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# On the Structure of the Schild Group in Relativity Theory

Published:2017-05-16
Printed: Dec 2017
• Gerd Jensen,
Sensburger Allee 22 a, D-14055 Berlin
• Christian Pommerenke,
Institut für Mathematik, Technische Universität, D-10623 Berlin
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## Abstract

Alfred Schild has established conditions that Lorentz transformations map world-vectors $(ct,x,y,z)$ with integer coordinates onto vectors of the same kind. These transformations are called integral Lorentz transformations. The present paper contains supplements to our earlier work with a new focus on group theory. To relate the results to the familiar matrix group nomenclature we associate Lorentz transformations with matrices in $\mathrm{SL}(2,\mathbb{C})$. We consider the lattice of subgroups of the group originated in Schild's paper and obtain generating sets for the full group and its subgroups.
 Keywords: Lorentz transformation, integer lattice, Gaussian integers, Schild group, subgroup
 MSC Classifications: 22E43 - Structure and representation of the Lorentz group 20H99 - None of the above, but in this section 83A05 - Special relativity

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