The canonical energy-momentum currents in cosmology

TL;DR

This work addresses the long-standing challenge of defining energy in General Relativity without relying on spacetime symmetries. It develops General Parallel Relativity (G_parallel_R), where a reference frame $\sigma_a{}^\mu$ is treated as an independent variational field and the frame is constrained to carry no intrinsic energy, yielding conserved on-shell currents $J_a{}^\mu$ and covariant Noether charges. In cosmology, the canonical time direction naturally aligns with the Kodama vector $K^\mu$, and the associated energy coincides with the Misner–Sharp mass $M_{MS}$, while the full set of currents $J_a^\mu$ provides a complete description of energy and momentum flows in FRW spacetimes. The framework unifies Einstein's two GR formulations and offers a geometric basis for quantum gravity, with potential extensions to black hole thermodynamics and quasilocal observables. $K^\mu$ and $M_{MS}$ appear as central objects in the canonical frame, and the formalism yields conserved quantities through surface charges $q_a$ that are independent of background Killing vectors, relying solely on the spacetime geometry and the frame field.

Abstract

The parallel theory of relativity predicts conserved energy-momentum currents for an arbitrary metric, without invoking Killing symmetries. By treating the reference frame as an independent variational field and requiring it to carry no energy, the theory naturally unifies Einstein's two formulations of gravity and yields uniquely defined covariant charges. In isotropic and homogeneous cosmology, the canonical time direction selected by the reference frame coincides with the Kodama vector, and the associated Noether energy reproduces the Misner-Sharp mass.