Square Torsion Theory

• Square torsion theory is a gravitational framework where the torsion tensor enters quadratically, introducing new dynamical degrees of freedom and effective dark stress–energy.
• It modifies Dirac field dynamics by yielding non-linear self-interactions through torsion-spin coupling, potentially observable at subatomic scales.
• The theory predicts anisotropic dark matter profiles, explains flat galactic rotation curves, and links to empirical relations like the baryonic Tully–Fisher law.

Square torsion theory refers to gravitational frameworks in which the spacetime torsion tensor enters quadratically into the action, supplementing or modifying the standard Einstein–Cartan or General Relativity (GR) treatments. By promoting the torsion field to an independent dynamical constituent with its own coupling, square-torsion models yield new degrees of freedom, effective stress–energy contributions, and non-trivial phenomenology for particle physics, astrophysics, and cosmology. Contemporary research has focused both on the construction of fundamental field equations and on applications to dark matter, baryonic rotation curves, cosmological fluids, and lattice field theory with defects.

1. Mathematical Structure and Action Principles

Square-torsion gravity augments the Einstein–Hilbert action by including a term quadratic in the irreducible torsion tensor $T^{I}{}_{JK}$: $$S = \frac{1}{16\pi G} \int d^4x\,\det(\theta)\left[\tfrac{1}{2} T^I{}_{JK} T_I{}^{JK} - R^{IJ}{}_{IJ}\right] + \int d^4x\,L_s$$ Here, $G$ is Newton's constant, $R^{IJ}{}_{IJ}$ the curvature scalar, and $L_s$ generically the matter Lagrangian. Under constraints of vanishing spin density, only one irreducible torsion component is retained. The quadratic torsion term introduces extra dynamical degrees of freedom; after variation, these appear as contributions to the gravitational field equations distinct from the usual Einstein tensor.

The field equations take the schematic form: $$\tfrac{1}{2} T^I{}_{ST} T^J{}^{ST} + G^{IJ} = 8\pi G P^{IJ}$$ where $P^{IJ}$ represents the matter stress–energy, and the torsion-square term can be recast as an effective “dark” stress–energy tensor $D^{IJ}$. Thus, the modified equations become: $G^{IJ} = 8\pi G \left( P_b^{IJ} + D^{IJ} \right)$ Constraints on the torsion tensor in spherical symmetry enforce particular forms for the "electric" and "magnetic" parts, encoding the allowed Lorentz and parity structures.

2. Coupling to Matter and Dirac Fields

Coupling square-torsion gravity to spinorial matter, notably Dirac fields, induces self-interaction terms in the Dirac equation through the torsion-spin coupling. The matter Lagrangian is: $$\mathcal{L}_M = \frac{i}{2} \left( \bar{\psi}\gamma^i D_i\psi - D_i\bar{\psi}\gamma^i\psi \right) - m \bar{\psi}\psi$$ where $D_i$ includes torsionary contributions. For totally antisymmetric spin tensors (characteristic of the Dirac field), the torsion can be eliminated algebraically, leading to non-linear potentials akin to the Nambu–Jona–Lasinio structure: $$i \gamma^\mu \tilde{D}_\mu \psi - \lambda(\bar{\psi}\Gamma\psi)\Gamma\psi - m\psi = 0$$ Here, the self-interaction coefficient $\lambda$ depends on the independent torsion coupling, commonly denoted $k$. By tuning $k$, the scale of nonlinearity may be rendered observable at subatomic rather than only Planckian energies. This mechanism offers novel perspectives for neutrino oscillation phenomena in the massless regime and analogies to superconductivity (e.g., emergent attractive contact interactions).

3. Structure Equations and Anisotropic Dark Stress–Energy

Square-torsion modifications generate an effective dark stress–energy tensor $D^{IJ}$, whose components satisfy an intrinsic anisotropic structure equation: $P - \rho = 4 P_\perp$ with $\rho$ the effective density, $P$ the radial pressure, and $P_\perp$ the tangential pressure. This equation arises from the quadratic form of torsion and the decomposition of its components under symmetry constraints. Its significance lies in modeling the distribution and stability of dark matter halos, as the anisotropy alters the effective potential and force balance in spherically symmetric structures.

