1P-H&S Model in Teleparallel Gravity

• The 1P-H&S model is a teleparallel gravity subfamily with a single free parameter that restricts the torsion to exclude its totally antisymmetric part.
• It unifies two formulations—Møller-Hayashi-Shirafuji and normalized NGR—thereby linking quadratic torsion invariants with covariant fracton gauge theory.
• Static spherically symmetric analyses reveal divergent torsion scalars at local horizons, challenging conventional black hole representations in teleparallel geometry.

The one-parameter Hayashi and Shirafuji model (1P-H&S) is a one-parameter subfamily of teleparallel gravity that appears in two closely related formulations. In the Møller-Hayashi-Shirafuji extension of teleparallel gravity, it is a restricted quadratic-torsion theory built from the Weitzenböck torsion with vanishing totally antisymmetric part; in New General Relativity (NGR), it is the normalized one-parameter deformation of the Teleparallel Equivalent of General Relativity (TEGR) obtained by keeping the tensorial and vector torsion couplings fixed while allowing the axial coupling to vary (Rovere, 27 May 2025, López et al., 27 Aug 2025). Recent work places the model at the intersection of covariant fracton gauge theory and teleparallel geometry: the linearized restricted torsion sector reproduces the covariant fracton field strength, while static spherically symmetric horizon analyses show unavoidable divergences in torsion scalars at the local horizon (Rovere, 27 May 2025, López et al., 27 Aug 2025).

1. Teleparallel setting and Weitzenböck torsion

Teleparallel gravity is a reformulation of General Relativity in which the fundamental variables are the vielbein fields $e_\mu{}^a$, and the connection is the Weitzenböck connection rather than the Levi-Civita connection. The Weitzenböck connection is

$$W_\mu{}^\rho{}_\nu = e^\rho_a\,\partial_\mu e_\nu{}^a ,$$

and the associated Weitzenböck torsion tensor is

$$T_\mu{}^\rho{}_\nu = W_{[\mu}{}^\rho{}_{\nu]} = e^\rho_a\,\partial_{[\mu} e_{\nu]}{}^a .$$

The connection is metric-compatible and flat, but it has non-zero torsion (Rovere, 27 May 2025).

Within this setting, the Møller-Hayashi-Shirafuji theory is the class of teleparallel models whose action is the most general quadratic function of the Weitzenböck torsion: $$S_{\mathrm{mhs}} = \frac{1}{2} \int \mathrm{d}^d x\, e\, \Big( \alpha_1\, T^{\mu\nu\rho} T_{\mu\nu\rho} + \alpha_2\, T^{\mu\nu\rho} T_{\mu\rho\nu} + \alpha_3\, T^{\mu\nu}{}_\nu T_\mu{}^\rho{}_\rho \Big),$$ with $e=\det(e_\mu{}^a)$. The three displayed contractions are, respectively, the full contraction, the contraction with the second and third indices exchanged in the second factor, and the contraction over the last index. The paper further notes that the Weitzenböck condition is imposed by taking vanishing spin connection, so local Lorentz symmetry is broken while global Lorentz transformations are preserved (Rovere, 27 May 2025).

2. Definition of the 1P-H&S sector

In the Møller-Hayashi-Shirafuji framework, the 1P-H&S model is obtained by restricting the torsion to have vanishing totally antisymmetric part. Equivalently, the components associated with the Young tableau $(1,1,1)$ are set to zero, leaving only the traceless part with Young shape $(2,1)$ and the overall trace. In this restricted sector there are only two independent quadratic invariants, and the action simplifies to

$$S = \frac{1}{2} \int \mathrm{d}^d x\, e\, \Big( \alpha\, T^{\mu\nu\rho} T_{\mu\nu\rho} + \beta\, T^{\mu\nu\rho}{}_\rho T_{\mu\sigma}{}^\sigma \Big),$$

with parameters $\alpha$ and $\beta$ (Rovere, 27 May 2025).

