Hi,

I'm trying to construct some coordinates on Minkowski spacetime based on a world line, $C$, ($\dot{C}\cdot\dot{C}=-1$) and forward light cone. I want the "time" coordinate of a point, $p$, to be the "retarded time", i.e the time $t(p)$ at the (unique when it exists) point $C(t(p))$ on $C$ joining $p$ by a null geodesic $\gamma$ (on the **forward** light cone of $C(t(p))$: $\gamma'\cdot\dot{C}<0$). I want the "distance" coordinate to then be the `light distance': $r(p)=-(p-C(t(p)))\cdot\dot{C}(t(p))$ and some angular coordinates defined by the "direction" (possibly via a projection onto an instantaneous 3-space) of $\gamma$ at $C$.

The problem I'm having is constructing the metric. Say I start with the vector field $\dot{C}$ and a triad ("forward") of linearly independent null vectors $N_i$ (so that $g(N_i,N_j)\neq 0$). If I Fermi-Walker transport this triad along $C$: \begin{equation} \nabla_{\dot{C}}N_i=(N_i\cdot \ddot{C})\dot{C}-(N_i\cdot \dot{C})\ddot{C}\quad (\ddot{C}=\nabla_{\dot{C}}\dot{C}) \end{equation} then they remain a null triad and do not rotate (I think). I could then choose to parallel translate $\dot{C}$ and $N_j$ along the cone to obtain a local basis field.

But seeing as the $N_i$ are null I'm not quite sure how to construct the metric (that is, assuming my previous steps are ethical).

The reason I would like these coordinates is to study solutions to Maxwell's equations \begin{equation} \mbox{d}F=0\qquad \mbox{d}\ast F = j \end{equation} and hopefully obtain something like the Liénard-Wiechert potential in the language of exterior calculus. So any references would also be nice (I've tried the standard texts, i.e. Jackson and Rohrlich).

Cheers,

Mat L