Torsion Tensor

A connection is the geometric equipment that tells you how to compare vectors at neighbouring points — how to parallel-transport, and hence how to differentiate covariantly. In a coordinate basis it is a table of numbers $\Gamma^\lambda{}_{\mu\nu}$ , and those numbers are famously not the components of a tensor: under a change of coordinates they pick up an inhomogeneous second-derivative term no tensor would tolerate. Yet hidden inside this non-tensorial object is a genuine tensor. Antisymmetrize the connection in its two lower indices and the offending term — symmetric in exactly those indices — cancels without residue, leaving what is, doubled by convention, the torsion tensor $T^\lambda{}_{\mu\nu}$ . Torsion answers one sharp geometric question. Take two short arrows at a point, slide each along the other by parallel transport, and ask whether the two resulting corners coincide. In flat space they do: the little parallelogram closes. With a general connection it does not, and the leading-order gap is governed entirely by torsion — a defect of translation, logically independent of curvature, which measures instead the rotation a vector suffers around a closed loop. General relativity disposes of torsion at the outset. The fundamental theorem of Riemannian geometry says that requiring the metric to be covariantly constant and the connection symmetric fixes the connection uniquely — the Levi-Civita connection built from the Christoffel symbols. Relativity adopts it, so $T^\lambda{}_{\mu\nu}=0$ by construction. Understanding torsion is understanding precisely which assumption you relax beyond Einstein's geometry — in Einstein–Cartan gravity, teleparallelism, or Cartan's geometry of defects.

Key takeaways

The torsion tensor $T^\lambda{}_{\mu\nu}=\Gamma^\lambda{}_{\mu\nu}-\Gamma^\lambda{}_{\nu\mu}$ is the antisymmetric part of an affine connection; despite being assembled from non-tensorial Christoffel symbols it transforms as a genuine $(1,2)$ tensor.
Geometrically it is the failure of two infinitesimal parallel-transported vectors to close a parallelogram: the corner-to-corner gap is $\Delta x^\lambda=-T^\lambda{}_{\mu\nu}\,a^\mu b^\nu$ — a translational defect, not a rotational one.
It surfaces algebraically in the derivative commutator: on a scalar $[\nabla_\mu,\nabla_\nu]\phi=-T^\lambda{}_{\mu\nu}\nabla_\lambda\phi$ , and on a vector torsion joins curvature, $[\nabla_\mu,\nabla_\nu]V^\rho=R^\rho{}_{\sigma\mu\nu}V^\sigma-T^\lambda{}_{\mu\nu}\nabla_\lambda V^\rho$ .
Torsion and curvature are independent: the flat Weitzenböck connection has $R=0,\ T\neq0$ , while standard GR has $R\neq0,\ T=0$ .
Any metric-compatible connection splits as $\Gamma^\lambda{}_{\mu\nu}=\mathring\Gamma^\lambda{}_{\mu\nu}+K^\lambda{}_{\mu\nu}$ , contorsion $K$ built entirely from $T$ ; setting $T=0$ returns the unique Christoffel connection.
In Einstein–Cartan gravity torsion is sourced algebraically by intrinsic spin, $T\propto\kappa\,S$ with $\kappa=8\pi G/c^4$ ; it does not propagate, so it vanishes outside spinning sources and the theory reduces to GR.
Key idea: torsion is the tensorial antisymmetric part of any affine connection; it measures the non-closure of infinitesimal parallelograms and is logically independent of curvature. Standard GR uses the unique metric-compatible torsion-free Levi-Civita connection, so the geometry is determined by the metric alone.

The core insight

Torsion is the tensorial antisymmetric part of any affine connection; it measures the non-closure of infinitesimal parallelograms and is logically independent of curvature. Standard GR uses the unique metric-compatible torsion-free Levi-Civita connection, so the entire geometry is fixed by the metric.

