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Sidelights on Relativity


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ETHER AND THE THEORY OF RELATIVITY

An Address delivered on May 5th, 1920, in the University of Leyden

GEOMETRY AND EXPERIENCE

An expanded form of an Address to the Prussian Academy of Sciences
in Berlin on January 27th, 1921.





ETHER AND THE THEORY OF RELATIVITY

An Address delivered on May 5th, 1920, in the University of Leyden



How does it come about that alongside of the idea of ponderable
matter, which is derived by abstraction from everyday life, the
physicists set the idea of the existence of another kind of matter,
the ether? The explanation is probably to be sought in those phenomena
which have given rise to the theory of action at a distance, and
in the properties of light which have led to the undulatory theory.
Let us devote a little while to the consideration of these two
subjects.

Outside of physics we know nothing of action at a distance. When
we try to connect cause and effect in the experiences which natural
objects afford us, it seems at first as if there were no other mutual
actions than those of immediate contact, e.g. the communication of
motion by impact, push and pull, heating or inducing combustion by
means of a flame, etc. It is true that even in everyday experience
weight, which is in a sense action at a distance, plays a very
important part. But since in daily experience the weight of bodies
meets us as something constant, something not linked to any cause
which is variable in time or place, we do not in everyday life
speculate as to the cause of gravity, and therefore do not become
conscious of its character as action at a distance. It was Newton's
theory of gravitation that first assigned a cause for gravity by
interpreting it as action at a distance, proceeding from masses.
Newton's theory is probably the greatest stride ever made in
the effort towards the causal nexus of natural phenomena. And yet
this theory evoked a lively sense of discomfort among Newton's
contemporaries, because it seemed to be in conflict with the
principle springing from the rest of experience, that there can be
reciprocal action only through contact, and not through immediate
action at a distance. It is only with reluctance that man's desire
for knowledge endures a dualism of this kind. How was unity to
be preserved in his comprehension of the forces of nature? Either
by trying to look upon contact forces as being themselves distant
forces which admittedly are observable only at a very small
distance--and this was the road which Newton's followers, who were
entirely under the spell of his doctrine, mostly preferred to
take; or by assuming that the Newtonian action at a distance is
only _apparently_ immediate action at a distance, but in truth is
conveyed by a medium permeating space, whether by movements or by
elastic deformation of this medium. Thus the endeavour toward a
unified view of the nature of forces leads to the hypothesis of an
ether. This hypothesis, to be sure, did not at first bring with it
any advance in the theory of gravitation or in physics generally,
so that it became customary to treat Newton's law of force as an
axiom not further reducible. But the ether hypothesis was bound
always to play some part in physical science, even if at first only
a latent part.

When in the first half of the nineteenth century the far-reaching
similarity was revealed which subsists between the properties of
light and those of elastic waves in ponderable bodies, the ether
hypothesis found fresh support. It appeared beyond question that
light must be interpreted as a vibratory process in an elastic, inert
medium filling up universal space. It also seemed to be a necessary
consequence of the fact that light is capable of polarisation that
this medium, the ether, must be of the nature of a solid body,
because transverse waves are not possible in a fluid, but only in
a solid. Thus the physicists were bound to arrive at the theory
of the "quasi-rigid" luminiferous ether, the parts of which can
carry out no movements relatively to one another except the small
movements of deformation which correspond to light-waves.

This theory--also called the theory of the stationary luminiferous
ether--moreover found a strong support in an experiment which is
also of fundamental importance in the special theory of relativity,
the experiment of Fizeau, from which one was obliged to infer
that the luminiferous ether does not take part in the movements of
bodies. The phenomenon of aberration also favoured the theory of
the quasi-rigid ether.

The development of the theory of electricity along the path opened
up by Maxwell and Lorentz gave the development of our ideas concerning
the ether quite a peculiar and unexpected turn. For Maxwell himself
the ether indeed still had properties which were purely mechanical,
although of a much more complicated kind than the mechanical
properties of tangible solid bodies. But neither Maxwell nor his
followers succeeded in elaborating a mechanical model for the ether
which might furnish a satisfactory mechanical interpretation of
Maxwell's laws of the electro-magnetic field. The laws were clear
and simple, the mechanical interpretations clumsy and contradictory.
Almost imperceptibly the theoretical physicists adapted themselves
to a situation which, from the standpoint of their mechanical
programme, was very depressing. They were particularly influenced
by the electro-dynamical investigations of Heinrich Hertz. For
whereas they previously had required of a conclusive theory that
it should content itself with the fundamental concepts which belong
exclusively to mechanics (e.g. densities, velocities, deformations,
stresses) they gradually accustomed themselves to admitting electric and
magnetic force as fundamental concepts side by side with those of
mechanics, without requiring a mechanical interpretation for them.
Thus the purely mechanical view of nature was gradually abandoned.
But this change led to a fundamental dualism which in the long-run
was insupportable. A way of escape was now sought in the reverse
direction, by reducing the principles of mechanics to those
of electricity, and this especially as confidence in the strict
validity of the equations of Newton's mechanics was shaken by the
experiments with beta-rays and rapid kathode rays.

