Vierbein

This article should be merged with Cartan formalism.

In differential geometry, Elie Cartan worked out an approach to the idea of connection that used his method of moving frames. 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 article 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.

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

Table of contents

1 The basic ingredients 2 Constructions 3 Advantages

The basic ingredients

Suppose given differential manifold M of dimension n, and fixed natural numbers p and q with p+q = n. 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, e:TM->V where TM is the tangent bundle of M.

Constructions

A (pseudo)Riemannian metric is defined over M as the push forward 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, A is defined as the unique connection 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 basically states that A can be extended to a connection over the SO(p,q) principal bundle)
• d_Ae=0. (this basically states that ∇ defined below is torsion-free)

Now that we've specified A, we can use it to define a connection over TM by the pullback (or is it push forward?) by 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.

Advantages

An advantages of the tetrad formalism, over the conventional metric tensor formulation, is in describing spinor bundles.

See also Riemannian geometry, General relativity, Palatini action

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