Einstein–Cartan Framework

• Einstein–Cartan framework is a gravitational theory extending GR by including spacetime torsion, which is algebraically linked to matter’s intrinsic spin density.
• It employs both metric-affine and tetrad-spin connection formalisms, leading to non-propagating torsion and effective four-fermion interaction terms in spinor dynamics.
• The theory predicts novel cosmological phenomena such as nonsingular bounces and spin-driven inflation, yet is tightly constrained by astrophysical and laboratory experiments.

The Einstein–Cartan (EC) framework is the standard extension of general relativity (GR) allowing spacetime with nonzero torsion, encoded in a metric-compatible but non-symmetric affine connection. In the EC theory, the torsion of spacetime is coupled algebraically to the intrinsic spin density of matter, leading to modifications of gravitational dynamics that become significant for high-spin densities, as encountered in the early universe, compact astrophysical objects, and certain quantum gravity scenarios. Structurally, the EC framework can be formulated either in the standard metric-affine formalism, or equivalently in the first-order (Palatini) tetrad and independent spin-connection formalism. The framework provides the unique theory of gravity with second-order field equations that is both diffeomorphism invariant and locally Lorentz invariant, with matter spin coupling realized consistently through the so-called Cartan equation. Below, the geometric structure, dynamical equations, quantum aspects, and leading cosmological and phenomenological consequences of EC gravity are surveyed in detail.

1. Geometric and Variational Structure

The Einstein–Cartan framework is defined on a four-dimensional Riemann–Cartan manifold with a (pseudo-)Riemannian metric $g_{\mu\nu}$ and an independent affine connection $\Gamma^\lambda_{\ \mu\nu}$, whose antisymmetric part is the torsion tensor: $$T^\lambda_{\ \mu\nu} = \Gamma^\lambda_{\ [\mu\nu]}.$$ Metric compatibility, $\nabla_\mu g_{\alpha\beta}=0$, remains imposed, but $\Gamma$ is not constrained to be symmetric. The connection can be decomposed as

$$\Gamma^\lambda_{\ \mu\nu} = \left\{^{\,\lambda}_{\mu\nu}\right\} + K^\lambda_{\ \mu\nu},$$

where $\left\{^{\,\lambda}_{\mu\nu}\right\}$ is the Levi–Civita connection, and $K^\lambda_{\ \mu\nu}$ is the contorsion tensor, algebraically related to torsion by

$$K^\lambda_{\ \mu\nu} = -T^\lambda_{\ \mu\nu} + T_\mu^{\ \lambda}{}_\nu + T_\nu^{\ \lambda}{}_\mu.$$

In the first-order formalism, the fundamental fields are an orthonormal coframe (tetrad) $e^a_\mu$, providing a soldering between spacetime and local Lorentz frames, and an independent Lorentz (spin) connection $\omega^{ab}_\mu=-\omega^{ba}_\mu$. The curvature $R^{ab}_{\ \ \mu\nu}(\omega)$ and torsion $T^a_{\ \mu\nu}$ are defined by usual structure equations.

The Einstein–Cartan gravitational action, including matter, is

$$S=\int d^4 x\, e\left[ \frac{1}{2\kappa} R(\Gamma) + \mathcal L_{\rm matter}(e, \omega, \Psi)\right],$$

where $e = \det(e^a_\mu)$, $\kappa=8\pi G$, and $R(\Gamma)$ is the Ricci scalar built from the full connection. In principal, additional invariants (e.g. Holst, Nieh–Yan, quadratic in torsion/curvature) can be included (Shaposhnikov et al., 2020, Shaposhnikov et al., 2020, Shah et al., 19 Nov 2025).

The defining feature is that variation with respect to $\Gamma$ (or $\omega$) yields algebraic (non-dynamical) equations for torsion in terms of the matter spin density tensor $S^{\lambda}{}_{\mu\nu}$: $$T^\lambda_{\ \mu\nu} + \delta^\lambda_\mu T^\sigma_{\ \nu\sigma} - \delta^\lambda_\nu T^\sigma_{\ \mu\sigma} = \kappa\, S_{\mu\nu}^{\ \ \lambda}$$ (Dereli et al., 2010, Khanapurkar, 2018, Ranjbar et al., 2024). Torsion is thus non-propagating and proportional to spin.

