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arxiv: 2604.19310 · v1 · submitted 2026-04-21 · 🌀 gr-qc · hep-th· math-ph· math.MP

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Extrinsic geometry and Hamiltonian analysis of symmetric teleparallel gravity

Salvatore Capozziello , Dario Sauro
Authors on Pith no claims yet

Pith reviewed 2026-05-10 02:30 UTC · model grok-4.3

classification 🌀 gr-qc hep-thmath-phmath.MP
keywords symmetric teleparallel gravitynon-metricityHamiltonian analysisdegrees of freedomextrinsic curvatureGauss-Codazzi relationsfoliations
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The pith

Symmetric teleparallel gravity shares the same number of degrees of freedom as general relativity.

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

The paper derives generalized Gauss-Codazzi relations for foliations that include non-metricity and applies them to the teleparallel framework. It identifies an extrinsic symmetric two-tensor that takes the place of extrinsic curvature and shows that no other geometric objects add independent dynamical degrees of freedom once the foliation and non-metricity constraints are in place. The variational principle supplies the necessary boundary terms for a well-posed Cauchy problem, and the resulting Hamiltonian for the symmetric teleparallel equivalent of general relativity is used to prove that the theory has exactly the same number of degrees of freedom as its Riemannian counterpart. A reader would care because this establishes dynamical equivalence, allowing the same initial-value formulation and constraint analysis to be carried over without extra modes.

Core claim

In the presence of non-metricity, the generalized Gauss-Codazzi relations constrain the extrinsic and intrinsic tensors such that only an extrinsic symmetric two-tensor contributes to the dynamics in the same way as the extrinsic curvature of Riemannian geometry; the Hamiltonian analysis of the symmetric teleparallel equivalent of general relativity then proves that the theory possesses the same number of degrees of freedom as its Riemannian counterpart.

What carries the argument

The extrinsic symmetric two-tensor, which replaces the role of the extrinsic curvature, together with the non-metricity and foliation constraints that close the constraint algebra.

If this is right

  • The boundary terms obtained from the variational principle guarantee a well-posed and well-defined Cauchy problem.
  • The Hamiltonian constraint algebra closes without additional secondary constraints.
  • Initial data can be specified using the same number of free functions as in general relativity.
  • The theory admits the same Dirac analysis and constraint counting as the Riemannian formulation.

Where Pith is reading between the lines

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

  • The equivalence suggests that existing numerical relativity codes based on the ADM formalism could be adapted to the teleparallel setting with minimal changes.
  • Similar extrinsic-tensor identifications might be tested in other non-metricity-based extensions to see whether the degree-of-freedom count remains unchanged.
  • The generalized Gauss-Codazzi relations provide a template for deriving initial-value formulations in broader classes of affine geometries.

Load-bearing premise

That no geometric object other than the identified extrinsic symmetric two-tensor can introduce new dynamical degrees of freedom once the foliation and non-metricity constraints are imposed.

What would settle it

A direct count of independent phase-space variables after all constraints are imposed that differs from the count obtained for general relativity.

read the original abstract

We analyze the properties of foliations in presence of non-metricity, deriving the generalized Gauss-Codazzi relations in full generality. These results are employed to study the teleparallel framework of non-metric geometry, obtaining constraints on the extrinsic and intrinsic tensors. In particular, an extrinsic symmetric two-tensor plays the role of the extrinsic curvature in Riemannian geometry, whereas no other geometric object can induce new dynamical degrees of freedom. Furthermore, we analyze the variational principle in presence of non-metricity, obtaining the boundary terms for the well-posed and well-defined Cauchy problem. Finally, we exploit the previous results to construct the Hamiltonian of the symmetric teleparallel equivalent of General Relativity, providing a proof that this theory shares the same number of degrees of freedom with its Riemannian counterpart.

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.

Referee Report

1 major / 2 minor

Summary. The manuscript analyzes foliations in the presence of non-metricity, deriving generalized Gauss-Codazzi relations in full generality. These are applied to the symmetric teleparallel equivalent of general relativity (STEGR), yielding constraints on extrinsic and intrinsic tensors; an extrinsic symmetric two-tensor is identified as the analogue of the extrinsic curvature, with the claim that no other geometric objects introduce new dynamical degrees of freedom. The variational principle is examined to obtain boundary terms ensuring a well-posed Cauchy problem, and the Hamiltonian is constructed, providing a proof that STEGR shares the same number of degrees of freedom as its Riemannian counterpart.

