Solution of Einstein's Field Equations for a Rotating, Stationary, and Dust-Filled Universe

The solution of Einstein's field equations, $R_{\mathrm{ij}}-\frac{1}{2}{g}_{\mathrm{ij}}{\text{R=8πκρu}}_i{u}_j{\mathrm{+\lambda g}}_{\mathrm{ij}}$ , for a line element of the form ${\mathrm{ds}}^2{\mathrm{=(dx}}^0{)}^2{\mathrm{-\delta }}^2{\mathrm{(x}}^1{\mathrm{)(dx}}^1{)}^2{\text{+2β(x}}^1{\mathrm{)dx}}^0{\mathrm{dx}}^2{\mathrm{-\gamma }}^2{\mathrm{(x}}^1{\mathrm{)(dx}}^2{\mathrm{)-\alpha }}^2{\mathrm{(x}}^1{\mathrm{)(dx}}^3{)}^2$ is found. The density, ρ, may be a function of position, and the cosmological constant λ is not necessary in order to have a finite density. The solution reduces to that of Gödel if the variable α is constant. If the requirement for an empty universe is made (R_ij = 0), the solution is conformally flat. The characteristics of the conformal curvature tensor are also obtained.