Bravais classes of two-dimensional relativistic lattices

Abstract

Abstract Repetition of motifs in space and time gives rise to regular patterns with symmetries described by relativistic crystallographic groups. This leads to natural generalizations of concepts familiar in Euclidean crystallography. Here only the two-dimensional relativistic case is considered. Conditions are derived for a Lorentz transformation to be crystallographic, i . e . to leave invariant a lattice in the two-dimensional Minkowskian space. The introduction of crystallographic transformations that change the sign of the indefinite metric tensor appears to be a necessary step in relativistic crystallography. The corresponding concepts of the theory of binary quadratic forms and of real quadratic fields are briefly discussed. A classification of all possible relativistic two-dimensional lattices is given and the corresponding Bravais classes are derived (at least in principle, as there are an infinite number of them). Isotropic lattices ( i . e with lattice points on the light cone) and incommensurable lattices ( i . e . with incommensurable metric tensor) have as holohedry a point group of finite order. The other ones, which are described essentially by metric tensors g ( B )=( a , b , c ) with relatively prime rational integers a , b , c and discriminant d = b 2 −4 ac not a square, always have a holohedry of infinite order. A number of lattice representatives of Bravais classes ordered according to the kinematical interpretation of the Lorentz transformation is given in the appendix.