In NGR, the same model is expressed through the irreducible quadratic torsion invariants

$$W_\mu{}^\rho{}_\nu = e^\rho_a\,\partial_\mu e_\nu{}^a ,$$0

where $$W_\mu{}^\rho{}_\nu = e^\rho_a\,\partial_\mu e_\nu{}^a ,$$1 is the axial torsion invariant, $$W_\mu{}^\rho{}_\nu = e^\rho_a\,\partial_\mu e_\nu{}^a ,$$2 the purely tensorial torsion invariant, and $$W_\mu{}^\rho{}_\nu = e^\rho_a\,\partial_\mu e_\nu{}^a ,$$3 the vector torsion invariant. The normalized 1P-H&S lagrangian is

$$W_\mu{}^\rho{}_\nu = e^\rho_a\,\partial_\mu e_\nu{}^a ,$$4

so the only free parameter is $$W_\mu{}^\rho{}_\nu = e^\rho_a\,\partial_\mu e_\nu{}^a ,$$5, while $$W_\mu{}^\rho{}_\nu = e^\rho_a\,\partial_\mu e_\nu{}^a ,$$6 and $$W_\mu{}^\rho{}_\nu = e^\rho_a\,\partial_\mu e_\nu{}^a ,$$7 are fixed (López et al., 27 Aug 2025).

Framework	1P-H&S specification	Parameters
Møller-Hayashi-Shirafuji	vanishing totally antisymmetric torsion; two quadratic invariants	$$W_\mu{}^\rho{}_\nu = e^\rho_a\,\partial_\mu e_\nu{}^a ,$$8
Normalized NGR	$$W_\mu{}^\rho{}_\nu = e^\rho_a\,\partial_\mu e_\nu{}^a ,$$9	$$T_\mu{}^\rho{}_\nu = W_{[\mu}{}^\rho{}_{\nu]} = e^\rho_a\,\partial_{[\mu} e_{\nu]}{}^a .$$0 free; $$T_\mu{}^\rho{}_\nu = W_{[\mu}{}^\rho{}_{\nu]} = e^\rho_a\,\partial_{[\mu} e_{\nu]}{}^a .$$1, $$T_\mu{}^\rho{}_\nu = W_{[\mu}{}^\rho{}_{\nu]} = e^\rho_a\,\partial_{[\mu} e_{\nu]}{}^a .$$2

A second NGR parameterization uses $$T_\mu{}^\rho{}_\nu = W_{[\mu}{}^\rho{}_{\nu]} = e^\rho_a\,\partial_{[\mu} e_{\nu]}{}^a .$$3 defined by

$$T_\mu{}^\rho{}_\nu = W_{[\mu}{}^\rho{}_{\nu]} = e^\rho_a\,\partial_{[\mu} e_{\nu]}{}^a .$$4

For the 1P-H&S model after normalization,

$$T_\mu{}^\rho{}_\nu = W_{[\mu}{}^\rho{}_{\nu]} = e^\rho_a\,\partial_{[\mu} e_{\nu]}{}^a .$$5

and this case is referred to in the literature as Type 2 NGR (López et al., 27 Aug 2025).

3. Relation to TEGR and the torsion-scalar basis

TEGR is the special NGR choice

$$T_\mu{}^\rho{}_\nu = W_{[\mu}{}^\rho{}_{\nu]} = e^\rho_a\,\partial_{[\mu} e_{\nu]}{}^a .$$6

which yields the torsion scalar

$$T_\mu{}^\rho{}_\nu = W_{[\mu}{}^\rho{}_{\nu]} = e^\rho_a\,\partial_{[\mu} e_{\nu]}{}^a .$$7

The 1P-H&S model generalizes TEGR by allowing $$T_\mu{}^\rho{}_\nu = W_{[\mu}{}^\rho{}_{\nu]} = e^\rho_a\,\partial_{[\mu} e_{\nu]}{}^a .$$8 to vary while fixing $$T_\mu{}^\rho{}_\nu = W_{[\mu}{}^\rho{}_{\nu]} = e^\rho_a\,\partial_{[\mu} e_{\nu]}{}^a .$$9 and $$S_{\mathrm{mhs}} = \frac{1}{2} \int \mathrm{d}^d x\, e\, \Big( \alpha_1\, T^{\mu\nu\rho} T_{\mu\nu\rho} + \alpha_2\, T^{\mu\nu\rho} T_{\mu\rho\nu} + \alpha_3\, T^{\mu\nu}{}_\nu T_\mu{}^\rho{}_\rho \Big),$$0 to their TEGR values (López et al., 27 Aug 2025).

