Video Upload Options

1. The Basic Ingredients

Suppose we are working on a differentiable manifold [math]\displaystyle{ M }[/math] of dimension [math]\displaystyle{ n }[/math], and have fixed natural numbers [math]\displaystyle{ p }[/math] and [math]\displaystyle{ q }[/math] with

[math]\displaystyle{ p + q = n . }[/math]

Furthermore, we assume that we are given an SO(p, q) principal bundle [math]\displaystyle{ B }[/math] over [math]\displaystyle{ M }[/math] and an SO(p, q)-vector bundle [math]\displaystyle{ V }[/math] associated to [math]\displaystyle{ B }[/math] by means of the natural [math]\displaystyle{ n }[/math]-dimensional representation of [math]\displaystyle{ \operatorname{SO}(p,q) }[/math]. Equivalently, [math]\displaystyle{ V }[/math] is a rank [math]\displaystyle{ n }[/math] real vector bundle over [math]\displaystyle{ M }[/math], equipped with a metric [math]\displaystyle{ \eta }[/math] with signature [math]\displaystyle{ (p,q) }[/math] (a.k.a. non-degenerate quadratic form).^[1]

The basic ingredient of the Cartan formalism is an invertible linear map [math]\displaystyle{ e\colon{V}\to \operatorname{T}M }[/math], between vector bundles over [math]\displaystyle{ M }[/math] where TM is the tangent bundle of [math]\displaystyle{ M }[/math]. The invertibility condition on [math]\displaystyle{ e }[/math] is sometimes dropped. In particular if [math]\displaystyle{ B }[/math] is the trivial bundle, as we can always assume locally, V has a basis of orthogonal sections [math]\displaystyle{ f_a = f_1 \ldots f_n }[/math]. With respect to this basis [math]\displaystyle{ \eta_{ab} = \eta(f_a, f_b) = \operatorname{diag}(1,\ldots 1, -1, \ldots, -1) }[/math] is a constant matrix. For a choice of local coordinates [math]\displaystyle{ x^\mu = x^{-1}, \ldots, x^{-n} }[/math] on [math]\displaystyle{ M }[/math] (the negative indices are only to distinguish them from the indices labeling the [math]\displaystyle{ f_a }[/math]) and a corresponding local frame [math]\displaystyle{ \textstyle \partial_\mu = \frac{\partial}{\partial x^\mu} }[/math] of the tangent bundle, the map [math]\displaystyle{ e }[/math] is determined by the images of the basis sections

[math]\displaystyle{ e_a := e(f_a) := e^\mu_a \partial_\mu. }[/math]

They determine a (non coordinate) basis of the tangent bundle (provided [math]\displaystyle{ e }[/math] is invertible and only locally if [math]\displaystyle{ B }[/math] is only locally trivialised). The matrix [math]\displaystyle{ e^\mu_a , \mu = -1, \dots, -n, a = 1, \dots, n }[/math] 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 [math]\displaystyle{ V \cong \operatorname{T}M }[/math] gives a reduction [math]\displaystyle{ B \to \operatorname{Fr}(M) }[/math] 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 [math]\displaystyle{ e }[/math], one cannot avoid that [math]\displaystyle{ e }[/math] becomes degenerate at some points of [math]\displaystyle{ M }[/math].

2. 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 [math]\displaystyle{ g_{\alpha\beta} }[/math] gives the inner product in the tangent space directly:

[math]\displaystyle{ \langle \mathbf{x},\mathbf{y} \rangle = g_{\alpha\beta} \, x^\alpha \, y^\beta.\, }[/math]

The tetrad [math]\displaystyle{ e_\alpha^i }[/math] 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:

[math]\displaystyle{ \langle \mathbf{x},\mathbf{y} \rangle = \eta_{ij} (e_\alpha^i \, x^\alpha) (e_\beta^j \, y^\beta).\, }[/math]

Here [math]\displaystyle{ \alpha }[/math] and [math]\displaystyle{ \beta }[/math] range over tangent-space coordinates, while [math]\displaystyle{ i }[/math] and [math]\displaystyle{ j }[/math] range over Minkowski coordinates. The tetrad field [math]\displaystyle{ e_\alpha^i(\mathbf{x}) }[/math] defines a metric tensor field via the pullback [math]\displaystyle{ g_{\alpha\beta}(\mathbf{x}) = \eta_{ij} \, e_\alpha^i(\mathbf{x}) \, e_\beta^j (\mathbf{x}) }[/math].

