A Re-Examination Of Foundational Elements Of Cosmology

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-Lema\^itre-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 \emph{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 \emph{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.