New General Relativity: Teleparallel Gravity

• New General Relativity is a teleparallel modification of gravity that uses quadratic torsion invariants instead of the Ricci scalar.
• It deviates from TEGR by altering Lorentz-sector constraints and potentially introducing additional propagating modes.
• Recent studies explore its weak-field dynamics, cosmological perturbations, and exact solutions while addressing issues like ghost instabilities.

New General Relativity (NGR) is a parity-preserving teleparallel modification of gravity in which the gravitational Lagrangian is built from quadratic invariants of the torsion tensor rather than from the Levi-Civita Ricci scalar. It contains the Teleparallel Equivalent of General Relativity (TEGR) as a special point, but away from that point it generally changes the Lorentz-sector constraint structure and can introduce additional propagating modes. Contemporary work studies NGR in both pure tetrad and covariant teleparallel formulations, and the resulting literature spans Hamiltonian classification, weak-field and cosmological perturbations, exact spherical solutions, and strong-field no-go results for black-hole horizons (Golovnev et al., 2023, López et al., 13 Feb 2026, Jiménez et al., 2019).

1. Geometric and conceptual foundations

NGR is formulated in teleparallel geometry, where the basic variable is a tetrad $e^a{}_\mu$ or $h^a{}_\mu$, and gravity is encoded in torsion with vanishing curvature. In the pure tetrad formulation, used in several perturbative analyses, the flat spin connection is set to zero and the Weitzenböck connection is written as

$$\Gamma^\alpha{}_{\mu\nu}=e_a{}^\alpha\,\partial_\mu e^a{}_\nu ,$$

with torsion

$$T^\alpha{}_{\mu\nu}=\Gamma^\alpha{}_{\nu\mu}-\Gamma^\alpha{}_{\mu\nu}, \qquad T_\mu \equiv T^\alpha{}_{\mu\alpha},$$

and metric

$g_{\mu\nu}=\eta_{ab}\,e^a{}_\mu e^b{}_\nu .$

In the covariant teleparallel formulation, by contrast, one retains a flat, metric-compatible inertial spin connection $\omega^a{}_{b\mu}$; this keeps local Lorentz covariance explicit and removes spurious “good tetrad/bad tetrad” issues (Golovnev et al., 2023, López et al., 13 Feb 2026).

A second standard organization uses the Lorentz-irreducible pieces of torsion. In the notation of the covariant spherical and Hamiltonian literature, these are the vector trace $\mathscr V_\mu$, the axial vector $\mathscr A_\mu$, and the purely tensorial part $\mathscr T_{\sigma\mu\nu}$, with quadratic scalars $\mathscr V$, $h^a{}_\mu$0, and $h^a{}_\mu$1. The TEGR torsion scalar is the specific combination

$h^a{}_\mu$2

This is the sense in which TEGR sits as a distinguished point inside a larger quadratic torsion family (López et al., 13 Feb 2026).

The status of local Lorentz symmetry depends on formulation. In the pure tetrad setting, local Lorentz invariance is generically broken away from TEGR, although remnant symmetries can survive in special cases. In the covariant tetrad-plus-spin-connection setting, local Lorentz invariance is preserved by construction, and the antisymmetric field equations are equivalent to the spin-connection equations (Golovnev et al., 2023, López et al., 27 Aug 2025).

2. Lagrangian structure and parameterizations

The parity-even NGR action is the most general quadratic torsion action built from three independent invariants. One common basis is

$h^a{}_\mu$3

with

$h^a{}_\mu$4

Another, linearly equivalent basis writes

$h^a{}_\mu$5

A third, heavily used in perturbation theory, introduces a linear superpotential

$h^a{}_\mu$6

and defines the torsion scalar by

$h^a{}_\mu$7

All three parameterizations are explicitly stated to be linearly equivalent in the literature (Golovnev et al., 2023, López et al., 13 Feb 2026, Blixt et al., 2019).

