A Class of Exact Solutions of Einstein's Field Equations

The work of Weyl on the gravitational field occasioned by an axially symmetric distribution of matter and charge is generalized to the case in which $g_{44}$ and $\phi$ for an electrostatic field are functionally related, with or without spatial symmetry. It is shown that the most general electrostatic field in which $g_{44}$ and $\phi$ are related by an equation of the form $g_{44}=\frac{1}{2}{(\phi +c)}^2$ can be represented by a line element of the form ${\mathrm{}\left(\mathrm{ds}\right)\mathrm{}}^2=-e^{-w}[{\left(d{x}^1\right)}^2+{\left(d{x}^2\right)}^2+{\left(d{x}^3\right)}^2]+e^w{\mathrm{}\left(\mathrm{dt}\right)\mathrm{}}^2$. Certain of the field equations are then identically satisfied while the remaining ones reduce to a single equation for $w$. The substitution $w=-2\mathrm{}log\mathrm{}(1+v)\mathrm{}$ transforms this into Laplace's equation for $v$, so that the solution can be expressed in terms of harmonic function.