Light-cone coordinates: Difference between revisions

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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. Sometimes, calculations simplify using light cone coordinates. See [[Newman-Penrose formalism]].