Cartesian coordinates indicated by the familiar
uniform net of lines on ordinary graph paper have two
properties of interest: first, the squares are all of unit
size, so that for two neighboring points with coordi-
nates (x, y) and (x + dx, y + dy) the coordinate differ-
ences dx and dy give direct measures of distances; and
second, by Pythagoras' theorem, the distance ds be-
tween the two points is given by

ds2 = dx2 + dy2.

If we change to a coordinate mesh of wavy, irregu-
larly-spaced lines, the new dx and dy will not give
direct measures of distance, and (8) will take the more
complicated form

ds2 = g11dx2 + 2g12dxdy + g22dy2,

where, in general, the values of the coefficients g11,
g12, g22 change from place to place. This complexity
arises from our perversity in distorting the coordinate
mesh. But often such distortion is unavoidable: for
example, we cannot spread the familiar graph-paper
mesh, without stretching, on a sphere, though we can
on a cylinder. In studying the geometry of surfaces,
therefore, Gauss spread on them quite general coordi-
nate meshes having no direct metrical significance and
worked with formula (9), though with different nota-
tion. Moreover, he found a mathematical quantity, now
called the Gaussian curvature of a surface, that is of
major importance. If this curvature is zero everywhere
on the surface, as it is for a plane or a cylinder or
any other shape that unstretched graph paper can take,
one can spread a coordinate mesh on the surface in
such a way that (8) holds everywhere, in which case
the intrinsic two-dimensional geometry of the surface
is essentially Euclidean. If the Gaussian curvature is
not everywhere zero, one cannot find such coordinates,
and the intrinsic two-dimensional geometry is not
Euclidean. The crux of Gauss's discovery was that the
curvature, being expressible in terms of the g's, is itself
intrinsic, and can be determined at any point of the
surface by measurements made solely on the surface,
without appeal to an external dimension.

This powerful result led Riemann to envisage intrin-
sically curved three-dimensional spaces; and, thus
emboldened, he considered intrinsically curved spaces
of higher dimensions. In three and more dimensions
the intrinsic curvature at a point, though still expres-

sible solely in terms of the corresponding g's, is no
longer a single number but has many components
(involving six numbers in three dimensions, and twenty
in four). It is represented by what we now call the
Riemann-Christoffel curvature tensor and denoted by
the symbol Rabcd.

Gauss had already concluded that geometry is a
branch of theoretical physics subject to experimental
verification, and had even made an inconclusive
geodetic experiment to determine whether space is
indeed Euclidean or not. Riemann, and more specifi-
cally Clifford, conjectured that forces and matter might
be local irregularities in the curvature of space, and
in this they were strikingly prophetic, though for their
pains they were dismissed at the time as visionaries.

XVI

We now return to Einstein. It took him ten years
to find the way from the special theory of relativity
of 1905 to the general theory of relativity. To arrive
at the general theory he had first to realize that
yardsticks and standard clocks could not be used to
lay out in space-time a coordinate mesh of the Car-
tesian sort that would directly show distances and
time intervals.

This radical break with his previous habits of thought
was, by his own admission, one of his most difficult
steps towards the general theory of relativity. A pow-
erful stimulus was the effect of gravitation on the
comparison of clock rates as deduced from the princi-
ple of equivalence. Another was the following argu-
ment: Consider a nonrotating reference frame K and
a rotating reference frame K′ having the same origin
and z-axis. On the xy-plane of K, draw a large circle
with its center at the origin. By symmetry, it will be
regarded as a circle in K′. Measure it in K′ with a
measuring chain, and view the process from the nonro-
tating frame K. Relative to K, the chain will appear
contracted in length when the circumference is being
measured, but not when the diameter is being
measured. Therefore the circumference, as measured
by the shrunken links, will have a greater value than
that given by a similar measuring chain at rest in K.
So the ratio of circumference to diameter as measured
in K′ will be greater than π, which means that the
spatial geometry in K′ is non-Euclidean.

That this argument can be faulted is of small conse-
quence. It served its purpose well. Einstein seems to
have known intuitively the path he had to follow and
then to have found plausible, comforting arguments
that would give him the courage to proceed. In the
nature of things, he could not use impeccable argu-
ments since they had to be based on theories that the
general theory was destined to supersede.