Where this lives today

Torsion is not a museum piece; it is an active experimental target and a working theoretical tool. Because a background torsion would couple to spin, its signatures — anomalous spin precession, tiny atomic frequency shifts — have been hunted with exquisite precision. Kostelecký, Russell and Tasson showed in 2008 how to reinterpret decades of spin-polarized torsion-pendulum, atomic-clock and comagnetometer data as bounds on the components of a constant background torsion, constraining many to fantastically small values; no torsion has been seen, tightening the empirical case for Levi-Civita. On the theory side, the teleparallel reformulation — gravity on Weitzenböck's flat, torsionful connection — is a laboratory for modified gravity: $f(T)$ theories built on the torsion scalar are studied as explanations of cosmic acceleration. And far from gravitation, Cartan's identification of torsion with dislocation density underlies the geometric theory of defects, where the closure gap of Example 2 is measured as a Burgers vector on the bench — the same tensor GR discards, at work in the lab.

The physical picture

Picture two short arrows, $a$ and $b$ , rooted at a point $P$ . March along $a$ , then parallel-transport $b$ to your new position and walk along that transported copy; now start over and do it in the opposite order, along $b$ first, then along the transported copy of $a$ . In flat space, and in ordinary curved space with the Levi-Civita connection, both routes deposit you at the same corner — the parallelogram closes. Torsion is precisely the leading-order vector by which the two endpoints refuse to coincide: a mismatch of translations, the two ways of "adding" $a$ and $b$ disagreeing. Contrast this with curvature, which is what you detect if you parallel-transport a single vector around a closed loop and find it comes back rotated. Torsion is two translations failing to commute; curvature is transport twisting a vector — separate attributes, neither implying the other. The cleanest example is a Lie group with its left-invariant connection: perfectly flat, yet its torsion equals the group's structure constants.

Figure

Two infinitesimal displacements a and b parallel-transported along each other fail to meet: the route a-then-b lands at a different corner from b-then-a, and the leading gap Δx = −T contracted on the two vectors vanishes exactly when the torsion T does

The same algebra shows up as GR's local flatness: at any point normal coordinates kill the symmetric part of $\Gamma$ — the equivalence principle in mathematical dress — but the antisymmetric part, being a tensor, cannot be removed in any frame. That indestructibility is torsion's operational signature.

Figure

Torsion versus curvature are independent defects of a connection: torsion is the translational gap from non-closing parallelograms, curvature the rotation a vector acquires around a closed loop, so standard GR sets T = 0 with R nonzero whereas teleparallel gravity sets R = 0 with T nonzero

The core equations

The definition is a single antisymmetrization. Torsion keeps only the part of the connection that flips sign under exchange of the two lower indices; the standard convention keeps the full difference, a factor of two over the antisymmetrizer.

T^\lambda{}_{\mu\nu}\;\equiv\;\Gamma^\lambda{}_{\mu\nu}-\Gamma^\lambda{}_{\nu\mu}\;=\;2\,\Gamma^\lambda{}_{[\mu\nu]} \qquad (1)

where: $T^\lambda{}_{\mu\nu}$ — the torsion tensor, antisymmetric in $\mu\nu$ ; $\Gamma^\lambda{}_{\mu\nu}$ — the affine connection coefficients; $[\mu\nu]$ — antisymmetrization, $\tfrac12(\,\cdot_{\mu\nu}-\cdot_{\nu\mu}\,)$ . Reading it physically: torsion is how much the connection fails to be symmetric in its lower indices, and a symmetric connection has none. The quickest place torsion shows itself is a second covariant derivative. On a scalar the commutator collapses to torsion alone, because a scalar has no index for curvature to rotate — the cleanest operational definition of $T$ .

\big(\nabla_\mu\nabla_\nu-\nabla_\nu\nabla_\mu\big)\phi\;=\;-\,T^\lambda{}_{\mu\nu}\,\nabla_\lambda\phi \qquad (2)

where: $\phi$ — any smooth scalar field; $\nabla_\mu$ — the covariant derivative of $\Gamma$ . Reading it physically: for Levi-Civita this commutator vanishes on scalars; with torsion it does not, and the failure directly measures $T$ . On a vector the commutator acquires the curvature term as well, and the two geometric defects sit side by side. This is the algebraic statement that torsion and curvature are the two independent obstructions to derivatives commuting.

