Unified field theory in a curvature-free five-dimensional manifold
Instead of identifying fields with the curvature of a metric, the present theory shows that they may be identified with the manner in which the four-way measuring system of the physical observer O is embedded in a flat five-dimensional manifold provided that due accouut is taken of the imperceptibility of the fifth dimension. In this system fields are introduced by treating the direction cosines, k l j , of the four directions of measurement and of the imperceptible direction as variable functions of position in the manifold. The track of an unconstrained body P is taken as a straight line (cosmodesic) in the manifold, but the ‘projection’ of it which O observes in his four-co-ordinate system is in general curved. Thus the equation describing the element ds of P 's cosmodesic in O 's four-co-ordinate system (∆ x µ ) is ds 2 cos 2 λ-2 ds sinλ{(∑ v =1 4 v l 5 v l µ ) ∆ x µ }={ 5 l µ 5 l v -2 5 l v ∑ k =1 5 k l 5 k l µ +∑ k =1 5 k l µ k l v }∆ x µ ∆ x v . When O applies the variational condition to ds which expresses the fact that the cosmodesic is straight, he concludes that it has a space-time curvature with two distinct components, one dependent upon λ which is the angle between the cosmodesic and an universal direction 5 Q and upon v l 5 , the other acting equally on all P bodies whatever the value of λ and depending only on 5 l µ . These ‘accelerations’ are shown to correspond to electromagnetic and gravitational fields respectively, and the inverse square law of force is shown to hold for spherically symmetrical fields of both types as a consequence of the condition of coherence of the measuring system. When the cause of the positional variation of the k l µ is a heavy body, having a constrained rotation, it is shown to give rise to the magnetic field that a body of charge equal to its gravitation mass would have, without the corresponding electrostatic field. The k l j 's are restricted by the requirement that the angles between the absolute fifth direction, the direction imperceptible for O , and the direction orthogonal to O 's four measuring directions, are all null.
