Symmetric teleparallel general relativity

H-J Yo (National Central U.); J.M. Nester

REVIEW

Symmetric teleparallel general relativity

J.M. Nester, H-J Yo (National Central U.)

General Relativity and Quantum Cosmology · Submitted 17 Sep 1998 · 23 references

Pith's verdict: A short, honest note that names symmetric teleparallel gravity and gets the algebra right; the geometric reframing is the contribution, and the authors are candid that no new calculation yet follows from it.

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arXiv IDgr-qc/9809049 (version 2)

Pith IDQDTL2OLM, Pith's permanent record of this paper and its review

CitationH-J Yo (National Central U.), J.M. Nester. "Symmetric teleparallel general relativity." arXiv:gr-qc/9809049 (1998). https://pith.science/paper/QDTL2OLM

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Editor's note

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Short version: this is a six-page note that puts a name and a clean geometric setting on what people now call symmetric teleparallel gravity, and the algebra checks out. It is not a new physical result, and the authors do not pretend it is. Read it as the origin point for a literature, not as a derivation of anything new about gravity. The reader's ACCEPT is right; the stress-test's worry is real but lands softly, for reasons below. One correction to the record: the classifier reports no stated limitations, which is wrong. The closing paragraphs list three explicitly, and they are the honest ones. The technical core is Eq. (7): in a teleparallel frame with vanishing connection and zero torsion, the coframe is exact, nonmetricity becomes minus the metric derivative, and the deformation tensor becomes minus the Christoffel symbol. Substituting into the general teleparallel Lagrangian (6) gives the old gamma-gamma density (8). That identity is Einstein 1916 and Tolman 1930; the new thing is the reading, and the paper says so, citing the hints in Hehl et al. and Wald. Credit where due: the derivation is short enough to check in an afternoon, the index algebra is consistent, and the paper correctly flags the five-parameter family of quadratic nonmetricity theories as the natural next step. Now the soft spots. The abstract's 'a true tensor' (abstract:s6) is unqualified; the body's 'covariant (but geometry gauge dependent) tensor' is the accurate version, and the abstract should match it. That is a minor fix. The sharper point is the stress-test's: the covariantization is definitional. Nothing in the paper exhibits a calculation that becomes possible, or a result that changes, under the new reading. The authors themselves concede this with 'we don't yet know what might be revealed.' That is an acknowledged limitation, not a hidden one, and it is the real ceiling on the paper's significance. The global-chart issue the reader flags is also acknowledged ('almost requires a global coordinate system') and is a genuine constraint on the representation, but not a flaw in the derivation. Who should read it: anyone working on energy-momentum localization, teleparallel gravity, or metric-affine formulations, and anyone who wants the citation for where symmetric teleparallel gravity starts. It deserves a serious referee, mainly to push on the abstract qualifier and to ask for one worked example where the STG reading buys something the coordinate calculation did not.

Referee report

Significance

If the construction holds, it opens a genuinely new corner of the space of teleparallel representations of GR: the standard metric-compatible, torsionful (Weitzenböck) case has been extensively studied, and the opposite extreme — flat, torsion-free, non-metric — has been largely overlooked apart from passing hints in the literature the authors cite. The derivation is short, self-contained, parameter-free, and checkable line by line; Eq. (7) is the correct specialization of the general decomposition of Eq. (2), and the reduction to Eq. (8) is the expected identity. The note also correctly identifies that the GR-equivalent theory is one member of a five-parameter family of quadratic-nonmetricity theories, which is a useful and concrete pointer for future work. The value here is conceptual and organizational rather than a new physical prediction, and the authors are candid about that: they state plainly that they do not yet know what the reformulation might reveal. The paper's honesty about its own limitations — including the non-commutation of covariant derivatives with index movement and the failure of naive minimal coupling — is a real strength and should be preserved in any redig.

