Noether and Hilbert (metric) energy-momentum tensors are not, in general, equivalent

Baker, Mark Robert; Kiriushcheva, N.; Kuzmin, S. V.

doi:10.1016/j.nuclphysb.2020.115240

Cited by 15 publications

(30 citation statements)

References 21 publications

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“…This result was determined independently by later authors ( [35][36][37], to name a few) and summarized in [13]. Furthermore, it has recently been shown that the Noether and Hilbert energy-momentum tensor are not, in general, equivalent [12], which further emphasizes the need for the investigation into the relationship between tensors which are derived directly from Noether's first theorem, and those which can only be obtained after adding the divergence of a superpotential and terms proportional to the equations of motion for a particular theory. This will be a subject that we address in this article.…”

Section: ) Motivationmentioning

confidence: 88%

“…Using these conditions on (12) and the identity A∂B = ∂(AB) − B∂A accordingly we are left with the general energy-momentum tensor,…”

Section: ) the Most General Fock Energy-momentum Tensor For Lin-earized Gravitymentioning

confidence: 99%

“…The following Appendix includes the system of linear equations corresponding to the most general energy-momentum tensor for linearized gravity (12) under the condition that the most general expression can be derived from the canonical Noether tensor supplemented by the ad-hoc addition of the divergence of a superpotential and terms proportional to the equations of motion given in (22). In addition to the equations given in this Appendix, (19) and ( 20) can be imposed to derive a symmetric energy-momentum tensor.…”

Section: Appendix a -General Energy-momentum Tensor System Of Lin-ear Equationsmentioning

confidence: 99%

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Canonical Noether and the energy–momentum non-uniqueness problem in linearized gravity

Baker¹

2021

Class. Quantum Grav.

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Recent research has highlighted the non-uniqueness problem of energy-momentum tensors in linearized gravity; many different tensors are published in the literature, yet for particular calculations a unique expression is required. It has been shown that (A) none of these spin-2 energy-momentum tensors are gauge invariant and (B) the Noether and Hilbert energy-momentum tensors are not, in general, equivalent; therefore uniqueness criteria is difficult to specify. Conventional wisdom states that the various published energy-momentum tensors for linearized gravity can be derived from the canonical Noether energy-momentum tensor of spin-2 Fierz-Pauli theory by adding ad-hoc 'improvement' terms (the divergence of a superpotential and terms proportional to the equations of motion), that these superpotentials are in some way unique or physically significant, and that this implies some meaningful connection to the Noether procedure. To explore this question of uniqueness, we consider the most general possible energy-momentum tensor for linearized gravity with free coefficients using the Fock method. We express this most general energy-momentum tensor as the canonical Noether tensor, supplemented by the divergence of a general superpotential plus all possible terms proportional to the equations of motion. We then derive systems of equations which we solve in order to prove several key results for spin-2 Fierz-Pauli theory, most notably that there are infinitely many conserved energy-momentum tensors derivable from the 'improvement' method, and there are infinitely many conserved symmetric energy-momentum tensors that follow from specifying the Belinfante superpotential alone. This disproves several recent claims that the Belinfante tensor is uniquely associated to the Hilbert tensor in spin-2 Fierz-Pauli theory. We give two new energy-momentum tensors of this form. Most importantly, since there are infinitely many energy-momentum tensors of this form, no meaningful or unique connection to Noether's first theorem can be claimed by application of the canonical Noether 'improvement' method.

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Section: ) Motivationmentioning

confidence: 88%

“…Using these conditions on (12) and the identity A∂B = ∂(AB) − B∂A accordingly we are left with the general energy-momentum tensor,…”

Section: ) the Most General Fock Energy-momentum Tensor For Lin-earized Gravitymentioning

confidence: 99%

Section: Appendix a -General Energy-momentum Tensor System Of Lin-ear Equationsmentioning

confidence: 99%

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Canonical Noether and the energy–momentum non-uniqueness problem in linearized gravity

Baker¹

2021

Class. Quantum Grav.

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“…In the recent literature these models were to prove that the Noether and Hilbert energymomentum tensor are not, in general, equivalent [4].…”

Section: Discussionmentioning

confidence: 99%

A connection between linearized Gauss–Bonnet gravity and classical electrodynamics II: Complete dual formulation

Baker

2021

Int. J. Mod. Phys. D

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In a recent publication, a procedure was developed which can be used to derive completely gauge invariant models from general Lagrangian densities with [Formula: see text] order of derivatives and [Formula: see text] rank of tensor potential. This procedure was then used to show that unique models follow for each order, namely classical electrodynamics for [Formula: see text] and linearized Gauss–Bonnet gravity for [Formula: see text]. In this paper, the nature of the connection between these two well-explored physical models is further investigated by means of an additional common property; a complete dual formulation. First, we give a review of Gauss–Bonnet gravity and the dual formulation of classical electrodynamics. The dual formulation of linearized Gauss–Bonnet gravity is then developed. It is shown that the dual formulation of linearized Gauss–Bonnet gravity is analogous to the homogenous half of Maxwell’s theory; both have equations of motion corresponding to the (second) Bianchi identity, built from the dual form of their respective field strength tensors. In order to have a dually symmetric counterpart analogous to the nonhomogenous half of Maxwell’s theory, the first invariant derived from the procedure in [Formula: see text] can be introduced. The complete gauge invariance of a model with respect to Noether’s first theorem, and not just the equation of motion, is a necessary condition for this dual formulation. We show that this result can be generalized to the higher spin gauge theories, where the spin-[Formula: see text] curvature tensors for all [Formula: see text] are the field strength tensors for each [Formula: see text]. These completely gauge invariant models correspond to the Maxwell-like higher spin gauge theories whose equations of motion have been well explored in the literature.

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“…This constitutes an example of a theory for which a locally-defined gauge-invariant Noetherian stressenergy tensor and the localised Hilbertian stress-energy tensor do not even agree up to a constant factor (cf. related examples in the context of Minkowski spacetime considered in Baker et al (2021)): Locally, the Hilbert stress-energy tensor for the theory under consideration is where i, j, k, … range only over spatial indices. By contrast, using that L local M = 1 2 i i + 4 , we obtain the following expression for the Noether stressenergy tensor: Notably though, both T Hilbert, local and T Noether, local are conserved just in case one of them is conserved, as they agree on-shell up to a constant factor.…”

Section: A Problem Case For Sus' Argumentmentioning

confidence: 99%

Miracles persist: a reply to Sus

Linnemann

Read

2022

Euro Jnl Phil Sci

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In a recent article in this journal, Sus purports to account for what have been identified as the ‘two miracles’ of general relativity—that (1) the local symmetries of all dynamical equations for matter fields coincide, and (2) the symmetries of the dynamical equations governing matter fields coincide locally with the symmetries of the metric field—by application of the familiar result that every symmetry of the action is also a symmetry of the resulting equations of motion. In this reply, we argue that, while otherwise exemplary in its clarity, Sus’ paper fails in this regard, for it rests upon a illegitimate application of the aforementioned result. Thus, we conclude, pace Sus, that these two miracles persist in general relativity.

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Noether and Hilbert (metric) energy-momentum tensors are not, in general, equivalent

Cited by 15 publications

References 21 publications

Canonical Noether and the energy–momentum non-uniqueness problem in linearized gravity

Canonical Noether and the energy–momentum non-uniqueness problem in linearized gravity

A connection between linearized Gauss–Bonnet gravity and classical electrodynamics II: Complete dual formulation

Miracles persist: a reply to Sus

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