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The Noether formalism for constructing conserved quantities in teleparallel equivalents of general relativity

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Pith's one-line read Applying the Noether theorem to the diffeomorphism invariance of the TEGR and STEGR actions yields conserved currents, superpotentials, and surface-integral charges that are covariant under coordinate transformations and, in TEGR…

desk verdict A useful unified exposition of the authors' Noether formalism for TEGR/STEGR conserved quantities, but the 'turning off gravity' step is underdetermined for multi-parameter spacetimes and the abstract overstates what is solved. read the letter →

arxiv 2505.18084 v1 pith:H6ENKZH6 submitted 2025-05-23 gr-qc hep-th

E. D. Emtsova , A. N. Petrov , A. V. Toporensky This is my paper · ORCID
classification gr-qchep-th MSC 83C4083D05 PACS 04.20.Fy04.20.Cv04.50.Kd
keywords NoethertheoremconservedquantitiesteleparallelgravityTEGRSTEGRsuperpotentialgravitationalenergy-momentumgaugefreedom
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper unites the authors' earlier work into a single tensor formalism for building conserved quantities in teleparallel equivalents of general relativity. It claims that applying the Noether theorem to diffeomorphism invariance of the TEGR and STEGR actions produces conserved currents and superpotentials that are covariant under coordinate changes, and in TEGR also invariant under local Lorentz rotations; integral charges become surface integrals. The teleparallel connections, which the field equations leave undetermined, are fixed by a "turning off gravity" principle, and the choice of initial tetrad in TEGR or coordinates in STEGR defines what the paper calls a gauge. A sympathetic reader would care because this offers a systematic, covariant answer to the long-standing problem of defining gravitational energy-momentum, while making explicit that the answer depends on a gauge choice.

What carries the argument

The machinery is the Noether second theorem in tensor form, organized through the Klein-Noether identities. For a diffeomorphism generated by $\xi^\alpha$, the Lagrangian's invariance forces the Noether current $I^\alpha(\xi)$ to be divergence-free and to equal the divergence of an antisymmetric superpotential $I^{\alpha\beta}(\xi)$; in a covariant theory this dual form makes conservation laws and charges coordinate-covariant by construction. The second piece of machinery is the "turning off gravity" principle: for a chosen GR solution one solves the zero-curvature condition for the Levi-Civita connection and then sets the flat teleparallel connection equal to the Levi-Civita connection at the flat limit. The pair of tetrad plus inertial spin connection in TEGR, or coordinates plus symmetric teleparallel connection in STEGR, is then organized into an equivalence class called a "gauge", and conserved quantities depend on which gauge is selected.

What would settle it

Take a single GR solution, such as Kerr, and apply the "turning off gravity" rule with two inequivalent initial tetrads in TEGR or two different coordinate systems in STEGR, then compute the Noether charge for the same timelike Killing vector using Eq. (4.49) or (4.69); if the two appropriate gauges give different energies, the paper's claim that gauges can be selected to give physically meaningful conserved quantities fails.

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Extended reading notes

Core claim

On the paper's own terms, the central result is that the Noether formalism, applied with the displacement vector $\xi^\alpha$ kept arbitrary, yields identically conserved currents that are exact divergences of antisymmetric superpotentials for both TEGR and STEGR. In TEGR the superpotential is $\overset{\bullet}{J}{}^{\alpha\beta}(\xi)=\frac{h}{\kappa}\,\overset{\bullet}{S}_a{}^{\alpha\beta} h^a{}_\sigma \xi^\sigma$, and after the field equations are used the current obeys $\partial_\alpha \overset{\bullet}{J}{}^\alpha(\xi)=\overset{\circ}{\nabla}_\alpha \overset{\bullet}{J}{}^\alpha(\xi)=0$; the associated charge $P(\xi)=\oint_{\partial\Sigma} ds_i\, \overset{\bullet}{J}{}^{0i}(\xi)$ is coordinate-covariant and Lorentz-invariant. In STEGR the superpotential is the standard Komar superpotential plus a non-metricity divergence term, $J^{\alpha\beta}_{\rm div}=\frac{\sqrt{-g}}{\kappa}\delta^{[\alpha}_\sigma (Q^{\beta]}-\hat Q^{\beta]})\xi^\sigma$, so the total current is coordinate-covariant. The flat teleparallel connections are not determined by the field equations; the paper fixes them by the "turning off gravity" rule and shows that different initial tetrads or coordinate systems select different gauges, which in general give different conserved quantities.