Major objections (2)

[Abstract, sentence 6, versus the paragraph following Eq. (8)] The abstract states without qualification that the energy-momentum density 'is a true tensor,' whereas the body states the accurate version: 'a covariant (but geometry gauge dependent) tensor.' The qualifier is not cosmetic. The object is a tensor with respect to the teleparallel connection, and that connection is fixed by declaring an arbitrary frame parallel, so the 'gauge' freedom being invoked is the old coordinate and frame freedom under a new name. Since this tensor claim is the paper's headline result, the abstract should carry the same qualification the body does, for instance by saying that the density is tensorial with respect to the symmetric teleparallel connection and that the residual gauge freedom corresponds to the choice of that connection. As written, a reader who stops at the abstract will take away a stronger claim than the manuscript defends.
[Paragraph following Eq. (6), and the closing liabilities paragraph] The promotion of coordinate choices to geometry choices is only as global as the teleparallel frame itself. The manuscript says the exact coframe 'at least locally' can be written ϑ^α = dx^α, and later concedes that the formulation 'almost requires a global coordinate system'; earlier it notes that parallelizability is 'a strong global topological restriction.' These three statements are correct individually but are never joined, and the tensor claim of the abstract depends on the join. Please state the precise hypothesis at the point where the covariantization is asserted — that the manifold admits a global parallelizable frame, equivalently a global chart in which the connection coefficients vanish — and say explicitly what survives when it fails, i.e. whether the energy-momentum object is then only patchwise tensorial with patching by transition functions. This is a framing and precision issue rather than an error in the algebra, but it is load-bearing for the central interpretive claim.

Minor objections (6)

[Eq. (7)] There is an index-position inconsistency in Eq. (7): the left-hand side is written A_{αβγ} with α down, while the final expression is -{^α_βγ} with α up. Please make the placement uniform and state once, explicitly, the convention relating A^α_β to A_{αβγ} via the frame, since the subsequent substitution into Eq. (6) depends on it.
[Eqs. (6) and (8)] The step from Eq. (6) to Eq. (8) passes through the antisymmetry of η_{αβ} and a particular ordering of the two deformation factors; the overall sign of the γγ density is convention-sensitive. Please add one line showing the intermediate contraction, or state the sign convention for η_{αβ} and for the wedge ordering, so that a reader can reproduce the sign of Eq. (8) without guesswork.
[Paragraph following Eq. (6)] The text says one 'can adjoin the vanishing curvature condition via a Lagrange multiplier,' and separately that the symmetry (torsion-free) condition can also be imposed by a multiplier, but the constrained action is never written. Since the note's claim is that the unconstrained Lagrangian of Eq. (6) plus constraints reproduces GR, a schematic display of the constrained 4-form, even in one line, would make the variational statement self-contained.
[Paragraph following Eq. (6), reference to [19,20]] The five independent quadratic nonmetricity invariants are invoked but not displayed, and the reader is not told which combination singles out the GR-equivalent member. Listing the five terms, or at least identifying the specific combination that yields Eq. (8), would substantially increase the usefulness of the note as a starting point for the investigation the authors announce.
[Closing liabilities paragraph] The remark that the source coupling reproducing standard GR 'is not teleparallel minimal coupling' is important and is stated in a single sentence. A brief indication of what the correct coupling prescription is, or a pointer to where it is worked out, would prevent the sentence from reading as an unresolved worry.
[Eq. (8) and surrounding text] The braces { } in Eq. (8) are evidently Christoffel symbols of the metric, but this is only implicit from Eq. (7). A short explicit statement of the notation at first use would help readers arriving at Eq. (8) directly.

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The paper

The paper claims

General relativity can be rewritten in a symmetric teleparallel geometry whose nonmetricity carries the gravitational force, turning the Einstein energy pseudotensor into a genuine tensor.

