Geometric Torsion in General Relativity

• Geometric torsion is the antisymmetric component of the affine connection that extends standard GR by incorporating spin through Riemann–Cartan geometry.
• It is implemented in theories like Einstein–Cartan and teleparallelism, where modified actions and quadratic torsion terms lead to new propagating degrees of freedom.
• Torsion influences physical phenomena by coupling to fermionic spin, contributing to topological invariants, and potentially impacting cosmological models such as early universe inflation.

Geometric torsion in general relativity (GR) refers to the antisymmetric component of the affine connection, extending the purely metric-based Riemannian geometry of standard GR to a broader framework where parallel transport and curvature incorporate the possibility of anholonomic translations. This generalization appears naturally in Riemann–Cartan manifolds and has significant implications for the coupling of gravity to spin, the structure of field equations, the nature of conserved currents, and the topological and holographic properties of gravitational dynamics.

1. Mathematical Foundations and Cartan’s Formulation

Cartan’s pioneering work introduced torsion as the translational part of the full Cartan connection, dualized via Grassmann (exterior) calculus, and expressed in terms of differential forms. For a manifold with frame 1-forms $\omega^i$ and Lorentz connection 1-forms $\omega^i_j$, Cartan’s structure equations are

$$\begin{align*} \text{Torsion 2-form:} && \Omega^i &= d\omega^i + \omega^i_j \wedge \omega^j \ \text{Curvature 2-form:} && \Omega^i_j &= d\omega^i_j + \omega^i_k \wedge \omega^k_j \end{align*}$$

Here, $\Omega^i$ vanishes for a Levi–Civita (torsionless) connection but is generally nonzero in the Riemann–Cartan geometry. This framework naturally unifies the geometric descriptions of elasticity (as in Cosserat media) and gravity, with torsion interpreted as translational curvature (failure of infinitesimal parallel transport to close).

Torsion enters the connection as

$$T^{\lambda}_{\phantom{\lambda}\mu\nu} = \Gamma^{\lambda}_{\phantom{\lambda}\mu\nu} - \Gamma^{\lambda}_{\phantom{\lambda}\nu\mu}$$

and in the gauge-theoretic context as the field strength corresponding to local translations.

2. Torsion in Theories of Gravity: Actions and Equations

Einstein–Cartan Theory

The canonical extension of GR including torsion is the Einstein–Cartan–Sciama–Kibble (ECSK) theory, with independent metric and connection. The gravitational action is a generalization of the Einstein–Hilbert term, built from the Ricci scalar constructed from a general (non-symmetric) connection: $$S = \int d^4x \sqrt{-g} \left[ -\frac{1}{\kappa^2} \widetilde{R} + \mathcal{L}_m \right]$$ where the tilde denotes calculation with general connection. Variation with respect to the connection yields the Cartan equation, relating torsion algebraically to the intrinsic spin $S^{\lambda\mu\nu}$ of matter: $T_{\lambda\mu\nu} \propto S_{\lambda\mu\nu}$ Torsion thus does not propagate in vacuum and vanishes outside spin sources.

Extensions with Dynamical Torsion

Dynamical torsion can be introduced through quadratic or higher-order invariants constructed from the torsion tensor (e.g., $Q_{\alpha\beta\gamma} Q^{\alpha\beta\gamma}, Q^{\lambda}_{\phantom{\lambda}\lambda\mu} Q^{\rho}_{\phantom{\rho}\rho}^\mu$ etc.), or nonminimal curvature–torsion couplings. This leads to modified field equations where torsion possesses its own equation of motion and can propagate in vacuum (Shabani et al., 2017).

New General Relativity and Teleparallelism

Alternative models such as “new general relativity” and the Teleparallel Equivalent of GR (TEGR) exploit the Weitzenböck connection (curvature-free, non-vanishing torsion) and formulate gravity with actions quadratic in the torsion tensor (Maluf, 2013, Blixt et al., 2019). The action generically takes the form: $$S = \int d^4x \, e\ [c_1\, T^{\rho}_{\phantom{\rho}\mu\nu} T_{\rho}^{\phantom{\rho}\mu\nu} + c_2\, T^{\rho}_{\phantom{\rho}\mu\nu} T^{\nu\mu}_{\phantom{\nu\mu}\rho} + c_3\, T^{\rho}_{\phantom{\rho}\mu\rho} T^{\sigma\mu}_{\phantom{\sigma\mu}\sigma}]$$ Tuning the $c_i$ coefficients recovers TEGR for a particular choice (equivalent to GR up to a boundary term) but generically leads to new propagating degrees of freedom, with ghost or strong coupling pathologies outside special parameter values (Jiménez et al., 2019).

3. Physical Consequences: Spin Coupling, Charges, and Cosmology

Spin–Torsion Coupling and Fermionic Matter

Spinors minimally couple to the full (torsionful) connection, and in ECSK theory, the axial-vector part of torsion is algebraically determined by the spin current $J^a_5 = \bar{\psi} \gamma^5 \gamma^a \psi$, leading effectively to contact four-fermion interactions of the type

$$\mathcal{L}_{4f} \sim G (\bar{\psi} \gamma^\mu \gamma^5 \psi)^2$$

These are extremely suppressed by the Planck scale under ordinary conditions but may play cosmological roles at very high density (Diakonov et al., 2011, Bonder, 2016).

