Dynamical Torsion & Nonmetricity Tensors

• Dynamical torsion and nonmetricity tensors are non-Riemannian fields that define the antisymmetric and non-metric properties of the affine connection in metric-affine gravity.
• Their quadratic and higher-order action terms lead to diverse propagating degrees of freedom, offering modified gravitational dynamics and novel black hole solutions.
• The interplay of these tensors establishes dualities and extended symmetries that inform both cosmological models and the behavior of high-spin fields.

Dynamical torsion and nonmetricity tensors are the fundamental non-Riemannian geometric fields in metric-affine gravity (MAG), where the affine connection $\Gamma^\lambda{}_{\mu\nu}$ is independent of the metric $g_{\mu\nu}$. Torsion, $T^\lambda{}_{\mu\nu}=2\Gamma^\lambda{}_{[\mu\nu]}$, encodes the antisymmetric part of the connection, while nonmetricity, $Q_{\lambda\mu\nu}=-\nabla_\lambda g_{\mu\nu}$, quantifies the failure of the connection to be metric-compatible. Theories incorporating these tensors exhibit a rich landscape of propagating degrees of freedom and symmetries, leading to novel gravitational dynamics, exact solutions, and potentially distinctive phenomenology beyond Riemannian general relativity.

1. Geometrical and Algebraic Structure

The affine connection in MAG is decomposable as

$$\Gamma^\lambda{}_{\mu\nu} = \{^\lambda{}_{\mu\nu}\} + K^\lambda{}_{\mu\nu} + L^\lambda{}_{\mu\nu},$$

where $\{^\lambda{}_{\mu\nu}\}$ is the Levi–Civita connection of $g$, $K$ is the contortion built from the torsion tensor, and $L$ is the disformation constructed from nonmetricity. These objects, especially $T^\lambda{}_{\mu\nu}$ and $g_{\mu\nu}$0, can be irreducibly decomposed under the Lorentz group (Bahamonde et al., 2021, Bahamonde et al., 2022, Aoki et al., 2023, Karahan et al., 2011):

• Torsion:

$g_{\mu\nu}$1

where $g_{\mu\nu}$2 (trace vector), $g_{\mu\nu}$3 (axial vector), and $g_{\mu\nu}$4 (pure tensor) are irreducible.

• Nonmetricity:

$g_{\mu\nu}$5

with $g_{\mu\nu}$6 (Weyl vector), $g_{\mu\nu}$7 (vector), $g_{\mu\nu}$8 (pseudo-trace), and $g_{\mu\nu}$9 (totally traceless).

The trace of torsion and the Weyl part of nonmetricity exhibit a duality: for specific actions, pure-trace torsion can be reinterpreted as a Weyl vector and vice versa (Klemm et al., 2018).

2. Dynamical Equations and Propagation

The dynamical content of torsion and nonmetricity depends on the theory’s action. In quadratic and higher-order MAG models, both tensors can propagate as genuine dynamical fields. For example, in parity-preserving quadratic MAG, the action schematically involves

$T^\lambda{}_{\mu\nu}=2\Gamma^\lambda{}_{[\mu\nu]}$0

where combinations of curvature and torsion/nonmetricity invariants provide kinetic terms. The field equations derived from metric and connection variations yield coupled systems for $T^\lambda{}_{\mu\nu}=2\Gamma^\lambda{}_{[\mu\nu]}$1, $T^\lambda{}_{\mu\nu}=2\Gamma^\lambda{}_{[\mu\nu]}$2, and $T^\lambda{}_{\mu\nu}=2\Gamma^\lambda{}_{[\mu\nu]}$3; in particular, the symmetrized part of the connection equation typically gives Maxwell-type equations for the Weyl vector or shear nonmetricity, e.g.,

$T^\lambda{}_{\mu\nu}=2\Gamma^\lambda{}_{[\mu\nu]}$4

for the Weyl vector in Weyl–Cartan geometry (Bahamonde et al., 2021, Bahamonde et al., 2022).

The irreducible components of $T^\lambda{}_{\mu\nu}=2\Gamma^\lambda{}_{[\mu\nu]}$5 and $T^\lambda{}_{\mu\nu}=2\Gamma^\lambda{}_{[\mu\nu]}$6 possess different kinetic and mass-like terms, leading to various massless and massive excitations (Proca-type for vectors, Fierz–Pauli–like for tensors) (Bahamonde et al., 2024).