The reading

The paper asks whether general relativity can be recast in a geometry that is flat in the strongest sense: both curvature and torsion vanish, while the metric is not parallel-transported, so the nonmetricity $ abla g$ carries the gravitational field. The authors show that in a suitable teleparallel frame the torsion-free condition makes the coframe exact, so the connection coefficients vanish and covariant derivatives reduce to ordinary coordinate partials. In that frame the connection deformation becomes minus the Christoffel symbol, and the teleparallel Lagrangian collapses to the familiar $��{-g}��g^{��}��(��{��}��{��}��{��}��{��}-��{��}��{��}��{��}��{��})$ expression long used in coordinate energy calculations. The payoff is that the Einstein energy-momentum pseudotensor is reinterpreted as a tensor in this geometry, with its old coordinate dependence recast as a choice of connection.

What carries the argument

The load-bearing device is the symmetric teleparallel connection: a connection with $R^��{}_��=0$ and $T^��=0$ but $Q_{��}=-Dg_{��}��0$, represented in a frame where its coefficients vanish. In that frame the coframe is closed, so locally $��^��=dx^��$ and the covariant derivative becomes $��_��$; the connection deformation $A^��{}_��$ reduces to minus the Christoffel symbol. This identity is what turns the standard $�� $ Lagrangian density into a covariant teleparallel Lagrangian and what promotes the Einstein pseudotensor to a tensor within the chosen geometry.

Extended reading notes

Core claim

The central claim is that general relativity admits a symmetric teleparallel representation: curvature and torsion vanish, nonmetricity does not, and the nonmetricity is the gravitational force. Starting from the decomposition of a general connection into the Levi-Civita connection plus contortion and nonmetricity terms, the authors set torsion to zero and choose a teleparallel frame with vanishing connection coefficients. Torsion-free flatness then forces the coframe to be a coordinate differential, $��^��=dx^��$ at least locally, so the deformation one-form is exactly $A^��{}_��=-��{��}��{��}��^��$. The Einstein-Hilbert Lagrangian minus a total derivative becomes the quadratic expression in Eq. (8), which is the standard coordinate Lagrangian but now read as a covariant object in the teleparallel geometry. Consequently the associated energy-momentum density, essentially the Einstein pseudotensor, is a true tensor: geometry-gauge dependent, but no longer a coordinate artifact.

Load-bearing premise

The construction needs a single teleparallel frame, equivalently a coordinate chart, in which the connection coefficients vanish and torsion is zero over the whole region where the energy-momentum tensor is claimed to be a true tensor; the paper establishes this only locally and notes that the formulation almost requires a global coordinate system.

Editorial extensions

If this is right

Old coordinate calculations of gravitational energy, such as Tolman's use of the Einstein pseudotensor, become covariant calculations in a specified symmetric teleparallel geometry.
The energy-momentum density is a tensor within that geometry, although changing the teleparallel connection changes the value, so the ambiguity is relocated from coordinates to a geometric gauge choice.
Removing the total derivative from the Einstein-Hilbert action leaves a covariant, asymptotically convergent Lagrangian that generates a covariant energy-momentum tensor.
The construction opens a five-parameter family of quadratic nonmetricity theories, of which the GR-equivalent case is only one member.

Reading between the lines

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

Because the paper's frame is guaranteed only locally, a globally nontrivial spacetime may force a patchwork of STGR charts; a natural extension is to ask which topologies admit a single torsion-free teleparallel frame and how the energy tensor glues across overlaps.
The energy density's dependence on the chosen connection suggests a test: compare STGR energy for a known exact solution with quasi-local energies such as the Brown-York or Bel-Robinson constructions; agreement in some gauge would give the tensor physical content beyond a relabelling.
The five-parameter quadratic nonmetricity family is likely to contain extra propagating modes; a linearized analysis around Minkowski space could identify whether any member has ghosts, tachyons, or additional polarizations.
Coupling STGR to fermions may be subtle: spinors need a teleparallel frame, but nonmetricity changes parallel transport, so the paper's warning that the GR interaction is not teleparallel minimal coupling could become a real consistency condition for matter.

