arxiv: 2602.01830 · v2 · submitted 2026-02-02 · 🌀 gr-qc

Recognition: no theorem link

Hyperbolicity analysis of the linearised 3+1 formulation in the Teleparallel Equivalent of General Relativity

Cheng cheng , Maria Jose Guzman

Authors on Pith no claims yet

Pith reviewed 2026-05-16 08:25 UTC · model grok-4.3

classification 🌀 gr-qc

keywords Teleparallel gravityhyperbolicity3+1 decompositionHamiltonian formulationgauge fixingwell-posednessnumerical relativity

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The pith

Linear analysis shows that gauge fixing removes imaginary eigenvalues from the Teleparallel Equivalent of General Relativity equations, making the system strongly hyperbolic.

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

The authors examine the principal symbol of the 3+1 evolution equations in the Teleparallel Equivalent of General Relativity using a Hamiltonian formulation. In a linearised one-dimensional setting, they find that certain sectors of the system produce imaginary eigenvalues, which would normally destroy hyperbolicity. These sectors form an isolated subsystem that corresponds to gauge degrees of freedom. By choosing a gauge condition that eliminates them, the remaining equations acquire a complete set of real eigenvectors and become strongly hyperbolic. The three-dimensional version of the system is presented with the same structure, providing the first concrete implementation of Hamilton's equations for numerical work in this theory.

Core claim

The principal symbol of the linearised one-dimensional 3+1 system in TEGR, constructed via the VAST decomposition of the canonical variables, contains a decoupled subsystem with imaginary eigenvalues. This subsystem is identified as pure gauge and can be removed by gauge fixing. The reduced system then possesses real eigenvalues with a complete set of eigenvectors, establishing strong hyperbolicity. The corresponding three-dimensional equations are written explicitly.

What carries the argument

The VAST decomposition of the canonical variables together with the principal symbol of the resulting first-order system, which isolates the gauge sector responsible for imaginary eigenvalues.

If this is right

The reduced system admits a well-posed initial-value formulation in one dimension.
Extension of the analysis to spherical symmetry becomes possible for proving well-posedness.
Practical numerical relativity simulations in TEGR can now be constructed from these equations.
The three-dimensional formulation shares the same hyperbolicity property after identical gauge fixing.

Where Pith is reading between the lines

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

This method of removing gauge modes may extend to the full nonlinear theory if suitable gauge conditions can be maintained dynamically.
Analogous hyperbolicity analyses could be performed for other teleparallel or modified gravity models that admit Hamiltonian formulations.
Successful numerical implementations would enable direct comparison of TEGR predictions with general relativity in strong-field regimes.

Load-bearing premise

The modes producing imaginary eigenvalues can be interpreted as pure gauge degrees of freedom whose removal leaves the physical evolution unchanged.

What would settle it

Evolving the gauge-fixed linearised system numerically and observing exponential instability or complex characteristics would disprove the strong hyperbolicity.

read the original abstract

We study the properties of the principal symbol of the 3+1 equations of motion in Teleparallel Equivalent of General Relativity (TEGR) and assess the conditions for hyperbolicity. We use the Hamiltonian formulation based on the vectorial, antisymmetric, symmetric trace-free, and trace (VAST) decomposition of the canonical variables in the Hamiltonian formalism, and the Hamilton's equations previously presented in the literature. We study the system of differential equations at the linear level in one dimension, and show that the principal symbol has a sector with imaginary eigenvalues, which renders the system not hyperbolic. This situation is circumvented by identifying the problematic sectors, which are an isolated system and can be removed by a gauge fixing. We prove that the remaining system of equations is strongly hyperbolic. We also present the system in three dimensions. This is the first practical use of Hamilton's equations in TEGR, and our work can be extended for proving well-posedness in spherical symmetry, and establish numerical relativity setups in TEGR.

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 shows strong hyperbolicity for the gauge-fixed linear 1D TEGR system but only writes down the 3D equations without the same principal-symbol check.

read the letter

The main thing to know is that this paper takes the existing Hamiltonian formulation of TEGR, applies the VAST decomposition, and finds that the linear 1D equations have a sector with imaginary eigenvalues. They isolate that sector, remove it by gauge fixing, and prove the rest is strongly hyperbolic. They also write out the 3D version of the system. This is the first time Hamilton's equations have been used this way for a hyperbolicity study in TEGR, which is a concrete technical step toward stable numerics in the theory.

Referee Report

1 major / 1 minor

Summary. The paper analyzes the principal symbol of the linearised 3+1 equations of motion in Teleparallel Equivalent of General Relativity (TEGR) using the VAST decomposition of canonical variables and previously derived Hamilton's equations. It performs the analysis at linear order in one spatial dimension, identifies an isolated sector with imaginary eigenvalues, removes this sector via gauge fixing, and proves that the remaining system is strongly hyperbolic. The corresponding 3D system is then written down explicitly. The work is presented as the first practical application of Hamilton's equations in TEGR and as a foundation for future well-posedness studies in spherical symmetry and numerical relativity.

