A general solution of Einstein's Equations for systems involving three variables

§ 1. In this paper we produce and discuss some general solutions of Einstein’s gravitational equations. For simplicity we restrict the group of solutions to those which involve only three of the space-time variables explicitly. It is immaterial which co-ordinate does not occur explicitly as we can, by transformation and appropriate boundary conditions, choose this co-ordinate to be whichever is desired. The transformation may be an imaginary one, but in the analysis no reality conditions are implied, so that choice of the time co-ordinate is purely a matter of the physical interpretation of the final results. We consider first the Cartesian ground-form in completely empty space, and modify it by the introduction of exponential factors, thusds 2 = —e ­λ dx 2 —e μ dy 2 —e γ dz 2 +e ρ dt 2 , where λ, μ, γ, ρ are to be explicit functions of (x, y, z ). This ground-form is to be thought of as a Cartesian ground-form in empty space which has been perturbed by the presence of the non-vanishing functions λ, μ, γ, ρ and we endeavour to interpret this perturbation as the result of the introduction of matter into the previously empty world. This interpretation may be permissible provided the components of the contracted Riemann tensor Gμγ , vanish at all points unoccupied by matter, and that the necessary boundary conditions and continuity also obtain. These equations Gμγ = 0 are then solved for λ, μ, γ, ρ. By regarding λ, μ, γ, ρ as perturbations it is not unreasonable to attempt to solve the gravitational equations, by a series of successive approximations, to be determined by selecting the terms in succession, according to their weight in λ, μ, γ, ρ.