Scalar Waves In An Almost Cylindrical Spacetime

The scalar wave equation is investigated for a scalar field propagating in a spacetime background ds²=e^{2a}(-dt²+dr²)+R(e^{-2ψ}dφ²+e^{2ψ}dz²). The metric is compactified in the radial direction. The spacetime slices of constant φ and z are foliated into outgoing null hypersurfaces by the null coordinate transformation u=t-r. The scalar field imitates the amplitude behavior of a light ray, or a gravitational wave, traveling along a null hypersurface when the area function R is a constant or is a function of u. These choices for R restrict the gravitational wave factor ψ to being an arbitrary function of u