Significance. If the central results hold, the work supplies a rigorous foliation-based Hamiltonian formulation for STEGR that confirms its dynamical equivalence to GR. The generalized Gauss-Codazzi identities and boundary-term analysis strengthen the geometric foundations of non-metric teleparallel theories, offering a concrete tool for constraint counting and well-posedness studies in this class of modified-gravity models.

major comments (1)
  1. [Hamiltonian analysis] The central DOF-equivalence claim rests on showing that all non-metricity components beyond the identified extrinsic symmetric two-tensor are either non-dynamical or fully constrained. The manuscript should supply the explicit constraint algebra (or at least the full set of primary/secondary constraints and their Poisson brackets) in the Hamiltonian section to allow independent verification of the final count.
minor comments (2)
  1. [Foliation identities] Notation for the non-metricity tensor and its irreducible parts under the foliation should be introduced with a compact table or explicit decomposition formulas to improve readability when the generalized Gauss-Codazzi relations are stated.
  2. [Variational principle] The boundary terms derived from the variational principle are stated without an accompanying check that they cancel under the chosen boundary conditions; a short paragraph confirming this cancellation would strengthen the well-posedness argument.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript, the positive assessment of its significance, and the recommendation for minor revision. We address the single major comment below.

read point-by-point responses

  1. Referee: The central DOF-equivalence claim rests on showing that all non-metricity components beyond the identified extrinsic symmetric two-tensor are either non-dynamical or fully constrained. The manuscript should supply the explicit constraint algebra (or at least the full set of primary/secondary constraints and their Poisson brackets) in the Hamiltonian section to allow independent verification of the final count.

    Authors: We agree that an explicit listing of the constraints and their algebra would strengthen the transparency of the Hamiltonian analysis and allow independent verification. In the revised manuscript we will expand the Hamiltonian section to include the complete set of primary and secondary constraints together with the relevant Poisson brackets. This addition will make the proof that STEGR possesses the same number of degrees of freedom as GR fully explicit and verifiable. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained via explicit foliation and constraint analysis

full rationale

The paper derives generalized Gauss-Codazzi relations from the non-metricity foliation, identifies the extrinsic symmetric two-tensor as the dynamical analogue of extrinsic curvature, shows all other geometric objects are constrained or non-dynamical, obtains boundary terms for the variational principle, and constructs the Hamiltonian explicitly. The DOF equivalence to GR follows from this constraint counting and phase-space analysis rather than from any self-definition, fitted input renamed as prediction, or load-bearing self-citation chain. The central claim is therefore independent of its inputs and does not reduce by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard axioms of differential geometry and foliation theory together with the established definition of symmetric teleparallel equivalent of GR; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • standard math Standard assumptions of differential geometry and foliation theory hold in the presence of non-metricity
    Invoked when deriving the generalized Gauss-Codazzi relations.

pith-pipeline@v0.9.0 · 5430 in / 1236 out tokens · 31370 ms · 2026-05-10T02:30:53.121711+00:00 · methodology

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Reference graph

Works this paper leans on

52 extracted references · 33 canonical work pages

  1. [1]

    As we have already noticed, this can be done in three different ways when the geometry has non-metricity

    Ricci relations The Ricci relations are found by contracting twice the full curvature tensor with the normal vectorn µ. As we have already noticed, this can be done in three different ways when the geometry has non-metricity. In particular, the first two Ricci relations are found by contracting one of the last two indices and one of the first twon µ. Due ...

  2. [2]

    Indeed, there is now a nontrivial contraction of the full curvature tensor with three normal vectors, which results in a rank-one tensor expression

    Rank-one relation As we have already observed, the presence of non-metricity gives rise to yet more complicated features with respect to the Riemannian case. Indeed, there is now a nontrivial contraction of the full curvature tensor with three normal vectors, which results in a rank-one tensor expression. For this reason, we shall dub this equation asrank...

  3. [3]

    Codazzi relations Let us now turn to the analysis of the rank-three relations. The first equation of this kind is thefirst generalized Codazzi relationthat we have derived before, whose expression in the present formalism is Rµ λρνnµhλ αhρ βhν σ = 2D[σKβ]α +ϵ θ [βKσ]α . (81) A very similar relation is found by contracting the normal vector with the second...

  4. [4]

    Palatini, Rend

    A. Palatini, Rend. Circ. Mat. Palermo43, no.1, 203-212 (1919)

    1919

  5. [5]

    Cartan, Annales Sci

    E. Cartan, Annales Sci. Ecole Norm. Sup.40, 325-412 (1923)

    1923

  6. [6]

    Geometry, Gauge, Gravity,

    R. Percacci and O. Zanusso, “Geometry, Gauge, Gravity,” SISSA Medialab, S.r.l. 2026, Trieste (Italy)

    2026

  7. [7]

    F. W. Hehl, J. D. McCrea, E. W. Mielke and Y. Ne’eman, Phys. Rept.258, 1-171 (1995) [arXiv:gr-qc/9402012 [gr-qc]]

    work page Pith review arXiv 1995

  8. [8]

    Percacci, Symmetry15, no.2, 449 (2023)

