Torsion, Nonmetricity, and Metric–Affine Gravity

Gravity need not be organized around the Levi-Civita connection alone. Torsion and nonmetricity let the same Einstein dynamics be rewritten in teleparallel variables, while independent connections and nonlinear actions can introduce genuinely new interactions or modes. This chapter builds the common geometric language first, proves the two boundary-term equivalences with general relativity, and then separates those exact reformulations from modified theories whose covariance, constraints, matter couplings, and stability require new analysis.

Helpful background. Einstein Equations supplies the dynamics to be recovered; Gravitational Actions and Boundary Terms explains variational equivalence; Formulations of General Relativity distinguishes variables from observables; Frames, Tetrads, and Spinors supplies tetrads and spin connections; and Differential Forms and Integration supplies curvature and torsion forms.

Enter the geometric trinity

For a metric $g$ and an affine connection $\Gamma$, the basic decomposition is

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

where $\mathring\Gamma$ is Levi-Civita, $K$ is determined by torsion, and $L$ by nonmetricity. General relativity can be represented with Levi-Civita curvature, with a flat metric-compatible torsionful connection, or with a flat torsion-free nonmetric connection. The equivalences are action identities up to divergences, not claims that curvature, torsion, and nonmetricity are the same tensor. Their derivation and boundary qualifications are reviewed in Bahamonde et al. 2023, §2.3, pp. 17–19, Eqs. (2.45)–(2.53), PDF.

Formulation	Connection conditions	Gravitational carrier	Relation to Einstein dynamics
Metric general relativity	Levi-Civita	Curvature	Baseline Einstein–Hilbert theory.
TEGR	Flat and metric-compatible	Torsion scalar	Equivalent bulk equations after the matched boundary divergence.
STEGR	Flat and torsion-free	Nonmetricity scalar	Equivalent bulk equations after the matched boundary divergence.
Einstein–Cartan	Metric-compatible; torsion varied	Curvature and auxiliary torsion	Spin sources algebraic torsion and contact interactions.
Metric–affine models	Metric and connection varied independently	Action-dependent mixture	May be an auxiliary reformulation or a theory with new affine modes.

The durable question is therefore not “which geometry is real?” but “which variables, constraints, matter couplings, and boundary data define the theory, and which observable dynamics survive after gauge and auxiliary fields are removed?”

Routes through the chapter

For the shortest route to the geometric equivalences, study the connection decomposition, then TEGR, then symmetric teleparallel gravity. Track the torsion and nonmetricity convention packages as wholes; this route culminates in two independent flat-FLRW checks.

For matter with spin or microscopic affine structure, move from the decomposition to Einstein–Cartan theory and then to metric–affine matter currents. This route distinguishes stress–energy from hypermomentum and shows when the connection is algebraic rather than propagating.

For modified-gravity model building, prove the TEGR identity before reading modified teleparallel gravity, then compare its covariance question with the projective-symmetry question in metric–affine gravity. Covariance, degree-of-freedom count, stability, and observational viability are four separate gates.

Chapter contents

1. Connections with Torsion and Nonmetricity defines $T^\rho{}_{\mu\nu}$ and $Q_{\alpha\mu\nu}$, derives contorsion and disformation, and checks how a general connection changes curvature.
2. Einstein–Cartan Gravity and Spin varies an independent Lorentz connection, solves its algebraic torsion equation, and obtains the induced axial four-fermion interaction.
3. Teleparallel Equivalent of General Relativity proves the curvature–torsion boundary identity under explicit flatness, metricity, global, and variational hypotheses.
4. Modified Teleparallel Gravity compares pure-tetrad and covariant tetrad–spin-connection formulations, then separates frame covariance from the unresolved generic mode count.
5. Symmetric Teleparallel and Nonmetricity Gravity constructs coincident gauge, derives the curvature–nonmetricity identity, and checks $\mathcal Q=6H^2$ directly.
6. Metric–Affine Dynamics and Matter Couplings decomposes hypermomentum into spin, dilation, and shear and uses projective symmetry to test admissible couplings and affine propagation.