Significance. If the central claim holds, the result supplies a concrete gauge-fixed formulation whose linearised 1D dynamics are strongly hyperbolic, which is a necessary step toward well-posed initial-value problems in TEGR. The explicit use of the VAST decomposition and the identification of removable gauge sectors constitute a technical advance that can be built upon for spherical-symmetry reductions and numerical implementations.

major comments (1)

[sections presenting the 3D equations and the hyperbolicity proof] The principal-symbol analysis, eigenvalue computation, and verification of a complete eigenvector basis are performed only for the linearised one-dimensional system. The three-dimensional formulation is presented (using the same VAST decomposition) without an explicit calculation of its principal symbol or confirmation that the gauge-fixed sector remains isolated and strongly hyperbolic once transverse spatial derivatives are restored. This gap is load-bearing for the claim that the 3+1 system is strongly hyperbolic.

minor comments (1)

[abstract] The abstract states that the remaining system is strongly hyperbolic but does not explicitly note that this statement applies to the one-dimensional reduction; a clarifying sentence would help readers distinguish the proven result from the 3D presentation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We appreciate the recognition of the technical contribution in applying the VAST decomposition to the linearised TEGR system and isolating the gauge sector. We address the major comment below and will revise the manuscript to strengthen the presentation.

read point-by-point responses

Referee: [sections presenting the 3D equations and the hyperbolicity proof] The principal-symbol analysis, eigenvalue computation, and verification of a complete eigenvector basis are performed only for the linearised one-dimensional system. The three-dimensional formulation is presented (using the same VAST decomposition) without an explicit calculation of its principal symbol or confirmation that the gauge-fixed sector remains isolated and strongly hyperbolic once transverse spatial derivatives are restored. This gap is load-bearing for the claim that the 3+1 system is strongly hyperbolic.

Authors: We agree that the explicit principal-symbol analysis, eigenvalue computation, and eigenvector verification are provided only for the linearised one-dimensional reduction. The three-dimensional equations are written explicitly using the VAST decomposition, but the full calculation of the principal symbol for a general spatial wave vector (including transverse derivatives) and the confirmation that the gauge-fixed sector remains isolated and strongly hyperbolic are not carried out in the current manuscript. This is a genuine gap in the supporting evidence for the 3+1 claim. In the revised version we will add the explicit 3D principal symbol, compute its eigenvalues for arbitrary propagation directions, and verify that the same gauge fixing isolates the problematic sector while yielding a complete set of eigenvectors, thereby establishing strong hyperbolicity in three dimensions. The 1D analysis already demonstrates the structure of the problematic sector; extending it to 3D is feasible because the transverse contributions enter the symbol in a manner that preserves the isolation of that sector. revision: yes

Circularity Check

0 steps flagged

No circularity; 1D principal-symbol analysis is an independent calculation

full rationale

The paper derives the principal symbol explicitly for the linearised 1D system, identifies an isolated gauge sector containing imaginary eigenvalues, removes that sector by gauge fixing, and then proves strong hyperbolicity for the remaining equations. The VAST decomposition and Hamilton's equations are taken from prior literature as external input; the hyperbolicity result is obtained by direct matrix analysis rather than by redefinition or fitting. The 3D system is written down but the hyperbolicity claim is confined to the 1D case. No step reduces the claimed proof to a self-citation, a fitted parameter renamed as prediction, or an ansatz smuggled in by definition. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard mathematical definitions of hyperbolicity for first-order PDE systems and the existing Hamiltonian formulation of TEGR from prior papers; no new free parameters or invented entities are introduced.

axioms (2)

standard math Standard definitions and criteria for strong hyperbolicity of linear first-order PDE systems via principal symbol eigenvalues
Invoked to assess the principal symbol and conclude strong hyperbolicity after gauge fixing.
domain assumption Validity of the VAST decomposition and the previously derived Hamilton's equations in TEGR
Forms the basis for the 3+1 formulation studied at the linear level.

pith-pipeline@v0.9.0 · 5481 in / 1395 out tokens · 68325 ms · 2026-05-16T08:25:00.347454+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

43 extracted references · 43 canonical work pages · 17 internal anchors

[1]

B. P. Abbottet al.(LIGO Scientific, Virgo), Phys. Rev. Lett.116, 061102 (2016), arXiv:1602.03837 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2016
[2]

B. P. Abbottet al.(LIGO Scientific, Virgo), Phys. Rev. Lett.119, 161101 (2017), arXiv:1710.05832 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2017
[3]