3. Constructions

A (pseudo-)Riemannian metric is defined over [math]\displaystyle{ M }[/math] as the pullback of [math]\displaystyle{ \eta }[/math] by [math]\displaystyle{ e }[/math]. To put it in other words, if we have two sections of [math]\displaystyle{ \mathrm{T}M }[/math], [math]\displaystyle{ \mathbf{X} }[/math] and [math]\displaystyle{ \mathbf{Y} }[/math],

[math]\displaystyle{ g(\mathbf{X},\mathbf{Y}) = \eta(e(\mathbf{X}),e(\mathbf{Y})) . }[/math]

A connection over [math]\displaystyle{ V }[/math] is defined as the unique connection [math]\displaystyle{ \mathbf{A} }[/math] satisfying these two conditions:

• [math]\displaystyle{ d\eta(a,b) = \eta(d_\mathbf{A}a,b) + \eta(a,d_\mathbf{A}b) }[/math] for all differentiable sections [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math] of [math]\displaystyle{ V }[/math] (i.e. [math]\displaystyle{ d_\mathbf{A}\eta = 0 }[/math]) where [math]\displaystyle{ d_\mathbf{A} }[/math] is the covariant exterior derivative. This implies that [math]\displaystyle{ \mathbf{A} }[/math] can be extended to a connection over the [math]\displaystyle{ \operatorname{SO}(p,q) }[/math] principal bundle.
• [math]\displaystyle{ d_\mathbf{A}e = 0 }[/math]. The quantity on the left hand side is called the torsion. This basically states that [math]\displaystyle{ \nabla }[/math] defined below is torsion-free. This condition is dropped in the Einstein-Cartan theory, but then we cannot define [math]\displaystyle{ \mathbf{A} }[/math] uniquely anymore.

This is called the spin connection.

Now that we have specified [math]\displaystyle{ \mathbf{A} }[/math], we can use it to define a connection [math]\displaystyle{ \nabla }[/math] over [math]\displaystyle{ \mathrm{T}M }[/math] via the isomorphism [math]\displaystyle{ e }[/math]:

[math]\displaystyle{ e(\nabla\mathbf{X}) = d_\mathbf{A}e(\mathbf{X}) }[/math] for all differentiable sections [math]\displaystyle{ \mathbf{X} }[/math] of [math]\displaystyle{ \mathrm{T}M }[/math].

Since what we now have here is a [math]\displaystyle{ \operatorname{SO}(p,q) }[/math] gauge theory, the curvature [math]\displaystyle{ \mathbf{F} }[/math] defined as [math]\displaystyle{ \bold{F}\ \stackrel{\mathrm{def}}{=}\ d\bold{A}+\bold{A}\wedge\bold{A} }[/math] is pointwise gauge covariant. This is simply the Riemann curvature tensor in a different form.

An alternate notation writes the connection form [math]\displaystyle{ \mathbf{A} }[/math] as [math]\displaystyle{ \omega }[/math], the curvature form [math]\displaystyle{ \mathbf{F} }[/math] as [math]\displaystyle{ \Omega }[/math], the canonical vector-valued 1-form [math]\displaystyle{ e }[/math] as [math]\displaystyle{ \theta }[/math], and the exterior covariant derivative [math]\displaystyle{ d_\mathbf{A} }[/math] as [math]\displaystyle{ D }[/math].

4. The Palatini Action

In the tetrad formulation of general relativity, the action, as a functional of the vierbein [math]\displaystyle{ e }[/math] and a connection form [math]\displaystyle{ \omega }[/math], with an associated field strength [math]\displaystyle{ \Omega = D\omega = d\omega + \omega \wedge \omega }[/math], over a four-dimensional differentiable manifold [math]\displaystyle{ M }[/math] is given by

[math]\displaystyle{ \begin{align} S \,\, & \stackrel{\mathrm{def}}{=}\ M^2_{pl}\int_M \varepsilon_{abcd}( e^a \wedge e^b \wedge \Omega^{cd}) = M^2_{pl}\int_M d^4x \, \varepsilon^{\mu\nu\rho\sigma} \varepsilon_{abcd} e^a_\mu e^b_\nu R^{cd}_{\rho\sigma}[\omega] \\[5pt] & = M^2_{pl}\int |e| d^4 x \, \frac{1}{2} e^{\mu}_a e^\nu_b R^{ab}_{\mu\nu} \\[5pt] & = \frac{c^4}{16 \pi G} \int d^4x \, \sqrt{-g} R[g] \end{align} }[/math]

where [math]\displaystyle{ \Omega_{\mu\nu}^{ab} = R_{\mu \nu}^{ab} }[/math] is the gauge curvature 2-form, [math]\displaystyle{ \varepsilon_{abcd} }[/math] is the antisymmetric Levi-Civita symbol, and that [math]\displaystyle{ |e| = \varepsilon^{\mu \nu \rho \sigma} \varepsilon_{abcd} e^a_\mu e^b_\nu e^c_\rho e^d_\sigma }[/math] is the determinant of [math]\displaystyle{ e_{\mu}^a }[/math]. Here we see that the differential form language leads to an equivalent action to that of the normal Einstein–Hilbert action, using the relations [math]\displaystyle{ |e| = \sqrt{-g} }[/math] and [math]\displaystyle{ R^{\lambda\sigma}_{\mu \nu}= e^\lambda_a e^\sigma_b R^{ab}_{\mu\nu} }[/math]. Note that in terms of the Planck mass, we set [math]\displaystyle{ \hbar = c =1 }[/math], whereas the last term keeps all the SI unit factors.

Note that in the presence of spinor fields, the Palatini action implies that [math]\displaystyle{ d\omega }[/math] is nonzero. So there's a non-zero torsion, i.e. that [math]\displaystyle{ \hat{\omega}^{ab}_\mu = \omega^{ab}_{\mu} + K^{ab}_\mu }[/math]. See Einstein–Cartan theory.