Modified Teleparallel Gravity

Modified teleparallel gravity promotes the torsion scalar of TEGR to a nonlinear function or couples it to additional fields. A tetrad accompanied by the corresponding flat inertial spin connection gives a transparently local-Lorentz-covariant formulation; declaring the spin connection zero for every tetrad does not. That kinematic repair is well established, but it neither proves that a model has only two gravitational degrees of freedom nor settles stability. Those dynamical questions remain background- and theory-dependent.

Required background. Frames, Tetrads, and Spinors supplies tetrads and Lorentz connections; Differential Forms and Integration supplies flat connections and torsion; and Gravitational Actions and Boundary Terms supplies the distinction between a boundary change and a bulk modification.

Helpful background. Einstein Equations fixes the general-relativistic limit, while Formulations of General Relativity helps separate a choice of variables from new dynamics.

Nonlinear torsion actions

Let $e^a{}_{\mu}$ be an invertible tetrad, let $\omega^a{}_{b\mu}$ be a flat Lorentz spin connection, and use the site torsion scalar $\mathcal T$ defined so that $\mathring R=-\mathcal T+2\mathring\nabla_\mu T^\mu$. A basic modified action is

$$S_f[e,\omega] =-\frac{1}{2\kappa}\int d^4x\,e\,f(\mathcal T) +S_{\mathrm m}[e,\omega,\Psi], \qquad \kappa=8\pi G.$$

The controlled GR limit is $f(\mathcal T)=\mathcal T+2\Lambda$, with matter coupled in the same way as in TEGR. The plus sign follows from the overall minus sign in $S_f$ and gives the Einstein–Hilbert integrand $\mathring R-2\Lambda$ after using the boundary identity. If $f''\ne0$, that identity no longer maps this action to an $f(\mathring R)$ action. Writing $\mathcal T=-\mathring R+B$ with $B=2\mathring\nabla_\mu T^\mu$ makes the obstruction visible: $f(-\mathring R+B)$ contains bulk powers and cross terms involving $B$.

Scalar–torsion models sharpen the same lesson. A coefficient $F(\phi)\mathcal T$ cannot be replaced by $-F(\phi)\mathring R$ and discarded boundary data, because integration by parts produces the bulk derivative coupling $-2T^\mu\partial_\mu F$. A field redefinition, a different matter metric, and a change of gravitational dynamics are therefore distinct comparison axes.

Covariance requires the tetrad–connection pair

Under a local Lorentz transformation,

$$e'^a{}_{\mu}=\Lambda^a{}_b e^b{}_{\mu}, \qquad \omega'=\Lambda\omega\Lambda^{-1}-(d\Lambda)\Lambda^{-1}.$$

Transforming both objects leaves the induced spacetime connection and $\mathcal T$ unchanged. Locally, flatness makes $\omega$ pure inertial, but it does not authorize the gauge choice $\omega=0$ simultaneously in every Lorentz-related tetrad. In a pure-tetrad presentation, a local Lorentz transformation changes $\mathcal T$ by a divergence. That is harmless for the linear TEGR action and generically enters the bulk equations when $f''\ne0$.

The distinction can be checked without gravity. In spherical coordinates on Minkowski spacetime, the diagonal tetrad

$$e^a{}_{\mu}=\operatorname{diag}(1,1,r,r\sin\theta)$$

has spurious torsion if one sets $\omega=0$. The corresponding inertial connection has the independent nonzero components

$$\omega^{\hat1}{}_{\hat2\theta}=-1, \qquad \omega^{\hat1}{}_{\hat3\phi}=-\sin\theta, \qquad \omega^{\hat2}{}_{\hat3\phi}=-\cos\theta,$$

plus those fixed by Lorentz antisymmetry. With this connection the torsion vanishes, just as it does for the Cartesian tetrad with zero connection. This explicit calculation is given in Krššák and Saridakis 2016, §IV.A, pp. 6–7, Eqs. (19)–(25), PDF. It tests frame independence; it does not count modes.

This example compares gauge-equivalent representatives of the torsion-free Minkowski configuration. The diagonal tetrad with zero connection is a different metric–connection configuration, even though its metric is also Minkowski. It is not forbidden as an off-shell field choice; whether it is a solution of a specified nonlinear model must be checked with both the connection and tetrad equations. A metric alone therefore does not select the teleparallel configuration.

The connection equation in covariant $f(\mathcal T)$ gravity is equivalent to the antisymmetric part of the tetrad equations once flatness is handled consistently. Thus the inertial connection is not an extra freely propagating Lorentz gauge field. It is nevertheless indispensable for stating the variational problem and identifying which tetrad–connection pairs represent the same physics.

Results and open questions through 2026-08-19

The evidence cutoff for this frontier assessment is 2026-08-19. “Covariant” in the table means covariance under simultaneous local Lorentz transformations of the tetrad and inertial connection; it is not a synonym for ghost-free or predictive.

