Gravitational field of a charged mass point

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RESUMO

Adopting, with Schwarzschild, the Einstein gauge (ǀμνǀ = -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 = (r3 + a3)⅓, where a is a constant. The Newtonian gravitational potential ψ ≡ (2/c2)(1 - g00) obeys exactly the Poisson equation (in the R variable), with the mass density equal to (E2/4πc2), E denoting the electric field. ψ also obeys a second linear equation in which the operator on ψ is the square root of the Laplacian operator. The electrostatic potential Φ (= Q/R), ψ, 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 (Q2/a). The charge Q defines a pivotal mass M* = (Q/G½). If M < M*, then the whole mass is electromagnetic. If M > M*, the electromagnetic part of the mass Mem equals [M - (M2 - M*2)½], whereas the material part of the mass Mmat equals (M2 - M*2)½. 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.

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