Light-cone coordinates

In relativity, light-cone coordinates is a special coordinate system where two of the coordinates, x⁺ and x^- are null coordinates and all the other coordinates are spatial.

Assume we're working with a (d,1) Lorentzian signature.

Instead of the standard coordinate system

$ds^{2}=-dt^{2}+\delta _{ij}dx^{i}dx^{j}$,

with $i,j=1,\dots ,d$ we have

$ds^{2}=-2dx^{+}dx^{-}+\delta _{ij}dx^{i}dx^{j}$

with $i,j=1,\dots ,d-1$, $x^{+}=\left[t+x\right]/{\sqrt {2}}$ and $x^{-}=\left[t-x\right]/{\sqrt {2}}$.

Both x⁺ and x^- can act as "time" coordinates.

One thing nice about light cone coordinates is that the causal structure is partially included into the coordinate system itself.

Light cone coordinates can also be generalized to curved spacetime in general relativity.