Sidelights on Relativity | Page 8

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