\big[\nabla_\mu,\nabla_\nu\big]V^\rho\;=\;R^\rho{}_{\sigma\mu\nu}\,V^\sigma\;-\;T^\lambda{}_{\mu\nu}\,\nabla_\lambda V^\rho \qquad (3)

where: $R^\rho{}_{\sigma\mu\nu}$ — the Riemann curvature tensor; $V^\rho$ — an arbitrary vector field. Reading it physically: curvature multiplies the vector (a rotation), torsion its derivative (a transport term); setting $T=0$ recovers the textbook Ricci identity of GR. Why torsion is a tensor at all is read off the connection's transformation law. Its inhomogeneous, non-tensorial term lives in the symmetric part and cannot survive antisymmetrization — Derivation 1 makes this a proof.

\Gamma'^\lambda{}_{\mu\nu}=\frac{\partial x'^\lambda}{\partial x^\alpha}\frac{\partial x^\beta}{\partial x'^\mu}\frac{\partial x^\gamma}{\partial x'^\nu}\,\Gamma^\alpha{}_{\beta\gamma}\;+\;\frac{\partial x'^\lambda}{\partial x^\alpha}\frac{\partial^2 x^\alpha}{\partial x'^\mu\partial x'^\nu} \qquad (4)

where: $x'^\mu$ — new coordinates; the first term is the tensorial rotation of indices; the second is the inhomogeneous piece $\partial^2 x^\alpha/\partial x'^\mu\partial x'^\nu$ , symmetric in $\mu\nu$ . Reading it physically: this symmetric term is what makes $\Gamma$ not a tensor, and what a coordinate change can cancel at a point — but only in the symmetric sector, never in torsion. Once you allow torsion yet still insist the metric be covariantly constant, the connection splits cleanly: the Christoffel part plus a correction called the contorsion, assembled from torsion alone.

K^\lambda{}_{\mu\nu}\;=\;\tfrac12\big(T^\lambda{}_{\mu\nu}+T_\mu{}^\lambda{}_\nu+T_\nu{}^\lambda{}_\mu\big),\qquad \Gamma^\lambda{}_{\mu\nu}\;=\;\mathring\Gamma^\lambda{}_{\mu\nu}+K^\lambda{}_{\mu\nu} \qquad (5)

where: $K^\lambda{}_{\mu\nu}$ — the contorsion tensor, the deviation of a metric-compatible connection from Levi-Civita; $\mathring\Gamma^\lambda{}_{\mu\nu}$ — the Christoffel symbols; $T_\mu{}^\lambda{}_\nu$ — torsion with indices raised and lowered by the metric. Reading it physically: all torsional information is repackaged into $K$ , so a metric-compatible geometry is fixed by the metric plus the torsion. Finally, the fundamental theorem of Riemannian geometry closes the door: metric compatibility with vanishing torsion pins the connection down uniquely and gives the Christoffel formula.

T^\lambda{}_{\mu\nu}=0\;\text{ and }\;\nabla_\rho g_{\mu\nu}=0\;\Longrightarrow\;\Gamma^\lambda{}_{\mu\nu}=\tfrac12\,g^{\lambda\sigma}\big(\partial_\mu g_{\sigma\nu}+\partial_\nu g_{\sigma\mu}-\partial_\sigma g_{\mu\nu}\big) \qquad (6)

where: $g_{\mu\nu}$ — the metric tensor; $g^{\lambda\sigma}$ — its inverse. Reading it physically: this is the theorem that lets GR speak of "the" connection of a spacetime — with torsion zero, the metric alone fixes parallel transport, geodesics and curvature.

Derivation AWhy the antisymmetric part of a connection is a genuine tensor

The goal is to prove that torsion transforms as a $(1,2)$ tensor even though the connection it is built from does not. The strategy is elementary: write the transformation law of $\Gamma$ , write it again with the lower indices swapped, and subtract. Because the non-tensorial term is a second partial derivative — and second partials commute — it is symmetric in the two lower indices and identical in both versions, so it cancels, leaving only the homogeneous tensorial part.

