This page covers notations and definitions, sometimes called the Cartan formalism, for the Cartan connection concept.

Vierbeins, et cetera

The vierbein or tetrad theory is the special case of a four-dimensional manifold. It applies to metrics of any signature. In any dimension, for a pseudo Riemannian geometry (with metric signature (p,q)), this Cartan connection theory is an alternative method in differential geometry. In different contexts it has also been called the orthonormal frame, repère mobile, soldering form or orthonormal nonholonomic basis method.

This section is an approach to tetrads, but written in general terms. In dimensions other than 4, words like triad, pentad, funfbein, elfbein etc. have been used. Vielbein covers all dimensions. (In German, vier stands for four and viel stands for many)

If you're looking for a basis-dependent index notation, see tetrad (index notation).

The basic ingredients

Suppose given a differential manifold M of dimension n, and fixed natural numbers p and q with p + q = n. Further, we suppose given a SO(p, q) principal bundle B over M (called the frame bundle), and a vector SO(p, q)-bundle V associated to B by means of with the natural n-dimensional representation of SO(p, q).

Suppose given also a SO(p, q)-invariant metric η of signature (p, q) over V; and an invertible linear map between vector bundles over M, , where TM is the tangent bundle of M.

Constructions

A (pseudo-)Riemannian metric is defined over M as the pullback of η by e. To put it in other words, if we have two sections of TM, X and Y,

g(X,Y) = η(e(X),e(Y)).

A connection over V is defined as the unique connection A satisfying these two conditions:

• dη(a,b) = η(d_Aa,b) + η(a,d_Ab) for all differentiable sections a and b of V (i.e. d_Aη = 0) where d_A is the covariant exterior derivative. This implies that A can be extended to a connection over the SO(p,q) principal bundle.
• d_Ae = 0. 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 can't define A uniquely anymore.

Now that we've specified A, we can use it to define a connection ∇ over TM via the isomorphism e:

e(∇X) = d_Ae(X) for all differentiable sections X of TM.

Since what we now have here is a SO(p,q) gauge theory, the Riemann curvature F defined as is pointwise gauge covariant. This is simply the Riemann tensor in a different guise.

See also connection form and curvature form.

Side note: the e here is often written as θ, the A here as ω and the F here as Ω and d_A as D.

The Palatini action

In the tetrad formulation of general relativity, the action, as a functional of the cotetrad e and a connection form A over a four dimensional differential manifold M is given by

where F is the gauge curvature 2-form and ε is the antisymmetric intertwiner of four "vector" reps of SO(3,1) normalized by η.

Note that in the presence of spinor fields, the Palatini action implies that d_Ae is nonzero! (i.e. we have torsion!!!) See Einstein-Cartan theory.