Reciprocal Static Metrics and Scalar Fields in the General Theory of Relativity

Abstract

In the first part of this paper a variant of the idea of reciprocal static solutions of the gravitational field equations is developed. Thus, given any static solution of Einstein's vacuum equations $R_{\mathrm{kl}}=0\mathrm{}$, a one-parameter family of pairs of solutions of the field equations with scalar field, viz., $R_{\mathrm{kl}}=-\mu V_{;k}{V}_{;l}$, $g^{\mathrm{kl}}{V}_{;kl}=0\mathrm{}$, can be written down by inspection. The special cases of spherical symmetry and axial symmetry are treated as explicit examples. In the former case all the solutions of the field equations are obtained in this way. In the second part of the paper the theory is discussed from a physical point of view, for which purpose the motion of a test particle in the spherically symmetric field is treated in some detail.