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 means four, zwei means two, fünf means five, elf means eleven and, in general, viel means many.)
The basic ingredient of the Cartan formalism is an invertiblelinear map, between vector bundles over where TM is the tangent bundle of . The invertibility condition on is sometimes dropped. In particular if is the trivial bundle, as we can always assume locally,
V has a basis of orthogonal sections
. With respect to this basis
is a constant matrix. For a choice of local coordinates on (the negative indices are only to distinguish them from the indices labeling the ) and a corresponding local frame
of the tangent bundle, the map is determined by the images
of the basis sections
They determine a (non coordinate) basis of the tangent bundle (provided is invertible and only locally if is only locally trivialised). The matrix is called the tetrad, vierbein, vielbein, etc.
Its interpretation as a local frame crucially depends on the implicit choice of local bases.
Note that an isomorphism
gives a reduction of the frame bundle, the principal bundle of the tangent bundle. In general, such a reduction is impossible for topological
reasons. Thus, in general for continuous maps , one cannot avoid that becomes degenerate at some points of .
The tetrad may be seen as a (linear) map from the tangent space to Minkowski space that preserves the inner product. This lets us find the inner product in the tangent space by mapping our two vectors into Minkowski space and taking the usual inner product there:
Here and range over tangent-space coordinates, while and range over Minkowski coordinates. The tetrad field defines a metric tensor field via the pullback .
. The quantity on the left hand side is called the torsion. This basically states that defined below is torsion-free. This condition is dropped in the Einstein–Cartan theory, but then we cannot define uniquely anymore.
where is the gauge curvature2-form, is the antisymmetric Levi-Civita symbol, and that is the determinant of . Here we see that the differential form language leads to an equivalent action to that of the normal Einstein–Hilbert action, using the relations and . Note that in terms of the Planck mass, we set , whereas the last term keeps all the SI unit factors.
^A variant of the construction uses reduction to a Spin(p, q) principal spin bundle. In that case, the principal bundle contains more information than the bundle V together with the metric η, which is needed to construct spinorial fields.