Load-bearing premise

The method assumes that switching off gravity in a chosen solution uniquely fixes the flat auxiliary connection, and that the Levi-Civita connection at that flat limit is the right one; if that convention is rejected or the flat limit is ambiguous, the conserved charges are not unique.

Editorial extensions

If this is right

  • For any solution of GR, the TEGR and STEGR constructions give conserved charges as surface integrals, so masses and angular momenta can be computed without choosing a non-covariant pseudotensor.
  • In TEGR, the resulting charges are invariant under local Lorentz rotations of the tetrad, resolving the covariance conflict noted for earlier tetrad-based energy definitions.
  • In STEGR, the charges are coordinate-covariant, and the non-metricity divergence in the Lagrangian contributes explicitly to the current and superpotential.
  • Because the flat connection is fixed only by an external principle, different gauges give different conserved quantities; selecting the physically meaningful gauge is part of solving a given model.
  • The formalism extends naturally to modifications of TEGR and STEGR, since it relies only on diffeomorphism invariance and the structure of flat connections.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the turning-off-gravity principle is right, then two gauges connected by a coordinate or local Lorentz transformation should produce identical Noether charges for the same Killing vector; checking this equality case-by-case would test the consistency of the gauge notion.
  • The framework suggests a selection rule the authors only gesture at: a gauge is appropriate when the Noether current vanishes for a freely falling observer, implementing the Einstein equivalence principle as a constraint on the flat connection.
  • One could apply the same Noether construction to f(T) or f(Q) gravity; since the extra terms alter the Lagrangian but not the flatness of the connection, the formalism would produce modified superpotentials whose physical values depend even more sensitively on the chosen gauge.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. This paper presents a unified Noether-based formalism for constructing conserved currents, superpotentials, and integral charges in both TEGR and STEGR. The authors start from the diffeomorphism invariance of the actions, apply the standard Noether/Klein-identity machinery of Section 3, and obtain covariant conserved quantities for TEGR in Section 4.1 and STEGR in Section 4.2. Since the teleparallel connections are non-dynamical, the paper introduces a "turning off gravity" principle in Section 5 to fix them, defines a gauge as an equivalence class of tetrad-connection pairs (TEGR) or coordinate-connection pairs (STEGR), and discusses the resulting solution-dependence of the charges. The paper is explicitly methodological and synthesizes the authors' earlier work, with applications reviewed in the conclusions.

Significance. The paper's main strength is its detailed, internally consistent derivation of covariant Noether currents and superpotentials, and its explicit treatment of the role of the non-dynamical teleparallel connection. If accepted, the formalism provides a unified reference for conserved quantities in teleparallel equivalents of GR, and it addresses a known deficiency of earlier observer-dependent definitions. The authors are transparent that the resulting charges depend on the chosen gauge and that the method is solution-dependent; Section 5.3 states this limitation directly. The principal weakness is that the "turning off gravity" principle is a postulate rather than a consequence of the field equations, and its application is not uniquely specified for multi-parameter solutions; this affects the predictive power of the formalism.

