A Re-Examination of Foundational Elements of Cosmology

Abstract

This paper undertakes a conceptual re-examination of several foundational elements of cosmology through the lens of spacetime symmetries. A new derivation of the Friedmann–Lemaître–Robertson–Walker metric is obtained by a careful conceptual examination of rotations and translations on generic manifolds, followed by solving the rotational and translational Killing equations, yielding both the metric and its translational generators for $k\in \{-1,0,1\}$ without any further assumptions. We then analyze how continuous symmetries are inherited by the Einstein tensor and the Hilbert energy–momentum tensor, proving two general propositions. Furthermore, we use the Maxwell and Kalb–Ramond fields to show that a homogeneous and isotropic energy–momentum tensor, in general, does not give rise to field configurations which share these symmetries. In particular, the Kalb–Ramond field we derive is significantly more general than what is usually encountered in the cosmological context. Finally, we provide a rigorous but accessible, elementary, and transparent derivation of the scalar–vector–tensor decomposition from the linearized Einstein equations. Together, these results highlight the value of multiple complementary formulations of the same cosmological physics.