Torsion-Based Gravity Technical Primer

1. Mathematical Foundations: What is Torsion?

1.1 Geometric Definition

In differential geometry, torsion measures the failure of a connection to be symmetric. For a general affine connection $\nabla$ on a manifold M, the torsion tensor T is defined by:

$$T(X,Y) = \nabla_X Y - \nabla_Y X - [X,Y]$$

where X,Y are vector fields and [X,Y] is their Lie bracket. In component notation:

$$T^\alpha{}{\beta\gamma} = \Gamma^\alpha{}{\beta\gamma} - \Gamma^\alpha{}_{\gamma\beta}$$

where $\Gamma^\alpha{}{\beta\gamma}$ are the connection coefficients. The torsion tensor is antisymmetric in its lower indices: $$T^\alpha{}{\beta\gamma} = -T^\alpha{}_{\gamma\beta}$$.

Note: Throughout this primer, indices are raised and lowered using the metric tensor $g_{\mu\nu}$ and its inverse $g^{\mu\nu}$.

1.2 Geometric Interpretation

Torsion has a clear geometric meaning: it measures how infinitesimal parallelograms fail to close. When parallel transporting vectors around an infinitesimal loop, torsion causes the loop to "twist out" of the original tangent plane. Mathematically, if we parallel transport vector V along X by $\delta x$, then along Y by $\delta y$, the failure to return to the starting point is:

$$\text{Gap} = T(X,Y)\delta x \delta y + O(\delta^3)$$

This contrasts with curvature, which measures the holonomy (rotation) of vectors around closed loops while remaining in the same tangent space.

2. Connection Decomposition and the Fundamental Theorem

2.1 General Affine Connection

Any affine connection on a (pseudo-)Riemannian manifold can be uniquely decomposed as:

$$\Gamma^\alpha{}{\beta\gamma} = \{\begin{matrix}\alpha\\beta\gamma\end{matrix}\} + K^\alpha{}{\beta\gamma}$$

where $\{\begin{matrix}\alpha\\beta\gamma\end{matrix}\}$ are the Christoffel symbols (Levi-Civita connection) and $K^\alpha{}_{\beta\gamma}$ is the contortion tensor:

$$K^\alpha{}{\beta\gamma} = \frac{1}{2}(T\beta{}^\alpha{}\gamma + T\gamma{}^\alpha{}\beta - T^\alpha{}{\beta\gamma})$$

The contortion is antisymmetric in its last two indices: $$K^\alpha{}{\beta\gamma} = -K^\alpha{}{\gamma\beta}$$ (or equivalently, $$K_{\alpha\beta\gamma} = -K_{\alpha\gamma\beta}$$ when all indices are lowered).

2.2 Metric Compatibility

A connection $\nabla$ is metric-compatible if $\nabla g = 0$. This condition, combined with the torsion specification, uniquely determines the connection via:

$$\Gamma^\alpha{}{\beta\gamma} = \frac{1}{2}g^{\alpha\delta}\left(\partial\beta g_{\gamma\delta} + \partial_\gamma g_{\beta\delta} - \partial_\delta g_{\beta\gamma}\right) + \frac{1}{2}g^{\alpha\delta}\left(T_{\beta\gamma\delta} + T_{\gamma\beta\delta} - T_{\delta\beta\gamma}\right)$$

where $T_\beta = T^\delta{}_{\beta\delta}$ is the torsion trace. Setting $T = 0$ recovers the Levi-Civita connection used in standard GR.

3. Curvature in the Presence of Torsion

3.1 Riemann Tensor with Torsion

The Riemann curvature tensor for a general connection is:

$$R^\alpha{}{\beta\gamma\delta} = \partial\gamma\Gamma^\alpha{}{\beta\delta} - \partial\delta\Gamma^\alpha{}{\beta\gamma} + \Gamma^\alpha{}{\epsilon\gamma}\Gamma^\epsilon{}{\beta\delta} - \Gamma^\alpha{}{\epsilon\delta}\Gamma^\epsilon{}_{\beta\gamma}$$

When torsion is present, this can be decomposed as:

$$R^\alpha{}{\beta\gamma\delta} = \overset{\circ}{R}{}^\alpha{}{\beta\gamma\delta} + \overset{\circ}{\nabla}{}\gamma K^\alpha{}{\beta\delta} - \overset{\circ}{\nabla}{}\delta K^\alpha{}{\beta\gamma} + K^\alpha{}{\epsilon\gamma}K^\epsilon{}{\beta\delta} - K^\alpha{}{\epsilon\delta}K^\epsilon{}{\beta\gamma}$$

where $\overset{\circ}{R}$ is the Riemann tensor of the Levi-Civita connection and $\overset{\circ}{\nabla}$ is the covariant derivative with respect to that connection.

