Cartan connection applications

This page covers applications of the Cartan formalism. For the general concept see Cartan connection.

Vierbeins, "et cetera"

The vierbein or tetrad theory much used in theoretical physics is a special case of the application of Cartan connection in four-dimensional manifold. It applies to metrics of any signature. 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 we are working on a differential manifold "M" of dimension "n", and have fixed natural numbers "p" and "q" with

:"p" + "q" = "n".

Furthermore, we assume that we are given a SO("p", "q") principal bundle "B" over "M" (called the frame bundle) This can be turned into a Spin("p","q") principal spin 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").

The basic ingredients are: η that is a SO("p", "q")-invariant metric with signature ("p", "q") over "V"; and an invertible linear map "e" between vector bundles over "M", $ecolon\{\; m\; T\}M\; o\; V$, where T"M" is the tangent bundle of "M".

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 $e\_\{alpha\}^i$ 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:

:

Here $alpha$ and 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 , 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.

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 T"M", 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:

* "d"η(a,b) = η("d"_A"a","b") + η("a","d"_A"b") 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"_A"e" = 0. The quantity on the left hand side is called the torsion. This basically states that $abla$ 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 T"M" via the isomorphism "e"::"e"(∇X) = "d"_A"e"(X) for all differentiable sections X of T"M".

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

An alternate notation writes the connection form A as ω, the curvature form F as Ω, the canonical vector-valued 1-form "e" as θ, and the exterior covariant derivative $d\_A$ 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

:$S\; stackrel\{mathrm\{def\{=\}\; frac\{1\}\{2\}int\_M\; epsilon(F\; wedge\; e\; wedge\; e)$

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, that is, have torsion. See Einstein-Cartan theory.

Notes