An Ordinary Short Gamma-Ray Burst with Extraordinary Implications: Fermi-GBM Detection of GRB 170817A

A. Goldsteinet al., Astrophys. J. Lett.848, L14 (2017), arXiv:1710.05446 [astro-ph.HE]

work page internal anchor Pith review Pith/arXiv arXiv 2017
[4]

Translation of Einstein's Attempt of a Unified Field Theory with Teleparallelism

A. Unzicker and T. Case, (2005), arXiv:physics/0503046

work page internal anchor Pith review Pith/arXiv arXiv 2005
[5]

F. W. Hehl, P. von der Heyde, G. D. Kerlick, and J. M. Nester, Rev. Mod. Phys.48, 393 (1976)

work page 1976
[6]

Hayashi and T

K. Hayashi and T. Nakano, Prog. Theor. Phys.38, 491 (1967)

work page 1967
[7]

Hayashi and T

K. Hayashi and T. Shirafuji, Phys. Rev. D19, 3524 (1979)

work page 1979
[8]

L. T. Buchman and J. M. Bardeen, Phys. Rev. D67, 084017 (2003), arXiv:gr-qc/0301072

work page internal anchor Pith review Pith/arXiv arXiv 2003
[9]

F. B. Estabrook, R. S. Robinson, and H. D. Wahlquist, Class. Quant. Grav.14, 1237 (1997), arXiv:gr-qc/9703072

work page internal anchor Pith review Pith/arXiv arXiv 1997
[10]

Numerical simulations of generic singuarities

D. Garfinkle, Phys. Rev. Lett.93, 161101 (2004), arXiv:gr-qc/0312117

work page internal anchor Pith review Pith/arXiv arXiv 2004
[11]

Alcubierre, Gen

M. Alcubierre, Gen. Rel. Grav.57, 122 (2025), arXiv:2503.03918 [gr-qc]

work page arXiv 2025
[12]

Capozziello, A

S. Capozziello, A. Finch, J. L. Said, and A. Magro, Eur. Phys. J. C81, 1141 (2021), arXiv:2108.03075 [gr-qc]

work page arXiv 2021
[13]

Peshkov, H

I. Peshkov, H. Olivares, and E. Romenski, Phys. Rev. D 112, 084070 (2025), arXiv:2211.13659 [gr-qc]

work page arXiv 2025
[14]

Olivares, I

H. Olivares, I. M. Peshkov, E. R. Most, F. M. Guercilena, and L. J. Papenfort, Phys. Rev. D105, 124038 (2022), arXiv:2111.05282 [gr-qc]

work page arXiv 2022
[15]

L. Pati, D. Blixt, and M.-J. Guzman, Phys. Rev. D107, 044071 (2023), arXiv:2210.07971 [gr-qc]

work page arXiv 2023
[16]

Hamiltonian formulation of teleparallel gravity

R. Ferraro and M. J. Guzm´ an, Phys. Rev. D94, 104045 (2016), arXiv:1609.06766 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2016
[17]

Blixt, M.-J

D. Blixt, M.-J. Guzm´ an, M. Hohmann, and C. Pfeifer, Int. J. Geom. Meth. Mod. Phys.18, 2130005 (2021), arXiv:2012.09180 [gr-qc]. 17

work page arXiv 2021
[18]

Golovnev and M.-J

A. Golovnev and M.-J. Guzman, Phys. Rev. D104, 124074 (2021), arXiv:2110.11273 [gr-qc]

work page arXiv 2021
[19]

Guzman, Gen

M.-J. Guzman, Gen. Rel. Grav.58, 27 (2026), arXiv:2311.01424 [gr-qc]

work page arXiv 2026
[20]

D’Ambrosio, M

F. D’Ambrosio, M. Garg, L. Heisenberg, and S. Zentarra, (2020), arXiv:2007.03261 [gr-qc]

work page arXiv 2020
[21]

T. W. Baumgarte and S. L. Shapiro, Phys. Rev. D59, 024007 (1998), arXiv:gr-qc/9810065

work page internal anchor Pith review Pith/arXiv arXiv 1998
[22]

Shibata and T

M. Shibata and T. Nakamura, Phys. Rev. D52, 5428 (1995)

work page 1995
[23]

Conformal and covariant Z4 formulation of the Einstein equations: strongly hyperbolic first-order reduction and solution with discontinuous Galerkin schemes

M. Dumbser, F. Guercilena, S. K¨ oppel, L. Rezzolla, and O. Zanotti, Phys. Rev. D97, 084053 (2018), arXiv:1707.09910 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2018
[24]

Campbell, J

M. Alcubierre, Introduction to 3+1 Numerical Relativity (2006), 10.1093/acprof:oso/9780199205677.001.0001

work page doi:10.1093/acprof:oso/9780199205677.001.0001 2006
[25]

