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Article

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

by
Christian Y. Cardall
Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6354, USA
Symmetry 2025, 17(8), 1245; https://doi.org/10.3390/sym17081245
Submission received: 22 May 2025 / Revised: 23 July 2025 / Accepted: 31 July 2025 / Published: 5 August 2025
(This article belongs to the Special Issue Recent Advance in Mathematical Physics II)

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 through a 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 but also G, such that G/c2 remains constant.
Keywords: relativity; Poincaré group; Galilei group; Bargmann group relativity; Poincaré group; Galilei group; Bargmann group

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MDPI and ACS Style

Cardall, C.Y. A Unified Perspective on Poincaré and Galilei Relativity: II. General Relativity: A. Kinematics. Symmetry 2025, 17, 1245. https://doi.org/10.3390/sym17081245

AMA Style

Cardall CY. A Unified Perspective on Poincaré and Galilei Relativity: II. General Relativity: A. Kinematics. Symmetry. 2025; 17(8):1245. https://doi.org/10.3390/sym17081245

Chicago/Turabian Style

Cardall, Christian Y. 2025. "A Unified Perspective on Poincaré and Galilei Relativity: II. General Relativity: A. Kinematics" Symmetry 17, no. 8: 1245. https://doi.org/10.3390/sym17081245

APA Style

Cardall, C. Y. (2025). A Unified Perspective on Poincaré and Galilei Relativity: II. General Relativity: A. Kinematics. Symmetry, 17(8), 1245. https://doi.org/10.3390/sym17081245

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