
"Relativity
and the
Problem of Space" Albert
Einstein (1952) English
translation published
1954
IT
is
characteristic of Newtonian physics that it has to ascribe independent
and real existence to space and time as well as to matter, for in Newton's
law of motion the idea of
acceleration appears. But in this
theory, acceleration can only denote "acceleration with respect to
space". Newton's space must thus be
thought of as "at
rest", or at
least as "unaccelerated", in order that one can consider the
acceleration, which appears in the law of motion, as being a magnitude
with any meaning. Much the same holds with time, which of course
likewise enters into the concept of acceleration. Newton
himself and
his most critical
contemporaries felt it to be
disturbing that one had to ascribe physical reality both to space
itself as well as to its state of motion; but there was at that time no
other alternative, if one wished to ascribe to mechanics a clear
meaning. It is indeed an exacting requirement to
have to
ascribe
physical
reality to space in general, and especially to empty space. Time and
again since remotest times philosophers have resisted such a
presumption. Descartes argued somewhat on
these
lines: space is
identical with extension, but extension is connected with bodies; thus
there is no space without bodies and hence no empty space. The weakness
of this argument lies primarily in what follows. It is certainly true
that the concept extension owes its origin to our experiences of laying
out or bringing into contact solid bodies. But from this it cannot be
concluded that the concept of extension may not be justified in cases
which have not themselves given rise to the formation of this concept.
Such an enlargement of concepts can be justified indirectly by its
value for the comprehension of empirical results. The
assertion that extension is confined to bodies
is
therefore of
itself certainly unfounded. We shall see later, however, that the
general theory of relativity confirms Descartes'
conception in a
roundabout way. What brought Descartes
to his
remarkably attractive view was certainly
the feeling that, without compelling necessity, one ought not to
ascribe reality to a thing like space, which is not capable of being
"directly experienced". The psychological origin
of the idea of space, or
of
the necessity for it, is far from being so obvious as it may appear to
be on the basis of our customary habit of thought. The old geometers
deal with conceptual objects (straight line, point, surface), but not
really with space as such, as was done later in analytical geometry.
The idea of space, however, is suggested by certain primitive
experiences. Suppose that a box has been constructed. Objects
can be arranged
in a
certain way inside
the box,
so that it
becomes full. The possibility of such arrangements is a property of the
material object "box", something that is given with the box, the "space
enclosed" by the box. This is something which is different for
different boxes, something that is thought quite naturally as being
independent of whether or not, at any moment, there are any objects at
all in the box. When there are no objects in the box, its space appears
to be "empty". So far, our concept
of space has been
associated
with the box. It turns out, however, that the storage possibilities
that make up the boxspace are independent of the thickness of the
walls of the box. Cannot this thickness be reduced to zero, without the
"space" being lost as a result? The naturalness of such a limiting
process is obvious, and now there remains for our thought the space
without the box, a selfevident thing, yet it appears to be so unreal
if we forget the origin of this concept. One can understand that it was
repugnant to Descartes to consider space
as
independent of material
objects, a thing that might exist without matter. (At the
same
time, this does not prevent him from treating space as a fundamental
concept in his analytical geometry.) The drawing of attention to the
vacuum in a mercury barometer has certainly disarmed the last of the
Cartesians. But it is not to be denied that, even at this primitive
stage, something unsatisfactory clings to the concept of space, or
to space thought of as an independent real thing. The
ways in which bodies can be packed into space
(e.g.
the box) are
the subject of threedimensional Euclidean geometry, whose axiomatic
structure readily deceives us into forgetting that it refers to
realisable situations. If now the concept of space
is formed
in the manner
outlined above, and following on from experience about the "filling" of
the box, then this space is primarily a bounded space. This limitation
does not appear to be essential, however, for apparently a larger box
can always be introduced to enclose the smaller one. In this way space
appears as something unbounded. I shall not
consider here how the
concepts of the
threedimensional and the Euclidean nature of space can be traced back
to relatively primitive experiences. Rather, I
shall consider first of all from other
points
of view the
rôle of the concept of space in the development of physical thought.
When a smaller box s is
situated, relatively at
rest,
inside the hollow
space of a larger box S, then
the hollow space of s is a part of the
hollow space of S, and the
same "space", which contains both of them,
belongs to each of the boxes. When s is in motion
with respect to S,
however, the concept is less simple. One is then inclined to think that
s encloses always the same space, but
a variable part of the space S.
