The Spherically Symmetrical Field in the Unified Theory

Abstract

In this paper a systematic investigation of the spherically symmetrical statical field in Einstein's unified theory of the electromagnetic and gravitational fields is given. The investigation is carried through in two cases: with and without time symmetry. By a time symmetrical field we understand one whose Riemann interval is not altered by changing $+dt$ into $\ensuremath{-}dt$.To begin with we develop a generalization of the formulas for parallel displacement and covariant affine differentiation in Einstein's geometry which enables us to introduce orthogonal transformations of the local coordinates, so as to bring them formally into agreement with the Gaussian coordinates. We then solve the Einstein field equations ${F}^{\ensuremath{\mu}\ensuremath{\alpha}}=0$ and ${G}^{\ensuremath{\mu}\ensuremath{\alpha}}=0$, first without the assumption of time symmetry. We determine the values and the physical significance of the constants of integration by considering the case of a very weak field, and obtain asymptotically the classical Coulomb field ($\frac{e}{{r}^{2}}$) and the classical Newtonian potential ($\ensuremath{-}\frac{m}{r}$). Positive and negative electricity appear in the unified theory as given empirical facts, just as in the classical theory. In the present theory, however, the Newtonian field and the Coulomb field can be separated only asymptotically. In the immediate vicinity of a charged mass particle they are inextricably bound together.Our solution gives a quadruple ("Vierbein") $_{1}h^{1}=1+\frac{{e}^{2}}{{r}^{4}},$ $_{2}h^{2}=1,$ $_{3}h^{3}=1,$ $_{4}h^{1}=\ensuremath{-}(\frac{\mathrm{ie}}{{r}^{2}}){(1+\frac{{e}^{2}}{{r}^{4}})}^{\frac{1}{2}},$ $_{4}h^{4}=1+m\ensuremath{\int}{r}^{\ensuremath{\infty}}{r}^{\ensuremath{-}2}{(1+\frac{{e}^{2}}{{r}^{4}})}^{\ensuremath{-}\frac{3}{2}}\mathrm{dr}.$ All the other components vanish. By a transformation of variables this is shown to reduce to the quadruple recently calculated by Einstein and Mayer.The field equations are then solved with the assumption of time symmetry. The electrostatic field vanishes, but the component ${g}_{44}$ of the metric form has approximately the Schwarzschild value, while the others have the Euclidean values. Because of the absence of a law of motion, no observable predictions having to do with the path of an exploring particle can be made. It is shown, however, that in the unified field the theory gives in the first approximation the same red shift as the 1916 theory. If it is assumed that the orbit of a planet is a Riemannian geodesic, as in the 116 theory, then the advance of the perihelion of Mercury comes out approximately 7" per century, that is, about the same as in the 1905 theory.