This dualism still confronts us in unextenuated form in the theory
of Hertz, where matter appears not only as the bearer of velocities,
kinetic energy, and mechanical pressures, but also as the bearer of
electromagnetic fields. Since such fields also occur _in vacuo_--i.e.
in free ether--the ether also appears as bearer of electromagnetic
fields. The ether appears indistinguishable in its functions from
ordinary matter. Within matter it takes part in the motion of matter
and in empty space it has everywhere a velocity; so that the ether
has a definitely assigned velocity throughout the whole of space.
There is no fundamental difference between Hertz's ether and
ponderable matter (which in part subsists in the ether).

The Hertz theory suffered not only from the defect of ascribing
to matter and ether, on the one hand mechanical states, and on the
other hand electrical states, which do not stand in any conceivable
relation to each other; it was also at variance with the result of
Fizeau's important experiment on the velocity of the propagation
of light in moving fluids, and with other established experimental
results.

Such was the state of things when H. A. Lorentz entered upon the
scene. He brought theory into harmony with experience by means of
a wonderful simplification of theoretical principles. He achieved
this, the most important advance in the theory of electricity since
Maxwell, by taking from ether its mechanical, and from matter its
electromagnetic qualities. As in empty space, so too in the interior
of material bodies, the ether, and not matter viewed atomistically,
was exclusively the seat of electromagnetic fields. According to
Lorentz the elementary particles of matter alone are capable of
carrying out movements; their electromagnetic activity is entirely
confined to the carrying of electric charges. Thus Lorentz succeeded
in reducing all electromagnetic happenings to Maxwell's equations
for free space.

As to the mechanical nature of the Lorentzian ether, it may be said
of it, in a somewhat playful spirit, that immobility is the only
mechanical property of which it has not been deprived by H. A.
Lorentz. It may be added that the whole change in the conception
of the ether which the special theory of relativity brought about,
consisted in taking away from the ether its last mechanical quality,
namely, its immobility. How this is to be understood will forthwith
be expounded.

The space-time theory and the kinematics of the special theory
of relativity were modelled on the Maxwell-Lorentz theory of the
electromagnetic field. This theory therefore satisfies the conditions
of the special theory of relativity, but when viewed from the latter
it acquires a novel aspect. For if K be a system of co-ordinates
relatively to which the Lorentzian ether is at rest, the
Maxwell-Lorentz equations are valid primarily with reference to K.
But by the special theory of relativity the same equations without
any change of meaning also hold in relation to any new system of
co-ordinates K' which is moving in uniform translation relatively
to K. Now comes the anxious question:--Why must I in the theory
distinguish the K system above all K' systems, which are physically
equivalent to it in all respects, by assuming that the ether
is at rest relatively to the K system? For the theoretician such
an asymmetry in the theoretical structure, with no corresponding
asymmetry in the system of experience, is intolerable. If we assume
the ether to be at rest relatively to K, but in motion relatively
to K', the physical equivalence of K and K' seems to me from the
logical standpoint, not indeed downright incorrect, but nevertheless
inacceptable.

The next position which it was possible to take up in face of this
state of things appeared to be the following. The ether does not
exist at all. The electromagnetic fields are not states of a medium,
and are not bound down to any bearer, but they are independent
realities which are not reducible to anything else, exactly like
the atoms of ponderable matter. This conception suggests itself
the more readily as, according to Lorentz's theory, electromagnetic
radiation, like ponderable matter, brings impulse and energy with
it, and as, according to the special theory of relativity, both
matter and radiation are but special forms of distributed energy,
ponderable mass losing its isolation and appearing as a special
form of energy.

More careful reflection teaches us, however, that the special theory
of relativity does not compel us to deny ether. We may assume the
existence of an ether; only we must give up ascribing a definite
state of motion to it, i.e. we must by abstraction take from it the
last mechanical characteristic which Lorentz had still left it. We
shall see later that this point of view, the conceivability of which
I shall at once endeavour to make more intelligible by a somewhat
halting comparison, is justified by the results of the general
theory of relativity.

Think of waves on the surface of water. Here we can describe two
entirely different things. Either we may observe how the undulatory
surface forming the boundary between water and air alters in the course
of time; or else--with the help of small floats, for instance--we
can observe how the position of the separate particles of water
alters in the course of time. If the existence of such floats for
tracking the motion of the particles of a fluid were a fundamental
impossibility in physics--if, in fact, nothing else whatever were
observable than the shape of the space occupied by the water as it
varies in time, we should have no ground for the assumption that
water consists of movable particles. But all the same we could
characterise it as a medium.

We have something like this in the electromagnetic field. For we may
picture the field to ourselves as consisting of lines of force.  If
we wish to interpret these lines of force to ourselves as something
material in the ordinary sense, we are tempted to interpret the
dynamic processes as motions of these lines of force, such that each
separate line of force is tracked through the course of time. It is
well known, however, that this way of regarding the electromagnetic
field leads to contradictions.