TEGR is recovered at specific parameter values in each basis. In the $h^a{}_\mu$8 superpotential basis, TEGR corresponds to $h^a{}_\mu$9. In the irreducible basis, TEGR corresponds to

$$\Gamma^\alpha{}_{\mu\nu}=e_a{}^\alpha\,\partial_\mu e^a{}_\nu ,$$0

In the convention used for the covariant Hamiltonian function,

$$\Gamma^\alpha{}_{\mu\nu}=e_a{}^\alpha\,\partial_\mu e^a{}_\nu ,$$1

TEGR is

$$\Gamma^\alpha{}_{\mu\nu}=e_a{}^\alpha\,\partial_\mu e^a{}_\nu ,$$2

These are convention-dependent presentations of the same distinguished TEGR locus (Golovnev et al., 2023, Bajardi et al., 2024).

Several special subfamilies recur. A one-parameter subfamily in the pure tetrad $$\Gamma^\alpha{}_{\mu\nu}=e_a{}^\alpha\,\partial_\mu e^a{}_\nu ,$$3 basis is defined by

$$\Gamma^\alpha{}_{\mu\nu}=e_a{}^\alpha\,\partial_\mu e^a{}_\nu ,$$4

and is referred to as “1-parameter New GR.” In covariant irreducible variables, one also encounters a normalized two-parameter description obtained by fixing $$\Gamma^\alpha{}_{\mu\nu}=e_a{}^\alpha\,\partial_\mu e^a{}_\nu ,$$5, leaving $$\Gamma^\alpha{}_{\mu\nu}=e_a{}^\alpha\,\partial_\mu e^a{}_\nu ,$$6 and $$\Gamma^\alpha{}_{\mu\nu}=e_a{}^\alpha\,\partial_\mu e^a{}_\nu ,$$7 as independent parameters after the relation $$\Gamma^\alpha{}_{\mu\nu}=e_a{}^\alpha\,\partial_\mu e^a{}_\nu ,$$8 (Golovnev et al., 2023, López et al., 13 Feb 2026).

Ordinary NGR is parity-even. A parity-violating extension adds a quadratic torsion term

$$\Gamma^\alpha{}_{\mu\nu}=e_a{}^\alpha\,\partial_\mu e^a{}_\nu ,$$9

which does not alter the spatially flat FRW background equations but does modify perturbations, especially tensor birefringence and vector-sector stability (Kang et al., 4 Jul 2025).

3. Field equations, constraints, and classification by type

Variation with respect to the tetrad yields field equations that split naturally into symmetric and antisymmetric parts. In the $$T^\alpha{}_{\mu\nu}=\Gamma^\alpha{}_{\nu\mu}-\Gamma^\alpha{}_{\mu\nu}, \qquad T_\mu \equiv T^\alpha{}_{\mu\alpha},$$0 superpotential formulation, the generalized gravitational tensor is

$$T^\alpha{}_{\mu\nu}=\Gamma^\alpha{}_{\nu\mu}-\Gamma^\alpha{}_{\mu\nu}, \qquad T_\mu \equiv T^\alpha{}_{\mu\alpha},$$1

whose symmetric part plays the role of the Einstein tensor, while the antisymmetric part encodes the Lorentz-sector equations. A useful identity is

$$T^\alpha{}_{\mu\nu}=\Gamma^\alpha{}_{\nu\mu}-\Gamma^\alpha{}_{\mu\nu}, \qquad T_\mu \equiv T^\alpha{}_{\mu\alpha},$$2

which controls much of the background and perturbative structure (Golovnev et al., 2023).

The Hamiltonian formulation reveals that the velocity–momentum map depends on four parameter combinations,

$$T^\alpha{}_{\mu\nu}=\Gamma^\alpha{}_{\nu\mu}-\Gamma^\alpha{}_{\mu\nu}, \qquad T_\mu \equiv T^\alpha{}_{\mu\alpha},$$3

or equivalently on four Hessian eigenvalue sets of multiplicities $$T^\alpha{}_{\mu\nu}=\Gamma^\alpha{}_{\nu\mu}-\Gamma^\alpha{}_{\mu\nu}, \qquad T_\mu \equiv T^\alpha{}_{\mu\alpha},$$4 beyond the four trivial temporal null directions. Their vanishing generates irreducible primary constraints and leads to nine non-trivial classes of NGR. Early ADM and premetric analyses identified these nine classes but left the full nonlinear constraint algebra and degree-of-freedom count open in the generic case (Blixt et al., 2019, Guzman et al., 2020).