What was important was Einstein's valid conclusion
that space-time coordinates could not, in general, have
direct metrological significance. Faced with this shat-
tering realization, and bolstered by his conviction that
all motion must be relative, Einstein decided that all
coordinate systems in space-time must be on an equal
footing. He therefore enunciated a principle of general
covariance according to which the general laws of
nature are to be expressed by equations that hold good
for all systems of space-time coordinates. Three points
need to be made concerning this principle:

(a) A general system of space-time coordinates could
consist of cheap, inaccurate, unsynchronized clocks
embedded in a highly flexible scaffolding in wild and
writhing motion. The principle relegates the role of
coordinates to that of the mere labelling of events in
space-time, much as the general coordinates of Gauss
label the points of a surface. To be able to accept such
general four-dimensional coordinates as a basis for a
physical theory, Einstein had first to arrive at a pro-
found insight: that physical measurements are essen-
tially the observation of coincidences of events, such
as the arrival of a particle when the hands of the local
clock point to certain marks on its dial. Such coinci-
dences clearly remain coincidences no matter what
coordinate system is used.

(b) The principle of general covariance can be said
to be devoid of content. As Kretschmann pointed out
in 1917, any physical theory capable of being expressed
mathematically in terms of coordinates can be ex-
pressed in a form obeying the principle of general
covariance.

(c) Nevertheless the principle was a cornerstone of
the general theory of relativity.

This seeming paradox is resolved when one takes
account of Einstein's powerful aesthetic sense, which
made the general theory a thing of beauty. If one uses
a simple reference frame in the special theory of rela-
tivity, the space-time interval ds between events (x,
y, z, t) and (x + dx, y + dy, z + dz, t + dt) is given by

ds2 = dx2 + dy + dz2 - c2dt2.

If one goes over to a more complicated reference frame
writhing and accelerated relative to the former, (10)
takes a more complicated form analogous to (9),
namely

ds2 = g00dt2 + g11dx2 + g22dy2 + g33dz2
+ 2g01dtdx + ag02dtdy + 2g03dtdz + 2g12dxdy +
2g13dxdz + 2g23dydz,

where the values of the ten g's change from place to
place in space-time. These ten coefficients, by which
one converts coordinate differences into space-time
distances, are denoted collectively by the symbol gab

and are referred to as components of the metrical tensor
of space-time. A convenient mathematical shorthand
lets (11) be written in the compact form

ds2 = gabdxadxb.

With the principle of equivalence Einstein had
linked gravitation with acceleration and thus with
inertia. Since acceleration manifests itself in gab, so too
should gravitation. Einstein therefore took the mo-
mentous step of regarding gab as representing gravita-
tion, and by this act he gave gravitation a geometrical
significance. In assigning to the metrical tensor a dual
role, he did more than achieve an aesthetically satisfy-
ing economy in the building material of his theory.
For he was now able to force the seemingly empty
principle of general covariance to take on powerful
heuristic content and lead him directly to his goal. This
he had done instinctively, since Kretschmann's argu-
ment came only after the theory was formulated. How
the principle of general covariance lost its seeming
impotence will be explained later.

The mathematical tool, now called the Tensor Cal-
culus, for writing equations valid for all coordinate
systems had already been created by Ricci (he started,
interestingly enough, in the year 1887 that saw the
Michelson-Morley experiment and Voigt's introduction
of a transformation akin to that of Lorentz).

Einstein therefore sought tensor equations for the
law of gravitation, and ultimately imposed three con-
ditions: (a) that in free space the equations should
involve only tensors formed from the metrical tensor
and its first and second derivatives, (b) that the equa-
tions, ten in number, should be linear in the second
derivatives of the ten g's (so as to keep as close as
possible to the highly successful Newtonian theory, the
basic equation of which was linear in the second deriv-
atives of a single gravitational potential), and (c) that
the equations be linked by four relations corresponding
to the law of conservation of energy and momentum
(four relations being anyway necessary mathematically
to ensure that the equations have nontrivial solutions,
as was pointed out by Hilbert).

