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

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Dictionary of the History of Ideas | ||

*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.

Dictionary of the History of Ideas | ||