Gravitational field of a charged mass point.
Adopting, with Schwarzschild, the Einstein gauge ((munu) = -1), a solution of Einstein's field equations for a charged mass point of mass M and charge Q is derived, which differs from the Reissner-Nordstrøm solution only in that the variable r is replaced by R = (r(3) + a(3))((1/3)), where a is a constant. The Newtonian gravitational potential psi identical with (2/c(2))(1 - g(00)) obeys exactly the Poisson equation (in the R variable), with the mass density equal to (E(2)/4pic(2)), E denoting the electric field. psi also obeys a second linear equation in which the operator on psi is the square root of the Laplacian operator. The electrostatic potential Phi (= Q/R), psi, and all the components of the curvature tensor remain finite at the origin of coordinates. The electromagnetic energy of the point charge is finite and equal to (Q(2)/a). The charge Q defines a pivotal mass M(*) = (Q/G((1/2))). If M < M(*), then the whole mass is electromagnetic. If M > M(*), the electromagnetic part of the mass M(em) equals [M - (M(2) - M(*2))((1/2))], whereas the material part of the mass M(mat) equals (M(2) - M(*2))((1/2)). When M > M(*), the constant a is determined, following Schwarzschild, by shrinking the "Schwarzschild radius" to zero. When M < M(*), a is determined so as to make the gravitational acceleration vanish at the origin.
Pekeris
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Department of Applied Mathematics, The Weizmann Institute, Rehovot, Israel.