Sidelights on Relativity | Page 5

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