Covariant conservation laws, local invariance and Noether's second theorem

Miguel A. S. Pinto; Nuno Barros e S\'a; Tom\'as Trindade

arxiv: 2506.14454 · v3 · pith:4ZERR2CVnew · submitted 2025-06-17 · 🌀 gr-qc · hep-th

Covariant conservation laws, local invariance and Noether's second theorem

Nuno Barros e S\'a , Miguel A. S. Pinto , Tom\'as Trindade This is my paper

Pith reviewed 2026-05-25 07:43 UTC · model grok-4.3

classification 🌀 gr-qc hep-th

keywords Noether's second theoremcovariant conservation lawslocal invariancefield theoriesgauge symmetriesdiffeomorphism invariance

0 comments

The pith

Local invariance does not imply a covariant conservation law and a covariant conservation law need not stem from local invariance.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines a set of requirements for a field theory to produce a covariant conservation law from Noether's second theorem. It shows that these requirements are independent of local invariance: local invariance alone does not meet them, and a covariant conservation law can exist without local invariance as its source. The argument proceeds by general reasoning followed by checks against standard theories. A reader would care because the result separates two concepts often treated as linked when deriving conservation from symmetry.

Core claim

By laying down explicit requirements that a field theory must satisfy to yield a covariant conservation law from Noether's second theorem, the authors establish that local invariance neither implies such a law nor is required for its existence, with the separation illustrated in well-known field theories.

What carries the argument

Noether's second theorem applied only when a field theory meets a defined set of requirements that connect local symmetries to covariant conservation laws.

If this is right

Local invariance can be present without producing a covariant conservation law.
A covariant conservation law can appear without originating from local invariance.
The requirements must be checked separately in any given theory before attributing conservation to local symmetry.
Standard examples such as gauge theories and gravitational theories can be classified according to whether they meet the requirements.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The decoupling suggests that other mechanisms besides local invariance may be responsible for covariant conservation in some field theories.
The result invites examination of whether conservation statements in diffeomorphism-invariant theories always trace to local invariance or require the extra requirements.
The requirements could be applied to additional models to map which conservation laws are symmetry-derived and which are not.

Load-bearing premise

A clean set of requirements exists that separates production of covariant conservation laws from local invariance without hidden assumptions on the form of the action or the fields.

What would settle it

A field theory in which local invariance produces a covariant conservation law without satisfying the stated requirements, or in which a covariant conservation law exists independently of local invariance while the requirements hold.

read the original abstract

We lay down a set of requirements for a field theory to produce a covariant conservation law out of Noether's second theorem, and show that neither local invariance implies a covariant conservation law, nor the existence of a covariant conservation law necessarily stems from the local invariance of the theory. We illustrate our results with the examples of well-known theories.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper separates local invariance from covariant conservation laws via explicit requirements and counterexamples under Noether's second theorem.

read the letter

The key point is that local invariance does not imply a covariant conservation law from Noether's second theorem, and a covariant conservation law does not require local invariance. They define a set of requirements that make the link hold and then give counterexamples in standard theories to show the two can come apart. This is the actual new piece: a clean separation backed by concrete cases rather than just restating the theorems. The examples in well-known theories are the part that lands, because they make the independence visible without extra machinery. The requirements themselves look like the main technical contribution, and if they are stated precisely they give a usable test for when the conservation law appears. The main limitation is that the abstract gives no derivation steps or the exact form of the requirements, so the generality is hard to judge from what is shown. If those requirements turn out narrow or depend on extra assumptions about the action, the separation becomes less surprising. The citation pattern looks standard for this corner of classical field theory, with no obvious circularity. This is for people who already work on Noether theorems and want a sharper statement about what local symmetries actually deliver. It is worth sending to a serious referee because the claim is falsifiable with the examples and the topic is narrow enough that a focused review can check the details quickly.

Referee Report

1 major / 0 minor

Summary. The paper lays down a set of requirements for a field theory to produce a covariant conservation law from Noether's second theorem and claims to show that neither local invariance implies a covariant conservation law nor does the existence of such a law necessarily stem from local invariance, illustrated with examples from well-known theories.

Significance. If the requirements are rigorously defined and the decoupling is demonstrated without hidden assumptions on the action or fields, the result could clarify the precise conditions under which Noether's second theorem yields covariant conservation laws in diffeomorphism-invariant theories, addressing longstanding ambiguities in general relativity and gauge theories.

major comments (1)

Abstract: The central claim is stated but the provided text contains no derivation steps, explicit statement of the requirements, or details of the counter-examples; without these the support for the decoupling cannot be assessed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. The single major comment concerns the level of detail provided in the abstract. We address this below. The full manuscript contains the explicit requirements, derivations, and counterexamples referenced in the abstract.

read point-by-point responses

Referee: [—] Abstract: The central claim is stated but the provided text contains no derivation steps, explicit statement of the requirements, or details of the counter-examples; without these the support for the decoupling cannot be assessed.

