Linear and Toroidal Geons

Abstract

It has been argued by Wheeler that the coupled equations of the electromagnetic field and the gravitational field of general relativity, $R_{\mathrm{ik}}-\frac{1}{2}{g}_{\mathrm{ik}}R=\left(\frac{8\pi G}{c^4}\right)T_{\mathrm{ik}},$ ${T_i}^k=\left(\frac{1}{4\pi }\right)(F_{\mathrm{ij}}{F}^{\mathrm{kj}}-\frac{1}{4}{F}_{\alpha \beta }{F}^{\alpha \beta }{{\delta }_i}^k),$ ${\mathrm{}(-g)\mathrm{}}^{-\frac{1}{2}}\left(\frac{\partial }{\partial x^j}\right){\mathrm{}(-g)\mathrm{}}^{\frac{1}{2}}{F}^{\mathrm{ij}}=0,$ $F_{\mathrm{ij}}=\frac{\partial A_j}{\partial x^i}-\frac{\partial A_i}{\partial x^j},$ should admit a set of completely singularity-free solutions, $A_i$, $g_{\mathrm{jk}} (i, j, k=1, 2, 3, 4)$, with the following properties: 1. The gravitational mass originates solely from the energy stored in the electromagnetic field. In particular there are no material masses present. 2. No charges or currents are present, and $A_4=0\mathrm{}$ everywhere. 3. The other components of the electromagnetic vector potential $A_i$ are vanishingly small except within a toroidal region of space. Physically the electromagnetic field consists of light waves circling the torus in either direction. Such a torus of electromagnetic field energy is called a toroidal geon. An exact and detailed mathematical treatment of the general toroidal geon problem would be extremely complicated, requiring the solution of a set of coupled nonlinear partial differential equations. However, in the present paper it is shown how toroidal geons of large major radius to minor radius ratio may be studied by a simple method of approximation, providing one has a complete knowledge of the so-called linear geons, the electromagnetic field energy of which is confined to an infinitely long circular cylinder rather than to a torus. A detailed mathematical treatment of linear geons proves to be possible, as is demonstrated in this paper. The electromagnetic field potentials $A_i i=(1, 2, 3)$ of a toroidal geon or of a linear geon possess the same general nature as the electromagnetic field potentials encountered in the solution of classical toroidal and cylindrical wave guide problems. In this paper the case is considered where the electromagnetic field of the linear geon is a monochromatic standing wave vibrating in the lowest transverse-electric mode of the system. The field equations are derived from a variational principle, and these equations are solved numerically. The results are not surprising, as the general form of the unknown functions can be ascertained by quite simple considerations. These results give the foundation material for a proposed later treatment of toroidal geons.