Original abstract

General relativity can be presented in terms of other geometries besides Riemannian. In particular, teleparallel geometry (i.e., curvature vanishes) has some advantages, especially concerning energy-momentum localization and its ``translational gauge theory'' nature. The standard version is metric compatible, with torsion representing the gravitational ``force''. However there are many other possibilities. Here we focus on an interesting alternate extreme: curvature and torsion vanish but the nonmetricity $\nabla g$ does not---it carries the ``gravitational force''. This {\it symmetric teleparallel} representation of general relativity covariantizes (and hence legitimizes) the usual coordinate calculations. The associated energy-momentum density is essentially the Einstein pseudotensor, but in this novel geometric representation it is a true tensor.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,479 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.09 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.02 · compares with prior results 0.76 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

General relativity can be presented in terms of other geometries besides Riemannian. In particular, teleparallel geometry (i.e., curvature vanishes) has some advantages, especially concerning energy-momentum localization and its ``translational gauge theory'' nature. The standard version is metric compatible, with torsion representing the gravitational ``force''. However there are many other possibilities. Here we focus on an interesting alternate extreme: curvature and torsion vanish but the nonmetricity $\nabla g$ does not---it carries the ``gravitational force''. This {\it symmetric teleparallel} representation of general relativity covariantizes (and hence legitimizes) the usual coordinate calculations. The associated energy-momentum density is essentially the Einstein pseudotensor, but in this novel geometric representation it is a true tensor.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	General relativity can be presented in terms of other geometries besides Riemannian.	background 1.00	0.05	0.04	asserted 0.74
s1	In particular, teleparallel geometry (i.e., curvature vanishes) has some advantages, especially concerning energy-momentum localization and its ``translational gauge theory'' nature.	background 1.00	0.05	0.04	asserted 0.93
s2	The standard version is metric compatible, with torsion representing the gravitational ``force''.	background 1.00	0.04	0.03	asserted 1.00
s3	However there are many other possibilities.	background 0.91	0.05	0.03	asserted 0.97
s4	Here we focus on an interesting alternate extreme: curvature and torsion vanish but the nonmetricity $\nabla g$ does not---it carries the ``gravitational force''.	contribution 0.94	0.26	0.30	asserted 1.00
s5	This {\it symmetric teleparallel} representation of general relativity covariantizes (and hence legitimizes) the usual coordinate calculations.	interpretation 0.59	0.59	0.36	asserted 1.00
s6	The associated energy-momentum density is essentially the Einstein pseudotensor, but in this novel geometric representation it is a true tensor.	result 0.71	0.69	0.30	asserted 0.99

What kind of paper the classifier reports this is: theoretical result 0.82 · evidence from proof or derivation 0.61.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

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Instabilities in Metric-Affine Theories of Gravity
gr-qc 2020-04 conditional novelty 5.0 of 10

Ricci-based metric-affine gravity actions that break projective symmetry contain an unavoidable ghost from the projective mode and a non-minimally coupled 2-form, and no choice of matter coupling removes it.
Cosmological Hyperfluids, Torsion and Non-metricity
gr-qc 2020-03 conditional novelty 5.0 of 10

The paper writes down the most general hypermomentum tensor compatible with an FLRW universe and uses it to derive conservation laws and modified Friedmann equations with torsion and non-metricity.
General Teleparallel Quadratic Gravity
gr-qc 2019-09 conditional novelty 5.0 of 10

The general teleparallel equivalent of General Relativity is the parent theory of both Weitzenbock and symmetric teleparallel formulations, and its quadratic extension propagates two rank-2 fields plus a 2-form unless...
Scale transformations in metric-affine geometry
gr-qc 2018-10 conditional novelty 5.0 of 10

The paper classifies metric-affine gravity actions by their behavior under three distinct notions of scale transformation and lists the parameter constraints that make each action invariant.
Coupling matter in modified $Q$-gravity
gr-qc 2018-06 conditional novelty 5.0 of 10