Step 1. Start from the connection's transformation law, $\Gamma'^\lambda{}_{\mu\nu}=\dfrac{\partial x'^\lambda}{\partial x^\alpha}\dfrac{\partial x^\beta}{\partial x'^\mu}\dfrac{\partial x^\gamma}{\partial x'^\nu}\Gamma^\alpha{}_{\beta\gamma}+\dfrac{\partial x'^\lambda}{\partial x^\alpha}\dfrac{\partial^2 x^\alpha}{\partial x'^\mu\partial x'^\nu}$ . Why: the second term is the inhomogeneous piece that disqualifies $\Gamma$ as a tensor; everything hinges on its index symmetry.

Step 2. Swap $\mu\leftrightarrow\nu$ throughout, giving $\Gamma'^\lambda{}_{\nu\mu}=\dfrac{\partial x'^\lambda}{\partial x^\alpha}\dfrac{\partial x^\beta}{\partial x'^\nu}\dfrac{\partial x^\gamma}{\partial x'^\mu}\Gamma^\alpha{}_{\beta\gamma}+\dfrac{\partial x'^\lambda}{\partial x^\alpha}\dfrac{\partial^2 x^\alpha}{\partial x'^\nu\partial x'^\mu}$ , and relabel dummies $\beta\leftrightarrow\gamma$ in the homogeneous term. Why: this lets the two homogeneous terms be compared index-for-index, differing only by $\Gamma^\alpha{}_{\beta\gamma}\to\Gamma^\alpha{}_{\gamma\beta}$ .

Step 3. Subtract. The inhomogeneous terms are equal because mixed partials commute, so they cancel exactly, leaving $\Gamma'^\lambda{}_{\mu\nu}-\Gamma'^\lambda{}_{\nu\mu}=\dfrac{\partial x'^\lambda}{\partial x^\alpha}\dfrac{\partial x^\beta}{\partial x'^\mu}\dfrac{\partial x^\gamma}{\partial x'^\nu}\big(\Gamma^\alpha{}_{\beta\gamma}-\Gamma^\alpha{}_{\gamma\beta}\big)$ . Why: the commuting of mixed partials is the whole content — it makes the non-tensorial term symmetric, so it dies.

Step 4. The left side is $T'^\lambda{}_{\mu\nu}$ and the parenthesis is $T^\alpha{}_{\beta\gamma}$ , so $T'^\lambda{}_{\mu\nu}=\dfrac{\partial x'^\lambda}{\partial x^\alpha}\dfrac{\partial x^\beta}{\partial x'^\mu}\dfrac{\partial x^\gamma}{\partial x'^\nu}\,T^\alpha{}_{\beta\gamma}$ . Why: verbatim the transformation rule of a $(1,2)$ tensor, completing the proof.

Go deeper · Derivation B · 1 minTorsion as the non-closure of an infinitesimal parallelogram

Derivation BTorsion as the non-closure of an infinitesimal parallelogram

The goal is to turn the abstract antisymmetric part of the connection into the concrete picture that opens this article: two small vectors slid along each other, and the gap between the resulting corners. We build the parallelogram by parallel-transporting each displacement along the other, keep everything to second order, and difference the two endpoints. What remains is exactly the torsion contracted with the two vectors, so vanishing torsion is equivalent to every infinitesimal parallelogram closing.

Step 1. Let $a^\mu$ and $b^\mu$ be infinitesimal displacements at $P$ . Parallel-transporting $b$ along $a$ shifts its components to first order: $\tilde b^\lambda=b^\lambda-\Gamma^\lambda{}_{\mu\nu}\,a^\mu b^\nu$ . Why: parallel transport is $\nabla_a b=0$ , which to leading order is exactly this shift by the connection.

Step 2. Route one — displace by $a$ , then by the transported $b$ — reaches $x_1^\lambda=a^\lambda+b^\lambda-\Gamma^\lambda{}_{\mu\nu}\,a^\mu b^\nu$ (positions relative to $P$ , to second order). Why: the second-order cross term is where the connection first enters, so it is what can distinguish the routes.

Step 3. Route two — displace by $b$ , then by the transported $a$ — reaches $x_2^\lambda=b^\lambda+a^\lambda-\Gamma^\lambda{}_{\nu\mu}\,b^\nu a^\mu$ , the mirror construction. Why: swapping $a$ and $b$ swaps the lower indices of the connection term, which is what fails to cancel.