It then becomes necessary to apportion to each box its particular
space, not thought of as bounded, and to assume that these two spaces
are in motion with respect to each other. Before
one has become aware of this
complication,
space appears as an unbounded medium or container in which
material objects swim around. But it must now be remembered that there
is an infinite number of spaces, which are in motion with respect to
each other. The concept of space as something
existing
objectively
and independent
of things belongs to prescientific thought, but not so the idea of the
existence of an infinite number of spaces in motion relatively to each
other. This latter idea is indeed logically
unavoidable,
but is
far from
having played a considerable rôle even in scientific thought.
But what about the psychological
origin of the
concept of time? This concept is undoubtedly associated with the fact
of "calling to mind", as well as with the differentiation between sense
experiences and the recollection of these. Of itself it is doubtful
whether the differentiation between sense experience and recollection
(or simple representation) is something psychologically directly given
to us. Everyone has experienced that he has been in doubt whether he
has actually experienced something with his senses or has simply dreamt
about it. Probably the ability to discriminate between these
alternatives first comes about as the result of an activity of the mind
creating order. An experience is associated with a
"recollection",
and
it is considered
as being "earlier" in comparison with present "experiences". This is a
conceptual ordering principle for recollected experiences, and the
possibility of its accomplishment gives rise to the subjective concept
of time, i.e. that concept of time which refers to the arrangement of
the experiences of the individual. What do we mean
by rendering
objective the concept
of time? Let us consider an example. A person A
("I") has the
experience "it is lightning". At the same time the person A
also
experiences such a behaviour of the person B as brings the
behaviour of B
into relation
with his own experience "it is lightning". Thus it
comes about that A
associates
with B the
experience "it is
lightning".
For the person A
the idea
arises that other persons also participate in
the experience "it is lightning". "It is lightning" is now no longer
interpreted as an exclusively personal experience, but as an experience
of other persons (or eventually only as a "potential experience"). In
this way arises the interpretation that "it is lightning", which
originally entered into the consciousness as an "experience", is now
also interpreted as an (objective) "event". It is just the sum total of
all events that we mean when we speak of the "real external world".
We have seen that we feel ourselves impelled to
ascribe
a temporal
arrangement to our experiences, somewhat as follows. If b is later than a and c later than b then c is also later than
a
("sequence of
experiences"). Now what is the position in this
respect with the
"events" which we
have associated with the experiences? At first sight it seems obvious
to assume that a temporal arrangement of events exists which agrees
with the temporal arrangement of the experiences. In general, and
unconsciously this was done, until sceptical doubts made themselves
felt. In order to arrive at the idea of an objective world,
an
additional constructive concept still is necessary: the event is
localised not only in time, but also in space. In
the previous paragraphs we have attempted to
describe
how the
concepts space, time and event can be put psychologically into relation
with experiences. Considered logically, they are free creations of the
human intelligence, tools of thought, which are to serve the purpose of
bringing experiences into relation with each other, so that in this way
they can be better surveyed. The attempt to become
conscious of the empirical
sources
of these
fundamental concepts should show to what extent we are actually bound
to these concepts. In this way we become aware of our freedom, of
which, in case of necessity, it is always a difficult matter to make
sensible use. We still have something essential to
add to this
sketch concerning the psychological origin of the concepts
spacetimeevent (we will call them more briefly "spacelike", in
contrast to concepts from the psychological sphere). We have linked up
the concept of space with experiences using boxes and the arrangement
of material objects in them. Thus this formation of concepts already
presupposes the concept of material objects (e.g. ''boxes"). In the
same way persons, who had to be introduced for the formation of an
objective concept of time, also play the rôle of material objects in
this connection. It appears to me, therefore, that the formation of the
concept of the material object must precede our concepts of time and
space. All these spacelike concepts already belong
to
prescientific thought,
along with concepts like pain, goal, purpose, etc. from the field of
psychology. Now it is characteristic of thought in physics, as of
thought in natural science generally, that it endeavours in principle
to make do with "spacelike" concepts alone, and strives to express
with their aid all relations having the form of laws. The
physicist seeks to reduce colours and tones to vibrations, the
physiologist thought and pain to nerve processes, in such a way that
the psychical element as such is eliminated from the causal nexus of
existence, and thus nowhere occurs as an independent link in the causal
associations. It is no doubt this attitude, which considers the
comprehension of all relations by the exclusive use of only spacelike
concepts as being possible in principle, that is at the present time
understood by the term "materialism" (since "matter" has lost its rôle
as a fundamental concept). Why is it necessary to
drag down from the Olympian
fields of Plato the
fundamental ideas of thought in natural science, and to attempt to
reveal their earthly lineage? Answer: in order to free these ideas from
the taboo attached to them, and thus to achieve greater freedom in the
formation of ideas or concepts. It is to the immortal credit of D.