Generalising we must say this:--There may be supposed to be extended
physical objects to which the idea of motion cannot be applied.
They may not be thought of as consisting of particles which allow
themselves to be separately tracked through time. In Minkowski's
idiom this is expressed as follows:--Not every extended conformation
in the four-dimensional world can be regarded as composed
of world-threads. The special theory of relativity forbids us to
assume the ether to consist of particles observable through time,
but the hypothesis of ether in itself is not in conflict with the
special theory of relativity. Only we must be on our guard against
ascribing a state of motion to the ether.

Certainly, from the standpoint of the special theory of relativity,
the ether hypothesis appears at first to be an empty hypothesis. In
the equations of the electromagnetic field there occur, in addition
to the densities of the electric charge, _only_ the intensities
of the field. The career of electromagnetic processes _in vacuo_
appears to be completely determined by these equations, uninfluenced
by other physical quantities. The electromagnetic fields appear as
ultimate, irreducible realities, and at first it seems superfluous
to postulate a homogeneous, isotropic ether-medium, and to envisage
electromagnetic fields as states of this medium.

But on the other hand there is a weighty argument to be adduced
in favour of the ether hypothesis. To deny the ether is ultimately
to assume that empty space has no physical qualities whatever. The
fundamental facts of mechanics do not harmonize with this view.
For the mechanical behaviour of a corporeal system hovering freely
in empty space depends not only on relative positions (distances)
and relative velocities, but also on its state of rotation, which
physically may be taken as a characteristic not appertaining to the
system in itself. In order to be able to look upon the rotation of
the system, at least formally, as something real, Newton objectivises
space.

Since he classes his absolute space together with real things, for
him rotation relative to an absolute space is also something real.
Newton might no less well have called his absolute space "Ether";
what is essential is merely that besides observable objects, another
thing, which is not perceptible, must be looked upon as real,
to enable acceleration or rotation to be looked upon as something
real.

It is true that Mach tried to avoid having to accept as real something
which is not observable by endeavouring to substitute in mechanics
a mean acceleration with reference to the totality of the masses in
the universe in place of an acceleration with reference to absolute
space. But inertial resistance opposed to relative acceleration of
distant masses presupposes action at a distance; and as the modern
physicist does not believe that he may accept this action at
a distance, he comes back once more, if he follows Mach, to the
ether, which has to serve as medium for the effects of inertia. But
this conception of the ether to which we are led by Mach's way of
thinking differs essentially from the ether as conceived by Newton,
by Fresnel, and by Lorentz. Mach's ether not only _conditions_ the
behaviour of inert masses, but _is also conditioned_ in its state
by them.

Mach's idea finds its full development in the ether of the general
theory of relativity. According to this theory the metrical
qualities of the continuum of space-time differ in the environment
of different points of space-time, and are partly conditioned by the
matter existing outside of the territory under consideration.  This
space-time variability of the reciprocal relations of the standards
of space and time, or, perhaps, the recognition of the fact that
"empty space" in its physical relation is neither homogeneous nor
isotropic, compelling us to describe its state by ten functions (the
gravitation potentials g_(mn)), has, I think, finally disposed of
the view that space is physically empty. But therewith the
conception of the ether has again acquired an intelligible content,
although this content differs widely from that of the ether of the
mechanical undulatory theory of light.  The ether of the general
theory of relativity is a medium which is itself devoid of _all_
mechanical and kinematical qualities, but helps to determine
mechanical (and electromagnetic) events.

What is fundamentally new in the ether of the general theory of
relativity as opposed to the ether of Lorentz consists in this, that
the state of the former is at every place determined by connections
with the matter and the state of the ether in neighbouring places,
which are amenable to law in the form of differential equations;
whereas the state of the Lorentzian ether in the absence of
electromagnetic fields is conditioned by nothing outside itself,
and is everywhere the same. The ether of the general theory of
relativity is transmuted conceptually into the ether of Lorentz if
we substitute constants for the functions of space which describe
the former, disregarding the causes which condition its state.
Thus we may also say, I think, that the ether of the general theory
of relativity is the outcome of the Lorentzian ether, through
relativation.

As to the part which the new ether is to play in the physics of
the future we are not yet clear. We know that it determines the
metrical relations in the space-time continuum, e.g. the configurative
possibilities of solid bodies as well as the gravitational fields;
but we do not know whether it has an essential share in the structure
of the electrical elementary particles constituting matter. Nor do
we know whether it is only in the proximity of ponderable masses
that its structure differs essentially from that of the Lorentzian
ether; whether the geometry of spaces of cosmic extent is approximately
Euclidean. But we can assert by reason of the relativistic equations
of gravitation that there must be a departure from Euclidean
relations, with spaces of cosmic order of magnitude, if there exists
a positive mean density, no matter how small, of the matter in the
universe. In this case the universe must of necessity be spatially
unbounded and of finite magnitude, its magnitude being determined
by the value of that mean density.