In $T^\lambda{}_{\mu\nu}=2\Gamma^\lambda{}_{[\mu\nu]}$7-type theories without explicit kinetic terms for $T^\lambda{}_{\mu\nu}=2\Gamma^\lambda{}_{[\mu\nu]}$8 and $T^\lambda{}_{\mu\nu}=2\Gamma^\lambda{}_{[\mu\nu]}$9, these fields are auxiliary—algebraically tied to derivatives of $Q_{\lambda\mu\nu}=-\nabla_\lambda g_{\mu\nu}$0 and do not propagate (0904.2774).

3. Exact Solutions and Black Holes

MAG supports exact solutions generalizing the classical black holes of general relativity by including independent spin (axial torsion), dilation (Weyl vector), and shear (traceless nonmetricity) charges:

• Kerr–Newman(-de Sitter) family: For scalar-flat Weyl–Cartan models, axisymmetric solutions with independent dynamical torsion and Weyl charges can be constructed. Both torsion (as the skewon two-form) and nonmetricity satisfy Maxwell-type field equations and produce Coulomb-like fields (Bahamonde et al., 2021).
• Reissner–Nordström-like solutions: The most general static, spherically symmetric black holes in quadratic and cubic MAG carry three conserved hypermomentum charges: spin, dilation, and shear. The metric function generically takes the form

$Q_{\lambda\mu\nu}=-\nabla_\lambda g_{\mu\nu}$1

where $Q_{\lambda\mu\nu}=-\nabla_\lambda g_{\mu\nu}$2 are the spin, dilation, and shear charges respectively (Bahamonde et al., 2022, Bahamonde et al., 2024, Bahamonde et al., 2020).

• Plebański–Demiański solutions: The full type D sector with mass, angular momentum, acceleration, NUT parameter, cosmological constant, electromagnetic, spin and dilation charges can be embedded in MAG. The propagation conditions require kinetic couplings for torsion and Weyl vector, and regularity places further constraints (Bahamonde et al., 2022).

Wave-like solutions (pp-waves) reveal that torsion and nonmetricity dynamically source scalar (helicity-0) gravitational wave polarization modes, a distinctive feature absent in GR (Bahamonde et al., 5 Nov 2025).

4. Symmetries and Gauge Structure

A significant feature of dynamical torsion and nonmetricity is their compatibility with extended internal and conformal symmetries:

• Projective invariance: Invariance under $Q_{\lambda\mu\nu}=-\nabla_\lambda g_{\mu\nu}$3 mixes torsion and nonmetricity, yielding a projectively invariant combination $Q_{\lambda\mu\nu}=-\nabla_\lambda g_{\mu\nu}$4 (antisymmetric nonmetricity minus torsion) which characterizes the physical degrees of freedom (Wheeler, 2023, Wheeler, 2024).
• Internal symmetries: Enlarged internal symmetries (e.g., SO(11,9)) associated with the space of 2-form-valued vectors $Q_{\lambda\mu\nu}=-\nabla_\lambda g_{\mu\nu}$5 are present, providing a structure sufficient to encode the Standard Model’s gauge group and ensuring decoupling from gravity in line with the Coleman–Mandula theorem (Wheeler, 2023).
• Conformal completion: The system integrates naturally into a conformal gauge algebra, with the mixed-symmetry part of nonmetricity identified with special conformal field strengths; conformal invariance can be explicitly realized and dynamically significant (Wheeler, 2024, Paci et al., 2023).

In special cases, the trace part of torsion and nonmetricity may be reinterpreted as one another via Weyl transformations, further linking these symmetries (Klemm et al., 2018).

