Classical Electron in General Relativity

The usual field equations of general relativity are modified to provide for the existence of electrons. The resulting equations, viz., $R_{\mu \nu }-\frac{1}{4}R{g}_{\mu \nu }=-k(T_{\mu \nu }-\frac{1}{4}T{g}_{\mu \nu })$, are considered to be microscopically true. The macroscopic averages of the equations are the usual Einstein field equations. On the basis of the field equations, it is shown that an electron may be constructed which is stable and has a momentum and energy which form a four-vector. The electron is held together by the gradient of the Gaussian curvature and is $\frac{3}{4}$ electro magnetic, ¼ gravitational energy. The radiation reaction problem is examined, and it is shown that the Cauchy problem for the field equations is nonregular in the sense that, as with the Lorentz-Dirac equation, one component of the initial acceleration of the electron must be specified. An interesting consequence of the field equations is that a purely electric (static) electron is not a regular solution. The future development of such a particle is not determined. Thus, a spherically symmetric electron must be dynamic in character, i.e., have currents and consequent magnetic moments.