Step 4. Subtract: $\Delta x^\lambda=x_1^\lambda-x_2^\lambda=-\big(\Gamma^\lambda{}_{\mu\nu}-\Gamma^\lambda{}_{\nu\mu}\big)a^\mu b^\nu=-T^\lambda{}_{\mu\nu}\,a^\mu b^\nu+\mathcal O(\epsilon^3)$ . Why: the symmetric parts of $\Gamma$ are common to both routes and cancel, isolating torsion; the parallelogram closes iff $T=0$ .

Go deeper · Derivation C · 1 minSplitting a metric-compatible connection into Levi-Civita plus contorsion

Derivation CSplitting a metric-compatible connection into Levi-Civita plus contorsion

The goal is to show what a connection looks like once you demand metric compatibility but permit torsion — the setting of Einstein–Cartan gravity. Imposing $\nabla_\rho g_{\mu\nu}=0$ gives three copies of one equation under cyclic permutation; the Christoffel combination isolates the symmetric part, and the leftover antisymmetric pieces reassemble into the contorsion, built entirely from torsion. Setting torsion to zero recovers the fundamental theorem, proving Levi-Civita is the unique torsion-free choice.

Step 1. Write metric compatibility as $\partial_\mu g_{\nu\rho}=\Gamma^\sigma{}_{\mu\nu}g_{\sigma\rho}+\Gamma^\sigma{}_{\mu\rho}g_{\nu\sigma}$ and its two cyclic permutations of $(\mu\nu\rho)$ . Why: three equations are what is needed to solve for the connection's symmetric and antisymmetric parts.

Step 2. Form the Christoffel combination $\partial_\mu g_{\nu\rho}+\partial_\nu g_{\rho\mu}-\partial_\rho g_{\mu\nu}$ . The symmetric-in-lower-index parts collect into the usual expression; the antisymmetric parts survive as torsion. Why: this signed sum is engineered so the symmetric pieces add and the cluttering metric derivatives cancel.

Step 3. Collecting remainders gives $2\,g_{\sigma\rho}\Gamma^\sigma{}_{(\mu\nu)}+g_{\sigma\rho}T^\sigma{}_{\mu\nu}+g_{\nu\sigma}T^\sigma{}_{\rho\mu}+g_{\mu\sigma}T^\sigma{}_{\rho\nu}$ , with $T^\sigma{}_{\mu\nu}=\Gamma^\sigma{}_{\mu\nu}-\Gamma^\sigma{}_{\nu\mu}$ . Why: the torsion terms appear only because $\Gamma$ is no longer symmetric, so they are the entire correction to the Riemannian result.

Step 4. Solve: $\Gamma^\lambda{}_{\mu\nu}=\tfrac12 g^{\lambda\sigma}\big(\partial_\mu g_{\sigma\nu}+\partial_\nu g_{\sigma\mu}-\partial_\sigma g_{\mu\nu}\big)+K^\lambda{}_{\mu\nu}$ , with $K^\lambda{}_{\mu\nu}=\tfrac12\big(T^\lambda{}_{\mu\nu}+T_\mu{}^\lambda{}_\nu+T_\nu{}^\lambda{}_\mu\big)$ . Why: the first term is $\mathring\Gamma^\lambda{}_{\mu\nu}$ and the second the contorsion, so $T=0$ gives $K=0$ and the unique Levi-Civita connection — the fundamental theorem.

Limiting cases

Torsion is understood best by cataloguing the regimes that switch it on and off — from standard relativity, where it is banished, to the theories built to revive it.

Case	Result	Meaning
General relativity (Levi-Civita)	$T^\lambda{}_{\mu\nu}=0$	The fundamental theorem makes this the unique metric-compatible Christoffel connection, so the metric determines everything.
Einstein–Cartan, vacuum or spinless matter	$T^\lambda{}_{\mu\nu}=0$	Torsion's field equation is algebraic, $T\propto$ spin density; with no spin source it vanishes and the theory reduces to GR.
Einstein–Cartan, spinning matter	$T^\lambda{}_{\mu\nu}\propto\kappa\,S^\lambda{}_{\mu\nu}$	Inside matter with spin density $S^\lambda{}_{\mu\nu}$ , torsion is nonzero but non-propagating (a contact interaction), $\kappa=8\pi G/c^4$ .
Teleparallel gravity (Weitzenböck)	$R^\rho{}_{\sigma\mu\nu}=0,\;T^\lambda{}_{\mu\nu}\neq0$	Gravity recast on a flat connection where all dynamics rides on torsion — empirically identical to GR for the simplest model.
Totally antisymmetric torsion	autoparallels $=$ metric geodesics	The contorsion drops out of the straightest-line equation, so freely-falling test particles trace the same paths as in GR.