Hume
and E. Mach that they, above all others,
introduced
this critical
conception. Science has taken over from
prescientific thought
the concepts space, time, and material object (with the important
special case "solid body") and has modified them and rendered them more
precise. Its first significant accomplishment was the development of
Euclidean geometry, whose axiomatic formulation must not be allowed to
blind us to its empirical origin (the possibilities of laying out or
juxtaposing solid bodies). In particular, the threedimensional nature
of space as well as its Euclidean character are of empirical origin (it
can be wholly filled by like constituted "cubes"). The
subtlety of the concept of space was enhanced
by the
discovery that
there exist no completely rigid bodies. All bodies
are elastically deformable and alter in
volume with
change in temperature. The structures, whose possible congruences are
to be described by Euclidean geometry, cannot therefore be represented
apart from physical concepts. But since physics after all must make use
of geometry in the establishment of its concepts, the empirical content
of geometry can be stated and tested only in the framework of the whole
of physics. In this connection atomistics
must also be
borne in mind, and its conception of finite divisibility; for spaces of
subatomic extension cannot be measured up. Atomistics
also compels us to give up, in
principle, the
idea of
sharply and statically defined bounding surfaces of solid bodies.
Strictly speaking, there are no precise laws, even in the macroregion,
for the possible configurations of solid bodies touching each other.
In spite of this, no one thought of giving up the
concept of space, for
it appeared indispensable in the eminently satisfactory whole system of
natural science. Mach,
in the
nineteenth century, was
the only one who thought seriously
of an elimination of the concept of space, in that he sought to replace
it by the notion of the totality of the instantaneous distances between
all material points. (He made this attempt in order to arrive at a
satisfactory understanding of inertia). The Field IN
Newtonian
mechanics, space and time play a dual rôle. First, they
play the part of carrier or frame for things that happen in physics, in
reference to which events are described by the space coordinates and
the time. In principle, matter is thought of as consisting of "material
points", the motions of which constitute physical happening. When
matter is thought of as being continuous, this is done as it were
provisionally in those cases where one does not wish to or cannot
describe the discrete structure. In this case small parts (elements of
volume) of the matter are treated similarly to material points, at
least in so far as we are concerned merely with motions and not with
occurrences which, at the moment, it is not possible or serves no
useful purpose to attribute to motions (e.g. temperature changes,
chemical processes). The second rôle of space and time was
that of
being an
"inertial
system". From all conceivable systems of reference, inertial systems
were considered to be advantageous in that, with respect to them, the
law of inertia claimed validity. In this, the
essential thing is that
"physical
reality", thought of as being independent of the subjects experiencing
it, was conceived as consisting, at least in principle, of space and
time on one hand, and of permanently existing material points, moving
with respect to space and time, on the other. The idea of the
independent existence of space and time can be expressed drastically in
this way: If matter were to disappear, space and time alone would
remain behind (as a kind of stage for physical happening). The
surmounting of
this standpoint
resulted from a
development which, in the first place, appeared to have nothing to do
with the problem of spacetime, namely, the appearance of the concept
of field and its final claim to replace, in principle, the idea of a
particle (material point). In the framework of classical physics, the
concept of field appeared as an auxiliary concept, in cases in which
matter was treated as a continuum. For example, in the consideration of
the heat conduction in a solid body, the state of the body is described
by giving the temperature at every point of the body for every
definite time. Mathematically, this means that the temperature T
is
represented as a mathematical expression (function) of the space
coordinates and the time t
(Temperature field). The law of heat conduction is
represented as a
local
relation
(differential equation), which embraces all special cases of the
conduction of heat. The temperature is here a simple example of the
concept of field. This is a quantity (or a complex of quantities),
which is a function of the coordinates and the time. Another example
is the description of the motion of a liquid. At every point there
exists at any time a velocity, which is quantitatively described by its
three "components" with respect to the axes of a coordinate system
