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.)

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

Contents

The basic ingredientsEdit

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

 

Furthermore, we assume that we are given an SO(p, q) principal bundle   over   and an SO(pq)-vector bundle   associated to   by means of the natural  -dimensional representation of  . Equivalently,   is a rank   real vector bundle over  , equipped with a metric   with signature   (a.k.a. non-degenerate quadratic form).[1]

The basic ingredient of the Cartan formalism is an invertible linear 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  .

Example: general relativityEdit

We can describe geometries in general relativity in terms of a tetrad field instead of the usual metric tensor field. The metric tensor   gives the inner product in the tangent space directly:

 

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  .

ConstructionsEdit

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

 

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

  •   for all differentiable sections   and   of   (i.e.  ) where   is the covariant exterior derivative. This implies that   can be extended to a connection over the   principal bundle.
  •  . 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.

This is called the spin connection.

Now that we have specified  , we can use it to define a connection   over   via the isomorphism  :

  for all differentiable sections   of  .

Since what we now have here is a   gauge theory, the curvature   defined as   is pointwise gauge covariant. This is simply the Riemann curvature tensor in a different form.

An alternate notation writes the connection form   as  , the curvature form   as  , the canonical vector-valued 1-form   as  , and the exterior covariant derivative   as  .

The Palatini actionEdit

In the tetrad formulation of general relativity, the action, as a functional of the vierbein   and a connection form  , with an associated field strength  , over a four-dimensional differentiable manifold   is given by

 

where   is the gauge curvature 2-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.

Note that in the presence of spinor fields, the Palatini action implies that   is nonzero. So there's a non-zero torsion, i.e. that  . See Einstein–Cartan theory.

NotesEdit

  1. ^ A variant of the construction uses reduction to a Spin(pq) 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.