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Search: MSC category 83 ( Relativity and gravitational theory )

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1. CMB Online first

Jensen, Gerd; Pommerenke, Christian
 On the structure of the Schild group in Relativity Theory 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, subgroupCategories:22E43, 20H99, 83A05

2. CMB 2015 (vol 59 pp. 123)

Jensen, Gerd; Pommerenke, Christian
 Discrete Space-time and Lorentz Transformations Alfred Schild has established conditions that Lorentz transformations map world-vectors $(ct,x,y,z)$ with integer coordinates onto vectors of the same kind. The problem was dealt with in the context of tensor and spinor calculus. Due to Schild's number-theoretic arguments, the subject is also interesting when isolated from its physical background. The paper of Schild is not easy to understand. Therefore we first present a streamlined version of his proof which is based on the use of null vectors. Then we present a purely algebraic proof that is somewhat shorter. Both proofs rely on the properties of Gaussian integers. Keywords:Lorentz transformation, integer lattice, Gaussian integersCategories:22E43, 20H99, 83A05
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