Light-cone coordinates: Difference between revisions

Lorentz transformations
(Lorentz transformations)
In [[relativity]], '''light-cone coordinates''' is a special coordinate system where two of the coordinates, x<sup>+</sup> and x<sup>-</sup> are [[null]] coordinates and all the other coordinates are spatial. Call them <math>x_\perp</math>.
Assume we're working with a (d,1) Lorentzian signature.
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 <math>x^+ \to e^\beta x^+</math>, <math>x^- \to e^{-\beta}x^-</math>, <math>x^i \to x^i</math>. A rotation in the ij-plane only affects <math>x_\perp</math>. The parabolic transformations show up as <math>x^+ \to x^+</math>, <math>x^- \to x^- + \delta_{ij}\alpha^i x^j + \frac{\alpha^2}{2} x^+</math>, <math>x^i \to x^i + \alpha^i x^+</math>. Another set of parabolic transformations show up as <math>x^+ \to x^+ + \delta_{ij}\alpha^i x^j + \frac{\alpha^2}{2} x^-</math>, <math>x^- \to x^-</math> and <math>x^i \to x^i + \alpha^i x^-</math>.
Light cone coordinates can also be generalized to curved spacetime in general relativity.