On the Natural Equivalence Between Canonical and Hilbert Energy Momentum Tensors via Noether's Theorem (2507.05780v1)

Abstract: In this work, we investigate the structure and properties of the canonical energy momentum tensor (EMT) across a range of field theories. We begin by developing a unified and systematic method that naturally yields the canonical EMT for gauge theory, without the need for artificial symmetrization or improvement terms. Our analysis highlights how Noether's theorem intrinsically emphasizes the 1-form nature of gauge potentials. We further extend to general relativity and demonstrate that the assumption of metric compatibility naturally implies a torsion free connection. Building upon variational symmetry principles, we establish the equivalence between the Einstein Hilbert EMT and the canonical EMT, thereby clarifying their respective roles in field dynamics and conservation laws. Lastly, we present an alternative derivation of the Einstein field equations via Noether's theorem.