1. Nonlinear transformations would producetor ϕ derivatives.
2. We are considering here only the local properties of the spacetime as determined by its line element ds2, and ignoring all questions of a global nature, such as the periodicity of the ϕ coordinate.
3. We are taking for the Levi-Civita symbols the values εabc = εabc = ±1, thereby avoiding many appearances of √2. The distinction between tensors and tensor densities is irrelevant here anyway, since the allowed transformations all have determinant unity. This notation implies that εabcGadGbeGcf = 12εdef.
4. Equation (3.10) may be identically satisfied by imposing the canonical coordinate condition x3 = ρ. However we shall not make this restriction.
5. Some special solutions of the equations of axially symmetric gravitational fields