4. Dark Coating Mechanism and Static Halo Solutions

A defining methodology in square-torsion gravity is the “dark coating” procedure (Editor's term), whereby any seed metric $g_0$ is rescaled conformally: $g = e^{2\sigma} g_0$ Here, the conformal factor $\sigma$ satisfies a wave equation, e.g. $\Box \sigma = 0$ in the static case. For spherically symmetric systems, this constructs static solutions that model halos: $$v^2(r) = \frac{G m(r)}{r} + \frac{r}{\mathcal{R}^*}$$

$\mathcal{R}^*$ is an integration constant related to the “dark coating”—it determines the strength of the deviation from the conventional Keplerian regime and dominates the velocity profile at large radii. The associated effective dark matter density is: $$\rho_{D}^{\text{Newton}} = \frac{1}{2\pi G \mathcal{R}^* r}$$

which matches the qualitative fall-off behavior of empirical profiles such as Navarro–Frenk–White.

Applied to central compact sources, the Schwarzschild metric can be dressed: $g = e^{2\sigma} g_\text{Schwarzschild}$ with $\sigma$ having a specific functional form resembling the tortoise coordinate. Notably, a geometric instability manifests in regions between the photon sphere and the event horizon—the “torsion sphere”—where the Komar dark mass density forces $\rho_D < 0$, signaling that static, dark-coated solutions are prohibited within this boundary.

5. Baryonic Tully–Fisher Relation and Galactic Rotation Curves

Square-torsion gravity provides a theoretical basis for the baryonic Tully–Fisher relation, the empirical law: $m_b \propto v_f^4$ where $m_b$ is the baryonic mass and $v_f$ the flat rotational velocity. Imposing the flat curve condition and using the Newtonian regime of the coated metric yields: $$G \mathcal{R}^* m_b = R_f^2 \implies m_b = \frac{\mathcal{R}^*}{4G} v_f^4$$ This connection is a notable achievement, suggesting that the integration constant $\mathcal{R}^*$—a universal parameter emerging from the torsion sector—governs the proportionality across galaxies. This feature is compatible with observed galactic dynamics and may explain the ubiquity of flat rotation curves and the scaling law itself.

6. Cosmological Implications: FLRW Fluids and Hubble Expansion

Beyond galactic structure, square-torsion theory modifies cosmological dynamics by incorporating spin fluids (e.g., Weyssenhoff fluids) whose averaged spin-squared contributions yield extra terms in the Friedmann equations: $$\ddot{a} = \left[-\frac{p+\rho}{2} + 4 C(k) S^2\right]\, a$$ Parameter $C(k) = (1-4k)(1+2k)(1-6k)$ encodes torsion coupling strength. The effective spin contributions may "open" universes otherwise closed in GR and avoid initial singularities by enforcing a minimum scale factor. The theory also accommodates Hubble-like expansion via the dark coating mechanism: $$\sigma = f(U) + g(V)\quad\text{with}\quad f' = -g' = H$$ leading to stress–energy tensor components exemplifying critical fluid behavior for linear expansion: $$\rho_H = \frac{3H^2}{8\pi G},\quad P_H = -\frac{H^2}{8\pi G}$$ The model further predicts a component of dark radiation, marked by tracelessness in the stress–energy tensor, potentially yielding observational signals distinct from standard Einsteinian gravitational waves.

7. Quantum and Lattice Realizations

Recent work (Imaki et al., 2019) demonstrates that torsion can be realized on a lattice via line defects (dislocations). A screw dislocation imparts local torsion, with explicit formulae: $T_{xy}^\tau(x) = -\frac{b}{2}\delta^{(2)}(x)$ Chiral fermions in such backgrounds experience torsion-induced vector and axial currents—a phenomenon designated the chiral torsional effect. This non-perturbative regularization not only mimics aspects of gravity in condensed matter systems but provides a platform for exploring higher order (quadratic) torsion terms and their observable consequences in quantum field theory.

Conclusion

Square torsion theory represents a substantive generalization of gravitational physics, wherein the torsion tensor is incorporated quadratically with independent coupling, yielding effective dark stress–energy, distinctive self-interactions for spinorial fields, and modified cosmological and galactic dynamics. The theory accommodates empirical facts such as the baryonic Tully–Fisher relation, explains flat rotation curves and halo stability, and introduces novel geometric entities (the torsion sphere) and radiation components. Quantum and condensed matter analogs offer further pathways for theoretical and observational exploration, positioning square-torsion frameworks as a flexible and potentially unifying approach within both astrophysics and high-energy theory.