    R. Percacci, Symmetry15, no.2, 449 (2023)

    2023

  9. [9]

    Melichev and R

    O. Melichev and R. Percacci, JHEP03, 133 (2024) [arXiv:2307.02336 [hep-th]]

    work page arXiv 2024

  10. [10]

    Martini, G

    R. Martini, G. Paci and D. Sauro, JHEP12, 138 (2024) [arXiv:2312.16681 [gr-qc]]

    work page arXiv 2024

  11. [11]

    Melichev, [arXiv:2406.14146 [hep-th]]

    O. Melichev, [arXiv:2406.14146 [hep-th]]

    work page arXiv

  12. [12]

    Melichev, JHEP08, 130 (2025) [arXiv:2504.13090 [hep-th]]

    O. Melichev, JHEP08, 130 (2025) [arXiv:2504.13090 [hep-th]]

    work page arXiv 2025

  13. [13]

    Teleparallel Gravity: An Introduction,

    R. Aldrovandi and J. G. Pereira,“Teleparallel Gravity: An Introduction,”Springer, 2013, Dordrecht

    2013

  14. [14]

    M. Adak, M. Kalay and O. Sert, Int. J. Mod. Phys. D15, 619-634 (2006) [arXiv:gr-qc/0505025 [gr-qc]]

    work page arXiv 2006

  15. [15]

    Adak, Turk

    M. Adak, Turk. J. Phys.30, 379-390 (2006) [arXiv:gr-qc/0611077 [gr-qc]]

    work page arXiv 2006

  16. [16]

    Adak, ¨O

    M. Adak, ¨O. Sert, M. Kalay and M. Sari, Int. J. Mod. Phys. A28, 1350167 (2013) [arXiv:0810.2388 [gr-qc]]

    work page arXiv 2013

  17. [17]

    Adak, Int

    M. Adak, Int. J. Geom. Meth. Mod. Phys.15, no.12, 1850198 (2018) [arXiv:1809.01385 [gr-qc]]

    work page arXiv 2018

  18. [18]

    M. Adak, T. Dereli, T. S. Koivisto and C. Pala, Int. J. Geom. Meth. Mod. Phys.20, no.Supp01, 2350215 (2023) [arXiv:2303.17812 [gr-qc]]

    work page arXiv 2023

  19. [19]

    Adak, [arXiv:2602.19194 [gr-qc]]

    M. Adak, [arXiv:2602.19194 [gr-qc]]

    work page arXiv

  20. [20]

    Coincident General Relativity

    J. Beltr´ an Jim´ enez, L. Heisenberg and T. Koivisto, Phys. Rev. D98, no.4, 044048 (2018) [arXiv:1710.03116 [gr-qc]]

    work page Pith review arXiv 2018

  21. [21]

    Teleparallel Palatini theories

    J. Beltr´ an Jim´ enez, L. Heisenberg and T. S. Koivisto, JCAP08, 039 (2018) [arXiv:1803.10185 [gr-qc]]

    work page Pith review arXiv 2018

  22. [22]

    The Geometrical Trinity of Gravity

    J. Beltr´ an Jim´ enez, L. Heisenberg and T. S. Koivisto, Universe5, no.7, 173 (2019) [arXiv:1903.06830 [hep-th]]

    work page Pith review arXiv 2019

  23. [23]

    Cosmology inf(Q)geometry,

    J. Beltr´ an Jim´ enez, L. Heisenberg, T. S. Koivisto and S. Pekar, Phys. Rev. D101, no.10, 103507 (2020) [arXiv:1906.10027 [gr-qc]]

    work page arXiv 2020

  24. [24]

    Teleparallel gravity: from theory to cosmology,

    S. Bahamonde, K. F. Dialektopoulos, C. Escamilla-Rivera, G. Farrugia, V. Gakis, M. Hendry, M. Hohmann, J. Levi Said, J. Mifsud and E. Di Valentino, Rept. Prog. Phys.86, no.2, 026901 (2023) [arXiv:2106.13793 [gr-qc]]

    work page arXiv 2023

  25. [25]

    Capozziello, V

    S. Capozziello, V. De Falco and C. Ferrara, Eur. Phys. J. C82(2022) no.10, 865 [arXiv:2208.03011 [gr-qc]]

    work page arXiv 2022

  26. [26]

    Capozziello, V

    S. Capozziello, V. De Falco and C. Ferrara, Eur. Phys. J. C83, no.10, 915 (2023) [arXiv:2307.13280 [gr-qc]]

    work page arXiv 2023

  27. [27]

    Capozziello, S

    S. Capozziello, S. Cesare and C. Ferrara, Eur. Phys. J. C85, no.9, 932 (2025) [arXiv:2503.08167 [gr-qc]]

    work page arXiv 2025

  28. [28]

