Special Relativity and its Newtonian Limit from a Group Theoretical Perspective

TL;DR

This paper uses group theory to explain how special relativity and its Newtonian limit are connected, emphasizing symmetry representations and contractions to understand the transition between the theories.

Contribution

It provides a systematic, pedagogical presentation of the group-theoretical framework underlying special relativity and its Newtonian limit, filling a gap in existing literature.

Findings

Relates Minkowski spacetime to Poincaré group representations

Shows how Newtonian mechanics emerges via symmetry contraction

Details the phase space and dynamics under symmetry transformations

Abstract

In this pedagogical article, we explore a powerful language for describing the notion of spacetime and particle dynamics intrinsic to a given fundamental physical theory, focusing on special relativity and its Newtonian limit. The starting point of the formulation is the representations of the relativity symmetries. Moreover, that seriously furnishes -- via the notion of symmetry contractions -- a natural way in which one can understand how the Newtonian theory arises as an approximation to Einstein's theory. We begin with the Poincar\'{e} symmetry underlying special relativity and the nature of Minkowski spacetime as a coset representation space of the algebra and the group. Then, we proceed to the parallel for the phase space of a spin zero particle, in relation to which we present the full scheme for its dynamics under the Hamiltonian formulation, illustrating that as essentially the symmetry feature of the phase space geometry. Lastly, the reduction of all that to the Newtonian theory as an approximation with its space-time, phase space, and dynamics under the appropriate relativity symmetry contraction is presented. While all notions involved are well established, the systematic presentation of that story as one coherent picture fills a gap in the literature on the subject matter.