Subsequent Dirac–Bergmann analyses completed the nonlinear count for the special types. Type 2 has $$T^\alpha{}_{\mu\nu}=\Gamma^\alpha{}_{\nu\mu}-\Gamma^\alpha{}_{\mu\nu}, \qquad T_\mu \equiv T^\alpha{}_{\mu\alpha},$$5 degrees of freedom, Type 3 has $$T^\alpha{}_{\mu\nu}=\Gamma^\alpha{}_{\nu\mu}-\Gamma^\alpha{}_{\mu\nu}, \qquad T_\mu \equiv T^\alpha{}_{\mu\alpha},$$6, Type 4 has $$T^\alpha{}_{\mu\nu}=\Gamma^\alpha{}_{\nu\mu}-\Gamma^\alpha{}_{\mu\nu}, \qquad T_\mu \equiv T^\alpha{}_{\mu\alpha},$$7, Type 5 has $$T^\alpha{}_{\mu\nu}=\Gamma^\alpha{}_{\nu\mu}-\Gamma^\alpha{}_{\mu\nu}, \qquad T_\mu \equiv T^\alpha{}_{\mu\alpha},$$8, Type 7 has null degrees of freedom, Type 8 has either $$T^\alpha{}_{\mu\nu}=\Gamma^\alpha{}_{\nu\mu}-\Gamma^\alpha{}_{\mu\nu}, \qquad T_\mu \equiv T^\alpha{}_{\mu\alpha},$$9 under a specific condition on the Lagrange multipliers or $g_{\mu\nu}=\eta_{ab}\,e^a{}_\mu e^b{}_\nu .$0 in the generic case, and Type 9 has $g_{\mu\nu}=\eta_{ab}\,e^a{}_\mu e^b{}_\nu .$1 (Tomonari et al., 2024, Tomonari, 2024).

Type	Defining irreducible condition	Nonlinear DoF reported
Type 2	$g_{\mu\nu}=\eta_{ab}\,e^a{}_\mu e^b{}_\nu .$2	6
Type 3	$g_{\mu\nu}=\eta_{ab}\,e^a{}_\mu e^b{}_\nu .$3	5
Type 4	$g_{\mu\nu}=\eta_{ab}\,e^a{}_\mu e^b{}_\nu .$4	5
Type 5	$g_{\mu\nu}=\eta_{ab}\,e^a{}_\mu e^b{}_\nu .$5	7
Type 7	$g_{\mu\nu}=\eta_{ab}\,e^a{}_\mu e^b{}_\nu .$6	0
Type 8	$g_{\mu\nu}=\eta_{ab}\,e^a{}_\mu e^b{}_\nu .$7	4 or 6
Type 9	$g_{\mu\nu}=\eta_{ab}\,e^a{}_\mu e^b{}_\nu .$8	3

The covariant Hamiltonian function for NGR has also been derived explicitly and compared with TEGR and $g_{\mu\nu}=\eta_{ab}\,e^a{}_\mu e^b{}_\nu .$9. In that formulation NGR reduces consistently to TEGR, while the canonical structure differs sharply from that of $\omega^a{}_{b\mu}$0 gravity (Bajardi et al., 2024).

4. Weak-field spectrum, gravitational waves, and cosmological perturbations

Linearized analysis around the trivial Minkowski tetrad shows that NGR can exhibit a rich spectrum. In the pure tetrad vacuum analysis, the tensor sector obeys

$\omega^a{}_{b\mu}$1

so there are two luminal spin-2 polarizations provided $\omega^a{}_{b\mu}$2. The pseudoscalar mode satisfies

$\omega^a{}_{b\mu}$3

and the generic vacuum theory admits additional scalar and vector modes. In that framework the generic theory carries $\omega^a{}_{b\mu}$4 dynamical linearized degrees of freedom, whereas TEGR carries only the two standard tensor modes (Golovnev et al., 2023).

The linear spectrum is, however, not the end of the story. A decoupling-limit and cubic-order analysis argued that the linear Kalb–Ramond gauge symmetry of the antisymmetric sector cannot be extended nonlinearly unless the theory is at the TEGR point. That work concluded that only TEGR yields a consistent, stable Minkowski background with gravity, and that non-TEGR models suffer a nonlinear obstruction and strong coupling around Minkowski (Jiménez et al., 2019).

