Excluded possibilities of geometrodynamical analog to electric charge

Abstract

in 1956, J. A. Wheeler and C. W. Misner [I] showed in some detail that the existence of charges in the world need not be associated with sources in the electromagnetic field equations, but rather could be described as a consequence of Maxwell's free field equations (i.e. Ffj,,:~,l =0 and F~'";,--0) in a space with a multiply connected topology [2]. The electric lines of force can be ' tra?ped' in the topological structure of space, with the number of trapped lines of force being directly proportional to the charge associated with that field. To make this idea clearer, let us examine a simple example of a multiply connected, or 'v, ormhole' topology trapping electric lines of force. If the interiors of two solid spheres of equal radius are removed from an ordinary three dimensional Euclidean space, and' the appropriate points On the surfaces of the two spheres are identified, we obtain a "wormhole' in our space. If now we imagine that the t**o spheres had been conducting, one having charge q and the other charge q . and that the resultant electric field were fixed during the above reconstruction of our space, we discover that the resultant electric field everywhere obeys the free field equations. However, for an)'one located outside the cutaway portions of the space, there wouk! be no way of telling that the charges were no longer there. For example, integrating the field over any closed surface surrounding one of the wormhole mouths would tell him that the surface still enclosed a charge q. We have created, in Wheeler's words, 'charge without charge'. Due to the symmetry in Maxwell's equations between E and B, magnetic lines of force could similarly be trapped by the topology, creating magnetic charges, ilowever, these have never been observed in nature [3l. Wc theretore make the natural assumption that magnetic lines of force cannot be trapped in a multiply-connected topology. The above concepts can bc put into a rigorous mathematical form [4]. The electromagnetic tensors/-'*~ and ~,,~,~,f'p" are curl free antisymmetric I~ensors, and as such coffespond to closed differential 2+formsfand *f(i.e