Fields in time

Parametrizing Null geodesics 

$$x=ct\cos\zeta\sin\eta$$$$y=ct\sin\zeta\sin\eta$$$$z=ct\cos\eta$$

$$d\textbf{t}=dt\hat{\textbf{t}}+td\zeta\hat{\boldsymbol{\zeta}}+t\sin\zeta d\eta\hat{\boldsymbol{\eta}}$$