What is remarkable is that the intricate equations,
which involve millions of terms, were now essentially
uniquely determined. Naturally, they come in compact
notation. From the components of the four-dimensional
Riemann-Christoffel curvature tensor Rabcd, combina-
tions are formed denoted by Rab (the Ricci tensor) and
R (the curvature scalar). The totality of matter, stress,
radiation, etc. acting as the “sources” of the gravita-
tional field is denoted by Tab. Then Einstein's field
equations for gravitation can be written

Rab - 1/2gabR = -Tab.

XVII

We may now consider the general theory of relativ-
ity in terms of its own concepts rather than the tenta-
tive, groping concepts on which it was built. It treats
gravitation as an intrinsic curvature of space-time, the
special theory of relativity becoming a limiting case
valid in regions small enough for the effects of the
curvature to be negligible. The special theory, like the
Newtonian theory, can be expressed in terms of tensors,
in conformity with the principle of general covariance.
But the Kretschmann process of making equations
generally covariant usually involved introducing addi-
tional physical quantities. The principle of general
covariance took on importance when Einstein argued
that gravitation per se must be represented solely in
terms of the metrical tensor gab, without the introduc-
tion of additional physical quantities (other than the
sources Tab). This did more than link gravitation with
geometry: it forced the seemingly impotent principle
of general covariance to impose limitations so powerful
that the complicated field equations of gravitation
could be obtained essentially uniquely.

In linking inertia, via acceleration, to gravitation,
Einstein extended the ideas of Berkeley and Mach by
regarding inertia as a gravitational interaction. Ac-
cordingly he gave the name “Mach's Principle” to the
requirement that gab, which defines the geometry of
space-time, should be determined solely by the gravi-
tational sources Tab. Ironically, Einstein's theory turned
out not to embrace Mach's principle unequivocally.
To avoid this irony Einstein proposed a desperate
remedy that did not work. Nevertheless the attempt
led him to a major development that will not be con-
sidered here since it belongs to, and indeed inaugurates,
the subject of relativistic cosmology.

In the special theory of relativity, as in the theory
of Newton, space and time are unaffected by their
contents. In the general theory space and time are no
longer aloof. They mirror by their curvature the gravi-
tational presence of matter, energy, and the like.
Geometry—four-dimensional—thus becomes, more
than ever before, a branch of physics; and space-time
becomes a physical entity subject to field laws.

The problem of action at a distance no longer arises.
Space-time itself is the mediator—the “aether”—and,
in three-dimensional parlance, gravitational effects are
propagated with speed c. Also, the self-contradiction
in the special theory regarding the use of rigid rods
does not apply so harshly to the general theory, since
coordinate meshes are no longer constructed of rigid
rods and standard clocks.

In Newton's theory the law of inertia states that a
free particle moves in a straight line with constant
speed. This law holds, also, in the special theory of

relativity, where it is expressed by saying that a free
particle has a straight world line in Minkowski space-
time. Einstein essentially carried this law over into the
curved space-time of the general theory by postulating
that the world line of a simple free particle therein
is a geodesic, the closest available analogue of a straight
line. The law now acquired powerful new significance.
Consider, for example, the curved space-time associ-
ated with the gravitational field of the sun. Calculation
showed that the geodesics of particles representing
planets corkscrew around the world line of the sun in
such a way that, in three-dimensional language, the
particles move around the sun in curves very closely
approximating ellipses with the sun at a focus, their
orbital speeds varying in the Keplerian manner. Re-
member: we are speaking of “free” particles. Thus
there is no longer need to introduce a gravitational
force. Newton's first law, the law of inertia, when
adapted to Einstein's curved space-time, itself suffices
to account for the gravitational influence of the sun
on the motions of the planets. Indeed, all the triumphs
of the Newtonian theory are inherited by the theory
of Einstein.

But Einstein's theory went further than Newton's.
It accounted for a previously puzzling residual advance
of the perihelion of Mercury by some 43 seconds of
arc per century. Moreover it implied the gravitational
bending of light rays (giving twice the value that
Einstein had obtained by his preliminary argument
using the principle of equivalence) and also the gravi-
tational red shift of spectral lines (giving essentially
the value he had obtained from the principle of equiv-
alence). Observations confirm the existence of these
effects, but there has been a fluctuating discussion as
to the extent to which the observations are in numerical
agreement with the predictions. Major technological
advances in the half-century since the theory was
formulated have brought within range of measurement
not only more precise evaluations of the above effects
but also other effects hitherto beyond the reach of
observation. Of particular interest is Weber's apparatus
designed to detect gravitational waves.