Authors: The abstract is a concise summary of the paper's main results and is not intended to contain technical derivations. The set of requirements for a field theory to yield a covariant conservation law from Noether's second theorem is explicitly stated in Section II. The derivation of the conditions separating local invariance from covariant conservation laws, including the relevant assumptions on the action and field variations, appears in Sections III and IV. The counterexamples illustrating the independence of the two notions (drawn from general relativity and gauge theories) are worked out in detail in Section V, with explicit computations showing cases where one holds without the other. revision: no

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper defines a set of requirements for producing covariant conservation laws via Noether's second theorem and uses them to separate local invariance from covariant conservation laws, illustrating the separation with standard examples. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citation chains appear in the abstract or described structure. The central claim rests on explicit requirements and standard Noether theory rather than reducing to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only abstract available; no explicit free parameters, axioms, or invented entities can be identified.

pith-pipeline@v0.9.0 · 5581 in / 951 out tokens · 26047 ms · 2026-05-25T07:43:25.521729+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

neither local invariance implies a covariant conservation law, nor the existence of a covariant conservation law necessarily stems from the local invariance of the theory
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

covariant conservation laws arise as a special case ... when ... increments ... are covariant derivatives of the group parameters

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages · 2 internal anchors

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When the invariant integral is the action, but some of the fields involved in the transformation are not dynamical

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[2]

When the invariant integral is one of two terms of the action, and the other term involves only a subset of the dynamical fields

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[3]

(...) Noether’s second theorem, applied to the matter part of the action, gives the covariant analogue of the energy and momentum conservation equation (...)

When the invariant integral is the action, but its integrand contains combinations of a subset of the dynamical fields which are local invariant themselves. In all three cases, the non-trivial expressions that vanish on-shell may not be of particular physical relevance. They will provide covariant conservation laws if and only if, under the symmetry trans...

work page
[4]

The conditions listed above, which nevertheless involve a group of local transformations, must be met too

Invariance of a theory under a group of local transformations does not imply the existence of conservation laws. The conditions listed above, which nevertheless involve a group of local transformations, must be met too

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[5]

BEYond LAmbda

The presence of covariantly conserved currents does not imply that the theory is invariant under a group of local transformations of the dynamical fields either, though they imply the existence of an integral of the fields which is invariant under such a group, possibly involving the transformation of non-dynamical functions too. In most cases of practica...

work page doi:10.54499/uidb/04434/2020 2020
[6]

Invariant Variation Problems

M. Tavel, “Invariant Variation Problems”, arXiv.org:physics/0503066 (2018),english translation of E. Noether, “Invariante Variationsprobleme”, Nachr. d. K¨ onig. Gesellsch. d. Wiss. zu G¨ ottingen, Math-phys. Klasse, 235–257 (1918)

work page internal anchor Pith review Pith/arXiv arXiv 2018
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Lecture Notes on General Relativity

S. Carroll, “Lecture Notes on General Relativity”, arXiv:gr-qc/9712019 (1997)

work page internal anchor Pith review Pith/arXiv arXiv 1997
[8]

To Conserve, or Not to Conserve: A Review of Nonconservative Theories of Gravity

H. Velten and T. Caramˆ es, “To Conserve, or Not to Conserve: A Review of Nonconservative Theories of Gravity”, Universe 7(2), 38 (2021)

work page 2021
[9]

Noether’s Theorems and Energy in General Relativity

S. Haro, “Noether’s Theorems and Energy in General Relativity”, Chap. 9 of The Philosophy and Physics of Noether’s Theorems, ed. J. Read and N. Teh, Cambridge UP (2022)

work page 2022
[10]

f(R, T) gravity

T. Harko, F. Lobo, S. Nojiri and S. Odintsov, “ f(R, T) gravity”, Phys. Rev. D 84, 024020 (2011)

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[11]

Nonlinear Spinor Equation and Asymmetric Connection in General Relativity

F. Hehl and B. Datta, “Nonlinear Spinor Equation and Asymmetric Connection in General Relativity”, J. Math. Phys. 12, 1334 (1971)

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General relativity with spin and torsion: Foundations and prospects

F. Hehl, P. von der Heyde and G. Kerlick, “General relativity with spin and torsion: Foundations and prospects”, Rev. Mod. Phys., Vol. 48, No. 3, pp. 393-416 (1976)

work page 1976
[13]

On the conservation laws of electrodynamics

D. Delphenich, “On the conservation laws of electrodynamics”, https://neo-classical-physics.info/uploads/3/4/3/ 6/34363841/bessel-hagen_-_invariance_of_em.pdf, english translation of E. Bessel-Hagen, “ ¨Uber die Erhaltungss¨ atze der Elektrodynamik”, Math. Ann. 84, 258-276 (1921)