On the covariance of teleparallel gravity theories

A. Golovnev, T. Koivisto, and M. Sandstad, Class. Quant. Grav.34, 145013 (2017), arXiv:1701.06271 [gr- qc]

work page internal anchor Pith review Pith/arXiv arXiv 2017
[26]

Castellani, P

L. Castellani, P. van Nieuwenhuizen, and M. Pilati, Phys. Rev. D26, 352 (1982)

work page 1982
[27]

Blixt, A

D. Blixt, A. Jim´ enez Cano, and A. Wojnar, Symmetry 16, 1384 (2024), arXiv:2408.16513 [gr-qc]

work page arXiv 2024
[28]

Sundermeyer,Symmetries in Fundamental Physics, Fundamental Theories of Physics, Vol

K. Sundermeyer,Symmetries in Fundamental Physics, Fundamental Theories of Physics, Vol. 176 (Springer, Cham, Switzerland, 2014)

work page 2014
[29]

Hamiltonian formalism for f(T) gravity

R. Ferraro and M. J. Guzm´ an, Phys. Rev. D97, 104028 (2018), arXiv:1802.02130 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2018
[30]

Quest for the extra degree of freedom in f(T) gravity

R. Ferraro and M. J. Guzm´ an, Phys. Rev. D98, 124037 (2018), arXiv:1810.07171 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2018
[31]

Ferraro and M

R. Ferraro and M. J. Guzm´ an, Phys. Rev. D101, 084017 (2020), arXiv:2001.08137 [gr-qc]

work page arXiv 2020
[32]

Guzman and S

M.-J. Guzman and S. Khaled Ibraheem, Int. J. Geom. Meth. Mod. Phys.18, 2140003 (2021), arXiv:2009.13430 [gr-qc]

work page arXiv 2021
[33]

Blixt, R

D. Blixt, R. Ferraro, A. Golovnev, and M.-J. Guzm´ an, Phys. Rev. D105, 084029 (2022), arXiv:2201.11102 [gr- qc]

work page arXiv 2022
[34]

Golovnev, (2024), arXiv:2411.14089 [gr-qc]

A. Golovnev, (2024), arXiv:2411.14089 [gr-qc]

work page arXiv 2024
[35]

Well-posedness of the scale-invariant tetrad formulation of the vacuum Einstein equations

D. Garfinkle and C. Gundlach, Class. Quant. Grav.22, 2679 (2005), arXiv:gr-qc/0501031

work page internal anchor Pith review Pith/arXiv arXiv 2005
[36]

Kreiss and J

H.-O. Kreiss and J. Lorenz,Initial-Boundary Value Prob- lems and the Navier-Stokes Equations, Classics in Ap- plied Mathematics, Vol. 47 (Society for Industrial and Applied Mathematics, Philadelphia, PA, 2004)

work page 2004
[37]

H¨ ormander,The Analysis of Linear Partial Dif- ferential Operators III: Pseudo-Differential Operators, Grundlehren der mathematischen Wissenschaften, Vol

L. H¨ ormander,The Analysis of Linear Partial Dif- ferential Operators III: Pseudo-Differential Operators, Grundlehren der mathematischen Wissenschaften, Vol. 274 (Springer-Verlag, Berlin, Heidelberg, 1985)

work page 1985
[38]

Well-posedness of formulations of the Einstein equations with dynamical lapse and shift conditions

C. Gundlach and J. M. Martin-Garcia, Phys. Rev. D74, 024016 (2006), arXiv:gr-qc/0604035

work page internal anchor Pith review Pith/arXiv arXiv 2006
[39]

The past attractor in inhomogeneous cosmology

C. Uggla, H. van Elst, J. Wainwright, and G. F. R. Ellis, Phys. Rev. D68, 103502 (2003), arXiv:gr-qc/0304002

work page internal anchor Pith review Pith/arXiv arXiv 2003
[40]

Luz and F

P. Luz and F. C. Mena, Class. Quant. Grav.42, 215011 (2025), arXiv:2505.22590 [gr-qc]

work page arXiv 2025
[41]

Introducing Cadabra: a symbolic computer algebra system for field theory problems

K. Peeters, (2007), arXiv:hep-th/0701238

work page internal anchor Pith review Pith/arXiv arXiv 2007
[42]

Peeters, J

K. Peeters, J. Open Source Softw.3, 1118 (2018)

work page 2018
[43]

Peeters, Computer Physics Communications176, 550 (2007)

K. Peeters, Computer Physics Communications176, 550 (2007). 18 Appendix A: 1-Dimensional System The principal symbol of the 1-dimensional analysis is Mx =   0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0...

work page 2007