(vector). The components of the velocity at a point (field components),
here also, are functions of the coordinates (x, y,
z)
and the time (t). It is
characteristic of the fields
mentioned that
they occur only within a ponderable mass; they serve only to describe a
state of this matter. In accordance with the historical development of
the field concept, where no matter was available there could also exist
no field. But in the first quarter of the nineteenth century it was
shown that the phenomena of the interference and motion of light could
be explained with astonishing clearness when light was regarded as a
wavefield, completely analogous to the mechanical vibration field in
an elastic solid body. It was thus felt necessary to introduce a field,
that could also exist in "empty space" in the absence of ponderable
matter. This state of affairs created a paradoxical
situation,
because, in
accordance with its origin, the field concept appeared to be restricted
to the description of states in the inside of a ponderable body. This
seemed to be all the more certain, inasmuch as the conviction was
held that every field is to be regarded as a state capable of
mechanical interpretation, and this presupposed the presence of matter.
One thus felt compelled, even in the space which had hitherto been
regarded as empty, to assume everywhere the existence of a form of
matter, which was called "aether". The emancipation
of the field concept
from the
assumption of its association with a mechanical carrier finds a place
among the psychologically most interesting events in the development of
physical thought. During the second half of the nineteenth century, in
connection with the researches of Faraday
and Maxwell it became more
and more clear that the description of electromagnetic processes in
terms of field was vastly superior to a treatment on the basis of the
mechanical concepts of material points. By the introduction of the
field concept in electrodynamics, Maxwell
succeeded in predicting the
existence of electromagnetic waves, the essential identity of which
with light waves could not be doubted because of the equality of their
velocity of propagation. As a result of this, optics was, in principle,
absorbed by electrodynamics. One psychological effect of this immense
success was that the field concept, as opposed to the mechanistic
framework of classical physics, gradually won greater independence.
Nevertheless, it was at first taken for granted
that
electromagnetic
fields had to be interpreted as states of the aether, and it was
zealously sought to explain these states as mechanical ones. But as
these efforts always met with frustration, science gradually became
accustomed to the idea of renouncing such a mechanical interpretation.
Nevertheless, the conviction still remained that electromagnetic
fields must be states of the aether, and this was the position at the
turn of the century. The aethertheory
brought with it the
question: How
does the aether behave from the mechanical point of view with respect
to ponderable bodies? Does it take part in the motions of the bodies,
or do its parts remain at rest relatively to each other? Many ingenious
experiments were undertaken to decide this question. The following
important facts should be mentioned in this connection: the
"aberration" of the fixed stars in consequence of the annual motion of
the earth, and the "Doppler effect", i.e. the
influence of the relative
motion of the fixed stars on the frequency of the light reaching us
from them, for known frequencies of emission. The results of all these
facts and experiments, except for one, the MichelsonMorley
experiment,
were explained by H. A. Lorentz on the
assumption
that the aether does
not take part in the motions of ponderable bodies, and that the parts
of the aether have no relative motions at all with respect to each
other. Thus the aether appeared, as it were, as the embodiment of a
space absolutely at rest. But the investigation of Lorentz
accomplished
still more. It explained all the electromagnetic and optical processes
within ponderable bodies known at that time, on the assumption that the
influence of ponderable matter on the electric field – and conversely
– is due solely to the fact that the constituent particles of matter
carry electrical charges, which share the motion of the particles.
Concerning the experiment of Michelson
and Morley, H. A. Lorentz
showed
that the result obtained at least does not contradict the theory of an
aether at rest. In spite of all these beautiful
successes the
state of
the theory was
not yet wholly satisfactory, and for the following reasons.
Classical mechanics, of which it could not be doubted that it holds
with a close degree of approximation, teaches the equivalence of all
inertial systems or inertial "spaces" for the formulation of natural
laws, i.e. the invariance of natural laws with respect to the
transition from one inertial system to another. Electromagnetic and
optical experiments taught the same thing with considerable accuracy.
