Connections with Torsion and Nonmetricity

A metric and an affine connection are independent pieces of geometric data. Curvature measures the failure of parallel transport around a loop, torsion is the antisymmetric part of the connection, and nonmetricity measures the failure of the connection to preserve the metric. Keeping all three separate is the cleanest way to decide whether an unfamiliar formulation changes gravitational dynamics or merely redistributes them among geometric variables.

Required background. Frames, Tetrads, and Spinors supplies the frame–coordinate conversion used for affine connections; Differential Forms and Integration supplies connection and curvature forms; and Gravitational Actions and Boundary Terms supplies the variational role of total divergences.

Helpful background. Einstein Equations provides the Levi-Civita baseline used in every limit, while Formulations of General Relativity helps distinguish a change of variables from a change of theory.

Curvature, torsion, and nonmetricity

Let the connection act on a vector as

$$\nabla_\mu V^\rho =\partial_\mu V^\rho+\Gamma^\rho{}_{\mu\sigma}V^\sigma.$$

The index order is consequential: the first lower index is the derivative index. Brackets and parentheses include weight $1/2$, for example $A_{[\mu\nu]}=(A_{\mu\nu}-A_{\nu\mu})/2$. In this chapter the three characteristic tensors are

$$\begin{aligned} R^\rho{}_{\sigma\mu\nu}(\Gamma) &=\partial_\mu\Gamma^\rho{}_{\nu\sigma} -\partial_\nu\Gamma^\rho{}_{\mu\sigma} +\Gamma^\rho{}_{\mu\lambda}\Gamma^\lambda{}_{\nu\sigma} -\Gamma^\rho{}_{\nu\lambda}\Gamma^\lambda{}_{\mu\sigma},\\ T^\rho{}_{\mu\nu} &=2\Gamma^\rho{}_{[\mu\nu]},\\ Q_{\alpha\mu\nu} &=-\nabla_\alpha g_{\mu\nu}. \end{aligned}$$

Thus $T^\rho{}_{\mu\nu}$ is antisymmetric in $\mu,\nu$, whereas $Q_{\alpha\mu\nu}$ is symmetric in its last two indices. The minus sign in the definition of $Q$ is a chapter-wide convention. Some sources use $+\nabla g$; their disformation, traces, scalar, boundary term, and action must then be translated together. These definitions and the corresponding metric-affine subclasses are developed explicitly in Bahamonde et al. 2023, §2.1.2–§2.1.5, pp. 9–12, PDF.

Curvature, torsion, and nonmetricity answer independent questions. A flat connection can still have torsion or nonmetricity. A metric-compatible connection has $Q=0$ but can have curvature and torsion. Imposing $T=Q=0$ selects the unique Levi-Civita connection of $g$; it does not follow merely from choosing coordinates at one point.

The unique distortion decomposition

Subtract the Levi-Civita connection $\mathring\Gamma$ and call the difference the distortion,

$$N^\rho{}_{\mu\nu} \equiv\Gamma^\rho{}_{\mu\nu}-\mathring\Gamma^\rho{}_{\mu\nu}.$$

Unlike either connection separately, $N^\rho{}_{\mu\nu}$ is a tensor. Its antisymmetric part in the two lower indices is fixed by torsion. Nonmetricity instead fixes $N_{\nu\alpha\mu}+N_{\mu\alpha\nu}$, with the first index lowered. These are different index symmetries: torsion also contributes to the lower-index symmetric part of the full distortion. Solving the two relations together gives the unique split

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

where the contorsion and disformation are

$$\begin{aligned} K^\rho{}_{\mu\nu} &=\frac12\left( T^\rho{}_{\mu\nu} +T_\mu{}^\rho{}_{\nu} +T_\nu{}^\rho{}_{\mu} \right),\\ L^\rho{}_{\mu\nu} &=\frac12\left( Q_\mu{}^\rho{}_{\nu} +Q_\nu{}^\rho{}_{\mu} -Q^\rho{}_{\mu\nu} \right). \end{aligned}$$

A quick derivation is clearest with the first index lowered. Metric compatibility of $K$ implies $K_{\rho\mu\nu}=-K_{\nu\mu\rho}$, and

$$T_{\rho\mu\nu}=K_{\rho\mu\nu}-K_{\rho\nu\mu}.$$

Cyclically permuting the indices and solving gives

$$K_{\rho\mu\nu} =\frac12\left( T_{\rho\mu\nu}-T_{\mu\nu\rho}+T_{\nu\rho\mu} \right).$$

Likewise, torsion freedom makes $L^\rho{}_{\mu\nu}$ symmetric in $\mu,\nu$, while $Q_{\alpha\mu\nu}=L_{\nu\alpha\mu}+L_{\mu\alpha\nu}$. Three permutations yield the displayed formula for $L$. This algebraic reconstruction, including its dimension-independent form, is also given in Iosifidis 2019, ch. 1, §§1.1–1.2, pp. 14–55, PDF.