Common trapTorsion must be non-tensorial because it is built from the non-tensorial Christoffel symbols

This common error inverts the truth. The connection $\Gamma^\lambda{}_{\mu\nu}$ is indeed not a tensor — its transformation law carries the inhomogeneous second-derivative term of equation (4). But that term is symmetric in the two lower indices, so it cancels identically the moment you antisymmetrize, and the surviving $T^\lambda{}_{\mu\nu}=\Gamma^\lambda{}_{\mu\nu}-\Gamma^\lambda{}_{\nu\mu}$ transforms as a bona fide $(1,2)$ tensor (Derivation 1 proves it line by line). A tensor built from non-tensors is not paradoxical — it happens whenever the non-tensorial part is confined to a symmetry channel you project out. Two neighbouring traps: torsion is a property of the connection, not the metric, and it is independent of curvature — flat can carry torsion (Weitzenböck) and curved can be torsion-free (every Riemannian manifold in GR). Never read " $\Gamma$ is not a tensor" as "torsion is not a tensor."

Worked examples

1 · How dense must matter be before torsion matters? (numeric)

In Einstein–Cartan gravity, intrinsic spin sources torsion algebraically, producing a spin–spin contact correction to the energy density. Estimate the density at which this correction becomes comparable to the rest-energy density for a neutron gas, and compare to nuclear and Planck densities.

Step 1. Set up the competing densities. For neutrons of number density $n$ and spin $\hbar/2$ , the spin density is $\sigma\sim n\hbar/2$ and the energy density is $u=n\,m c^2$ . The torsional contact correction — quadratic in spin, one factor of $G$ — is of order $\Delta u\sim (G/c^2)\,\sigma^2$ . It is quadratic in spin because torsion couples spin to spin, a contact interaction.

Step 2. Equate them: $(G/c^2)(n\hbar/2)^2\sim n\,mc^2$ gives the critical number density $n_c\sim m c^4/(G\hbar^2)$ (up to an $\mathcal O(1)$ factor). With $m=1.675\times10^{-27}\,\text{kg}$ , $c=2.998\times10^8\,\text{m/s}$ , $G=6.674\times10^{-11}$ , $\hbar=1.055\times10^{-34}$ , this is $n_c\approx\mathbf{1.8\times10^{85}\ m^{-3}}$ .

Step 3. Convert to mass density: $\rho_c=n_c\,m\approx\mathbf{3\times10^{58}\ kg/m^3}$ (order $10^{58}$ ). An order-of-magnitude estimate — the exact ECSK coefficient shifts it by a small factor, not orders of magnitude.

Step 4. Put it in context. Nuclear density is $\rho_\text{nuc}\approx2.3\times10^{17}\,\text{kg/m}^3$ , so $\rho_c$ sits about 41 orders of magnitude above it yet still 38 orders below the Planck density $\rho_\text{Pl}\approx5.2\times10^{96}\,\text{kg/m}^3$ . The verdict: torsion is dynamically irrelevant in every laboratory and astrophysical setting, competing only in the extreme early universe — which is why Einstein–Cartan passes all current tests of GR while still offering a possible bounce in place of the Big Bang singularity.

2 · The closure gap and its condensed-matter twin, the Burgers vector (numeric)

Take a three-dimensional space with a single constant torsion component $T^z{}_{xy}=\theta$ (all others zero) and two straight edges of length $a$ and $b$ along $\hat x$ and $\hat y$ . Compute the failure of the parallelogram to close, and interpret it.