If we consider the gravitational field and the electromagnetic field
from the stand-point of the ether hypothesis, we find a remarkable
difference between the two. There can be no space nor any part
of space without gravitational potentials; for these confer upon
space its metrical qualities, without which it cannot be imagined
at all. The existence of the gravitational field is inseparably
bound up with the existence of space. On the other hand a part of
space may very well be imagined without an electromagnetic field;
thus in contrast with the gravitational field, the electromagnetic
field seems to be only secondarily linked to the ether, the formal
nature of the electromagnetic field being as yet in no way determined
by that of gravitational ether. From the present state of theory
it looks as if the electromagnetic field, as opposed to the
gravitational field, rests upon an entirely new formal _motif_,
as though nature might just as well have endowed the gravitational
ether with fields of quite another type, for example, with fields
of a scalar potential, instead of fields of the electromagnetic
type.

Since according to our present conceptions the elementary particles
of matter are also, in their essence, nothing else than condensations
of the electromagnetic field, our present view of the universe
presents two realities which are completely separated from each other
conceptually, although connected causally, namely, gravitational ether
and electromagnetic field, or--as they might also be called--space
and matter.

Of course it would be a great advance if we could succeed in
comprehending the gravitational field and the electromagnetic field
together as one unified conformation. Then for the first time the
epoch of theoretical physics founded by Faraday and Maxwell would
reach a satisfactory conclusion. The contrast between ether and
matter would fade away, and, through the general theory of relativity,
the whole of physics would become a complete system of thought,
like geometry, kinematics, and the theory of gravitation.  An
exceedingly ingenious attempt in this direction has been made by
the mathematician H. Weyl; but I do not believe that his theory will
hold its ground in relation to reality. Further, in contemplating
the immediate future of theoretical physics we ought not unconditionally
to reject the possibility that the facts comprised in the quantum
theory may set bounds to the field theory beyond which it cannot
pass.

Recapitulating, we may say that according to the general theory of
relativity space is endowed with physical qualities; in this sense,
therefore, there exists an ether. According to the general theory
of relativity space without ether is unthinkable; for in such space
there not only would be no propagation of light, but also no possibility
of existence for standards of space and time (measuring-rods and
clocks), nor therefore any space-time intervals in the physical
sense. But this ether may not be thought of as endowed with the
quality characteristic of ponderable media, as consisting of parts
which may be tracked through time. The idea of motion may not be
applied to it.




GEOMETRY AND EXPERIENCE

An expanded form of an Address to the Prussian Academy of Sciences
in Berlin on January 27th, 1921.



One reason why mathematics enjoys special esteem, above all other
sciences, is that its laws are absolutely certain and indisputable,
while those of all other sciences are to some extent debatable and
in constant danger of being overthrown by newly discovered facts.
In spite of this, the investigator in another department of science
would not need to envy the mathematician if the laws of mathematics
referred to objects of our mere imagination, and not to objects
of reality. For it cannot occasion surprise that different persons
should arrive at the same logical conclusions when they have already
agreed upon the fundamental laws (axioms), as well as the methods
by which other laws are to be deduced therefrom. But there is another
reason for the high repute of mathematics, in that it is mathematics
which affords the exact natural sciences a certain measure of
security, to which without mathematics they could not attain.

At this point an enigma presents itself which in all ages has agitated
inquiring minds. How can it be that mathematics, being after all
a product of human thought which is independent of experience, is
so admirably appropriate to the objects of reality? Is human reason,
then, without experience, merely by taking thought, able to fathom
the properties of real things.

In my opinion the answer to this question is, briefly, this:--As far
as the laws of mathematics refer to reality, they are not certain;
and as far as they are certain, they do not refer to reality.
It seems to me that complete clearness as to this state of things
first became common property through that new departure in mathematics
which is known by the name of mathematical logic or "Axiomatics."
The progress achieved by axiomatics consists in its having neatly
separated the logical-formal from its objective or intuitive
content; according to axiomatics the logical-formal alone forms
the subject-matter of mathematics, which is not concerned with the
intuitive or other content associated with the logical-formal.

Let us for a moment consider from this point of view any axiom of
geometry, for instance, the following:--Through two points in space
there always passes one and only one straight line. How is this
axiom to be interpreted in the older sense and in the more modern
sense?

The older interpretation:--Every one knows what a straight line
is, and what a point is. Whether this knowledge springs from an
ability of the human mind or from experience, from some collaboration
of the two or from some other source, is not for the mathematician
to decide. He leaves the question to the philosopher. Being based
upon this knowledge, which precedes all mathematics, the axiom
stated above is, like all other axioms, self-evident, that is, it
is the expression of a part of this _a priori_ knowledge.

The more modern interpretation:--Geometry treats of entities which
are denoted by the words straight line, point, etc. These entities
do not take for granted any knowledge or intuition whatever, but
they presuppose only the validity of the axioms, such as the one
stated above, which are to be taken in a purely formal sense, i.e.
as void of all content of intuition or experience. These axioms are
free creations of the human mind. All other propositions of geometry
are logical inferences from the axioms (which are to be taken in
the nominalistic sense only). The matter of which geometry treats
is first defined by the axioms. Schlick in his book on epistemology has
therefore characterised axioms very aptly as "implicit definitions."

This view of axioms, advocated by modern axiomatics, purges mathematics
of all extraneous elements, and thus dispels the mystic obscurity
which formerly surrounded the principles of mathematics.