Formulation or calculation	Evidence and supported result	Regime or assumptions	Limitation or negative evidence
Covariant $f(\mathcal T)$ formulation	Krššák and Saridakis 2016, §§II–IV, pp. 2–9, PDF construct the tetrad–spin-connection equations and obtain frame-independent Minkowski, FLRW, and spherical descriptions.	Flat inertial Lorentz connection paired with each tetrad; classical field equations.	Establishes transparent local Lorentz covariance, not the nonlinear constraint count, hyperbolicity, or stability.
Broad theory review	Bahamonde et al. 2023, §§5.1–5.4, pp. 75–88, PDF organize $f(\mathcal T)$ and scalar–torsion actions, field equations, and cosmological perturbations.	Several distinct Lagrangians and backgrounds; source conventions must be translated as a package.	A catalogue of models and linear results does not supply a universal degree-of-freedom theorem.
Constraint and perturbation critique	Golovnev 2024, §§5–7, pp. 6–8, PDF compares contradictory Hamiltonian counts and relates rank-changing or accidental symmetries to missing linear modes.	Primarily $f(\mathcal T)$, with emphasis on Minkowski and cosmological perturbations and phase-space rank.	Gives strong negative evidence against reading a symmetric-background quadratic action as a generic count; a complete all-background well-posedness result remains absent.
Homogeneous-background perturbations	Danieli and De Felice 2025, §§III–V, pp. 6–16, PDF find only the two tensor polarizations in the vacuum gravity sector on flat FLRW and their anisotropic Bianchi I branch.	Quadratic perturbations about specified homogeneous vacuum backgrounds and a particular $f(\mathcal T)$ class.	The authors explicitly leave inhomogeneous backgrounds and possible strong coupling open; absence at quadratic order is not proof that nonlinear modes do not exist.

Competing interpretations and an adverse test

The conservative position is that a mode absent on Minkowski or FLRW can reappear away from a singular constraint surface. On this reading, accidental linear gauge symmetry and a rank-changing kinetic matrix signal strong coupling: the symmetric-background calculation has lost access to dynamics present nonlinearly or on less special data. The permissive position gives more weight to analyses finding only two propagating tensor modes and asks whether some covariant $f(\mathcal T)$ subclasses genuinely possess additional constraints. The Bianchi I result strengthens that possibility because anisotropy alone did not release another vacuum mode, but it does not decide inhomogeneous backgrounds or the full Hamiltonian problem.

An adverse test must therefore vary more than the tetrad gauge. Start with a fixed nonlinear $f$ satisfying $f''\ne0$, choose an inhomogeneous background with nonconstant $\mathcal T$, compute the complete principal symbol and constraint rank, and approach a symmetric background continuously. A jump in rank, a vanishing kinetic coefficient, or a cutoff collapsing along that limit supports the strong-coupling interpretation. A constant rank with positive kinetic and gradient matrices, propagated constraints, and a well-posed Cauchy problem would support the genuinely constrained interpretation. No result in the table establishes all of those properties for generic data.

What is settled and what remains open

It is settled that retaining the appropriate flat spin connection removes the artificial frame dependence created by imposing zero connection in every tetrad. It is also settled that nonlinear $f(\mathcal T)$ dynamics are not TEGR in disguise. What remains unresolved is a universal count and stability classification across theory class, background, solution branch, matter coupling, and strong-coupling scale. Second-order field equations alone do not answer those questions, because constraints and kinetic rank rather than derivative order determine the physical modes.

Phenomenological parameter bounds are not assessed here: likelihood construction, data systematics, and observational inference belong to Tests of Gravity. Likewise, an apparent cosmological fit cannot substitute for a controlled initial-value problem below a stated cutoff.

What could change these conclusions

The conclusions could change after a complete nonlinear constraint theorem fixes the generic rank, an independent inhomogeneous-background calculation changes the stability conclusion, a strong-coupling or cutoff analysis invalidates a claimed regime, or a new covariant prediction survives its separate empirical appraisal with matter couplings held fixed.

Common pitfalls

Calling a tetrad “bad.” A tetrad is not intrinsically bad because its inertial spin connection is nonzero. The error is pairing it with the wrong connection and then treating inertial torsion as gravitational.

Equating covariance with health. A covariant action may still propagate a ghost, lose hyperbolicity, or become strongly coupled on a special background.

Exercises

In spherical coordinates on Minkowski spacetime, compare two configurations: (A) the diagonal tetrad with $\omega=0$, and (B) the same tetrad with the inertial connection displayed above. Which is gauge-equivalent to the Cartesian tetrad with zero connection? What can be concluded if A and B give different field equations in a nonlinear $f(\mathcal T)$ model?

Solution

Choice B gives $T^a{}_{\mu\nu}=0$ and hence $\mathcal T=0$, matching the Cartesian pair after the coordinate and frame changes. Choice A leaves nonzero torsion and is not that transformation of the Cartesian pair, because the connection was not transformed. Different equations are therefore compatible with covariance: the configurations are physically different in their affine data. It would be inconsistent to claim that they were merely two gauges of one configuration. Neither comparison by itself counts extra modes; that requires the full constraints and perturbations of an actual solution.

References

• Bahamonde, S., et al., “Teleparallel Gravity: From Theory to Cosmology,” Reports on Progress in Physics 86 (2023) 026901. DOI. Open PDF
• Danieli, V., and A. De Felice, “$f(T)$ Gravity: Background Dependence and Propagating Degrees of Freedom,” Physical Review D 111 (2025) 104023. DOI. Open PDF
• Golovnev, A., “Degrees of Freedom in Modified Teleparallel Gravity,” Ukrainian Journal of Physics 69 (2024) 456. DOI. Open PDF
• Krššák, M., and E. N. Saridakis, “The Covariant Formulation of $f(T)$ Gravity,” Classical and Quantum Gravity 33 (2016) 115009. DOI. Open PDF