Dictionary of the History of Ideas Studies of Selected Pivotal Ideas |

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I. | RELATIVITY |

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Dictionary of the History of Ideas | ||

#### 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

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]

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

that space is at rest, or moves uniformly forward in a

straight line without any circular motion.

refer to this as

*the Newtonian principle of relativity,*

though a better phrase might well be

*the Newtonian*

dilemma.It troubled Newton.

dilemma.

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):

Shut yourself up with some friend in the largest room below

decks of some large ship and there procure gnats, flies, and

such other small winged creatures. Also get a great tub full

of water and within it put certain fishes; let also a certain

bottle be hung up, which drop by drop lets forth water

into another narrow-necked bottle placed underneath.

Then, *the ship lying still,* observe how these small winged

animals fly with like velocities towards all parts of the room;

how the fishes swim indifferently towards all sides; and how

the distilling drops all fall into the bottle placed underneath.

And casting anything towards your friend, you need not

throw it with more force one way than another, provided

the distances be equal... [Now] make the ship move with

what velocity you please, *so long as the motion is uniform.*

... You shall not be able to discern the least alteration

in all the forenamed effects, *nor can you gather by any
of them whether the ship moves or stands still.*...

Galileo then has Sagredo drive the point home by

remarking:

... I remember that being in my cabin I have wondered

a hundred times whether the ship moved or stood still; and

sometimes I have imagined that it moved one way, when

it moved the other way....

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,

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 *v*2/*c*2. 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.

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 *v*2/*c*2

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:

*xʹ*=

*x*-

*vt*,

*yʹ*=

*y*,

*zʹ*=

*z*

. 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:

*x*′ = (

*x*-

*vt*)/√1 -

*v*2/

*c*2,

*y′*=

*y*,

*z*=

*z*

*t′*= (

*t*-

*vx*/

*c*2/√1 -

*v*/

*c*2.

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

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 - *v*2/*c*2 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

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

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

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*
/*c*2. 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*/*c*2, 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

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/c*2. 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 = mc*2.

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 *mc*2, 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

*x*2 +

*y*2 +

*z*2.

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 = *x*2 + *y*2 + *z*2.

Under the Lorentz transformation (2) there is an

analogous quantity *s* such that

s2 = *x*′2 + *y*′2 + *z*′2 - *c*2*t*′2
= *x*2 + *y*2 + *z*2 - *c*2*t*2.

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
= *x*2 + *y*2 + *z*2 + 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

*ds*2 = (spatial distance)2 - (time interval)2

he will get the same result as the other. The large

small discrepancy in the time intervals, this latter being

greatly magnified by the factor

*c*2.

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 *ds*2 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

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

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

of relativity,

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

*ds*2 = *dx*2 + *dy*2.

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

*ds*2 = *g*11*dx*2 + 2*g*12*dxdy* + *g*22*dy*2,

where, in general, the values of the coefficients *g*11,

*g*12, *g*22 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-

*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

*R*abcd.

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

*ds*2 = *dx*2 + *dy* + *dz*2 - *c*2*dt*2.

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

*ds*2 = *g*00*dt*2 + *g*11*dx*2 + *g*22*dy*2 + *g*33*dz*2

+ 2*g*01*dtdx* + a*g*02*dtdy* + 2*g*03*dtdz* + 2*g*12*dxdy* +

2*g*13*dxdz* +
2*g*23*dydz*,

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 *g*ab

*metrical tensor*

of space-time. A convenient mathematical shorthand

lets (11) be written in the compact form

*ds*2 = *g*ab*dx*a*dx*b.

With the principle of equivalence Einstein had

linked gravitation with acceleration and thus with

inertia. Since acceleration manifests itself in *g*ab, so too

should gravitation. Einstein therefore took the mo-

mentous step of regarding *g*ab 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 *R*abcd, combina-

tions are formed denoted by *R*ab (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 *T*ab. Then Einstein's field

equations for gravitation can be written

*R*ab - 1/2*g*ab*R* = -*T*ab.

*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 *g*ab, without the introduc-

tion of additional physical quantities (other than the

sources *T*ab). 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 *g*ab, which defines the geometry of

space-time, should be determined solely by the gravi-

tational sources *T*ab. 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

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

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

*g*ab, 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 *g*ab 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 = h*v* (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 *g*ab.

From the two basic constants, the speed of light and

the Newtonian gravitational constant, that enter the

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

ciple of Relativity

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),

tion of Scientific Thought

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,

tation

*Einstein: His Life and*

Times(New York, 1947; 1953), contains both biographical

Times

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).

Einstein, Creator and Rebel

BANESH HOFFMANN

[See also Causation; Cosmic Images; Cosmology since 1850;Mathematical Rigor; Matter; Space; Time and Measure-

ment; Unity of Science.]

Dictionary of the History of Ideas | ||