major comments (3)
  1. [Section 5.1, Section 5.2] The "turning off gravity" algorithm is underdetermined for multi-parameter solutions, as the Kerr family illustrates: in Section 5.1 steps 1-4 (and analogously Section 5.2 steps 1-3), one solves the flatness condition R=0 for the parameters of the chosen GR solution, but for Kerr the condition holds on the entire slice M=0 with arbitrary rotation parameter a, because the M=0 Kerr line element is flat in oblate-spheroidal coordinates. The algorithm therefore does not select a unique flat connection, and since every current and charge in Eqs. (4.49) and (4.69) is evaluated with this connection, different choices yield different conserved quantities. This is not merely the acknowledged coordinate/tetrad ambiguity; it is an underdetermination inside the very step intended to fix the connection. Please specify a unique additional rule (for example, a prescribed flat-space coordinate limit or a continuity condition) or explicitly state that the principle must be supplemented by further conventions, and discuss how the Kerr angular-momentum application in Ref. [22] resolves this.
  2. [Eq. (4.68)] The displayed conservation statement "J^alpha(xi) = partial_beta J^{alpha beta}(xi) = nabla_beta J^{alpha beta}(xi) = 0" is incorrect: the current is not zero; rather its divergence vanishes. This contradicts Eq. (3.20), Eq. (4.46), and the interpretation in Section 4.3. The correct statements are partial_alpha J^alpha = 0 and J^alpha = partial_beta J^{alpha beta} as separate identities; please fix this equation.
  3. [Section 5.3] The notion of an "appropriate gauge" is not defined by any algorithmic criterion; Section 5.3 only says that physically meaningful results are needed. Because the formalism permits infinitely many gauges, and the paper concedes that it is "highly solution-dependent, and thus not generally applicable", the practical prescription for selecting a gauge remains incomplete. Please either provide a principled selection rule or state more precisely the conditions under which the construction yields unique physical charges.
minor comments (5)
  1. [Eq. (4.44)] The index structure in "partial_alpha J^{alpha beta}" is inconsistent with Eq. (3.20); it should be "partial_beta J^{alpha beta}".
  2. [Throughout] The name "Wietzenbock" should be spelled "Weitzenbock" (for example, in Section 5.1).
  3. [Eq. (4.60)] The term written "-2 nabla_beta nabla_beta xi^alpha" should be clarified as "-2 nabla_beta nabla^beta xi^alpha" or the analogous form, to avoid index ambiguity.
  4. [Abstract] The phrase "what gives different conserved quantities" should be rephrased to "which gives different conserved quantities" or "leading to different conserved quantities" for grammatical correctness.
  5. [Section 4.3] The statement that in a freely falling frame all current components vanish is a local statement, whereas the integral charges are quasi-local, so the wording should make the domain of validity explicit.

Circularity Check

1 steps flagged · score 4.0 of 10

Noether core is self-contained, but the teleparallel connection is fixed by the very conserved quantities it is used to produce, and the Kerr flat-connection choice is additionally underdetermined.

  1. fitted input called prediction [Section 2.1; Section 5.1; Section 5.3]
    "Then, if it is necessary, it has to be defined by additional requirements, for example, by a construction of acceptable conserved quantities for a concrete solution as it will be shown below. ... one of the main purposes in constructing conserved quantities is to find appropriate the gauges in TEGR in which we would have physically meaningful results for the concrete solutions. ... it is highly solution-dependent, and thus not generally applicable because gauges have to be determined for each concrete solution separately."

    The conserved charges P(ξ) in Eqs. (4.49) and (4.69) are evaluated using the flat teleparallel connection, which the theory itself does not determine (varying the action with respect to the connection gives 0 = 0). The paper fixes this connection by demanding 'acceptable' or 'physically meaningful' conserved quantities and by subsequently selecting 'appropriate' gauges. Thus the charges are used as the criterion for choosing the connection that produces them, so the numerical results are fitted inputs rather than independent predictions. Section 5.3's admission that the method is 'solution-dependent' and 'not generally applicable' confirms that the gauge choice is an external selection, not a derived output.

full rationale

The central Noether derivation in Section 3 (Eqs. 3.15-3.21) is self-contained: currents, superpotentials, and charges are constructed algebraically from the Lagrangian and the diffeomorphism vector, and the covariance statements in Section 4 follow from explicit covariant rewriting plus the standard Klein-Noether identities. This part is not circular. The problematic step is the determination of the teleparallel connection/gauge. Sections 2.1 and 2.2 state that the connection is not fixed by the equations of motion, and Section 5 fixes it by the 'turning off gravity' convention. For one-parameter families such as Schwarzschild this is a well-defined background subtraction, but for Kerr the flatness condition R=0 leaves the rotation parameter a arbitrary at M=0, so the stated algorithm does not select a unique flat connection. More importantly, Sections 5.1 and 5.3 choose 'appropriate' gauges by requiring physically meaningful values, which makes the finally obtained charges a selected rather than predicted output. This is an admitted and visible gauge-dependence, yet it does undercut the claim of deriving specific conserved quantities from first principles alone. The many self-citations [14]-[22] document the method's history and applications, including the claimed first Kerr angular-momentum calculation, but they are not load-bearing for the formal Noether identities; the circularity lies in the gauge-selection step, not in the self-citation chain. Overall, the core formalism is independent, but the application-level charge values are partially fitted, giving a score of 4 rather than 0.