Curvature as a distortion identity

Substituting $\Gamma=\mathring\Gamma+N$ into the curvature definition and grouping tensorial terms gives

$$R^\rho{}_{\sigma\mu\nu}(\Gamma) =\mathring R^\rho{}_{\sigma\mu\nu} +2\mathring\nabla_{[\mu}N^\rho{}_{\nu]\sigma} +2N^\rho{}_{[\mu|\lambda|}N^\lambda{}_{\nu]\sigma}.$$

This is the master identity behind the chapter. It is exact: no field equation, weak-field expansion, or choice of dimension has been used. Contractions relate the Ricci tensors and scalars, but the ordering matters because a general Ricci tensor need not be symmetric.

Three restrictions illuminate its content:

• $K=L=0$ gives ordinary pseudo-Riemannian geometry.
• $R(\Gamma)=0$ and $Q=0$ give metric teleparallel geometry; the vanishing left side converts $\mathring R$ into a torsion-quadratic term plus a divergence.
• $R(\Gamma)=0$ and $T=0$ give symmetric teleparallel geometry; now $\mathring R$ becomes a nonmetricity-quadratic term plus a divergence.

Those last two identities support formulations equivalent to general relativity only for their specially tuned quadratic scalars. Arbitrary functions of torsion or nonmetricity are new theories, not automatic reformulations.

Independent checks and interpretation

The decomposition passes several checks that are useful for catching sign errors.

First, take Weyl-type nonmetricity $Q_{\alpha\mu\nu}=2A_\alpha g_{\mu\nu}$ and zero torsion. Direct substitution gives

$$L^\rho{}_{\mu\nu} =\delta^\rho_\nu A_\mu +\delta^\rho_\mu A_\nu -g_{\mu\nu}A^\rho,$$

which is symmetric in $\mu,\nu$ and reproduces $-\nabla_\alpha g_{\mu\nu}=2A_\alpha g_{\mu\nu}$. Second, setting $A_\mu=0$ and $T=0$ makes both distortion pieces vanish. Third, $\Gamma$, $K$, and $L$ all have inverse-length dimension, so every term in the curvature identity has inverse-length-squared dimension.

The connection used by matter is part of the physical theory. Spinors require a spin connection, and matter with dilation or shear currents can couple to other affine components. Empirical likelihoods and numerical bounds are studied in Tests of Gravity, while proposed microscopic quantum completions require additional assumptions; this page establishes the classical geometry needed to pursue either question.

Common pitfalls

A vanishing connection is not a tensorial statement. If every coordinate coefficient vanishes at a point, torsion vanishes there, but curvature can remain because derivatives of the connection need not vanish. Nonmetricity can remain through metric derivatives. In a noncoordinate frame, the frame’s anholonomy also enters torsion, so vanishing spin-connection coefficients alone do not imply vanishing torsion. A connection vanishing throughout a coordinate patch, in contrast, is both flat and torsion-free there.

Nonmetricity is not determined by the connection alone. It compares $\Gamma$ with a specified metric. Changing the physical matter metric changes the interpretation even if the connection coefficients are held fixed.

Exercises

Under the projective shift $\Gamma^\rho{}_{\mu\nu}\mapsto\Gamma^\rho{}_{\mu\nu}+\delta^\rho_\nu A_\mu$, find the changes in torsion, nonmetricity, and curvature. Show that the Ricci scalar is invariant.

Solution

Substitution gives $\delta T^\rho{}_{\mu\nu}=\delta^\rho_\nu A_\mu-\delta^\rho_\mu A_\nu$ and $\delta Q_{\alpha\mu\nu}=2A_\alpha g_{\mu\nu}$. The quadratic connection terms cancel against the covariantized derivatives, leaving $\delta R^\rho{}_{\sigma\mu\nu}=2\delta^\rho_\sigma\partial_{[\mu}A_{\nu]}$. Hence $\delta R_{\sigma\nu}=2\partial_{[\sigma}A_{\nu]}$, which is antisymmetric; contraction with the symmetric metric gives $\delta R=0$. This check also identifies the undetermined projective vector in the Einstein–Hilbert metric-affine connection equation.

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