    Van Nieuwenhuizen, Phys

    P. Van Nieuwenhuizen, Phys. Rept.68, 189-398 (1981)

    1981

  29. [29]

    Casadio, I

    R. Casadio, I. Kuntz and G. Paci, Eur. Phys. J. C82, no.3, 186 (2022) [arXiv:2110.04325 [hep-th]]

    work page arXiv 2022

  30. [30]

    Y. F. Cai, S. Capozziello, M. De Laurentis and E. N. Saridakis, Rept. Prog. Phys.79(2016) no.10, 106901 [arXiv:1511.07586 [gr-qc]]

    work page Pith review arXiv 2016

  31. [31]

    Heisenberg, Phys

    L. Heisenberg, Phys. Rept.1066(2024), 1-78 [arXiv:2309.15958 [gr-qc]]

    work page arXiv 2024

  32. [32]

    Golovnev, Ukr

    A. Golovnev, Ukr. J. Phys.69(2024) no.7, 456 [arXiv:2405.14184 [gr-qc]]

    work page arXiv 2024

  33. [33]

    Blixt, A

    D. Blixt, A. Golovnev, M. J. Guzman and R. Maksyutov, Phys. Rev. D109(2024) no.4, 044061 [arXiv:2306.09289 [gr-qc]]

    work page arXiv 2024

  34. [34]

    Hamiltonian analysis of curvature-squared gravity with or without conformal invariance,

    J. Klusoˇ n, M. Oksanen and A. Tureanu, Phys. Rev. D89, no.6, 064043 (2014) [arXiv:1311.4141 [hep-th]]

    work page arXiv 2014

  35. [35]

    Barker and D

    W. Barker and D. Glavan, [arXiv:2510.08201 [gr-qc]]

    work page arXiv

  36. [36]

    Glavan, T

    D. Glavan, T. Zlosnik and C. Lin, JCAP04, 072 (2024) [arXiv:2311.17459 [gr-qc]]

    work page arXiv 2024

  37. [37]

    P. A. M. Dirac, Can. J. Math.2, 129-148 (1950)

    1950

  38. [38]

    J. L. Anderson and P. G. Bergmann, Phys. Rev.83, 1018-1025 (1951)

    1951

  39. [39]

    Quantization of gauge systems

    M. Henneaux and C. Teitelboim, “Quantization of gauge systems”, Princeton University Press, 1994, Princeton (USA)

    1994

  40. [40]

    R. L. Arnowitt, S. Deser and C. W. Misner, Phys. Rev.116, 1322-1330 (1959)

    1959

  41. [41]

    A Relativist’s Toolkit: The Mathematics of Black-Hole Mechanics,

    E. Poisson,“A Relativist’s Toolkit: The Mathematics of Black-Hole Mechanics,”Cambridge University Press, 2009, Cam- bridge, UK

    2009

  42. [42]

    Capozziello, A

    S. Capozziello, A. Finch, J. L. Said and A. Magro, Eur. Phys. J. C81(2021) no.12, 1141 [arXiv:2108.03075 [gr-qc]]

    work page arXiv 2021

  43. [43]

    D’Ambrosio, L

    F. D’Ambrosio, L. Heisenberg and S. Zentarra, Fortsch. Phys.71, no.12, 2300185 (2023) [arXiv:2308.02250 [gr-qc]]

    work page arXiv 2023

  44. [44]

    Tomonari and S

    K. Tomonari and S. Bahamonde, Eur. Phys. J. C84, no.4, 349 (2024) [erratum: Eur. Phys. J. C84, no.5, 508 (2024)] [arXiv:2308.06469 [gr-qc]]

    work page arXiv 2024

  45. [45]

    A. A. Starobinsky, Phys. Lett. B91, 99-102 (1980)

    1980

  46. [46]

    K. S. Stelle, Gen. Rel. Grav.9, 353-371 (1978)

    1978

  47. [47]

    K. S. Stelle, Phys. Rev. D16, 953-969 (1977)

    1977

  48. [48]

    S. M. Christensen and S. A. Fulling, Phys. Rev. D15, 2088-2104 (1977)

    2088

  49. [49]

    R. J. Riegert, Phys. Lett. B134, 56-60 (1984)

    1984

  50. [50]

    Deser and A

    S. Deser and A. Schwimmer, Phys. Lett. B309, 279-284 (1993) [arXiv:hep-th/9302047 [hep-th]]

    work page arXiv 1993

  51. [51]

    On the quantum field theory of the gravitational interactions

    D. Anselmi, JHEP06, 086 (2017) [arXiv:1704.07728 [hep-th]]

    work page Pith review arXiv 2017

  52. [52]

    Anselmi and M

    D. Anselmi and M. Piva, Phys. Rev. D96, no.4, 045009 (2017) [arXiv:1703.05563 [hep-th]]. 29

    work page arXiv 2017