3.2 Modified Bianchi Identities

With torsion, the Bianchi identities become:

First Bianchi: $$R^\alpha{}{[\beta\gamma\delta]} + \nabla{[\beta}T^\alpha{}{\gamma]\delta} + T{[\beta\gamma|}{}^\epsilon T^\alpha{}_{\delta]\epsilon} = 0$$

Second Bianchi: $$\nabla_{[\epsilon}R^\alpha{}{\beta|\gamma\delta|]} + T{[\epsilon|\gamma|}{}^f R^\alpha{}_{\beta|\delta|]f} = 0$$

These reduce to the standard identities when $T = 0$.

4. Three Approaches to Torsion in Gravity

4.1 Einstein-Cartan Theory ($U_4$)

Geometry: Riemann-Cartan space with both curvature and torsion
Connection: Metric-compatible with torsion coupled to spin
Field equations:

$$G_{\mu\nu}(g) = \kappa(T_{\mu\nu} + U_{\mu\nu}(S))$$

$$T^\alpha{}{\beta\gamma} + \delta^\alpha\beta T_\gamma - \delta^\alpha_\gamma T_\beta = \kappa S^\alpha{}_{\beta\gamma}$$

where $U_{\mu\nu} \sim \kappa S^2$ is quadratic in the spin tensor $S^\alpha{}_{\beta\gamma}$. Key features:

  • Torsion is algebraically related to spin (no propagation)

  • Reduces to GR in vacuum or for spinless matter

  • Modifications significant only at nuclear densities

4.2 Teleparallel Gravity ($T_4$)

Geometry: Weitzenböck space with torsion but zero curvature
Connection: The teleparallel connection satisfies $R^\alpha{}{\beta\gamma\delta} \equiv 0$
Fundamental variables: Tetrad fields $e^\alpha\mu$ (vierbein)

The torsion tensor in terms of tetrads:

$$T^\alpha{}{\mu\nu} = e^\alpha_a\left(\partial\mu e^a_\nu - \partial_\nu e^a_\mu + \omega^a{}{b\mu}e^b\nu - \omega^a{}{b\nu}e^b\mu\right)$$

where $\omega^a{}_{b\mu}$ is the spin connection. In pure teleparallel gravity, we set $\omega = 0$ (Weitzenböck gauge).

Field equations:

$$\partial_\nu(eS_a{}^{\mu\nu}) - e\left(S_\beta{}^{\rho\mu} T^\beta{}{\rho a} - \frac{1}{4}e_a^\mu T\right) = \frac{\kappa}{2} ee_a^\rho T\rho^\mu$$

where $T = T^\alpha{}{\beta\gamma}S\alpha{}^{\beta\gamma}$ is the torsion scalar and $S_a^{\mu\nu}$ is the superpotential:

$$S_a^{\mu\nu} = \frac{1}{2}\left(K^{\mu\nu}{}a - g^{\mu\nu}T\beta{}^\beta{}_a + g^{\mu}{}a T\beta{}^{\beta\nu}\right)$$

4.3 Metric-Affine Gravity (MAG)

Geometry: General metric-affine space with curvature, torsion, and non-metricity Q
Connection: Most general linear connection
Field equations: Derived from action:

$$S = \int d^4x \sqrt{-g} \left[f(g,R,T,Q) + L_{\text{matter}}\right]$$

This framework includes GR, Einstein-Cartan, and teleparallel gravity as special cases. For example, $$f(R,T,Q) = R/(2\kappa) + \alpha T^2 + \beta Q_{\mu\nu\alpha}Q^{\mu\nu\alpha}$$.

5. Key Differences: Riemannian vs Torsion Approaches

5.1 Physical Interpretation

Riemannian (GR):

  • Gravity = curved spacetime

  • Free fall = geodesic motion in curved space

  • No absolute parallelism

  • Energy-momentum curves spacetime

Teleparallel:

  • Gravity = force field in flat spacetime

  • Free fall = force equation in flat space with torsion

  • Absolute parallelism exists (distant parallelism)

  • Energy-momentum generates torsion field

5.2 Mathematical Structure

Property Comparisons

Curvature:

  • Riemannian GR: $R \neq 0$

  • Teleparallel GR: $R = 0$

  • Einstein-Cartan: $R \neq 0$

Torsion:

  • Riemannian GR: $T = 0$

  • Teleparallel GR: $T \neq 0$

  • Einstein-Cartan: $T \neq 0$

Connection:

  • Riemannian GR: Levi-Civita

  • Teleparallel GR: Weitzenböck

  • Einstein-Cartan: Metric-compatible

Gauge group:

  • Riemannian GR: Diff(M)

  • Teleparallel GR: Global Lorentz*

  • Einstein-Cartan: Local Lorentz

Degrees of freedom:

  • Riemannian GR: 2

  • Teleparallel GR: 2

  • Einstein-Cartan: 2**

*In pure Weitzenböck gauge; with non-trivial spin connection, local Lorentz is preserved
**Torsion is algebraic in Einstein-Cartan, so no extra propagating degrees of freedom

5.3 Field Equations Structure

Standard GR: I$R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R = \kappa T_{\mu\nu}$$

  • 4th order in metric derivatives for vacuum

  • No well-defined gravitational energy-momentum

Teleparallel GR: Field equations as above

  • 2nd order in tetrad derivatives always

  • Well-defined gravitational energy-momentum tensor $t^{\mu\nu}$

Modified theories:

  • $f(R)$: 4th order, Ostrogradsky ghosts

  • $f(T)$: 2nd order, no ghosts but frame-dependent

5.4 Equivalence Principle

GR: Strong equivalence principle fundamental

  • Gravity eliminated locally by coordinate choice

  • Christoffel symbols vanish in local inertial frame

Teleparallel: Weak equivalence principle only

  • Gravity cannot be eliminated locally

  • Torsion tensor frame-independent but non-zero

5.5 Conservation Laws

GR: Pseudo-tensor for gravitational energy

  • Not covariant

  • No local conservation law

Teleparallel: True tensor for gravitational energy

  • Covariant under global Lorentz

  • Local conservation: $\partial_\nu(T^{\mu\nu} + t^{\mu\nu}) = 0$

6. Quantum and Cosmological Implications

6.1 Quantization

Riemannian approach:

  • Non-renormalizable due to dimensional coupling

  • Higher derivatives in quantum corrections

Torsion approach:

  • Better UV behavior in some formulations

  • Natural coupling to fermion spin

  • Ashtekar variables in LQG incorporate torsion

6.2 Cosmology

Standard cosmology: Based on FLRW metric

  • Singularities unavoidable classically

  • Dark energy as cosmological constant

Torsion cosmology:

  • Big Bounce possible in Einstein-Cartan

  • $f(T)$ provides dynamical dark energy

  • Modified continuity equations

6.3 Gravitational Waves

GR: Two polarizations (+ and ×)

  • Propagate at speed c

  • Quadrupole radiation

Teleparallel: Same observational predictions

  • Different theoretical framework

  • Energy carried by torsion waves

Recommended Reading List on Torsion-based Approaches

I. Foundational Papers

1. Sur une généralisation de la notion de courbure de Riemann et les espaces à torsion, Élie Cartan, Comptes Rendus de l'Académie des Sciences (Paris), Vol. 174, pp. 593-595 (1922), Available through academic libraries, Cartan's original introduction of torsion in geometry, establishing the mathematical foundation for non-Riemannian geometry.

2. General relativity with spin and torsion: Foundations and prospects, Friedrich W. Hehl, Paul von der Heyde, G. David Kerlick, James M. Nester, Reviews of Modern Physics, Vol. 48, pp. 393-416 (1976), https://doi.org/10.1103/RevModPhys.48.393, The definitive comprehensive review of Einstein-Cartan theory, establishing it as a viable gauge theory for the Poincaré group.

3. The physical structure of general relativity, Dennis W. Sciama, Reviews of Modern Physics, Vol. 36, pp. 463-469 (1964), https://doi.org/10.1103/RevModPhys.36.463, Sciama's revival of Einstein-Cartan theory establishing the modern physical interpretation of torsion as coupled to intrinsic spin.

4. Translation of Einstein's Attempt of a Unified Field Theory with Teleparallelism, Alexander Unzicker, Timothy Case, arXiv:physics/0503046 (2005), https://arxiv.org/abs/physics/0503046, The first complete English translation of Einstein's original teleparallel papers (1928-1930), making these foundational works accessible.

5. New general relativity, K. Hayashi, T. Shirafuji, Physical Review D, Vol. 19, pp. 3524-3553 (1979), https://doi.org/10.1103/PhysRevD.19.3524, Establishment of the teleparallel equivalent of general relativity (TEGR), showing exact equivalence between Einstein's GR and a torsion-based formulation.