But a presentation of its principles thus clarified makes it also
evident that mathematics as such cannot predicate anything about
perceptual objects or real objects. In axiomatic geometry the words
"point," "straight line," etc., stand only for empty conceptual
schemata. That which gives them substance is not relevant to
mathematics.

Yet on the other hand it is certain that mathematics generally,
and particularly geometry, owes its existence to the need which
was felt of learning something about the relations of real things
to one another. The very word geometry, which, of course, means
earth-measuring, proves this. For earth-measuring has to do with
the possibilities of the disposition of certain natural objects
with respect to one another, namely, with parts of the earth,
measuring-lines, measuring-wands, etc. It is clear that the system
of concepts of axiomatic geometry alone cannot make any assertions
as to the relations of real objects of this kind, which we will
call practically-rigid bodies. To be able to make such assertions,
geometry must be stripped of its merely logical-formal character
by the co-ordination of real objects of experience with the empty
conceptual frame-work of axiomatic geometry. To accomplish this,
we need only add the proposition:--Solid bodies are related, with
respect to their possible dispositions, as are bodies in Euclidean
geometry of three dimensions. Then the propositions of Euclid contain
affirmations as to the relations of practically-rigid bodies.

Geometry thus completed is evidently a natural science; we may in
fact regard it as the most ancient branch of physics. Its affirmations
rest essentially on induction from experience, but not on logical
inferences only. We will call this completed geometry "practical
geometry," and shall distinguish it in what follows from "purely
axiomatic geometry." The question whether the practical geometry
of the universe is Euclidean or not has a clear meaning, and its
answer can only be furnished by experience. All linear measurement
in physics is practical geometry in this sense, so too is geodetic
and astronomical linear measurement, if we call to our help the
law of experience that light is propagated in a straight line, and
indeed in a straight line in the sense of practical geometry.

I attach special importance to the view of geometry which I
have just set forth, because without it I should have been unable
to formulate the theory of relativity. Without it the following
reflection would have been impossible:--In a system of reference
rotating relatively to an inert system, the laws of disposition of
rigid bodies do not correspond to the rules of Euclidean geometry
on account of the Lorentz contraction; thus if we admit non-inert
systems we must abandon Euclidean geometry. The decisive step in
the transition to general co-variant equations would certainly not
have been taken if the above interpretation had not served as a
stepping-stone. If we deny the relation between the body of axiomatic
Euclidean geometry and the practically-rigid body of reality,
we readily arrive at the following view, which was entertained by
that acute and profound thinker, H. Poincare:--Euclidean geometry
is distinguished above all other imaginable axiomatic geometries
by its simplicity. Now since axiomatic geometry by itself contains
no assertions as to the reality which can be experienced, but can
do so only in combination with physical laws, it should be possible
and reasonable--whatever may be the nature of reality--to retain
Euclidean geometry. For if contradictions between theory and
experience manifest themselves, we should rather decide to change
physical laws than to change axiomatic Euclidean geometry. If we
deny the relation between the practically-rigid body and geometry,
we shall indeed not easily free ourselves from the convention
that Euclidean geometry is to be retained as the simplest. Why
is the equivalence of the practically-rigid body and the body of
geometry--which suggests itself so readily--denied by Poincare and
other investigators? Simply because under closer inspection the
real solid bodies in nature are not rigid, because their geometrical
behaviour, that is, their possibilities of relative disposition,
depend upon temperature, external forces, etc. Thus the original,
immediate relation between geometry and physical reality appears
destroyed, and we feel impelled toward the following more general
view, which characterizes Poincare's standpoint. Geometry (G)
predicates nothing about the relations of real things, but only
geometry together with the purport (P) of physical laws can do so.
Using symbols, we may say that only the sum of (G) + (P) is subject
to the control of experience. Thus (G) may be chosen arbitrarily,
and also parts of (P); all these laws are conventions. All that
is necessary to avoid contradictions is to choose the remainder of
(P) so that (G) and the whole of (P) are together in accord with
experience. Envisaged in this way, axiomatic geometry and the part
of natural law which has been given a conventional status appear
as epistemologically equivalent.

_Sub specie aeterni_ Poincare, in my opinion, is right. The idea
of the measuring-rod and the idea of the clock co-ordinated with it
in the theory of relativity do not find their exact correspondence
in the real world. It is also clear that the solid body and the
clock do not in the conceptual edifice of physics play the part of
irreducible elements, but that of composite structures, which may
not play any independent part in theoretical physics. But it is my
conviction that in the present stage of development of theoretical
physics these ideas must still be employed as independent ideas;
for we are still far from possessing such certain knowledge
of theoretical principles as to be able to give exact theoretical
constructions of solid bodies and clocks.

Further, as to the objection that there are no really rigid bodies
in nature, and that therefore the properties predicated of rigid
bodies do not apply to physical reality,--this objection is by
no means so radical as might appear from a hasty examination. For
it is not a difficult task to determine the physical state of a
measuring-rod so accurately that its behaviour relatively to other
measuring-bodies shall be sufficiently free from ambiguity to allow
it to be substituted for the "rigid" body. It is to measuring-bodies
of this kind that statements as to rigid bodies must be referred.