2. Field Equations, Spin–Torsion Coupling, and Four-Fermion Interactions

The EC field equations are obtained by independent variation of metric/tetrad and connection/spin-connection. The symmetric variation yields

$$G_{\mu\nu}(g) = \kappa\,T_{\mu\nu} + \kappa^2\,\Sigma_{\mu\nu}(S),$$

where $T_{\mu\nu}$ is the canonical energy–momentum tensor, and $\Sigma_{\mu\nu}$ contains terms quadratic in spin (Ranjbar et al., 2024). The algebraic Cartan equation implies that torsion is determined locally and instantaneously by the spin current, so it does not propagate.

For Dirac (or Weyssenhoff-type spin fluids) matter, the minimally coupled Dirac action involves covariant derivatives with the full spin connection. Algebraic elimination of torsion leads to an effective four-fermion current–current interaction. Explicitly, in the standard minimal coupling the resulting interaction is an axial–axial contact term,

$$\Delta S_{4\psi} = + \frac{\kappa^2}{8}\int d^4x\, e\,(\bar\psi \gamma^5\gamma_\mu\psi)^2$$

(Khanapurkar, 2018, Chishtie, 8 Sep 2025). The Dirac equation accordingly acquires a cubic self-interaction (the Hehl–Datta equation). For more general matter content, or in cosmological settings, the effective stress tensor and spin source in Friedmann equations or perturbation theory contain additional spin–torsion corrections (Medina et al., 2018, Palle, 2022, Razina et al., 2010).

In generalizations involving higher-derivative or quadratic curvature/torsion terms (e.g., quadratic Riemann–Cartan invariants), some components of torsion can become dynamical. With appropriate constraints, this can realize healthy propagating torsion (e.g., an effective Proca vector field from the trace vector of torsion) (Barker et al., 2023).

3. Renormalization, Quantum Structure, and Experimental Exclusion

Quantization of the first-order Einstein–Cartan action, making full use of its principal gauge symmetries (diffeomorphism × local Lorentz), can be systematically undertaken via the Batalin–Vilkovisky (BV) and background-field methods (Brandt et al., 2024). At one loop, the UV divergences are of the same structure as in the effective field theory of gravity: all divergences can be absorbed into the redefinition of a finite set of operators (including $\int e\,R$, possible $\int T^2$, and $\int \Lambda e$). Dynamical torsion mixes with the tetrad in loops but does not propagate at tree level, unless further curvature-squared invariants are admitted (Barker et al., 2023).

However, once fermions are included, the induced four-fermion contact term is dimension-6 and non-renormalizable; loop diagrams with two such vertices generate quartic ($\sim \Lambda^4$) divergences that cannot be absorbed into the original action (Chishtie, 8 Sep 2025). This limits the EC framework, minimally coupled to spinors, to an effective theory with breakdown at energies far below the Planck scale.

Empirically, classical spin–torsion couplings introduce universal-free-fall violations at the $10^{-12}$ level for Earth-laboratory or satellite experiments (e.g. polarised-matter torsion effects), three orders of magnitude above current MICROSCOPE bounds ($\delta a/a < 10^{-15}$) and at/larger than limits from torsion-balance and lunar laser ranging (Chishtie, 8 Sep 2025). This, combined with collider and astrophysical constraints on $(\bar\psi \gamma^5\gamma_\mu\psi)^2$ operators, effectively excludes standard EC gravity in its minimal form as a viable description of fundamental physics. Torsionless GR as realized in the USMEG–EFT is empirically viable (Chishtie, 8 Sep 2025).

4. Cosmological Consequences and Structure Formation

The Einstein–Cartan framework modifies cosmological dynamics in the presence of nonzero spin density:

• For homogeneous isotropic universes, the EC-modified Friedmann equations acquire additional spin–spin contributions which scale rapidly with the scale factor ($\sim a^{-6}$), acting as stiff-matter with repulsive effect (Medina et al., 2018, Cabral et al., 2020, Palle, 2022). This generically leads to nonsingular cosmological bounces, eliminating the big bang singularity—at high densities, torsion-induced four-fermion repulsion halts collapse and reverses the universe (Medina et al., 2018, Cabral et al., 2020).
• For early-universe inflationary scenarios, nonmininal couplings to the Higgs (and possible Holst/Nieh–Yan/Barbero–Immirzi terms) expand the phenomenological possibilities. The scalar spectral tilt $n_s$ takes the universal value $n_s = 1-2/N_*$, while the tensor-to-scalar ratio $r$ can vary continuously between $\sim 10^{-10}$ and 1 depending on the couplings, interpolating between metric and Palatini limits for Higgs inflation (Shaposhnikov et al., 2020, Shaposhnikov et al., 2020).
• In cosmic perturbation theory, nonzero background spin and torsion generate a gravitational slip ($\eta\equiv\Psi/\Phi\neq 1$) and small 'mass' corrections for gravitational waves. These modifications are suppressed at late times, restoring concordance cosmology, but introduce distinctive signatures at high redshift (Ranjbar et al., 2024).
• Generalizations with quadratic curvature/torsion invariants allow for richer dynamical behavior, including effective dark energy, cyclic cosmologies, and even regular traversable wormholes supported by exotic matter consistent with the EC field equations (Shah et al., 19 Nov 2025, Sarkar et al., 1 Sep 2025).