For static, spherically symmetric analyses, the relevant irreducible torsion scalars are

$$S_{\mathrm{mhs}} = \frac{1}{2} \int \mathrm{d}^d x\, e\, \Big( \alpha_1\, T^{\mu\nu\rho} T_{\mu\nu\rho} + \alpha_2\, T^{\mu\nu\rho} T_{\mu\rho\nu} + \alpha_3\, T^{\mu\nu}{}_\nu T_\mu{}^\rho{}_\rho \Big),$$1

together with the torsion scalar $$S_{\mathrm{mhs}} = \frac{1}{2} \int \mathrm{d}^d x\, e\, \Big( \alpha_1\, T^{\mu\nu\rho} T_{\mu\nu\rho} + \alpha_2\, T^{\mu\nu\rho} T_{\mu\rho\nu} + \alpha_3\, T^{\mu\nu}{}_\nu T_\mu{}^\rho{}_\rho \Big),$$2 above. The normalized NGR lagrangian for $$S_{\mathrm{mhs}} = \frac{1}{2} \int \mathrm{d}^d x\, e\, \Big( \alpha_1\, T^{\mu\nu\rho} T_{\mu\nu\rho} + \alpha_2\, T^{\mu\nu\rho} T_{\mu\rho\nu} + \alpha_3\, T^{\mu\nu}{}_\nu T_\mu{}^\rho{}_\rho \Big),$$3 always includes $$S_{\mathrm{mhs}} = \frac{1}{2} \int \mathrm{d}^d x\, e\, \Big( \alpha_1\, T^{\mu\nu\rho} T_{\mu\nu\rho} + \alpha_2\, T^{\mu\nu\rho} T_{\mu\rho\nu} + \alpha_3\, T^{\mu\nu}{}_\nu T_\mu{}^\rho{}_\rho \Big),$$4 with fixed weight (López et al., 27 Aug 2025).

A specific structural simplification arises for spherically symmetric solutions with $$S_{\mathrm{mhs}} = \frac{1}{2} \int \mathrm{d}^d x\, e\, \Big( \alpha_1\, T^{\mu\nu\rho} T_{\mu\nu\rho} + \alpha_2\, T^{\mu\nu\rho} T_{\mu\rho\nu} + \alpha_3\, T^{\mu\nu}{}_\nu T_\mu{}^\rho{}_\rho \Big),$$5: in that ansatz, the parameter $$S_{\mathrm{mhs}} = \frac{1}{2} \int \mathrm{d}^d x\, e\, \Big( \alpha_1\, T^{\mu\nu\rho} T_{\mu\nu\rho} + \alpha_2\, T^{\mu\nu\rho} T_{\mu\rho\nu} + \alpha_3\, T^{\mu\nu}{}_\nu T_\mu{}^\rho{}_\rho \Big),$$6, and hence $$S_{\mathrm{mhs}} = \frac{1}{2} \int \mathrm{d}^d x\, e\, \Big( \alpha_1\, T^{\mu\nu\rho} T_{\mu\nu\rho} + \alpha_2\, T^{\mu\nu\rho} T_{\mu\rho\nu} + \alpha_3\, T^{\mu\nu}{}_\nu T_\mu{}^\rho{}_\rho \Big),$$7, drops out of the field equations, so the solutions are locally equivalent to TEGR. The paper states this as an effective reduction of the 1P-H&S field equations to those of TEGR in the relevant sector (López et al., 27 Aug 2025).

4. Linearized formulation and embedding of covariant fracton gauge theory

The 2025 analysis of covariant fractons identifies the linearized restricted torsion sector of 1P-H&S with a free gauge theory of a symmetric rank-2 tensor $$S_{\mathrm{mhs}} = \frac{1}{2} \int \mathrm{d}^d x\, e\, \Big( \alpha_1\, T^{\mu\nu\rho} T_{\mu\nu\rho} + \alpha_2\, T^{\mu\nu\rho} T_{\mu\rho\nu} + \alpha_3\, T^{\mu\nu}{}_\nu T_\mu{}^\rho{}_\rho \Big),$$8 transforming as

$$S_{\mathrm{mhs}} = \frac{1}{2} \int \mathrm{d}^d x\, e\, \Big( \alpha_1\, T^{\mu\nu\rho} T_{\mu\nu\rho} + \alpha_2\, T^{\mu\nu\rho} T_{\mu\rho\nu} + \alpha_3\, T^{\mu\nu}{}_\nu T_\mu{}^\rho{}_\rho \Big),$$9

for scalar gauge parameter $e=\det(e_\mu{}^a)$0. Its gauge-invariant field strength is

$e=\det(e_\mu{}^a)$1

and the most general quadratic action is

$e=\det(e_\mu{}^a)$2

Because of the Young symmetries, only two independent contractions exist (Rovere, 27 May 2025).