Assumptions & free parameters 3 free parameters · 6 assumptions · 1 invented entities

The central claim rests on the Noether theorem, the equivalence of TEGR and STEGR to GR, and the ad hoc 'turning off gravity' principle used to fix flat teleparallel connections. The gauge choice, meaning the initial tetrad in TEGR or initial coordinates in STEGR, and the observer displacement vector are free inputs that determine the values of conserved quantities. No new physical particles, forces, or dimensions are introduced; the 'gauge' notion is a classification of choices rather than a physical entity.

free parameters (3)
  • Gauge choice: initial tetrad in TEGR = No numerical value; solution-dependent, e.g., a static diagonal tetrad for Schwarzschild
    Section 5.1 states that different initial tetrads lead to different gauges and different conserved quantities. The tetrad is chosen by hand to produce physically meaningful results.
  • Gauge choice: initial coordinates in STEGR = No numerical value; solution-dependent, e.g., Schwarzschild coordinates
    Section 5.2 states that different initial coordinates lead to different gauges and different conserved quantities in STEGR.
  • Observer displacement vector xi^alpha = Arbitrary timelike vector; often a Killing vector
    Section 4.3 defines charges relative to this vector. Different observers yield different current components, and the equivalence principle is used to interpret vanishing currents.
assumptions (6)
  • standard math Diffeomorphism invariance of the TEGR and STEGR actions, with the Lagrangian as a scalar density of weight +1
    Used in Section 3 to derive the Klein-Noether identities (3.9)-(3.13) and the Noether current (3.15).
  • domain assumption The Lie derivative determines how fields vary under diffeomorphisms, including the non-dynamical flat connections
    Section 4.1, Eqs. (4.32)-(4.33): the inertial spin connection in TEGR is treated as transforming under diffeomorphisms even though it is not dynamical.
  • domain assumption Teleparallel connections are flat and non-dynamical; they are not determined by the field equations
    Section 2, Eqs. (2.3) and (2.23), and Section 5 state these are external structures fixed by additional requirements.
  • domain assumption TEGR and STEGR are dynamically equivalent to GR, so their field equations are the Einstein equations
    Section 2, Eqs. (2.18)-(2.19) and (2.28)-(2.30).
  • ad hoc to paper The 'turning off gravity' principle: the flat teleparallel connection is obtained by solving R=0 for parameters of a GR solution and taking the Levi-Civita connection at those parameters, so all conserved quantities vanish in flat spacetime
    Section 5.1 steps 1-4 and Section 5.2 steps 1-3. This is a postulate, not a consequence of the field equations, and it does not uniquely fix the connection or the gauge.
  • domain assumption Observers are represented by displacement vectors xi^alpha, and conserved currents are interpreted as energy-momentum measured by such observers
    Section 4.3. This interpretation guides the physical meaning of charges and the equivalence-principle condition that currents vanish for freely falling observers.
invented entities (1)
  • Gauge, an equivalence class of tetrad and connection pairs in TEGR or coordinate and connection pairs in STEGR
    purpose: Labels the family of equivalent choices of the non-dynamical teleparallel connection and tetrad or coordinates that yield covariant conserved quantities; different gauges give different charge values.
    This is a terminological and classificatory construct introduced in Section 5, not a physical entity with observable consequences. Its selection is solution-dependent and cannot be independently verified.

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Pith. "Pith review of The Noether formalism for constructing conserved quantities in teleparallel equivalents of general relativity." pith.science (2026). https://pith.science/paper/H6ENKZH6

@misc{pith2026250518084,
  author       = {Pith},
  title        = {Pith review of: The Noether formalism for constructing conserved quantities in teleparallel equivalents of general relativity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H6ENKZH6}},
  note         = {Machine review of arXiv:2505.18084}
}
read the original abstract

This paper brings a methodological character where we present a comprehensive formalism for constructing conserved quantities in the Teleparallel Equivalent of General Relativity (TEGR) and Symmetric Teleparallel Equivalent of General Relativity (STEGR). It was developed in series of our earlier works and, here, we unite it into a complete form. By employing the Noether method within a tensor formalism, conserved currents, superpotentials, and charges are constructed. These are shown to be covariant under coordinate transformations and local Lorentz rotations in TEGR, while in STEGR, they are covariant under coordinate transformations. The teleparallel (flat) connections in both theories are defined using the "turning off gravity" principle. Uniting such defined flat connections with tetrad in TEGR and metric in STEGR a new fruitful in applications notion "gauge" is introduced. The choice of various initial tetrads in TEGR or initial coordinates in STEGR leads to different gauges, what gives different conserved quantities. Finally, we discuss an appropriate choice of gauges from a possible set of them.

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Reference graph

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