Later work reached a different linear conclusion. A gauge-invariant weak-field study reported three instability-free branches beyond TEGR: the decoupling branch often called Type 2, a Type 3 branch with two tensors plus one massless scalar, and a Type 8 branch with only the two tensor modes. In that account the generic theory is ill-behaved because the vector sector contains an Ostrogradsky ghost unless the parameters make the vector Lagrangian vanish (Bahamonde et al., 2024). Subsequent discussion of Type 3 emphasized that mode counting must be done from the equations of motion or Hamiltonian constraints, not by substituting constraints back into the Lagrangian (Golovnev, 21 May 2026).

Cosmological perturbations provide a partially independent test. In spatially flat FLRW, the pure tetrad analysis found that the background equations reduce to those of GR up to a rescaling of the effective gravitational constant by $\omega^a{}_{b\mu}$5, and that the tensor mode equation becomes

$\omega^a{}_{b\mu}$6

The graviton speed remains $\omega^a{}_{b\mu}$7 when $\omega^a{}_{b\mu}$8. For generic $\omega^a{}_{b\mu}$9, the vector sector contains two dynamical transverse vectors and one constrained transverse vector, while the scalar sector generically contains one dynamical Lorentz-sector mode $\mathscr V_\mu$0, one dynamical acoustic mode, and one constraint. The $\mathscr V_\mu$1 limit is smooth in the robust three-parameter case (Golovnev et al., 2023).

That same cosmological analysis isolates special loci. The one-parameter condition $\mathscr V_\mu$2 yields metric perturbation equations identical to GR, with $\mathscr V_\mu$3 and no observable deviations in the linear metric sector for an ideal fluid; the theory is therefore consistent but “boring” observationally at that level. By contrast, Types 3, 4, and 8 exhibit discontinuities between Minkowski and FLRW, with accidental gauge modes in flat space becoming constraints in cosmology, which the paper interprets as strong coupling around Minkowski (Golovnev et al., 2023).

For Type 3 specifically, a later gauge-invariant FLRW analysis reported $\mathscr V_\mu$4 physical degrees of freedom—$\mathscr V_\mu$5 tensors, $\mathscr V_\mu$6 Lorentz-vector modes, and $\mathscr V_\mu$7 Lorentz-scalar mode—and derived background-hierarchy bounds. In the notation $\mathscr V_\mu$8, the combined ghost-free and viability window was

$\mathscr V_\mu$9

with Region III supporting well-behaved tensor, vector, and scalar propagation (Tomonari et al., 16 May 2026).

The parity-violating extension changes this picture. In a spatially flat FRW background it leaves the background unchanged but modifies perturbations. The tensor action acquires a helicity-dependent gradient term, leading to chiral birefringence, and the paper identifies a parameter region with $\mathscr A_\mu$0 in which tensor, vector, and scalar perturbations are all propagating and ghost-free (Kang et al., 4 Jul 2025).

5. Spherical symmetry, exact solutions, and the horizon problem

Spherically symmetric vacuum solutions occupy a central but disputed place in NGR. In one-parameter NGR, using the most general spherically symmetric tetrad in the Weitzenböck gauge, the field equations split into three branches. Branch 1 gives the Schwarzschild metric with a free tetrad boost. Branch 2 gives

$\mathscr A_\mu$1

with constant

$\mathscr A_\mu$2

so that the horizon remains at $\mathscr A_\mu$3 while the spacetime is not asymptotically flat unless $\mathscr A_\mu$4. Branch 3 again yields the Schwarzschild metric, now with nontrivial tetrad functions $\mathscr A_\mu$5 and $\mathscr A_\mu$6 (Asuküla et al., 2023).

In that Branch 2 geometry, the photon sphere radius and the ISCO remain at the Schwarzschild values $\mathscr A_\mu$7 and $\mathscr A_\mu$8, but null and timelike geodesic integrals are rescaled by $\mathscr A_\mu$9. Consequently, light deflection, Shapiro delay, and perihelion advance are rescaled relative to Schwarzschild, and ray tracing shows that positive and negative deformations can enlarge or shrink the observed shadow (Asuküla et al., 2023).

