Cartan connection applications

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

1 Vierbeins, et cetera

[edit] 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).

[edit] 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) (this can be turned into a Spin(p,q) principal bundle via the associated bundle construction if there are spinorial fields), and a vector SO(p, q)-bundle V associated to B by means of 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, $e\colon{\rm T}M\to V$ , where TM is the tangent bundle of M.

[edit] Example

We can describe geometries in general relativity in terms of a tetrad field instead of the usual metric tensor field. The metric tensor $g_{\alpha\beta}\!$ gives the inner product in the tangent space directly:

$\langle \mathbf{x},\mathbf{y} \rangle = g_{\alpha\beta} \, x^{\alpha} \, y^{\beta}$ .

The tetrad $e_{\alpha}^i$ may be seen as a (linear) map from the tangent space to Minkowski space which 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:

$\langle \mathbf{x},\mathbf{y} \rangle = \eta_{ij} (e_{\alpha}^i \, x^{\alpha}) (e_{\beta}^j \, y^{\beta})$ .

Here $α$ and $β$ range over tangent-space coordinates, while $i$ and $j$ range over Minkowski coordinates. The tetrad field is less general than the metric tensor field: given any tetrad field $e_{\alpha}^i(\mathbf{x})$ there is an equivalent metric tensor field $g_{\alpha\beta}(\mathbf{x}) = \eta_{ij} \, e_{\alpha}^i(\mathbf{x}) \, e_{\beta}^j(\mathbf{x})$ , but a metric tensor field cannot be expressed using tetrads unless it defines a Minkowskian inner product. Normally this is no limitation because we require solutions of general relativity to be locally Minkowskian everywhere.

[edit] 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 $\nabla$ defined below is torsion-free. This condition is dropped in the Einstein-Cartan theory, but then we can't define A uniquely anymore.

This is called the spin connection.

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 curvature F defined as $\bold{F}\ \stackrel{\mathrm{def}}{=}\ d\bold{A}+\bold{A}\wedge\bold{A}$ is pointwise gauge covariant. This is simply the Riemann curvature 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.

[edit] 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

$S\ \stackrel{\mathrm{def}}{=}\ \frac{1}{2}\int_M \epsilon(F \wedge e \wedge e)$