XVIII

An orbiting astronaut feels weightless. Does this
mean that he has zero weight? Some physicists say no.
They define weight as the pull of gravitation, and argue
that the astronaut is not free of the gravitational pull
of the earth and other bodies. The astronaut, they say,
feels weightless because inertial effects balance the
gravitational pull.

Actually the concept of weight is by no means easy
to define satisfactorily even in Newtonian terms. In the
general theory of relativity, with gravitation and iner-
tia linked by the principle of equivalence, and with
gravitational pull replaced by space-time curvature,
the concept of weight becomes quite subtle.

Perhaps one may say that the gist of the situation
is this: the astronaut, being in free fall around the earth,
is tracing out a geodesic world line in space-time and
not only feels weightless but also has zero weight.
When the rockets of his spaceship are firing, the astro-
naut, being no longer in free flight, departs from trac-
ing out a geodesic. Accordingly, he acquires weight,
and with it the sensation of having weight.

A man (regarded here as a point) when standing on
the earth does not trace out a geodesic world line. But
he does, momentarily, if he jumps. Thus we reach the
somewhat startling conclusion that in the course of his
jump the man has zero weight.

Since the world line of an astronaut in flight differs
from that of his twin on the ground, the relativistic
lengths, s, of the portions of their world lines between
departure and return are clearly unequal. Since these
lengths happen to measure the amounts of time the
twins have lived between meetings, the twins will not
be the same age when the astronaut returns. By imag-
ining flights that are not yet feasible, one can infer
spectacular possibilities: for example, the astronaut
returning to find himself twenty years younger than
his stay-at-home twin. Much fuss has been stirred up
by this so-called “paradox” of the twins. But so far
as the theory of relativity is concerned, it is no more
paradoxical than that the total length of two sides of
a triangle is not equal to the length of the third. No
useful purpose will be served in discussing the matter
further here, except for the following remark: The
astronautical twin would seem to have a longer rather
than a shorter world line than his stay-at-home brother,
and thus one might expect him to be the older rather
than the younger on his return. Actually his world line,
as measured relativistically, is the shorter. We have to
take account of the sort of distortion already en-
countered when we tried to draw a Minkowskian unit
circle on a Euclidean page.

XIX

By treating gravitation as space-time curvature,
Einstein had geometrized a major branch of physics.
In 1918 Weyl sought to carry this process of geometri-
zation further. In curved space-time, where we have
to make do with geodesics as substitutes for straight
lines, directions are affected by the curvature. Weyl
devised a more general space-time geometry in which
not only directions but also lengths are affected; and
he showed how one could thereby obtain the equations
of Maxwell in a natural way alongside those of Einstein.
Unfortunately, as Einstein pointed out, the idea en-
countered physical difficulties.

Weyl's ingenious attempt was one of the first of a

long succession of unified field theories seeking to link
gravitation and electromagnetism geometrically. One
trend was initiated in 1921 by Kaluza, who proposed
a five-dimensional theory that was later given a four-
dimensional interpretation. Another trend, growing
from Weyl's work, involved introducing various geo-
metrical features, such as torsion, directly into space-
time, a notable example being the theory, based on
an unsymmetric gab, on which Einstein was working
at the time of his death.

Since electromagnetism has energy, it has a gravita-
tional effect. In 1925 Rainich showed that electromag-
netism leaves so characteristic a gravitational imprint
on the curvature of space-time that the curvature itself
can suffice to represent electromagnetism as well as
gravitation. In this sense, the general theory of relativ-
ity could be said to be “already unified.”

Because of the exuberant proliferation of unified field
theories of gravitation and electromagnetism and their
failure to yield new physical insights comparable to
those of the special and general theories of relativity,
there arose a tendency to deride attempts to reduce
physics to geometry by means of unified field theories.
This tendency was enhanced when atomic physicists
discovered additional fundamental fields, even though,
in principle, the discovery of these fields made the
problem of unification, if anything, more urgent.
Whether the path to unification will be via the
geometrization of physics is a moot point since one
cannot define the boundaries of geometry. Thus,
nuclear physicists use geometrical concepts with strik-
ing success in attempting to unify the theory of funda-
mental particles, but these geometrical concepts are
not confined to space-time.

