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. Call them $x_{\perp }$.

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.

A boost in the tx plane shows up as $x^{+}\to e^{\beta }x^{+}$, $x^{-}\to e^{-\beta }x^{-}$, $x^{i}\to x^{i}$. A rotation in the ij-plane only affects $x_{\perp }$. The parabolic transformations show up as $x^{+}\to x^{+}$, $x^{-}\to x^{-}+\delta _{ij}\alpha ^{i}x^{j}+{\frac {\alpha ^{2}}{2}}x^{+}$, $x^{i}\to x^{i}+\alpha ^{i}x^{+}$. Another set of parabolic transformations show up as $x^{+}\to x^{+}+\delta _{ij}\alpha ^{i}x^{j}+{\frac {\alpha ^{2}}{2}}x^{-}$, $x^{-}\to x^{-}$ and $x^{i}\to x^{i}+\alpha ^{i}x^{-}$.

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