1. The basic ingredients

2. Example: general relativity

3. Constructions

4. The Palatini action

5. Notes

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).

The basic ingredients

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(p, q)-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 ({{aka}} 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 relativity

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 .

Constructions

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 action

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.

Notes

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