II. Comprehensive Review Articles

6. Einstein-Cartan Theory, Andrzej Trautman, arXiv:gr-qc/0606062 (2006), https://arxiv.org/abs/gr-qc/0606062, An authoritative modern review of Einstein-Cartan theory by one of the leading experts, providing historical context and theoretical foundations.

7. Teleparallel Gravity: From Theory to Cosmology, Sebastian Bahamonde et al., Reports on Progress in Physics, arXiv:2106.13793 (2021, updated 2023), https://arxiv.org/abs/2106.13793, The most comprehensive modern review covering teleparallel gravity from theoretical foundations to cosmological applications, including machine learning approaches.

8. The teleparallel equivalent of general relativity, José W. Maluf, Annalen der Physik, Vol. 525, pp. 339-357 (2013), arXiv:1303.3897, https://arxiv.org/abs/1303.3897, A pedagogical review emphasizing how general relativity can be formulated using tetrad fields and torsion tensors.

9. f(T) teleparallel gravity and cosmology, Yi-Fu Cai, Salvatore Capozziello, Mariafelicia De Laurentis, Emmanuel N. Saridakis, Reports on Progress in Physics, Vol. 79, 106901 (2016), arXiv:1511.07586, https://arxiv.org/abs/1511.07586, A comprehensive review of f(T) modified teleparallel gravity, covering torsional constructions and cosmological applications.

10. Teleparallel Theories of Gravity: Illuminating a Fully Invariant Approach, Martin Krššák, Rogier J. van den Hoogen, J.G. Pereira, Christian G. Böhmer, Alan A. Coley, Classical and Quantum Gravity, Vol. 36, 183001 (2019), arXiv:1810.12932, https://arxiv.org/abs/1810.12932, A pedagogical review clarifying misconceptions about local Lorentz invariance and demonstrating fully invariant formulations.

III. Gauge Theory Formulations

11. Metric-affine gauge theory of gravity: Field equations, Noether identities, world spinors, and breaking of dilation invariance, Friedrich W. Hehl, J. D. McCrea, Eckehard W. Mielke, Yuval Ne'eman, Physics Reports, Vol. 258, pp. 1-171 (1995), https://doi.org/10.1016/0370-1573(94)00111-F, Comprehensive treatment of metric-affine gravity theories, providing the broader geometric context for torsion-based theories.

12. Einstein Lagrangian as the Translational Yang-Mills Lagrangian, Y. M. Cho, Physical Review D, Vol. 14, pp. 2521-2525 (1976), https://doi.org/10.1103/PhysRevD.14.2521, First fiber bundle formulation of teleparallel gravity as a gauge theory for spacetime translations.

13. A pedagogical review of gravity as a gauge theory, Jason Bennett, arXiv:2104.02627 (2021), https://arxiv.org/abs/2104.02627, A pedagogical introduction to viewing gravity as gauge theory, covering both general relativity as Poincaré gauge theory and Newton-Cartan gravity.

14. Gauge Theories of Gravitation, Milutin Blagojević, Friedrich W. Hehl, arXiv:1210.3775 (2012, updated 2022), https://arxiv.org/abs/1210.3775, A comprehensive guide to gauge theories of gravitation encompassing both Einstein's theory and teleparallel gravity as subcases.

IV. Modified Teleparallel Theories

15. Extended Theories of Gravity, Salvatore Capozziello, Mariafelicia De Laurentis, Physics Reports, Vol. 509, pp. 167-321 (2011), arXiv:1108.6266, https://arxiv.org/abs/1108.6266, A broad review of extended gravity theories including torsional approaches, comparing f(R), f(T), and other modifications.

16. Torsion Gravity: a Reappraisal, H.I. Arcos, J.G. Pereira, International Journal of Modern Physics D, Vol. 13, pp. 2193-2240 (2004), arXiv:gr-qc/0501017, https://arxiv.org/abs/gr-qc/0501017, A critical review comparing different approaches to torsion in gravity, analyzing whether curvature and torsion represent independent degrees of freedom.

V. Recent Theoretical Developments

17. Geometry and Covariance of Symmetric Teleparallel Theories of Gravity, Alexey Golovnev, María José Guzmán, Konrad Kuijper, Alejandro Kuijken, arXiv:2306.09289 (2023, updated 2024), https://arxiv.org/abs/2306.09289, Establishes rigorous geometric foundations and covariant frameworks for symmetric teleparallel gravity.

