A unified perspective on Poincar\'e and Galilei relativity: I. Special relativity

A semantic adjustment to what physicists mean by the terms `special relativity' and `general relativity' is suggested, which prompts a conceptual shift to a more unified perspective on physics governed by the Poincar\'e group and physics governed by the Galilei group. After exploring the limits of a unified perspective available in the setting of 4-dimensional spacetime, a particular central extension of the Poincar\'e group -- analogous to the Bargmann group that is a central extension of the Galilei group -- is presented that deepens a unified perspective on Poincar\'e and Galilei physics in a 5-dimensional spacetime setting. The immediate focus of this paper is classical physics on affine 4-dimensional and 5-dimensional spacetimes (`special relativity' as redefined here), including the electrodynamics that gave rise to Poincar\'e physics in the first place; but the results here may suggest the existence of a `Galilei general relativity' more extensive than generally known, to be pursued in the sequel.