All practical geometry is based upon a principle which is accessible
to experience, and which we will now try to realise. We will
call that which is enclosed between two boundaries, marked upon a
practically-rigid body, a tract. We imagine two practically-rigid
bodies, each with a tract marked out on it. These two tracts are
said to be "equal to one another" if the boundaries of the one tract
can be brought to coincide permanently with the boundaries of the
other. We now assume that:

If two tracts are found to be equal once and anywhere, they are
equal always and everywhere.

Not only the practical geometry of Euclid, but also its nearest
generalisation, the practical geometry of Riemann, and therewith
the general theory of relativity, rest upon this assumption. Of the
experimental reasons which warrant this assumption I will mention
only one. The phenomenon of the propagation of light in empty space
assigns a tract, namely, the appropriate path of light, to each
interval of local time, and conversely. Thence it follows that
the above assumption for tracts must also hold good for intervals
of clock-time in the theory of relativity. Consequently it may be
formulated as follows:--If two ideal clocks are going at the same
rate at any time and at any place (being then in immediate proximity
to each other), they will always go at the same rate, no matter where
and when they are again compared with each other at one place.--If
this law were not valid for real clocks, the proper frequencies
for the separate atoms of the same chemical element would not be
in such exact agreement as experience demonstrates. The existence
of sharp spectral lines is a convincing experimental proof of the
above-mentioned principle of practical geometry. This is the ultimate
foundation in fact which enables us to speak with meaning of the
mensuration, in Riemann's sense of the word, of the four-dimensional
continuum of space-time.

The question whether the structure of this continuum is Euclidean,
or in accordance with Riemann's general scheme, or otherwise,
is, according to the view which is here being advocated, properly
speaking a physical question which must be answered by experience,
and not a question of a mere convention to be selected on practical
grounds. Riemann's geometry will be the right thing if the laws
of disposition of practically-rigid bodies are transformable into
those of the bodies of Euclid's geometry with an exactitude which
increases in proportion as the dimensions of the part of space-time
under consideration are diminished.

It is true that this proposed physical interpretation of geometry
breaks down when applied immediately to spaces of sub-molecular
order of magnitude. But nevertheless, even in questions as
to the constitution of elementary particles, it retains part of
its importance. For even when it is a question of describing the
electrical elementary particles constituting matter, the attempt
may still be made to ascribe physical importance to those ideas
of fields which have been physically defined for the purpose
of describing the geometrical behaviour of bodies which are large
as compared with the molecule. Success alone can decide as to the
justification of such an attempt, which postulates physical reality
for the fundamental principles of Riemann's geometry outside of the
domain of their physical definitions. It might possibly turn out
that this extrapolation has no better warrant than the extrapolation
of the idea of temperature to parts of a body of molecular order
of magnitude.

It appears less problematical to extend the ideas of practical
geometry to spaces of cosmic order of magnitude. It might, of course,
be objected that a construction composed of solid rods departs more
and more from ideal rigidity in proportion as its spatial extent
becomes greater. But it will hardly be possible, I think, to assign
fundamental significance to this objection. Therefore the question
whether the universe is spatially finite or not seems to me
decidedly a pregnant question in the sense of practical geometry.
I do not even consider it impossible that this question will be
answered before long by astronomy. Let us call to mind what the
general theory of relativity teaches in this respect. It offers
two possibilities:--

1. The universe is spatially infinite. This can be so only if the
average spatial density of the matter in universal space, concentrated
in the stars, vanishes, i.e. if the ratio of the total mass of the
stars to the magnitude of the space through which they are scattered
approximates indefinitely to the value zero when the spaces taken
into consideration are constantly greater and greater.

2. The universe is spatially finite. This must be so, if there is
a mean density of the ponderable matter in universal space differing
from zero. The smaller that mean density, the greater is the volume
of universal space.

I must not fail to mention that a theoretical argument can be adduced in
favour of the hypothesis of a finite universe. The general theory
of relativity teaches that the inertia of a given body is greater as
there are more ponderable masses in proximity to it; thus it seems
very natural to reduce the total effect of inertia of a body to
action and reaction between it and the other bodies in the universe,
as indeed, ever since Newton's time, gravity has been completely
reduced to action and reaction between bodies.  From the equations
of the general theory of relativity it can be deduced that this
total reduction of inertia to reciprocal action between masses--as
required by E. Mach, for example--is possible only if the universe
is spatially finite.

On many physicists and astronomers this argument makes no impression.
Experience alone can finally decide which of the two possibilities
is realised in nature. How can experience furnish an answer? At first
it might seem possible to determine the mean density of matter by
observation of that part of the universe which is accessible to our
perception. This hope is illusory. The distribution of the visible
stars is extremely irregular, so that we on no account may venture
to set down the mean density of star-matter in the universe as
equal, let us say, to the mean density in the Milky Way. In any
case, however great the space examined may be, we could not feel
convinced that there were no more stars beyond that space. So it
seems impossible to estimate the mean density. But there is another
road, which seems to me more practicable, although it also presents
great difficulties. For if we inquire into the deviations shown
by the consequences of the general theory of relativity which are
accessible to experience, when these are compared with the consequences
of the Newtonian theory, we first of all find a deviation which
shows itself in close proximity to gravitating mass, and has been
confirmed in the case of the planet Mercury. But if the universe
is spatially finite there is a second deviation from the Newtonian
theory, which, in the language of the Newtonian theory, may be
expressed thus:--The gravitational field is in its nature such as
if it were produced, not only by the ponderable masses, but also by
a mass-density of negative sign, distributed uniformly throughout
space. Since this factitious mass-density would have to be enormously
small, it could make its presence felt only in gravitating systems
of very great extent.