18. L_∞-Algebras of Einstein-Cartan-Palatini Gravity, Marija Dimitrijević Ćirić, Grigorios Giotopoulos, Voja Radovanović, Richard J. Szabo, arXiv:2212.07289 (2022), https://arxiv.org/abs/2212.07289, Complete BV-BRST formulation using cyclic L_∞-algebra, establishing connections to topological gauge theories.

19. On the Viability of f(Q) Gravity Models, Avik De, Tee-How Loo, Classical and Quantum Gravity, Vol. 40, 115007 (2023), https://iopscience.iop.org/journal/0264-9381, Proves that energy conservation in f(Q) theory requires special conditions, showing most non-linear models violate conservation unless Q is constant.

VI. Cosmological Applications

20. Bouncing Cosmology in f(Q) Symmetric Teleparallel Gravity, Francesco Bajardi, Salvatore Capozziello, Luca Fabbri, arXiv:2011.01248 (2020), https://arxiv.org/abs/2011.01248, Uses order reduction method to find bouncing cosmological solutions and develops the corresponding quantum cosmological Wave Function.

21. Teleparallel Gravity and Quintessence: The Role of Nonminimal Boundary Couplings, S. A. Kadam, Bivudutta Mishra, Jackson Levi Said, arXiv:2408.03417 (2024), https://arxiv.org/abs/2408.03417, Develops autonomous dynamical systems for scalar-tensor gravity with non-minimal couplings to torsion and boundary terms.

22. Constraining the Modified Symmetric Teleparallel Gravity Using Cosmological Data, Shambel Sahlu, Alnadhief H.A. Alfedeel, arXiv:2412.20831 (2024), https://arxiv.org/abs/2412.20831, Uses extensive observational data to constrain f(Q) gravity models, showing promising results in addressing cosmological tensions.

23. Cosmological Bounces, Cyclic Universes, and Effective Cosmological Constant in Einstein-Cartan-Dirac-Maxwell Theory, Francisco Cabral, Francisco S. N. Lobo, Diego Rubiera-Garcia, arXiv:2003.07463 (2020), https://arxiv.org/abs/2003.07463, Demonstrates bouncing cosmologies and effective cosmological constant generation with matter-antimatter asymmetry.

VII. Connections to Dark Matter and Particle Physics

24. Einstein-Cartan Portal to Dark Matter, Mikhail Shaposhnikov, Andrei Shkerin, Inar Timiryasov, Sebastian Zell, arXiv:2008.11686 (2020), https://arxiv.org/abs/2008.11686, Demonstrates how torsion-induced four-fermion interactions provide a novel mechanism for dark matter production across a wide mass range.

25. Signature of Einstein-Cartan Theory, Bruno Arderucio Costa, Yuri Bonder, arXiv:2309.11536 (2023), https://arxiv.org/abs/2309.11536, Proposes experimental signatures for detecting spacetime torsion through spin polarization measurements in beam experiments.

VIII. Textbooks and Pedagogical Resources

26. Teleparallel Gravity: An Introduction, Ruben Aldrovandi, José Geraldo Pereira, Fundamental Theories of Physics, Vol. 173, Springer (2013), https://doi.org/10.1007/978-94-007-5143-9, The first comprehensive book dedicated exclusively to teleparallel gravity with pedagogical treatment of foundations.

27. Gravitation and Gauge Symmetries, Milutin Blagojević, Institute of Physics Publishing (2002), ISBN: 0-7503-0767-6, Systematic pedagogical treatment of gravity as spacetime gauge theory, covering Poincaré gauge theory and teleparallel gravity.

IX. Special Issues and Collections

28. Torsion-Gravity and Spinors in Fundamental Theoretical Physics, Luca Fabbri (Editor), Universe Special Issue (2022-2023), https://www.mdpi.com/books/reprint/7472-torsion-gravity-and-spinors-in-fundamental-theoretical-physics, Dedicated collection addressing torsion-gravity coupling with spinning matter across various applications.

29. Teleparallel Gravity: From Foundations to Observational Constraints, Sebastian Bahamonde, Jackson Levi Said (Editors), Universe Special Issue (2021), https://www.mdpi.com/journal/universe/special_issues/TeleGrav, Collection on theoretical foundations and observational constraints of teleparallel theories.

X. Historical Context

30. Teleparallel Gravity: An Overview, V.C. de Andrade, L.C.T. Guillen, J.G. Pereira, arXiv:gr-qc/0011087 (2000), https://arxiv.org/abs/gr-qc/0011087, Overview of teleparallel gravity fundamentals including field equations and energy-momentum definitions.