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I'm currently taking classes on Special Relativity and Minkowski space. I have one question: how would future-oriented, contravariant time-like, space-like and light-like vectors transform under a Lorentz transformation?

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  • $\begingroup$ Have you tried a calculation of a boost applied to such vectors? You can start with the 1+1 case. You should find that the tips move along distinct hyperboloids, where the light cone is a degenerate hyperboloid of radius [interval] zero. $\endgroup$ – robphy Jan 4 at 12:27
  • $\begingroup$ Light-like vectors are eigenvectors, that is point. Just visualise the skew. $\endgroup$ – Abhimanyu Pallavi Sudhir Jan 4 at 15:12
  • $\begingroup$ @AbhimanyuPallaviSudhir Though in n+1 for n > 1 dimensions, not all lightlike vectors are eigenvectors. $\endgroup$ – Ricky Tensor Jan 5 at 1:58
  • $\begingroup$ Oh yes. Light-like vectors in the plane of a Lorentz boost are eigenvectors, sorry. Also worth asking if the OP means just "Lorentz boost" by "Lorentz transformation", or if he's including rotations. $\endgroup$ – Abhimanyu Pallavi Sudhir Jan 5 at 8:03

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