Recurring conventions and decisive checks

The site uses signature $(-,+,+,+)$,

$$T^\rho{}_{\mu\nu}=2\Gamma^\rho{}_{[\mu\nu]}, \qquad Q_{\alpha\mu\nu}=-\nabla_\alpha g_{\mu\nu},$$

and the derivative index is the middle lower index of $\Gamma^\rho{}_{\mu\sigma}$. Transposing a formula from a source that puts the derivative index last changes the visible form of the projective shift and its hypermomentum trace. Reversing the metric signature or nonmetricity definition requires translating the scalar, action, and boundary term together.

Two identities organize the chapter:

$$\mathring R=-\mathcal T+2\mathring\nabla_\mu T^\mu, \qquad \mathring R=-\mathcal Q+\mathring\nabla_\mu(Q^\mu-\widetilde Q^\mu).$$

For the site definitions, both a Cartesian flat-FLRW tetrad in TEGR and a Cartesian coincident-gauge metric in STEGR give the positive scalar value $6H^2$. That shared number is not enough to establish equivalence: the divergence, matter coupling, inertial connection, and boundary variation must also match.

The most useful adverse checks recur across pages. Does a supposedly tensorial conclusion survive a frame or coordinate change when the connection transforms too? Does a connection equation contain derivatives or remain algebraic? Does a missing perturbative mode reflect a true constraint or a rank-changing strong-coupling limit? Does projective invariance leave a matter trace condition that the chosen coupling violates?

Reformulation versus new dynamics

Three operations mark the boundary. First, replacing a scalar linearly while retaining its exact divergence can preserve the Einstein bulk equations; applying a nonlinear function generally cannot. Second, changing a matter coupling can distinguish formulations that were equivalent in pure gravity. Third, adding connection kinetic terms can turn auxiliary or projective pieces into particles, after which constraint counting, kinetic signs, characteristics, and cutoff are mandatory.

The Einstein–Hilbert Palatini example makes the distinction concrete. Its projective invariance leaves one connection vector undetermined and imposes a traceless connection Euler tensor. A compatible matter current must share that trace identity, or the theory must be changed explicitly. The calculation appears in Iosifidis 2019, ch. 2, §2.1, pp. 56–66, Eqs. (2.1)–(2.55), PDF.

Empirical likelihoods and numerical bounds are studied in Tests of Gravity. Low-energy quantum-gravity EFT is developed in Quantum Gravity; this chapter introduces quantum-field-theory concepts only as needed for the classical calculations. The geometries, actions, and consistency tests developed here provide a basis for both observational and quantum questions.

Review the chapter

Use these questions to test whether the formulations remain distinct in your reasoning:

• Why does setting a flat inertial spin connection to zero in one tetrad not authorize setting it to zero after every local Lorentz transformation?
• Which hypotheses turn a scalar identity with a divergence into equivalence of Euler–Lagrange equations?
• Why can spin source torsion in Einstein–Cartan theory without producing a propagating torsion wave?
• What does the projective trace of the metric–affine connection equation require of matter hypermomentum?

As a transfer problem, consider three modifications of an Einstein–Hilbert Palatini action: (A) no connection-dependent matter; (B) algebraic terms quadratic in torsion and nonmetricity plus a fermion spin current; and (C) a curvature-squared term that contains derivatives of the independent connection. Classify what can be concluded about the connection in each case.

Solution

In A, the connection equation fixes Levi-Civita geometry only up to the projective vector, which is gauge for the action; the metric equation is Einstein’s equation. In B, the quadratic algebraic operator can determine torsion and nonmetricity in terms of hypermomentum. Substitution produces local matter contact terms, but no connection wave follows unless a kinetic derivative was added. In C, curvature contains derivatives of $\Gamma$, so a connection sector can propagate. That observation alone does not establish health: one must count constraints on the chosen background, diagonalize kinetic terms, test the principal symbol, and state the cutoff. Thus A is an equivalent formulation, B can give new interactions without new gravitational propagation, and C is a candidate new dynamical theory requiring a full stability analysis.

References

• Bahamonde, S., et al., “Teleparallel Gravity: From Theory to Cosmology,” Reports on Progress in Physics 86 (2023) 026901. DOI. Open PDF
• Iosifidis, D., Metric-Affine Gravity and Cosmology: Aspects of Torsion and Non-Metricity in Gravity Theories, PhD thesis, Aristotle University of Thessaloniki (2019). Stable record. Open PDF