Noether’s First Theorem and the Energy-Momentum Tensor Ambiguity Problem

The general set of Noetherian energy-momentum tensors in linearized gravity: mathematical framework

“…In section 3 we compared our results to numerous common linearized pseudotensor expressions of Einstein, Landau-Lifshitz, Goldberg, Weinberg, Papapetrou, Bergmann-Thomson and Møller. From this comparison we have shown that only Einstein's linearized pseudotensor can be classified as Noetherian; a result which is expected as it is connected to the canonical Noether tensor which is derived from only a restricted set of (canonical) action symmetries [7,17], therefore can be classfied as Noetherian without requiring results from section 4.7. We then compared our results to several energy-momentum tensors of linearized gravity including Hilbert, Fierz, Butcher and Padmanabhan, and showed that none of these expressions can be classified as Noetherian.…”

“…Noether's first theorem is often praised as one of the most significant theorems in physics; the ability for one to simultaneously obtain the equations of motion, conservation laws (and force densities) directly from a Lagrangian in one compact identity is an idealistic presentation of the most foundational equations in a given theory. In field theory, the Bessel-Hagen method [18,19] allows one to derive standard physical conserved tensors, such as the energy-momentum tensor, directly from the Noether identity [7,17]. Our current accepted theory of gravitation, general relativity, to some degree conflicts with Noether's first theorem, primarily due to the lack of a finite global group of symmetry transformations such as the Poincaré group in Minkowski spacetime; the 4-parameter Poincaré translation of this group is the coordinate transformation from which an energy-momentum tensor derived from Noether's first theorem is defined.…”

“…3 There is another distinct approach to Noetherʼs first theorem that does not require the ad-hoc introduction of any 'improvements', known as the Bessel-Hagen method [17][18][19]: by considering only Noetherʼs first theorem [15,16] and the complete set of action symmetries, one can directly obtain the correct, unique, physical energy-momentum tensor directly from Noetherʼs first theorem in cases such as electrodynamics, Yang-Mills theory, and linearized Gauss-Bonnet gravity [7]. The Bessel-Hagen method uses Noetherʼs first theorem directly, therefore results from this method are completely results of Noetherʼs first theorem, as no additional ad-hoc terms must be added to the Noether identity to obtain known physical expressions [17]. For this reason, alongside the results in [1], a clear question arises: which gravitational energy-momentum expressions (if any) can be derived directly from Noetherʼs first theorem without the introduction of ad-hoc terms not present in the Noether identity?…”

“…We refer the reader to [1] for significant discussion on the topic of energy-momentum in linearized gravity, the non-uniqueness problem, and the issues that this raises with the superpotential 'improvement' method. 3 There is another distinct approach to Noetherʼs first theorem that does not require the ad-hoc introduction of any 'improvements', known as the Bessel-Hagen method [17][18][19]: by considering only Noetherʼs first theorem [15,16] and the complete set of action symmetries, one can directly obtain the correct, unique, physical energy-momentum tensor directly from Noetherʼs first theorem in cases such as electrodynamics, Yang-Mills theory, and linearized Gauss-Bonnet gravity [7]. The Bessel-Hagen method uses Noetherʼs first theorem directly, therefore results from this method are completely results of Noetherʼs first theorem, as no additional ad-hoc terms must be added to the Noether identity to obtain known physical expressions [17].…”

“…As discussed above, this is one of the two possible freedoms that exist explicitly in the Noether identity, 3 The superpotential 'improvement' method has long been used to add desired terms ad-hoc to an energy-momentum tensor derived from Noetherʼs first theorem [2,13,14]; these 'improvements' are terms which do not exist in the Noether identity [15,16]. Typically this is performed after only the 4-parameter Poincaré translation symmetry (the symmetry associated with deriving an energy-momentum tensor from the Noether identity) is used without the complete set of symmetries of the action, leading to the so-called canonical Noether energymomentum tensor, which often does not correspond to the known physical energy-momentum tensor [13,17]; therefore 'improvement' terms are added via superpotentials to obtain the desired result. A major issue with this approach is that if the desired result is not known as in linearized gravity (unlike in the case of e.g.…”