But the foundation of electromagnetic theory taught that a particular
inertial system must be given preference, namely that of the
luminiferous aether at rest. This view of the theoretical foundation
was much too unsatisfactory. Was there no modification that, like
classical mechanics, would uphold the equivalence of inertial systems
(special principle of relativity)? The answer to
this question is the special theory
of
relativity. This
takes over from the theory of MaxwellLorentz
the assumption of the
constancy of the velocity of light in empty space. In order to bring
this into harmony with the equivalence of inertial systems (special
principle of relativity), the idea of the absolute character of
simultaneity must be given up; in addition, the Lorentz
transformations
for the time and the space coordinates follow for the transition from
one inertial system to another. The whole content of the special theory
of relativity is included in the postulate: The laws of Nature are
invariant with respect to the Lorentz transformations. The important
thing of this requirement lies in the fact that it limits the possible
natural laws in a definite manner. What is the
position of the special
theory of
relativity in regard to the problem of space? In the first place we
must guard against the opinion that the fourdimensionality of
reality has been newly introduced for the first time by this theory.
Even in classical physics the event is localised by four numbers, three
spatial coordinates and a time coordinate; the totality of physical
"events" is thus thought of as being embedded in a fourdimensional
continuous manifold. But on the basis of classical mechanics this
fourdimensional continuum breaks up objectively into the
onedimensional time and into threedimensional spatial sections, only
the latter of which contain simultaneous events. This resolution is the
same for all inertial systems. The simultaneity of two definite events
with reference to one inertial system involves the simultaneity of
these events in reference to all inertial systems. This is what is
meant when we say that the time of classical mechanics is absolute.
According to the special theory of relativity it is otherwise.
The sum total of events which are simultaneous
with a
selected event
exist, it is true, in relation to a particular inertial system, but no
longer independently of the choice of the inertial system. The
fourdimensional continuum is now no longer resolvable objectively into
sections, all of which contain simultaneous events; "now" loses for the
spatiaIly extended world its objective meaning. It is because of this
that space and time must be regarded as a fourdimensional continuum
that is objectively unresolvable, if it is desired to express the
purport of objective relations without unnecessary conventional
arbitrariness. Since the special
theory of
relativity revealed the
physical equivalence of all inertial systems, it proved the
untenability of the hypothesis of an aether at rest. It was therefore
necessary to renounce the idea that the electromagnetic field is
to be regarded as a state of a material carrier. The field thus becomes
an irreducible element of physical description, irreducible in the same
sense as the concept of matter in the theory of Newton.
Up to now we have directed our attention to
finding in
what respect the
concepts of space and time were modified by the special theory of
relativity. Let us now focus our attention on those elements which this
theory has taken over from classical mechanics. Here also, natural laws
claim validity only when an inertial system is taken as the basis of
spacetime description. The principle of inertia and the principle of
the constancy of the velocity of light are valid only with respect to
an inertial system. The fieldlaws also can claim to have a meaning and
validity only in regard to inertial systems. Thus,
as in classical mechanics, space is here
also an
independent
component in the representation of physical reality. If we imagine
matter and field to be removed, inertialspace or, more accurately,
this space together with the associated time remains behind. The
fourdimensional structure (Minkowskispace)
is
thought of as being the
carrier of matter and of the field. Inertial spaces, with their
associated times, are only privileged fourdimensional coordinate
systems, that are linked together by the linear Lorentz
transformations. Since there exist in this fourdimensional structure
no longer any sections which represent "now" objectively, the concepts
of happening and becoming are indeed not completely suspended, but yet
complicated. It appears therefore more natural to think of physical
reality as a fourdimensional existence, instead of, as hitherto, the
evolution of a threedimensional existence. This
rigid fourdimensional space of the special
theory
of relativity
is to some extent a fourdimensional analogue of H. A.