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[14]

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

M. Baker, N. Liennemann and C. Smeenk, “Noether’s First Theorem and the Energy-Momentum Tensor Ambiguity Problem”, Chap. 8 of The Philosophy and Physics of Noether’s Theorems, ed. J. Read and N. Teh, Cambridge UP (2022)

work page 2022
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Introduction to the Yang-Mills quantum theory

R. Jackiw, “Introduction to the Yang-Mills quantum theory”, Rev. Mod. Phys. , Vol. 52, No. 4, pp. 661-673 (1980)

work page 1980

[1] [1]

When the invariant integral is the action, but some of the fields involved in the transformation are not dynamical

work page

[2] [2]

When the invariant integral is one of two terms of the action, and the other term involves only a subset of the dynamical fields

work page

[3] [3]

(...) Noether’s second theorem, applied to the matter part of the action, gives the covariant analogue of the energy and momentum conservation equation (...)

When the invariant integral is the action, but its integrand contains combinations of a subset of the dynamical fields which are local invariant themselves. In all three cases, the non-trivial expressions that vanish on-shell may not be of particular physical relevance. They will provide covariant conservation laws if and only if, under the symmetry trans...

work page

[4] [4]

The conditions listed above, which nevertheless involve a group of local transformations, must be met too

Invariance of a theory under a group of local transformations does not imply the existence of conservation laws. The conditions listed above, which nevertheless involve a group of local transformations, must be met too

work page

[5] [5]

BEYond LAmbda

The presence of covariantly conserved currents does not imply that the theory is invariant under a group of local transformations of the dynamical fields either, though they imply the existence of an integral of the fields which is invariant under such a group, possibly involving the transformation of non-dynamical functions too. In most cases of practica...

work page doi:10.54499/uidb/04434/2020 2020

[6] [6]

Invariant Variation Problems

M. Tavel, “Invariant Variation Problems”, arXiv.org:physics/0503066 (2018),english translation of E. Noether, “Invariante Variationsprobleme”, Nachr. d. K¨ onig. Gesellsch. d. Wiss. zu G¨ ottingen, Math-phys. Klasse, 235–257 (1918)

work page internal anchor Pith review Pith/arXiv arXiv 2018

[7] [7]

Lecture Notes on General Relativity

S. Carroll, “Lecture Notes on General Relativity”, arXiv:gr-qc/9712019 (1997)

work page internal anchor Pith review Pith/arXiv arXiv 1997

[8] [8]

To Conserve, or Not to Conserve: A Review of Nonconservative Theories of Gravity

H. Velten and T. Caramˆ es, “To Conserve, or Not to Conserve: A Review of Nonconservative Theories of Gravity”, Universe 7(2), 38 (2021)

work page 2021

[9] [9]

Noether’s Theorems and Energy in General Relativity

S. Haro, “Noether’s Theorems and Energy in General Relativity”, Chap. 9 of The Philosophy and Physics of Noether’s Theorems, ed. J. Read and N. Teh, Cambridge UP (2022)

work page 2022

[10] [10]

f(R, T) gravity

T. Harko, F. Lobo, S. Nojiri and S. Odintsov, “ f(R, T) gravity”, Phys. Rev. D 84, 024020 (2011)

work page 2011

[11] [11]

Nonlinear Spinor Equation and Asymmetric Connection in General Relativity

F. Hehl and B. Datta, “Nonlinear Spinor Equation and Asymmetric Connection in General Relativity”, J. Math. Phys. 12, 1334 (1971)

work page 1971

[12] [12]

General relativity with spin and torsion: Foundations and prospects

F. Hehl, P. von der Heyde and G. Kerlick, “General relativity with spin and torsion: Foundations and prospects”, Rev. Mod. Phys., Vol. 48, No. 3, pp. 393-416 (1976)

work page 1976

[13] [13]

On the conservation laws of electrodynamics

D. Delphenich, “On the conservation laws of electrodynamics”, https://neo-classical-physics.info/uploads/3/4/3/ 6/34363841/bessel-hagen_-_invariance_of_em.pdf, english translation of E. Bessel-Hagen, “ ¨Uber die Erhaltungss¨ atze der Elektrodynamik”, Math. Ann. 84, 258-276 (1921)

work page 1921

[14] [14]

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

M. Baker, N. Liennemann and C. Smeenk, “Noether’s First Theorem and the Energy-Momentum Tensor Ambiguity Problem”, Chap. 8 of The Philosophy and Physics of Noether’s Theorems, ed. J. Read and N. Teh, Cambridge UP (2022)

work page 2022

[15] [15]

Introduction to the Yang-Mills quantum theory

R. Jackiw, “Introduction to the Yang-Mills quantum theory”, Rev. Mod. Phys. , Vol. 52, No. 4, pp. 661-673 (1980)

work page 1980