A new class of modified gravity theories couples nonmetricity to the matter Lagrangian, producing second-order field equations and late-time accelerating cosmological solutions.
Teleparallel Palatini theories
gr-qc 2018-03 conditional novelty 5.0 of 10

The Palatini formalism with Lagrange multipliers reproduces general relativity and its teleparallel equivalents from a general quadratic action in flat geometry, and yields new cosmological equations with a dynamical lapse.
Coincident General Relativity
gr-qc 2017-10 conditional novelty 5.0 of 10

General Relativity can be recast without any affine connection in a flat, torsion-free geometry, and the resulting f(Q) extension yields new cosmological models that accelerate at early and late times.
Equatorial Periodic Orbits and Gravitational Wave Signatures in Weyl Geometric Gravity Black Holes
gr-qc 2026-11 accept novelty 4.0 of 10

Periodic orbits and gravitational wave signatures around Weyl geometric gravity black holes are computed, showing that the nonmetricity parameters systematically shift orbital thresholds and waveform features.
ACT-DR6 consistent inflation in generalised entropic cosmology and $f(Q)$ gravity
gr-qc 2026-07 conditional novelty 4.0 of 10

Reconstruction produces explicit f(Q) and generalised-entropic inflation models (and scalar-coupled versions) whose slow-roll parameters match ACT-DR6 + Planck-BAO constraints on n_s and r.
Physical nonviability of $f(\mathbb{Q})$ in the scalar-tensor representation
gr-qc 2026-06 unverdicted novelty 4.0 of 10

The scalar-tensor representation of f(Q) gravity reproduces the known ghost/strong-coupling obstruction, so the pathology is not an artifact of the original variables.
Dynamical Dark Energy or Modified Gravity? Signatures in Gravitational Wave Propagation
gr-qc 2025-09 conditional novelty 4.0 of 10

Reconstructing the dark energy density from DESI BAO and DESyr5 supernovae, then recasting it as f(Q) gravity, predicts a low-redshift gravitational wave damping ν≈0.18 (≳2σ from GR) only for the DESyr5 dataset.
Fixed-Time Voltage Regulation for Boost Converters via Unit-Safe Saturating Functions
eess.SY 2025-08 unverdicted novelty 4.0 of 10

A fixed-time feedback controller for boost converters uses a new class of saturating functions, observers, and adaptive gains to regulate output voltage within a bounded settling time under unknown load resistance.
Properties of compact objects in quadratic non-metricity gravity
gr-qc 2025-07 reject novelty 4.0 of 10

Quadratic non-metricity gravity with a fitted coupling ξ can match PSR J0740+6620 while keeping the radial sound speed below c²/3, according to the paper's chosen metric ansatz.
Quintessence dark energy model in non-linear $f(Q)$ theory with bulk-viscosity
gr-qc 2025-06 reject novelty 4.0 of 10

A quadratic f(Q) gravity model with bulk viscosity is fitted to expansion data, but the reported Hubble-law formula contradicts the paper's own scale-factor solution.
Maxwell-$f(Q)$ theory
gr-qc 2025-01 conditional novelty 4.0 of 10

A proposed new charged AdS black hole in f(Q) gravity, but the claimed absence of an uncharged limit and the charge-induced cosmological constant conflict with the paper's own formulas.
Covariant formulation of f(Q) theory
gr-qc 2021-04 conditional novelty 4.0 of 10

Setting the affine connection to zero in f(Q) gravity is inconsistent with spherically symmetric and FRW coordinates unless f(Q) is linear, and a covariant connection ansatz repairs this.
The coupling of matter and spacetime geometry
hep-th 2020-04 conditional novelty 4.0 of 10

Under minimal coupling, a general affine connection with torsion spoils gauge invariance and Dirac consistency, whereas symmetric teleparallel (integrable, torsion-free) geometry remains consistent.
Bayesian and Machine-Learning Analyses of Nonminimal $f(Q)$ Gravity and $H_0$ Tension
gr-qc 2025-11 reject novelty 3.0 of 10