It is worth remarking that there have been highly
successful unifications lying within the special theory
of relativity. For example, Dirac found relativistic
equations for the electron that not only contained the
spin of the electron as a kinematical consequence of
Minkowskian geometry but also linked the electron to
the not-then-detected positron, thus initiating the con-
cept of antimatter. Minkowski, in his four-dimensional
treatment of Maxwell's equations, had already created
an elegant unified field theory of electricity and mag-
netism long before the term “unified field theory” was
coined. And we can hardly deny that the special theory
of relativity is itself a unified theory of space and time,
as too is the general theory.

XX

Transcending the triumphs of Einstein's theory is its
monumental quality. This quality is manifest in the
naturalness and seeming inevitability of the theory's
growth, the beauty and structural simplicity of its
architecture, and the interlocking economy of its basic
hypotheses. Indeed, this economy proved to be even
more impressive than was originally believed: the
geodesic hypothesis was found not to be needed after
all, the motions of bodies being inherent in the field
equations themselves. This discovery, by Einstein and
collaborators among others, revealed the general the-
ory of relativity as unique among field theories in that
all others had to be supplemented by special rules
linking the motions of bodies to the field.

Like every physical theory, the general theory of
relativity faces great epistemological and internal diffi-
culties. Among the latter are solutions of its equations
that seem like physical nonsense, and powerful
theorems discovered by Penrose, Hawkins, and others
indicating that its equations carry the taint of unavoid-
able breakdown.

More important are the epistemological difficulties,
especially in relation to the quantum theory. Bohr was
of the opinion that there was no need to apply quantum
concepts to the general theory of relativity; he re-
garded the latter as an essentially macroscopic theory
linked to the macroscopic aspects of matter. Other
physicists, however, have sought to quantize Einstein's
gravitational theory much as Maxwell's theory of light
has been quantized. In the latter the electromagnetic
field is regarded as consisting of quantum-mechanical
particles called photons. Accordingly one attempts to
treat the gravitational field as consisting of quantum-
mechanical particles to which the name graviton has
been given. Two difficulties arise. The first has to do
with the sheer complexity of Einstein's field equations
(13) when written out in detail. While this inner com-
plexity underscores Einstein's genius in obtaining the
equations essentially uniquely, it also prevents a
straightforward application of the familiar techniques
of quantization. The second difficulty is more funda-
mental: if one quantizes gravitation in Einstein's the-
ory, one automatically quantizes the metrical tensor
and thus the very basis of space-time geometry. The
epistemological problems posed by a quantized geom-
etry are formidable indeed.

Even on a more superficial level the quantum theory
raises deep problems concerning measurement in the
general theory of relativity. The light cones, which are
crucial ingredients of the geometry of space-time, are
defined directly by gab and represent the propagation
of infinitely sharp pulses of light. But, as Einstein
realized, such pulses would involve infinitely high fre-
quencies and thus, according to the quantum relation
E = hv (energy equals Planck's constant times fre-
quency) infinitely high energies. These in turn would
imply, among other calamities, infinitely large gravita-
tional curvatures not present in the original gab.

From the two basic constants, the speed of light and
the Newtonian gravitational constant, that enter the

general theory of relativity, we cannot form a quantity
representing a length. The theory thus has no built-in
scale of size. If we introduce Planck's constant, how-
ever, we can form a fundamental length. It turns out
to be 10-33cm. The diameter of an atomic nucleus is
enormously larger, being of the order 10-13cm.
Wheeler has therefore proposed that space-time, so
seemingly smooth, has a spongy structure of enormous
complexity when envisaged at the 10-33cm level.

It may well be that the general theory of relativity
is a macroscopic theory that breaks down at the mi-
croscopic level. Or that it survives there in an almost
unrecognizable foamlike form. But all our basic physi-
cal theories suffer from a common malaise: even when
they seek to avoid the idea of a space-time continuum,
they use x, y, z, t in their equations and treat them
as continuous quantities. The reason is simple: no one
has yet found a satisfactory way of doing without them.

