aequationes mathematicae

, Volume 72, Issue 3, pp 288–298

A characterization of Lorentz boosts

Authors

  • Walter Benz
    • Mathematsches SeminarUniversität Hamburg
    • Institut für MathematikKarl-Franzens Universität Graz
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

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).

39B52 51F25 51P05 83A05

Keywords.

Real inner product spaces Lorentz transformations Lorentz boosts functional equations

Copyright information

© Birkhäuser Verlag, Basel 2006