Unified field theory in a curvature-free five-dimensional manifold

Abstract

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.