Sidelights on Relativity | Page 7

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