Lorentz's
rigid
threedimensional aether. For this theory also the following statement
is valid: The description of physical states postulates space as being
initially given and as existing independently. Thus even this theory
does not dispel Descartes' uneasiness
concerning the
independent, or
indeed, the a priori existence of "empty space". The real aim of the
elementary discussion given here is to show to what extent these doubts
are overcome by the general theory of relativity. The Concept of
Space in the
General Theory of Relativity THIS
theory
arose primarily from the endeavour to
understand the equality of inertial and gravitational mass. We start
out from an inertial system S_{1},
whose space is, from the physical point
of view, empty. In other words, there exists in the part of space
contemplated neither matter (in the usual sense) nor a field (in the
sense of the special theory of relativity). With reference to S_{1}
let
there be a second system of reference S_{2}
in uniform acceleration. Then S_{2}
is thus not an
inertial system. With respect to S_{2}
every test mass
would move with an acceleration, which is independent of its physical
and chemical nature. Relative to S_{2},
therefore, there exists a state
which, at least to a first approximation, cannot be distinguished from
a gravitational field. The following concept is thus compatible with
the observable facts: S_{2} is
also equivalent to an "inertial system";
but with respect to S_{2} a
(homogeneous) gravitational field is present
(about the origin of which one does not worry in this connection). Thus
when the gravitational field is included in the framework of the
consideration, the inertial system loses its objective significance,
assuming that this "principle of equivalence" can be
extended to any relative motion whatsoever of the systems of reference.
If it is possible to base a consistent theory on these fundamental
ideas, it will satisfy of itself the fact of the equality of inertial
and gravitational mass, which is strongly confirmed empirically.
Considered fourdimensionally, a nonlinear
transformation of the four
coordinates corresponds to the transition from S_{1}
to S_{2}.
The question
now arises: What kind of nonlinear transformations are to be
permitted, or, how is the Lorentz transformation to be generalised? In
order to answer this question, the following consideration is decisive.
We ascribe to the
inertial system of
the earlier
theory this property: Differences in coordinates are measured by
stationary "rigid" measuring rods, and differences in time by clocks at
rest. The first assumption is supplemented by another, namely, that for
the relative laying out and fitting together of measuring rods at rest,
the theorems on "lengths" in Euclidean geometry hold. From
the results of the special theory of
relativity it
is then
concluded, by elementary considerations, that this direct physical
interpretation of the coordinates is lost for systems of reference
(S_{2}) accelerated relatively
to
inertial systems (S_{1}). But if
this is
the case, the coordinates now express only the order or rank of the
"contiguity" and hence also the dimensional grade of the space, but do
not express any of its metrical properties. We are thus led to extend
the transformations to arbitrary continuous transformations.
This
implies the general principle of relativity: Natural laws must be
covariant with respect to arbitrary continuous transformations of the
coordinates. This requirement (combined with that of the greatest
possible logical simplicity of the laws) limits the natural laws
concerned incomparably more strongly than the special principle of
relativity. This train of ideas is based
essentially on the
field as
an independent
concept. For the conditions prevailing with respect to S_{2}
are
interpreted as a gravitational field, without the question of the
existence of masses which produce this field being raised. By virtue of
this train of ideas it can also be grasped why the laws of the pure
gravitational field are more directly linked with the idea of general
relativity than the laws for fields of a general kind (when, for
instance, an electromagnetic field is present). We have, namely, good
ground for the assumption that the "fieldfree" Minkowskispace
represents a special case possible in natural law, in fact, the
simplest conceivable special case. With respect to its metrical
character, such a space is characterised by the fact that dx_{1}²
+
dx_{2}² +
dx_{3}² is the square of the spatial
separation, measured with a
unit
gauge, of two infinitesimally neighbouring points of a
threedimensional "spacelike" cross section (Pythagorean theorem),
whereas dx_{4} is the temporal
separation, measured with a suitable time
gauge, of two events with common (x_{1},
x_{2}, x_{3}).
All this simply means
that an objective metrical significance is attached to the quantity
ds² =
dx_{1}²
+
dx_{2}² + dx_{3}² 
dx_{4}² (1)
as is readily shown with the aid of the Lorentz
transformations.
