aequationes mathematicae

, Volume 72, Issue 3, pp 288–298 | Cite as

A characterization of Lorentz boosts

  • Walter Benz
  • Jens Schwaiger
Research paper


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).

39B52 51F25 51P05 83A05 


Real inner product spaces Lorentz transformations Lorentz boosts functional equations 


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Copyright information

© Birkhäuser Verlag, Basel 2006

Authors and Affiliations

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

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