5. Gauge Symmetries, Gauge Fields, and Covariant Hamiltonian Structure

The full local gauge symmetry group of the Einstein–Cartan theory is $\Diff(M)\ltimes \mathrm{SO}(1,3)_\mathrm{loc}$ (spacetime diffeomorphisms and local Lorentz), as manifest in the covariant Hamiltonian (multisymplectic) formalism (Pilc, 2016). Gauge-invariant local observables, covariant Poisson brackets on the space of covariant observables, and closure of the complete constraint algebra without space-time splitting have been constructed explicitly.

Minimal coupling of Maxwell or Yang-Mills gauge fields to EC gravity results in minimal but gauge-noninvariant torsion–gauge couplings through the canonical spin density of the gauge field. Local gauge invariance is broken at the level of the Cartan equation, but gauge-invariant observables can be constructed locally in normal frames and in the $\delta$-function (momentum) approximation, showing that physical polarization and helicity remain well-defined at macroscopic scales (Socolovsky, 2012, Cabral et al., 2020).

6. Quantum, Emergent, and Nonstandard Extensions

The EC framework forms the foundation for a variety of quantum gravity, emergent gravity, and nonstandard mass-scale extensions.

• Strong-coupling, lattice, and Regge-calculus analyses reveal nontrivial order–disorder phase transitions at critical couplings, with nonperturbatively defined correlation lengths and possible continuum limits for quantum EC gravity (Xue, 2011).
• In the Diakonov model and its generalizations, the tetrad and Lorentz connection are expressed as composites built from primordial spinor condensates, yielding self-interacting nonlinear Heisenberg–Pauli–type spinor equations and emergent metric and connection structures (Obukhov et al., 2012).
• No-scale or scale-invariant generalizations produce metric theories with calculable six-dimensional operators after integrating out torsion, with phenomenological consequences for inflation and dark matter production (Shaposhnikov et al., 2020, Shaposhnikov et al., 2020).
• Quadratic curvature extensions (e.g., $R_{[\mu\nu]} R^{[\mu\nu]}$) allow for dynamical, healthy propagating torsion modes (e.g., a Proca vector field), with implications for the gravitational origin of vector fields and the production of gauge-singlet fermionic dark matter (Barker et al., 2023).

7. Summary and Outlook

The Einstein–Cartan framework is a uniquely determined geometric extension of general relativity incorporating torsion and local spin–matter couplings via algebraic field equations. Algebraic elimination of torsion implies that, in the minimally coupled theory, all new effects are suppressed by the Planck mass and manifest through contact four-fermion interactions and matter-induced corrections to gravitational dynamics. The framework produces rich cosmological structure—cosmological bounces, singularity avoidance, spin-driven inflation, and new tensor/scalar signatures—offers a technically robust gauge structure, and underlies a variety of extended models with robust quantum/classical duality.

However, the theory is experimentally constrained to parameter regions already nearly indistinguishable from torsionless GR (Chishtie, 8 Sep 2025), and, once quantized with matter, is non-renormalizable due to uncontrolled quartic divergences from the induced four-fermion terms. Within modern effective-field-theory approaches to gravity, the EC framework does not appear as a low-energy completion compatible with observed constraints, though it remains central in the study of alternative quantum gravity scenarios, early-universe dynamics, and possible emergent spacetime models (Brandt et al., 2024, Obukhov et al., 2012, Shaposhnikov et al., 2020).

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References:

(Khanapurkar, 2018, Dereli et al., 2010, Ranjbar et al., 2024, Razina et al., 2010, Medina et al., 2018, Xue, 2011, Shah et al., 19 Nov 2025, Socolovsky, 2012, Cabral et al., 2020, Pasmatsiou et al., 2016, Chishtie, 8 Sep 2025, Brandt et al., 2024, Pilc, 2016, Obukhov et al., 2012, Sarkar et al., 1 Sep 2025, Barker et al., 2023, Palle, 2022, Shaposhnikov et al., 2020, Shaposhnikov et al., 2020).