A Beckman Quarles Type Theorem for Plane Lorentz Transformations

rvlathematische Zeitschrift

Math. Z. 177, 101-106 (1981)

9 Springer-Vertag t981

A Beckman Quarles Type Theorem for Plane Lorentz Transformations Walter Benz Mathematisches Seminar der Universitiit, Bundesstr. 55, D-2000 Hamburg, Federal Republic of Germany

1. The result of this note is

Theorem A. Given a fixed real number p ~=0 and given a mapping ~ of IR 2 into the set Po(IR2) of all non empty subsets of IR2 such that AB = p implies A'B' = p for all A, B M R 2, A ' ~ A ~, B'6BL Then a must be a Lorentz transformation. Here AB denotes the pseudo euclidean distance ( b l - a l ) 2 - ( b 2 - a 2 ) 2 for A = ( a l , a 2 ) , B =(bl,b2).

2. Our Theorem A corresponds to a result of F.S. Beckman and D.A. Quarles [1], concerning euclidean isometries. A similar characterization for hyperbolic isometries was established by A.D. Kuz'minyh [2]. The best known result before Theorem A for plane Lorentz transformations is due to E.M. Schr6der [4] (p. 144, Satz 6):

TheoremB. Given f i x e d real numbers a,b such that a~=O and b/a