Light-cone coordinates: Difference between revisions

Some uses.
(Lorentz transformations)
(Some uses.)
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. Sometimes, calculations simplify using light cone coordinates.
Light cone coordinates are sometimes used to describe relativistic collisions, especially if the relative velocity is very close to the speed of light. It's also used in the [[light cone gauge]] of string theory.