Mathematically, this fact corresponds to the condition that ds²
is
invariant with respect to Lorentz transformations. If
now, in the sense of the general principle of
relativity, this space
(cf. eq. (1) )
is subjected
to an arbitrary continuous transformation
of the coordinates, then the objectively significant quantity ds is
expressed in the new system of coordinates by the relation
ds² = g_{ik}
dx_{i}
dx_{k}
(1a) which
has to be summed up over the indices i
and k
for
all combinations 11, 12, . . .
up to 44 . The
terms g_{ik} now are not
constants, but
functions of the coordinates, which are determined by the arbitrarily
chosen transformation. Nevertheless, the terms g_{ik}
are not
arbitrary
functions of the new coordinates, but just functions of such a kind
that the form (1a) can
be
transformed back again into the form (1)
by a
continuous transformation of the four coordinates. In order that this
may be possible, the functions g_{ik}
must satisfy certain general
covariant equations of condition, which were derived by B.
Riemann more
than half a century before the formulation of the general theory of
relativity ("Riemann condition"). According to the principle of
equivalence, (1a)
describes in general covariant form a gravitational field of a special
kind, when the functions g_{ik}
satisfy the Riemann condition. It follows that the
law for the pure gravitational
field
of a general
kind must be satisfied when the Riemann condition is satisfied; but it
must be weaker or less restricting than the Riemann condition. In this
way the field law of pure gravitation is practically completely
determined, a result which will not be justified in greater detail here.
We are now in a position to see how
far the
transition to the general theory of relativity modifies the concept of
space. In accordance with classical mechanics and according to the
special theory of relativity, space (spacetime) has an existence
independent of matter or field. In order to be able
to describe at all that which
fills
up space and is
dependent on the coordinates, spacetime or the inertial system with
its metrical properties must be thought of at once as existing, for
otherwise the description of "that which fills up space" would have no
meaning. On the basis of the general theory of relativity, on
the
other hand, space as opposed to "what fills space", which is dependent
on the coordinates, has no separate existence. Thus a pure
gravitational field might have been described in terms of the g_{ik}
(as
functions of the coordinates), by solution of the gravitational
equations. If we imagine the gravitational field, i.e. the functions g_{ik},
to be removed, there does not remain a space of the type (1),
but
absolutely nothing, and also no "topological space". For the functions g_{ik}
describe not
only the field, but at the same time also the
topological and metrical structural properties of the manifold.
A space of the type (1),
judged from the standpoint of
the general
theory of relativity, is not a space without field, but a special case
of the g_{ik} field, for which
– for the coordinate system used, which
in itself has no objective significance – the functions g_{ik}
have
values that do not depend on the coordinates. There is no such thing
as an empty space, i.e. a space without field. Spacetime
does not claim existence on its own,
but only
as a
structural quality of the field. Thus Descartes
was
not so far from the truth when he
believed he must exclude the existence of an empty space.
The notion indeed appears absurd, as long as physical reality is seen
exclusively in ponderable bodies. It requires the
idea of the field as the
representative
of reality, in
combination with the general principle of relativity, to show the true
kernel of Descartes' idea; there exists
no space
"empty of field".
Generalized
Theory of
Gravitation THE
theory of the pure gravitational field on the basis of the general
theory of relativity is therefore readily obtainable, because we may be
confident that the "fieldfree" Minkowski
space with
its metric in
conformity with (1)
must
satisfy the general laws of field. From this
special case the law of gravitation follows by a generalisation which
is practically free from arbitrariness. The further
development of the theory is not so
unequivocally
determined by the general principle of relativity; it has been
attempted in various directions during the last few decades. It is
common to all these attempts, to conceive physical reality as a field,
and moreover, one which is a generalisation of the gravitational field,
and in which the field law is a generalisation of the law for the pure
gravitational field. After long probing I believe that I have now
found the most natural form for this generalisation, but I
have
not yet been able to find out whether this generalised law can stand up
against the facts of experience. The question of
the particular field law is
secondary in
the preceding
general considerations. At the present time, the main question is
whether a field theory of the kind here contemplated can lead to the
goal at all. By this is meant a theory which describes exhaustively
physical reality, including fourdimensional space, by a field. The
presentday generation of physicists is inclined to answer this
question in the negative. In conformity with the present form of the
quantum theory, it believes that the state of a system cannot be
specified directly, but only in an indirect way by a statement of the
statistics of the results of measurement attainable on the system. The
conviction prevails that the experimentally assured duality of nature
(corpuscular and wave structure) can be realised only by such a
weakening of the concept of reality. I think that such a farreaching
theoretical renunciation is not for the present justified by our actual
knowledge, and that one should not desist from pursuing to the end the
path of the relativistic field theory.  