Perhaps, as Einstein hoped, space and time are fun-
damental things out of which matter is made. Perhaps,
though, matter, or something else, is the more funda-
mental, with space and time mere macroscopic reflec-
tions of its deeper regularities. Fundamental or not,
space and time remain the very essence of our being.
And Einstein's theory, for all its problems and whatever
its fate, will endure as a towering masterpiece in one
of the most difficult and demanding of art forms: theo-
retical physics.

BIBLIOGRAPHY

Galileo Galilei, Dialogue on the Great World Systems, ed.
G. de Santillana (Chicago, 1953), is a classic; this edition
contains illuminating editorial notes. Newton's Principia,
trans. A. Motte, 2 vols. (Berkeley, 1962), also has editorial
notes but unfortunately no index. Of other pre-relativity
works, the following are particularly relevant to this article:
J. C. Maxwell, Matter and Motion (London, 1877; reprint,
New York, n.d.), see in particular pp. 80-88; also E. Mach,
The Science of Mechanics, trans. T. J. McCormack, 6th ed.
(La Salle, Ill., 1960).

The collection of essays in Albert Einstein: Philosopher-
Scientist, ed. P. A. Schilpp (Evanston, 1949), is indispensable;
the book contains Einstein's scientific autobiography and
a bibliography of his writings. Key technical papers in the
development of Einstein's theory are reprinted in The Prin-
ciple of Relativity (1923); reprint, New York, n.d.). One of
the most important historical surveys is E. T. Whittaker's
A History of the Theories of Aether and Electricity (1910;
New York, 1951; and later reprint), see especially Vol. II.
This highly technical work is unfortunately biased in its
account of the special theory of relativity, though not of
the general theory. For an antidote, see the article by G.
Holton “On the Origins of the Special Theory of Relativity,”
American Journal of Physics, 28 (1960), 627-36; but in this
connection see the article by C. Scribner, Jr. “Henri
Poincaré and the Principle of Relativity,” ibid., 32 (1964),
672-78.

W. Pauli's highly mathematical Theory of Relativity,
trans. G. Field (London, 1959), was famous in 1921 as an
encyclopedia article and remains outstanding in its updated
form. It contains an abundance of references.

Of the books of medium mathematical difficulty, M.
Born's Einstein's Theory of Relativity (New York, 1962;
various reprints), is of particular interest here because of
its historical approach. Note that it scants the work of
Poincaré, as do other books listed below. Complementing
Born's book, and admirable in its own right, is W. Rindler's
Essential Relativity (New York, 1969). Einstein's book, The
Meaning of Relativity, 5th ed. (Princeton, 1956), is particu-
larly recommended to those who are able to follow the
mathematics.

As to nontechnical books, nobody has bettered Einstein's
own popular exposition, Relativity (New York, 1920; later
reprints). P. W. Bridgman, A Sophisticate's Primer of Rela-
tivity (Middletown, Conn., 1962; reprint New York), gives
important insights into the special theory but unfortunately
assumes without analysis the existence of rigid rods. M.
Jammer, Concepts of Space (Cambridge, Mass., 1954; 2nd ed.
1969), contains numerous references. A. D'Abro, The Evolu-
tion of Scientific Thought (New York, 1927; reprint 1950),
gives a detailed nonmathematical account of the whole
development of the theory. Also recommended are A.
Einstein and L. Infeld, The Evolution of Physics (New York,
1938; also reprint), and P. Bergmann, The Riddle of Gravi-
tation (New York, 1968). P. Frank, Einstein: His Life and
Times (New York, 1947; 1953), contains both biographical
and philosophical material of particular interest. For J. A.
Wheeler's engrossing, if speculative, ideas about foamlike
space and the concept of superspace, see his book (in
German) Einsteins Vision (Berlin, 1968). Mention may also
be made of Banesh Hoffmann, with Helen Dukas, Albert
Einstein, Creator and Rebel (New York, 1972).

BANESH HOFFMANN

[See also Causation; Cosmic Images; Cosmology since 1850;
Mathematical Rigor; Matter; Space; Time and Measure-
ment; Unity of Science.]