Assuming that we know, let us say, the statistical distribution
of the stars in the Milky Way, as well as their masses, then by
Newton's law we can calculate the gravitational field and the mean
velocities which the stars must have, so that the Milky Way should
not collapse under the mutual attraction of its stars, but should
maintain its actual extent. Now if the actual velocities of the stars,
which can, of course, be measured, were smaller than the calculated
velocities, we should have a proof that the actual attractions
at great distances are smaller than by Newton's law. From such a
deviation it could be proved indirectly that the universe is finite.
It would even be possible to estimate its spatial magnitude.

Can we picture to ourselves a three-dimensional universe which is
finite, yet unbounded?

The usual answer to this question is "No," but that is not the right
answer. The purpose of the following remarks is to show that the
answer should be "Yes." I want to show that without any extraordinary
difficulty we can illustrate the theory of a finite universe by
means of a mental image to which, with some practice, we shall soon
grow accustomed.

First of all, an observation of epistemological nature. A
geometrical-physical theory as such is incapable of being directly
pictured, being merely a system of concepts. But these concepts
serve the purpose of bringing a multiplicity of real or imaginary
sensory experiences into connection in the mind. To "visualise"
a theory, or bring it home to one's mind, therefore means to give
a representation to that abundance of experiences for which the
theory supplies the schematic arrangement. In the present case we
have to ask ourselves how we can represent that relation of solid
bodies with respect to their reciprocal disposition (contact) which
corresponds to the theory of a finite universe. There is really
nothing new in what I have to say about this; but innumerable
questions addressed to me prove that the requirements of those who
thirst for knowledge of these matters have not yet been completely
satisfied.

So, will the initiated please pardon me, if part of what I shall
bring forward has long been known?

What do we wish to express when we say that our space is infinite?
Nothing more than that we might lay any number whatever of bodies
of equal sizes side by side without ever filling space. Suppose
that we are provided with a great many wooden cubes all of the
same size. In accordance with Euclidean geometry we can place them
above, beside, and behind one another so as to fill a part of space
of any dimensions; but this construction would never be finished;
we could go on adding more and more cubes without ever finding
that there was no more room. That is what we wish to express when
we say that space is infinite. It would be better to say that space
is infinite in relation to practically-rigid bodies, assuming that
the laws of disposition for these bodies are given by Euclidean
geometry.

Another example of an infinite continuum is the plane. On a plane
surface we may lay squares of cardboard so that each side of any
square has the side of another square adjacent to it. The construction
is never finished; we can always go on laying squares--if their laws
of disposition correspond to those of plane figures of Euclidean
geometry. The plane is therefore infinite in relation to the
cardboard squares. Accordingly we say that the plane is an infinite
continuum of two dimensions, and space an infinite continuum of
three dimensions. What is here meant by the number of dimensions,
I think I may assume to be known.

Now we take an example of a two-dimensional continuum which is
finite, but unbounded. We imagine the surface of a large globe and
a quantity of small paper discs, all of the same size. We place
one of the discs anywhere on the surface of the globe. If we move
the disc about, anywhere we like, on the surface of the globe,
we do not come upon a limit or boundary anywhere on the journey.
Therefore we say that the spherical surface of the globe is an
unbounded continuum. Moreover, the spherical surface is a finite
continuum. For if we stick the paper discs on the globe, so that
no disc overlaps another, the surface of the globe will finally
become so full that there is no room for another disc. This simply
means that the spherical surface of the globe is finite in relation
to the paper discs. Further, the spherical surface is a non-Euclidean
continuum of two dimensions, that is to say, the laws of disposition
for the rigid figures lying in it do not agree with those of the
Euclidean plane. This can be shown in the following way. Place
a paper disc on the spherical surface, and around it in a circle
place six more discs, each of which is to be surrounded in turn
by six discs, and so on. If this construction is made on a plane
surface, we have an uninterrupted disposition in which there are
six discs touching every disc except those which lie on the outside.

[Figure 1: Discs maximally packed on a plane]

On the spherical surface the construction also seems to promise
success at the outset, and the smaller the radius of the disc
in proportion to that of the sphere, the more promising it seems.
But as the construction progresses it becomes more and more patent
that the disposition of the discs in the manner indicated, without
interruption, is not possible, as it should be possible by Euclidean
geometry of the the plane surface. In this way creatures which
cannot leave the spherical surface, and cannot even peep out from
the spherical surface into three-dimensional space, might discover,
merely by experimenting with discs, that their two-dimensional
"space" is not Euclidean, but spherical space.

