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

TL;DR

This paper explores the fundamental relationship between canonical and Hilbert energy-momentum tensors using Noether's theorem, providing a unified approach and clarifying their roles in gauge theories and general relativity.

Contribution

It introduces a systematic method to derive the canonical EMT in gauge theories and proves the equivalence between Einstein Hilbert and canonical EMTs through variational symmetries.

Findings

Unified derivation of canonical EMT for gauge theories

Demonstration of equivalence between Einstein Hilbert and canonical EMTs

Alternative derivation of Einstein field equations via Noether's theorem

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.