The identification with the restricted teleparallel sector is made through the linearized Weitzenböck torsion: $e=\det(e_\mu{}^a)$3 The parameters are related by

$e=\det(e_\mu{}^a)$4

With this identification, every solution of the linear covariant fracton gauge theory corresponds to a solution of the linearized Møller-Hayashi-Shirafuji equations in the restricted sector. The solution space of covariant fracton theory is therefore a subset of the solution space of the linearized 1P-H&S model; the converse is not necessarily true, because the 1P-H&S model includes more general torsions (Rovere, 27 May 2025).

The same analysis states that the linearized 1P-H&S theory contains both the symmetric and antisymmetric parts of the vielbein perturbation, whereas the covariant fracton theory arises by restricting to longitudinal diffeomorphisms and to configurations for which the antisymmetric part is cohomologically trivial, i.e. an exact form. The variation with respect to the antisymmetric part of the linearized vielbein yields constraints that are automatically satisfied by the structure of the fracton field strength (Rovere, 27 May 2025).

5. Horizon diagnostics and black-hole sector in NGR

In NGR, matter couples minimally to the metric. As a result, test particles follow geodesics and null congruence expansions can be used to detect local horizons. In static, spherically symmetric coordinates, the apparent or local horizon is defined by the vanishing of the expansion scalar of outgoing null geodesics, $e=\det(e_\mu{}^a)$5; for the tetrad ansatz used in the black-hole analysis, this becomes

$e=\det(e_\mu{}^a)$6

in terms of tetrad functions $e=\det(e_\mu{}^a)$7 and $e=\det(e_\mu{}^a)$8 (López et al., 27 Aug 2025).

The teleparallel interpretation of a black hole requires that the torsion invariants appearing in the action remain finite at the horizon. Otherwise, the horizon is a genuine singularity of the teleparallel geometry rather than a coordinate artifact. For the 1P-H&S model, the paper finds that at the local horizon the vector and tensorial torsion scalars diverge, and so does the torsion scalar: $e=\det(e_\mu{}^a)$9 as $(1,1,1)$0, where $(1,1,1)$1 measures distance to the horizon. The axial invariant $(1,1,1)$2 is regular or zero at the horizon, but this does not prevent the divergence of the other invariants (López et al., 27 Aug 2025).

The same work states that this singular behavior is unavoidable in the physically viable NGR models, including TEGR and 1P-H&S. In the 1P-H&S case, the conclusion is that the theory is unable to describe a well-defined black hole configuration in teleparallel geometric terms. The manifold is not extendable across the horizon within teleparallel geometry, so the horizon is treated as a true singularity rather than a removable coordinate singularity (López et al., 27 Aug 2025).

6. Theoretical position and implications

The 1P-H&S model occupies a distinctive place within teleparallel gravity because it can be described both as a restricted quadratic-torsion sector of the Møller-Hayashi-Shirafuji theory and as a one-parameter NGR subfamily. In one direction, it provides a geometric realization of covariant fracton gauge theory: the linearized restricted Weitzenböck torsion matches the fracton field strength, and the fracton equations of motion coincide with the corresponding projection of the linearized teleparallel equations (Rovere, 27 May 2025). In the other direction, it remains among the physically viable NGR models discussed in the black-hole analysis, alongside TEGR, yet inherits the same obstruction at the local horizon through divergences of $(1,1,1)$3, $(1,1,1)$4, and $(1,1,1)$5 (López et al., 27 Aug 2025).

This suggests a sharp separation between two uses of the model. In the restricted linearized sector, 1P-H&S supports a precise embedding of a symmetric rank-2 gauge theory with scalar double-derivative gauge symmetry. In the static spherically symmetric horizon problem, the same teleparallel structure produces singular torsion invariants even when the metric sector is locally equivalent to TEGR. A plausible implication is that the model is especially informative where torsion, rather than curvature alone, is taken as the primary geometric object.

The papers therefore frame 1P-H&S not merely as a one-parameter deformation of TEGR, but as a test case for two broader questions: how teleparallel torsion sectors encode higher-rank gauge structures, and how far teleparallel geometry can reproduce standard gravitational intuitions about horizons and black holes without introducing genuine torsion singularities (Rovere, 27 May 2025, López et al., 27 Aug 2025).