Cartan formalism (physics)

The vierbein or tetrad theory much used in theoretical physics is a special case of the application of Cartan connection in four-dimensional manifolds. It applies to metrics of any signature. (See metric tensor.) This section is an approach to tetrads, but written in general terms. In dimensions other than 4, words like triad, pentad, zweibein, fünfbein, elfbein etc. have been used. Vielbein covers all dimensions. (In German, vier stands for four and viel stands for many.)

For a basis-dependent index notation, see tetrad (index notation).

Suppose we are working on a differentiable manifold M of dimension n, and have fixed natural numbers p and q with

Furthermore, we assume that we are given a SO(p, q) principal bundle B over M and a SO(p, q)-vector bundle V associated to B by means of the natural n-dimensional representation of SO(p, q). Equivalently, V is a rank n real vector bundle over M, equipped with a metric η with signature (p, q) (aka non degenerate quadratic form).

The basic ingredient of the Cartan formalism is an invertible linear map ${\displaystyle e\colon {V}\to {\rm {T}}M}$, between vector bundles over M where TM is the tangent bundle of M. The invertibility condition on e is sometimes dropped. In particular if B is the trivial bundle, as we can always assume locally, V has a basis of orthogonal sections ${\displaystyle f_{a}=f_{1}\ldots f_{n}}$. With respect to this basis ${\displaystyle \eta _{ab}=\eta (f_{a},f_{b})={\rm {diag}}(1,\ldots 1,-1,\ldots ,-1)}$ is a constant matrix. For a choice of local coordinates ${\displaystyle x^{\mu }=x^{-1},\ldots ,x^{-n}}$ on M (the negative indices are only to distinguish them from the indices labeling the ${\displaystyle f_{a}}$) and a corresponding local frame ${\displaystyle \partial _{\mu }={\frac {\partial }{\partial x^{\mu }}}}$ of the tangent bundle, the map e is determined by the images of the basis sections

...
Wikipedia