Characterisations of Lorentz transformations

Abstract

A.D. Alexandrov in 1953, and E.C. Zeeman in 1964 independently published papers proving that the assumption of linearity is unnecessary for characterising Lorentz Transformations. This thesis investigates and generalises that theorem, surveying work by other mathematicians on the theorem, and providing an alternative approach to its proof. Real four-dimensional Minkowski spacetime, with inner product x · y = x1y1 + x2y2 + x3y3 - x4y4, is the background space for the original versions of the theorem. This is extended to the (n+1)-dimensional metric affine space, M n+1, with Minkowski inner product, over a field F where F is ordered and every positive element has a square root in F. In this context, automorphisms of F other than the identity exist, so semilinearity is possible. This leads to the extension of the definition of Lorentz transformation to a Generalised Lorentz transformation: a bijection, g, of M n+1 which is semilinear with respect to some automorphism μ, and which preserves the inner product up to the automorphism, that is, If we define a vector x to be lightlike iff x · x = 0, then this enables definition of a binary relation λ which holds for any two points connected by a lightlike vector. A λ-automorphism is a bijection f satisfying a λ b iff f(a) λf (b) and the original Alexandrov-Zeeman Theorem proves that any such function is a Lorentz transformation, up to translation and dilation. But if instead we set up an axiom system for M n+1 based on the relation λ, we can easily prove a λ-automorphism of M n+1 maps lines onto lines and hence, by standard deductions prove a generalised version of the Alexandrov-Zeeman Theorem : a λ-automorphism is a Generalised Lorentz transformation, up to a translation and a dilation.