aequationes mathematicae

, Volume 72, Issue 3, pp 288–298

A characterization of Lorentz boosts

Research paper

DOI: 10.1007/s00010-006-2827-9

Cite this article as:
Benz, W. & Schwaiger, J. Aequ. math. (2006) 72: 288. doi:10.1007/s00010-006-2827-9
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Summary.

Suppose that X is a real inner product space of (finite or infinite) dimension at least 2. The following result will be proved in this note. A bijection λ ≠ id of the space-time \(Z = X \oplus {\user2{{\mathbb{R}}}}\) is an orthochronous Lorentz boost if, and only if,
  1. (i)
    There exists e ≠  0 in X and \(\tau :X \to {\user2{{\mathbb{R}}}}\backslash \{ 0\}\) with
    $$ \lambda {\left( {x,{\sqrt {1 + x^{2} } }} \right)} = {\left( {x + \tau (x)e,{\sqrt {1 + (x + \tau (x)e^{2} )} }} \right)} $$
    for all xX, and
     
  2. (ii)

    l(v,w)  =  0 implies l (λ(v), λ(w))  =  0 for all v,wZ where l(z1, z2) designates the Lorentz–Minkowski distance of z1, z2Z.

     
Moreover, we characterize (general) Lorentz boosts by distance invariance and the behavior on certain subspaces of Z.

Mathematics Subject Classification (2000).

39B5251F2551P0583A05

Keywords.

Real inner product spacesLorentz transformationsLorentz boostsfunctional equations

Copyright information

© Birkhäuser Verlag, Basel 2006

Authors and Affiliations

  1. 1.Mathematsches SeminarUniversität HamburgHamburgGermany
  2. 2.Institut für MathematikKarl-Franzens Universität GrazGrazAustria