Topological and Holographic Terms

Torsion offers new topological invariants, most notably:

• Holst term: $S_H \sim \alpha \int \epsilon^{abcd} R_{abcd}$, important in loop quantum gravity, which is trivial in the absence of torsion.
• Nieh–Yan term: A total derivative $\int \partial_a (\epsilon^{abcd} T_{bcd})$; only when spin-producing torsion is present can this contribute physically.

However, only the Nieh–Yan term is a true topological invariant in the presence of torsion; the Holst term affects the dynamics if matter with spin is present, modifying torsion–spin relations and introducing Barbero–Immirzi–parameter dependence (Banerjee, 2010). Notably, in a Weyssenhoff fluid sphere exterior to a black hole, the Nieh–Yan surface “torsion charge” vanishes as the horizon forms, indicating a loss of “torsional hair” at horizons.

Gauss Law, Torsion Charge and Dark Matter

In Einstein–Cartan theory, the “weak Gauss law” fails: the asymptotically measured gravitational mass can exceed the integrated interior mass due to torsion contributions, suggesting a possible geometric explanation for dark matter effects (Schucker et al., 2012).

Cosmology: Torsion and the Early Universe

Geometric torsion modifies cosmological dynamics significantly in certain frameworks. In AP geometry, the torsion scalar can mimic a cosmological constant or drive an inflationary epoch: $$\mathcal{T} = 3H,\qquad p_T = \frac{1}{24\pi} \mathcal{T}^2$$ This induces exponential expansion and evades singularities and particle horizon problems, providing an alternative to scalar-field-driven inflationary scenarios (Wanas et al., 2012).

4. Symmetries and Conserved Quantities

Symmetry Groups and Cartan Transformations

Allowing for a trace (scalar) torsion enables new local symmetries (“Cartan transformations”), mixing conformal rescaling of the metric and shifts of a scalar torsion field: $$g_{\mu\nu} \to e^{f(x)} g_{\mu\nu},\quad \phi \to \phi + f(x)$$ The resultant theory possesses invariance under Cartan gauge choices and retains all classical GR tests (i.e., orbits, redshifts, etc.), despite the presence of torsion (Fonseca-Neto et al., 2012).

Conservation Laws, Energy–Momentum, and Spin Tensors

In the presence of torsion, stress–energy–momentum (SEM) and spin tensors arise naturally from metric/tetrad and connection variations, respectively. The Belinfante–Rosenfeld symmetrization relates canonical (non-symmetric) and Hilbert (symmetric) SEM tensors, with conservation law

$\partial_\mu (T^{\mu\nu} + \text{spin terms}) = 0$

The canonical Hamiltonian analysis of (tele)parallel gravity demonstrates that energy–momentum and angular momentum satisfy the Poincaré algebra in phase space, reflecting the underlying spacetime symmetries (Maluf, 2013).

5. Observational and Theoretical Implications, Limitations, and Spectral Geometry Arguments

Experimental Status and Theoretical Sufficiency

Despite the formal and conceptual richness, all empirical tests of gravity uphold the Einstein equivalence principle and the predictions of standard (torsionless) GR. Generic theories with torsion can often be reformulated as GR plus additional matter fields, and no unequivocal experimental evidence for torsion exists. Many operational and conceptual difficulties—such as ambiguous normalization of charges, modified parallel transport, and multiple types of autoparallels—further challenge the necessity of torsion in a fundamental gravitational theory (Garecki, 2011, Bonder, 2016).

Spectral Geometry and Obstruction

Spectral geometry provides a powerful framework for characterizing geometric structures via the spectrum of Dirac–type operators. A recent result demonstrates that, in the presence of torsion, it is impossible to construct a spectral Einstein functional (using the Wodzicki residue and Dirac operator with torsion) that yields a well-defined tensor density; extra derivative terms associated with torsion’s algebraic structure cannot be cancelled. This result provides a structural justification for the exclusion of geometric torsion in physically acceptable gravity models (Bochniak et al., 27 Dec 2024).

6. Advanced Constructions: Biconformal and Schrödinger Geometries

Biconformal gauge theory “doubles” the structure of spacetime, introducing both a solder form $e^a$ and a co–solder form $f_a$. Imposing vanishing torsion $T^a = 0$ but not co–torsion allows recovery of locally scale-invariant general relativity; setting both to zero yields only trivial geometries. This links biconformal geometry to double field theory and offers a geometric mechanism for unification of gravitation with other interactions, depending on the properties of $f_a$ (Wheeler, 2018).

In non-relativistic limits (Newton–Cartan geometry), arbitrary spatial torsion can be engineered via gauging procedures or special reductions, but standard GR only reproduces torsionless (twistless) Newton–Cartan theory on shell (Bergshoeff et al., 2017).

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In conclusion, geometric torsion broadens the landscape of possible spacetime geometries and connects gravity intimately with spin, parity-violating effects, and topological invariants. While it leads to rich mathematical structures and has profound implications for the coupling of fundamental fields, cosmology, and conserved quantities, robust physical, observational, and spectral-analytic arguments currently favor the simplest, torsionless (Levi–Civita) formulation as the most satisfactory model for gravitational interactions. Nonetheless, the paper of geometric torsion continues to inform the search for extensions or alternatives to general relativity, with relevance for quantum gravity, cosmology, and exotic matter couplings.