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XIII

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

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

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

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

OP2 = x′2 + y′2 + z′2 = x2 + y2 + z2.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

XIV

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

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

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

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

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

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

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

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

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

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

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

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

XV

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

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

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

ds2 = dx2 + dy2.

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

ds2 = g11dx2 + 2g12dxdy + g22dy2,

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

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

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

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

XVI

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

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

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

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

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

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

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

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

ds2 = dx2 + dy + dz2 - c2dt2.

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

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

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

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

ds2 = gabdxadxb.

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

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

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

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

Rab - 1/2gabR = -Tab.

XVII

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

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

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

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

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

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

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

XVIII

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

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

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

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

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

XIX

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

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

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

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

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

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

XX

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

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

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

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

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

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

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

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