Einstein-Cartan theory
Einstein-Cartan theory is a modification of general relativity that incorporates spacetime torsion — the antisymmetric part of the affine connection — as a dynamical variable coupled to the intrinsic angular momentum (spin) of matter. Developed by Élie Cartan in the 1920s and later connected to gravity by independent work in the 1970s, it represents the minimal geometric extension of general relativity that accounts for the spin content of matter fields. In standard general relativity, the connection is the Levi-Civita connection, which is torsion-free by construction. Einstein-Cartan theory relaxes this assumption, allowing torsion to be generated by spin density and propagate as a gravitational degree of freedom.
The theory is not an alternative to general relativity in the same way that MOND or braneworld scenarios are alternatives. It is a minimal extension: in the limit of vanishing spin density, Einstein-Cartan theory reduces exactly to general relativity. The differences appear only in regimes where matter has significant spin polarization — extremely dense matter such as neutron stars, or the early universe at densities approaching the Planck scale.
The Geometry of Torsion
In differential geometry, an affine connection defines how vectors are transported between nearby points. The connection has two independent parts: the symmetric part, which determines geodesics and is encoded in the Christoffel symbols; and the antisymmetric part, the torsion tensor Tλμν = Γλμν − Γλνμ. The Levi-Civita connection sets torsion to zero, but this is a choice, not a necessity. In Einstein-Cartan theory, torsion is determined algebraically by the spin density of matter through a field equation that is the gravitational analog of how matter curves spacetime.
The key insight is that spin, like mass, is a source of gravity. But while mass curves spacetime through the symmetric part of the energy-momentum tensor, spin twists spacetime through the antisymmetric part. The torsion tensor is not a propagating field in the usual sense — it does not satisfy a wave equation and does not carry independent degrees of freedom away from matter sources. It is algebraically related to spin density, much as the Newtonian gravitational potential is algebraically related to mass density. This makes Einstein-Cartan theory a theory with contact interactions rather than long-range spin-spin forces.
Field Equations and Structure
The Einstein-Cartan field equations consist of two sets. The first is the Einstein equation generalized to include torsion: the Einstein tensor (now constructed from a connection with torsion) is proportional to the symmetric part of the energy-momentum tensor. The second equation relates torsion to spin density: the contortion tensor, which measures the deviation from metric-compatibility, is proportional to the spin tensor of matter.
For standard matter — scalar fields, unpolarized fermions, electromagnetic fields — the spin density vanishes, torsion vanishes, and the theory reproduces general relativity exactly. The differences emerge for spinor fields with macroscopic polarization. In the early universe, when fermion densities were enormous and spins might have been aligned, torsion could have prevented the formation of singularities, replacing the Big Bang singularity with a bounce. In rotating neutron stars, spin alignment could generate measurable torsion effects.
Status and Predictions
Despite its theoretical elegance, Einstein-Cartan theory faces a challenge: the effects of torsion are suppressed by the Planck scale. For any conceivable terrestrial experiment, the spin density required to produce detectable torsion is far beyond what can be achieved. The theory's primary observational window is cosmological: the early universe, where densities approached the Planck scale and spin effects might have been significant.
Some researchers have explored whether Einstein-Cartan torsion could resolve the cosmological constant problem or generate inflation without an inflaton field. These proposals are speculative but illustrate the theory's role as a minimal test case: before invoking extra dimensions, modified gravity, or string theory, one should ask what the most conservative extension of general relativity predicts. Einstein-Cartan theory is that conservative extension.
Einstein-Cartan theory is the reminder that general relativity was not derived from first principles but constructed from aesthetic assumptions — among them, that spacetime is torsion-free. Cartan showed that this assumption is dispensable, and that dispensing it reveals a new coupling between geometry and matter that general relativity ignores. The theory may never be experimentally distinguished from general relativity at observable scales, but its existence changes the epistemic status of Einstein's equations: they are not the unique geometric theory of gravity, but the torsion-free special case of a broader framework. In a field crowded with speculative alternatives, Einstein-Cartan theory stands out for its modesty: it changes only what must be changed, and it changes it in the direction that geometry itself suggests.