A Unified Perspective on Poincaré and Galilei Relativity: II. General Relativity: A. Kinematics

Abstract

Building on the first paper in this series (Paper~I), a unified perspective on Poincaré and Galilei physics in a 5-dimensional spacetime setting is further pursued by consideration of the kinematics of general relativity, with the gravitational dynamics to be addressed separately. The metric of the 5-dimensional affine spacetimes governed by the Bargmann groups considered in Paper~I (central extensions of the Poincaré and Galilei groups) is generalized to curved spacetime by extending the usual 1+3 (traditionally `3+1') formalism of general relativity on 4-dimensional spacetime to a 1+3+1 formalism, whose spacetime kinematics is shown to be consistent with that of the usual 1+3 formalism. Spacetime tensor laws governing the motion of an elementary classical material particle and the dynamics of a simple fluid are presented, along with their 1+3+1 decompositions; these reference the foliation of spacetime in a manner that partially reverts the Einstein perspective (accelerated fiducial observers, and geodesic material particles and fluid elements) to a Newton-like perspective (geodesic fiducial observers, and accelerated material particles and fluid elements subject to a gravitational force). These spacetime laws of motion for particles and fluids also suggest that a strong-field Galilei general relativity would involve a limit in which not only $c \rightarrow \infty$, but also $G \rightarrow \infty$, such that $G / c^2$ remains constant.