In SR (Special Relativity) we use the Lorentztransformation (LT) to relate one frame of reference to another. Say S to S' via a LT that depends only on v where v is the relative velocity between S and S'.

In Mathematics (and QM) we seem to be always interessted in the Kernel, the Range and the Eigenvalues and vectors of a matrix.

What do these things mean with respect to the LT?

After some thinking I concluded: The Kernel represents stuff that also moves with v relative to S or stuff that rests in S'. The Range should be anything that does not move with v.

I have no Idea what the Eigenvalues and Eigenvectors do.

Am I right so far? What is the meaning of the eigenstuff?

  • $\begingroup$ Lorenz transformations are invertible (and they must be, since the inverse transforms the frame back), hence the kernel is always trivial and the range is always the whole space. $\endgroup$ – Martin May 6 '15 at 13:30

It sounds like you're a little rusty on the pure mathematics definitions. Here's a refresher, as applied to the Lorentz transformation:

  • Kernel: the set of vectors that is mapped under the transformation to the zero vector. The LT's kernel is the empty set; all nonzero vectors are mapped to other nonzero vectors under the transformation.
  • Range: the set of all vectors that the transformation could map to. In essence, what the whole four-volume of spacetime maps to. Since the kernel is empty, all of spacetime maps to all of spacetime under the transformation. The range is all Minkowski spacetime. (Note: this does not mean that every vector must map to itself, only that every vector has a unique, nonzero pre-image.)
  • Eigenvectors: "characteristic directions" that do not change (are invariant) under the transformation. For the Lorentz transformation, the lightlike directions are invariant, and are thus eigenvectors. Any vector orthogonal to the plane of the boost is also invariant, and thus also an eigenvector.
  • Eigenvalues: how much a particular characteristic direction is strethed or shrunk.. Transformations can leave directions invariant while still dilating or shrinking the vectors along those directions by some factor. However, the Lorentz transformation is orthogonal, preserving all inner products (spacetime intervals). That includes the magnitude of any vector, so even the vectors along characteristic directions retain the same magnitude as before. That means the eigenvalues associated with those directions are $+1$. In the plane of the boost, there are no real eigenvectors (and therefore no associated real eigenvalues), but some authors might consider "complex eigenvalues" or "complex eigenvectors". I personally prefer instead to consider "eigenplanes" or such instead, which would still have completely real eigenvalues.

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