5. Cosmological Dynamics and Perturbations

In cosmology, dynamical torsion and nonmetricity are sourced by spin, dilation, and shear components of the matter hypermomentum. The covariant model of cosmological hyperfluids yields the most general homogeneous and isotropic forms for these tensors in Friedmann-Lemaître-Robertson-Walker (FLRW) backgrounds (Iosifidis, 2020):

• Modified Friedmann equations acquire extra terms depending explicitly on the dynamical torsion and nonmetricity components, e.g.,

$Q_{\lambda\mu\nu}=-\nabla_\lambda g_{\mu\nu}$6

for pure torsion, and

$Q_{\lambda\mu\nu}=-\nabla_\lambda g_{\mu\nu}$7

for pure Weyl nonmetricity, where $Q_{\lambda\mu\nu}=-\nabla_\lambda g_{\mu\nu}$8 is the Hubble rate and $Q_{\lambda\mu\nu}=-\nabla_\lambda g_{\mu\nu}$9, $$\Gamma^\lambda{}_{\mu\nu} = \{^\lambda{}_{\mu\nu}\} + K^\lambda{}_{\mu\nu} + L^\lambda{}_{\mu\nu},$$0 parameterize torsion and nonmetricity.

• Non-Riemannian effects can induce or impede cosmic acceleration, leading to new phenomenology in early- and late-time dynamics.
• The spin-3 (helicity-3) sector of nonmetricity propagates in cosmological perturbation theory as transverse-traceless rank-3 modes and can source new gravitational wave phenomena (Aoki et al., 2023).

Dualities exist: torsion and nonmetricity can sometimes be exchanged (e.g., under $$\Gamma^\lambda{}_{\mu\nu} = \{^\lambda{}_{\mu\nu}\} + K^\lambda{}_{\mu\nu} + L^\lambda{}_{\mu\nu},$$1).

6. Physical Implications and Model Building

The variety of coupling terms and algebraic structures in MAG enables the construction of gravitational models interpolating between and generalizing Einstein–Cartan, Weyl, teleparallel, and symmetric teleparallel geometries (Karahan et al., 2011). Dynamical torsion and nonmetricity tensors, depending on the choice of couplings, can behave as massive vector/tensor fields, inducing modifications to gravitational dynamics and wave propagation. In models with higher-order curvature invariants (cubic MAG), ghost and gradient instabilities can be eliminated, and massive tensor degrees of freedom for both torsion and nonmetricity can propagate stably (Bahamonde et al., 2024, Bahamonde et al., 5 Nov 2025).

The effective field dynamics admit TeVeS-like modified gravity, vector-inflation models, and gravitational aether frameworks by suitable restriction or rearrangement of the dynamical variables and couplings (Karahan et al., 2011). In addition, conformally covariant actions for mixed-symmetry tensors (hook-type, spin-3, etc.) are consistent with the quadratic and cubic MAG constructions and clarify the nature of higher-spin propagation in these backgrounds (Paci et al., 2023).

7. Constraints, Dualities, and Limiting Cases

• In Palatini-type $$\Gamma^\lambda{}_{\mu\nu} = \{^\lambda{}_{\mu\nu}\} + K^\lambda{}_{\mu\nu} + L^\lambda{}_{\mu\nu},$$2 models, unless the connection couples directly to matter, both torsion and nonmetricity are non-dynamical, appearing only as auxiliary fields fixed by algebraic relations involving derivatives of $$\Gamma^\lambda{}_{\mu\nu} = \{^\lambda{}_{\mu\nu}\} + K^\lambda{}_{\mu\nu} + L^\lambda{}_{\mu\nu},$$3—no kinetic terms nor propagation (0904.2774).
• For minimal quadratic models (e.g., Einstein–Cartan or Weyl in isolation), either torsion or nonmetricity can only carry a pure gauge mode or be eliminated.
• The “torsion trace $$\Gamma^\lambda{}_{\mu\nu} = \{^\lambda{}_{\mu\nu}\} + K^\lambda{}_{\mu\nu} + L^\lambda{}_{\mu\nu},$$4 Weyl vector” duality is concrete: models with only trace torsion or only Weyl nonmetricity are equivalent up to transformation of variables (Klemm et al., 2018).
• Full dynamical propagation (physical degrees of freedom) requires explicit kinetic terms for the desired irreducible components, realized in general by quadratic or higher-order invariants in curvature, torsion, and nonmetricity (Bahamonde et al., 2024, Bahamonde et al., 2022).

In summary, the dynamics of torsion and nonmetricity tensors in metric-affine and general post-Riemannian gravity frameworks provide an enlarged scope for gravitational phenomenology, new families of exact solutions, and a rich algebraic and geometric structure that tightly intertwines internal symmetries, conformal invariance, and the propagation of higher-spin fields. The precise physical content is highly model-dependent, governed by the tensor irreducible content selected as dynamical by the underlying action.