Vierbein

Vierbein, or tetrad, is a field that provides a local orthonormal frame at each point of a four-dimensional spacetime. It consists of four vector fields e_μ^a, where μ is a curved spacetime index and a is a flat local Lorentz index. The spacetime metric g_μν is recovered from the Minkowski metric η_ab via g_μν = η_ab e_μ^a e_ν^b. The inverse tetrad e_a^μ satisfies e_a^μ e_μ^b = δ_a^b and e_μ^a e_a^ν = δ_μ^ν. Thus, at each point the e_μ^a form an orthonormal basis for the tangent space, providing a bridge between curved spacetime and a locally flat frame.

The tetrad formalism is particularly important for coupling fermions to gravity, since spinor fields transform under local Lorentz transformations in the tangent space. This requires a spin connection ω_μ^{ab}, enabling a covariant derivative D_μ acting on fields with local Lorentz indices, e.g., D_μ ψ = ∂_μ ψ + 1/2 ω_μ^{ab} Σ_ab ψ. The spin connection is determined (in the absence of torsion) by the tetrad through the tetrad postulate ∇_μ e_ν^a = 0, and can be expressed in terms of e_μ^a and its derivatives.

Related relations include det e_μ^a = √(-g). The formalism exhibits local Lorentz invariance: e_μ^a transforms under local Lorentz transformations on the a-index, while g_μν remains invariant. The tetrad formulation is dynamically equivalent to the metric formulation of general relativity and is widely used in contexts requiring spinor fields, in Cartan’s lattice of gravity, and in certain approaches to quantum gravity. In four dimensions it is common to refer to it as the vierbein; in three dimensions, the spatial analogue is called the dreibein.