From the latest results of the theory of relativity it is probable
that our three-dimensional space is also approximately spherical,
that is, that the laws of disposition of rigid bodies in it are
not given by Euclidean geometry, but approximately by spherical
geometry, if only we consider parts of space which are sufficiently
great. Now this is the place where the reader's imagination boggles.
"Nobody can imagine this thing," he cries indignantly. "It can be
said, but cannot be thought. I can represent to myself a spherical
surface well enough, but nothing analogous to it in three dimensions."

[Figure 2: A circle projected from a sphere onto a plane]

We must try to surmount this barrier in the mind, and the patient
reader will see that it is by no means a particularly difficult
task. For this purpose we will first give our attention once more to
the geometry of two-dimensional spherical surfaces. In the adjoining
figure let _K_ be the spherical surface, touched at _S_ by a plane,
_E_, which, for facility of presentation, is shown in the drawing as
a bounded surface. Let _L_ be a disc on the spherical surface.  Now
let us imagine that at the point _N_ of the spherical surface,
diametrically opposite to _S_, there is a luminous point, throwing a
shadow _L'_ of the disc _L_ upon the plane _E_. Every point on the
sphere has its shadow on the plane. If the disc on the sphere _K_ is
moved, its shadow _L'_ on the plane _E_ also moves. When the disc
_L_ is at _S_, it almost exactly coincides with its shadow.  If it
moves on the spherical surface away from _S_ upwards, the disc
shadow _L'_ on the plane also moves away from _S_ on the plane
outwards, growing bigger and bigger. As the disc _L_ approaches the
luminous point _N_, the shadow moves off to infinity, and becomes
infinitely great.

Now we put the question, What are the laws of disposition of the
disc-shadows _L'_ on the plane _E_? Evidently they are exactly the
same as the laws of disposition of the discs _L_ on the spherical
surface. For to each original figure on _K_ there is a corresponding
shadow figure on _E_. If two discs on _K_ are touching, their
shadows on _E_ also touch. The shadow-geometry on the plane agrees
with the the disc-geometry on the sphere. If we call the disc-shadows
rigid figures, then spherical geometry holds good on the plane _E_
with respect to these rigid figures. Moreover, the plane is finite
with respect to the disc-shadows, since only a finite number of
the shadows can find room on the plane.

At this point somebody will say, "That is nonsense. The disc-shadows
are _not_ rigid figures. We have only to move a two-foot rule about
on the plane _E_ to convince ourselves that the shadows constantly
increase in size as they move away from _S_ on the plane towards
infinity." But what if the two-foot rule were to behave on the
plane _E_ in the same way as the disc-shadows _L'_? It would then
be impossible to show that the shadows increase in size as they
move away from _S_; such an assertion would then no longer have
any meaning whatever. In fact the only objective assertion that can
be made about the disc-shadows is just this, that they are related
in exactly the same way as are the rigid discs on the spherical
surface in the sense of Euclidean geometry.

We must carefully bear in mind that our statement as to the growth
of the disc-shadows, as they move away from _S_ towards infinity,
has in itself no objective meaning, as long as we are unable to
employ Euclidean rigid bodies which can be moved about on the plane
_E_ for the purpose of comparing the size of the disc-shadows. In
respect of the laws of disposition of the shadows _L'_, the point
_S_ has no special privileges on the plane any more than on the
spherical surface.

The representation given above of spherical geometry on the
plane is important for us, because it readily allows itself to be
transferred to the three-dimensional case.

Let us imagine a point _S_ of our space, and a great number
of small spheres, _L'_, which can all be brought to coincide with
one another. But these spheres are not to be rigid in the sense
of Euclidean geometry; their radius is to increase (in the sense
of Euclidean geometry) when they are moved away from _S_ towards
infinity, and this increase is to take place in exact accordance
with the same law as applies to the increase of the radii of the
disc-shadows _L'_ on the plane.

After having gained a vivid mental image of the geometrical
behaviour of our _L'_ spheres, let us assume that in our space there
are no rigid bodies at all in the sense of Euclidean geometry, but
only bodies having the behaviour of our _L'_ spheres. Then we shall
have a vivid representation of three-dimensional spherical space,
or, rather of three-dimensional spherical geometry. Here our spheres
must be called "rigid" spheres. Their increase in size as they
depart from _S_ is not to be detected by measuring with
measuring-rods, any more than in the case of the disc-shadows on
_E_, because the standards of measurement behave in the same way as
the spheres. Space is homogeneous, that is to say, the same
spherical configurations are possible in the environment of all
points.* Our space is finite, because, in consequence of the
"growth" of the spheres, only a finite number of them can find room
in space.

* This is intelligible without calculation--but only for the
two-dimensional case--if we revert once more to the case of the disc
on the surface of the sphere.

In this way, by using as stepping-stones the practice in thinking
and visualisation which Euclidean geometry gives us, we have acquired
a mental picture of spherical geometry. We may without difficulty
impart more depth and vigour to these ideas by carrying out special
imaginary constructions. Nor would it be difficult to represent the
case of what is called elliptical geometry in an analogous manner.
My only aim to-day has been to show that the human faculty of
visualisation is by no means bound to capitulate to non-Euclidean
geometry.