Light-cone coordinates: Difference between revisions

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In [[Theory of relativity|special relativity]], '''light-cone coordinates''' is a special coordinate system where two of the coordinates, x<sup>+</sup> and x<sup>−</sup> are [[null (mathematics)|null]] coordinates and all the other coordinates are spatial. Call them <math>x_\perp</math>.
 
Assume we're are working with a (d,1) Lorentzian signature.
 
Instead of the standard coordinate system (using [[Einstein notation]])
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]].
 
Light cone coordinates are sometimes used to describe relativistic collisions, especially if the relative velocity is very close to the speed of light. It'sThey are also used in the [[light cone gauge]] of string theory.
 
== See also ==