FOUR-DIMENSIONAL TUBES
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XI. 6]
It will be evident that we have done no more than touch upon
the fringe of a region as yet unexplored. The experimental
physicist finds a difficulty in working in this unfamiliar world,
but it is clear that to the mathematician there is a wide field of
investigation open in examining the properties of such discrete
tubes of force in four dimensions. In particular it is desirable
to study the character of the calamoids in the immediate neigh
bourhood of an electron or magneton. It will be necessary to
take into account both “ electricity ” and “ magnetism/' but we
may recall in this connection McLaren’s conception of matter
as a “ hole in the aether,” i.e. a region where the electromagnetic
field equations cease to apply.
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6. A Five-Dimensional Theory
In the general theory of relativity there is an apparent dualism
of gravitation and electricity. It has been suggested that the
inter-connection between the electromagnetic and the gravita
tional field may be considered from a single standpoint by
employing a five-dimensional space. Kaluza * has given a theory
in which the ten Einstein gravitational potentials and the four
electromagnetic potentials are expressed in terms of the
coefficients of the line element in a Riemannian space of five
dimensions. The equations of motion of electrical particles in
an electromagnetic field take the form of the equations of the
geodesics. Klein | has developed this theory and shown that
it may be put into relation with the undulatory form of the
quantum theory given by de Broglie and Schrôdinger (Chapter
XVII). He further suggests that the origin of the quantum is
to be found in the periodicity of the fifth dimension thus intro
duced, and that the atomicity of electricity may be interpreted
as a quantum theory law.
Although such a mode of representation may be found con
venient by the mathematician, it does not appear that it is able
to throw much light on the physical aspect of the interpretation
of the quantum.
* Kaluza, Preuss. Akad. Wiss., Berlin, Ber., vol. 54, p. 966, 1921.
t Klein, Zeits. f. Physik, vol. 37, p. 895, 1926; Nature, vol. 118, p.
516, 1926.
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