Scalar meson field in a conformally flat space

TL;DR

This paper revisits scalar meson fields in conformally flat spacetimes, identifying overlooked solutions and solving related equations under broader assumptions, expanding understanding of such fields in general relativity.

Contribution

It uncovers an important overlooked case in scalar meson field solutions and generalizes equations related to gravitational instantons.

Findings

Identified a previously overlooked solution case.

Solved equations related to gravitational instantons with fewer restrictions.

Abstract

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 $e^{\psi}\left({dx^1}^2 + {dx^2}^2 + {dx^3}^2 - {dx^4}^2\right)$ were made by Ray for massive and massless mesons and Gursay for massless mesons. Both of them concluded that $\psi$ 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.