Scalar meson field in a conformally flat space

Among several authors, who studied massive and massless scalar meson fields in general relativity, attempts to obtain a complete set of solutions for a conformally flat metric $e^ψ\left({dx^1}^2 + {dx^2}^2 + {dx^3}^2 - {dx^4}^2\right)$ were made by Ray for massive and massless mesons and Gursay for massless mesons. Both of them concluded that $ψ$ must be a function of $K_0\left({dx^1}^2 + {dx^2}^2 + {dx^3}^2 -{dx^4}^2\right) + K_1 x^1 + K_2 x^2 + K_3 x^3 + K_4 x^4$, where, where $K_0,~K_1,~ K_2,~K_3,~K_4$ are all constants. Both Ray and Gursay, however, overlooked an important particular case, which is studied here. As a by-product certain equations obtained by Auria and Regge in connection with "Gravitational theories with asymptotic flat Instantons," are solved under less restrictive assumptions.

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