Teleparallel

Teleparallelism, or teleparallel gravity, is a formulation of gravitation in which the geometric description of gravity is provided by torsion instead of curvature. The theory uses a tetrad field e^a_mu that defines a local orthonormal frame at each point and a curvature-free Weitzenböck connection whose nonzero torsion encodes gravitational effects. In this view, parallel transport preserves the tetrad field, and gravity acts as a force arising from torsion rather than from spacetime curvature.

Teleparallel gravity contrasts with general relativity, which is built on the Levi-Civita connection that is torsion-free and encodes gravity entirely in curvature. The teleparallel equivalent of general relativity (TEGR) constructs a Lagrangian from the torsion scalar T and yields field equations that are dynamically equivalent to Einstein's equations, reproducing the same phenomenology up to a total derivative boundary term.

Because torsion provides a natural framework for translational gauge symmetry, TEGR allows defining gravitational energy-momentum via torsion surface terms. It also serves as a gauge-theoretic reformulation of GR and provides a basis for modified gravity, notably f(T) gravity, where the Lagrangian is a function of T and can yield different cosmological behavior.

Historically, teleparallel ideas trace to Einstein's attempts to unify gravity with electromagnetism, with a fully developed formulation by Hayashi and Shirafuji in 1979 and subsequent refinements. Teleparallel frameworks are investigated in cosmology, black hole physics, and quantum gravity as alternatives or extensions to GR. Current work also addresses covariant formulations that restore local Lorentz invariance and reconcile f(T) theories with observational constraints.