Step 1. Use the closure formula from Derivation 2: $\Delta x^\lambda=-T^\lambda{}_{\mu\nu}\,a^\mu b^\nu$ . With $a^\mu=a\,\delta^\mu_x$ and $b^\nu=b\,\delta^\nu_y$ , only the $\lambda=z$ component survives: $\Delta x^z=-T^z{}_{xy}\,a\,b=-\theta\,a\,b$ . The gap points out of the $xy$ -plane — the two in-plane translations fail to commute and the mismatch is a perpendicular step, a "screw".

Step 2. Plug in numbers: torsion $\theta=0.02\ \text{m}^{-1}$ , edges $a=b=0.10\ \text{m}$ , so $\Delta x^z=-(0.02)(0.10)(0.10)=\mathbf{-2.0\times10^{-4}\ m}=-0.20\ \text{mm}$ — and, being second order, the gap scales as the product $a\,b$ , dropping fourfold if both edges are halved.

Step 3. Read the physics. In Cartan's geometry of defects, torsion is the continuous dislocation density of an elastic medium, and this closure gap is precisely the Burgers vector enclosed by the circuit — the net lattice step accumulated on a loop around dislocation lines. A crystallographer measuring a Burgers vector of $0.20\ \text{mm}$ per circuit is measuring torsion — the tensor GR discards, seen in the lab.

Physics of the portrait

Torsion has a single unmistakable author, and he saw its physical meaning decades before the physics caught up.

Origins · 1 minÉlie Cartan and how this was discovered

ÉC

Élie Cartan

1869–1951 · geometry of connections with torsion and Einstein–Cartan gravity

Élie Cartan was the geometer who gave the connection its modern shape and, in doing so, discovered torsion. Trained in the French tradition of Lie groups and differential systems, he spent the 1920s generalizing Levi-Civita's parallel transport into the "method of moving frames," and in a short 1922 note to the Comptes Rendus — "Sur une généralisation de la notion de courbure de Riemann et les espaces à torsion" — he isolated the antisymmetric part of an affine connection and named it torsion. Extraordinarily, Cartan proposed on physical grounds that this torsion should couple to the intrinsic angular momentum of matter, three years before spin was established in quantum mechanics. His intuition was vindicated four decades later when Sciama and Kibble built exactly that coupling into a gauge theory of gravity — Einstein–Cartan–Sciama–Kibble theory. Cartan corresponded with Einstein between 1929 and 1932, teaching him teleparallel geometry during Einstein's unified-field attempts. His legacy reaches far beyond relativity — exterior calculus, Lie groups, spinors — but torsion is the invariant that carries his name past Einstein's geometry.

History

1917

— Levi-Civita introduces parallel transport, defining the symmetric torsion-free connection

1922

— Cartan isolates torsion as the antisymmetric part of an affine connection and links it to intrinsic angular momentum

1925

— Uhlenbeck and Goudsmit establish electron spin, later recognized as the physical source Cartan had anticipated

1928

— Weitzenböck constructs the flat connection with nonzero torsion, the basis of teleparallelism

1961

— Kibble formulates gravity as a gauge theory of the Poincaré group, with torsion sourced by spin

1976

— Hehl, von der Heyde, Kerlick and Nester publish the definitive review of Einstein–Cartan theory

Going further

Texts. Misner, Thorne and Wheeler, Gravitation, §10 (covariant derivative and connection) · Wald, General Relativity, §3.1 (derivative operators and torsion) · Carroll, Spacetime and Geometry, §3.2 (Christoffel connection and torsion) · Schutz, A First Course in General Relativity, Ch. 5 · Nakahara, Geometry, Topology and Physics, §7.3 and §7.8 (torsion, curvature, contorsion) · Hehl, von der Heyde, Kerlick and Nester, Rev. Mod. Phys. 48, 393 (1976) (Einstein–Cartan theory).