A later exact vacuum solution, written in isotropic coordinates, introduces a second dimensionless parameter $\mathscr T_{\sigma\mu\nu}$0 fixed by the NGR couplings through

$\mathscr T_{\sigma\mu\nu}$1

and reduces to Schwarzschild at $\mathscr T_{\sigma\mu\nu}$2. The paper then maps weak-field observables to $\mathscr T_{\sigma\mu\nu}$3 and $\mathscr T_{\sigma\mu\nu}$4 and quotes representative bounds such as

$\mathscr T_{\sigma\mu\nu}$5

from perihelion data, together with similar $\mathscr T_{\sigma\mu\nu}$6-level bounds from light bending and Shapiro delay. That work explicitly states that it focuses on the weak-field regime and that a detailed strong-field analysis is deferred (Vandeev et al., 6 May 2026).

Recent covariant analyses challenge the black-hole interpretation of such solutions. One study considered vacuum and non-vacuum static spherically symmetric configurations under the assumption of a local black-hole horizon and finite torsion invariants. It found that horizon regularity enforces algebraic constraints fixing the parameters to physically pathological models, such as Type I branches with ghost instabilities or Type II branches without propagating spin-2 modes, leaving only the TEGR-equivalent case as physically meaningful (López et al., 13 Feb 2026).

A subsequent covariant analysis sharpened the point by arguing that, once local horizons are detected through null-congruence expansions and matter is minimally coupled to the metric, all physically viable NGR models—including TEGR and the one-parameter Hayashi–Shirafuji model—exhibit divergences in torsion scalars at the local horizon. In that sense, the horizon itself is a teleparallel geometric singularity, and the manifold cannot include it without making the action ill-defined (López et al., 27 Aug 2025).

Within a different strong-field setting, the $\mathscr T_{\sigma\mu\nu}$7 sector with $\mathscr T_{\sigma\mu\nu}$8 was used to study a generalized McVittie spacetime in a positive-curvature bouncing cosmology. Near the bounce, the background behaves like GR with a curvature renormalization

$\mathscr T_{\sigma\mu\nu}$9

while the local horizon evolves as

$\mathscr V$0

with a linear term in $\mathscr V$1 that breaks time-reversal symmetry across the bounce. This analysis is explicitly perturbative and near-bounce in scope (Yildirim et al., 5 Apr 2026).

6. Relation to GR, present status, and open directions

TEGR remains the unique point at which NGR reproduces GR exactly. At that point the conformal transformation laws collapse to the familiar GR ones, the cosmological perturbation equations reduce to those of GR, and the covariant Hamiltonian reduces to the TEGR constraint structure (Golovnev et al., 2023, Bajardi et al., 2024).

Away from TEGR, the observational situation is uneven. At the FLRW background level, NGR can mimic GR up to a rescaling of the effective gravitational coupling, and in the one-parameter cosmological model the linear metric sector is indistinguishable from GR for ideal-fluid matter (Golovnev et al., 2023). In spherical symmetry, some branches admit exact or near-Schwarzschild metrics whose weak-field parameters can be constrained tightly by classical tests (Vandeev et al., 6 May 2026). Parity-odd extensions enlarge the phenomenology further by allowing chiral tensor propagation and, in specific regions, all scalar, vector, and tensor perturbations to be ghost-free (Kang et al., 4 Jul 2025).

At the same time, the theory’s consistency remains actively debated. One line of work identifies nonlinear obstructions around Minkowski and argues that TEGR is the only consistent NGR with a stable Minkowski background that includes gravity (Jiménez et al., 2019). Another line, combining gauge-invariant weak-field analysis and full Dirac–Bergmann counting for the special types, finds non-TEGR branches with controlled linear spectra and specific nonlinear degree-of-freedom counts (Bahamonde et al., 2024, Tomonari et al., 2024, Tomonari, 2024). The black-hole sector is likewise unsettled: exact vacuum solutions and weak-field phenomenology coexist with recent no-go results stating that physically meaningful non-TEGR black holes are excluded by local-horizon regularity, or even that viable NGR horizons are necessarily torsion-singular (López et al., 13 Feb 2026, López et al., 27 Aug 2025).

The current literature therefore presents NGR less as a single settled alternative to GR than as a structured family of teleparallel theories whose viability depends sharply on parameter choice, formulation, and regime. A recurrent conclusion is that the robust questions are now nonlinear: detailed Hamiltonian analysis beyond linear order, systematic stability across backgrounds, and the role of parity-odd torsion terms in enlarging the viable parameter space remain central open problems (Golovnev et al., 2023).