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Page 7
_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.
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