Provenance. The definition and tensorial-cancellation argument follow Wald §3.1 and Carroll §3.2; a verifier should reproduce equation (4) and confirm the symmetric second-derivative term cancels under antisymmetrization. The commutator identities and parallelogram non-closure are standard (MTW §10, Nakahara §7.3); the contorsion decomposition is checked against metric compatibility (Nakahara §7.8, Hehl et al. 1976). The coupling $\kappa=8\pi G/c^4$ is the standard Einstein constant; the Example 1 critical density is an order-of-magnitude estimate ( $\rho_c\sim m^2c^4/G\hbar^2\approx3\times10^{58}\,\text{kg/m}^3$ ) from textbook constants, and the torsion bounds are from Kostelecký, Russell and Tasson (2008).

Builds on. The Affine Connection · Covariant Derivative · Parallel Transport · Christoffel Symbols. Leads to. The Levi-Civita Connection · Metric Compatibility · The Geodesic Equation.

Ask the tutor: Ask me to walk you through the parallelogram construction step by step, or to add torsion to a specific GR calculation by swapping in the contorsion.

Is torsion the same thing as the antisymmetric part of the metric?

No — this conflates two different structures. The metric $g_{\mu\nu}$ is symmetric by definition. Torsion is the antisymmetric part of the connection, a separate piece of equipment specifying parallel transport. A single metric is in fact compatible with infinitely many connections differing by their torsion, all preserving lengths and angles. General relativity uses the metric to build one particular torsion-free connection (Levi-Civita), but that is a choice enforced by the fundamental theorem, not an identity.

If torsion vanishes in GR, why bother learning it?

Because knowing what you have set to zero is how you understand what a theory assumes. Torsion is the exact object whose vanishing is one of the two axioms — with metric compatibility — that single out the Levi-Civita connection; every statement beginning "the connection is symmetric" is a statement that $T=0$ . In Einstein–Cartan gravity, teleparallelism, supergravity, or the geometry of crystal defects, torsion is precisely the dial turned back on. It also sharpens the equivalence principle: a freely-falling frame erases gravity because the symmetric connection can be gauged away at a point, but torsion, being a tensor, cannot.

How is torsion different from curvature, in one sentence?

Torsion measures a failure of translations to commute — the gap by which an infinitesimal parallelogram fails to close — while curvature measures a failure of transport to preserve direction — the rotation a vector picks up around a closed loop. In the shared commutator, equation (3), curvature multiplies the vector while torsion multiplies its derivative. Crucially they are independent: the Weitzenböck connection is flat yet torsionful, and every Riemannian manifold in GR is curved yet torsion-free.

What is the contorsion tensor, and why the funny name?

The contorsion $K^\lambda{}_{\mu\nu}$ is the exact difference between a metric-compatible connection and Levi-Civita: $\Gamma=\mathring\Gamma+K$ . The name signals it is built from torsion, $K^\lambda{}_{\mu\nu}=\tfrac12(T^\lambda{}_{\mu\nu}+T_\mu{}^\lambda{}_\nu+T_\nu{}^\lambda{}_\mu)$ — the same information repackaged, contorsion antisymmetric in its first and third indices where torsion is antisymmetric in its last two. It is the practically useful form: to add torsion to any GR calculation, replace $\mathring\Gamma$ by $\mathring\Gamma+K$ . Setting $T=0$ gives $K=0$ and ordinary relativity.

Do particles fall differently when there is torsion?

It depends on which "straight line" they follow. Autoparallels — the tracks of a freely transported tangent vector — depend only on the symmetric part of the connection; metric geodesics, the extremals of proper length, depend on $\mathring\Gamma$ . With generic torsion these families split, and deciding which a real spinless particle follows needs the matter's equation of motion, not geometry alone. One clean case: if torsion is totally antisymmetric, its contorsion drops out of the autoparallel equation and the two families coincide — freely-falling particles trace exactly the GR paths, which is why totally antisymmetric torsion is the hardest kind to detect.

Has spacetime torsion ever been measured?

No torsion has been detected, but stringent upper bounds exist. Because torsion couples to the spin of test particles, experiments watching spins for anomalous precession — torsion pendula, atomic magnetometers, atomic clocks — can be reinterpreted as torsion searches. Kostelecký, Russell and Tasson (2008) did exactly this, mining existing data to bound the components of a constant background torsion to extraordinarily small values. Any torsion that does exist must either be confined inside spinning matter (Einstein–Cartan, far too weak to see) or be a totally antisymmetric background that has escaped every spin experiment.