A nonminimal f(Q) gravity model fitted to CC, DESI BAO and three supernova samples gives H0 ≈ 68 km/s/Mpc, similar to ΛCDM, and is disfavored by BIC.
Dynamics of late time universe in $f(Q)$ gravity
gr-qc 2025-04 reject novelty 3.0 of 10

A constant-jerk Hubble law is fitted to cosmic chronometer and Pantheon data, then used to evaluate three f(Q) gravity models, yielding acceleration, quintessence or phantom EoS, and SEC violation.
Comparing Equivalent Gravities: common features and differences
gr-qc 2022-08 conditional novelty 3.0 of 10

The paper reviews the equivalence of General Relativity, teleparallel gravity, and symmetric teleparallel gravity, showing the same dynamics arise from curvature, torsion, and non-metricity respectively.
Cosmology with the Laser Interferometer Space Antenna
astro-ph.CO 2022-04 accept novelty 3.0 of 10

LISA should measure the Hubble constant to a few percent and probe modified gravity and early-universe physics through gravitational waves, according to this community roadmap.
Teleparallel Gravity: From Theory to Cosmology
gr-qc 2021-06 conditional novelty 3.0 of 10

A comprehensive review that organizes teleparallel gravity from metric-affine geometry through modified theories to cosmological observations, and lists corrections to several published equations along the way.
Observational constraints of $f(Q)$ gravity
gr-qc 2019-07 conditional novelty 3.0 of 10

Polynomial f(Q) gravity models fitted to a broad set of background cosmological data are statistically indistinguishable from ΛCDM, with the extra deviation terms consistent with zero.
The Geometrical Trinity of Gravity
hep-th 2019-03 accept novelty 3.0 of 10

General relativity admits three equivalent geometric formulations, based on curvature, torsion, or non-metricity, and the review lays out how each reproduces the same massless spin-2 dynamics.

See all 64 Pith citations

How the literature uses it

167 arXiv papers cite this in 223 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 63 sentences without a confident label

Builds on it 2 papers

A possible action one can conceive in analogy to the teleparallel axion model [ref] as generalization of newer general relativity [ref] is then given by [equation] In the case that the parameters take the values [equation] one obtains the symmetric teleparallel equivalent of relativity [ref] (STEGR), together with symmetric teleparallel axion, which is non-minimally coupled to the unique parity-odd invariant of non-metricity.

Teleparallel axions and cosmology · Symmetric teleparallel axions · 0.96
This will be the generalization of the symmetric teleparallel gravity [ref] .

Metric-Connection Geometries on Pre-Leibniz Algebroids: A Search for Geometrical Structure in String Models · Concluding Remarks · 0.88

Cites it as support 1 paper

In symmetric teleparallel gravity [ref] , MCP is viable for all standard fields, including spinors [ref] .

The coupling of matter and spacetime geometry · Introduction · 0.99

Cites it as background 127 papers

Among other possibilities, general relativity can be equivalently formulated in terms of the nonmetricity tensor [ref] , which measures the departure from zero of the covariant derivative of the metric tensor (see [ref] for an extensive review).

G\"odel-type spacetimes in $f(Q)$ gravity · Introduction · 0.99
In this work we seek yet another explanation for the discrepancy as we study the evolution of matter perturbations in modified $F(Q)$ gravity [ref] , the nonlinear generalisation of the improved version of General Relativity [ref] on a flat and torsion-free spacetime continuum known as Symmetric Teleparallel Gravity (STG) [ref] .

Testing $F(Q)$ gravity with redshift space distortions · Introduction · 0.85
A third, dynamically rich method has just been proposed: the symmetric teleparallel formulation [ref] , where gravity is fully given by the non-metricity scalar $Q$ and both curvature and torsion vanish, i.e. the $f(Q)$ gravity.

Cosmology of f(Q,L_m) gravity with Holographic Ricci Dark Energy: Early-Time Inflation and Late-Time Acceleration and RGUP Corrected Observables · Introduction · 0.97

Discussion

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