Tetrads

With the aim of introducing some concepts and notations, we consider first a theory based on a pseudo-Riemannian connected 4-dimensional spacetime manifold $\mathcal{M}$. In order to define the components of vector and tensor fields, we have to define a basis in the tangent spaces $T_x \mathcal{M}$ at all the points $x \in \mathcal{M}$. We can introduce local coordinates $x^{\mu}$ and the basis provided by the vectors $\partial_{\mu} = \partial / \partial x^{\mu}$ (remember that there is a one-to-one correspondence between vector fields and first order linear differential operators). We obtain in this way the holonomic components.

In particular, we indicate by $g_{\mu \nu}$ the holonomic components of the covariant metric tensor and by $g^{\mu \nu}$ the elements of the inverse matrix, which are the components of a contravariant tensor. These tensors can be used to raise and lower the holonomic indices of other tensors.

We can also introduce at every point $x \in \mathcal{M}$ an orthonormal tetrad (also called, in German, Vierbein) of four-vectors $e_i(x)$, with the property

$$e_i \cdot e_k = g_{\mu \nu} e^{\mu}_i e^{\nu}_k = g_{ik}.$$ (1.1)

By means of these bases we obtain the anholonomic components of vector and tensor fields. An explanation of the notations and the values of the constant anholonomic components $g_{ik} = g^{ik}$ of the metric tensor, which can be used to raise and lower the anholonomic indices, is given in Section 0.2. The use of a tetrad field (also called moving frame or repère mobile in French) is a powerful instrument in differential geometry, extensively used by E. Cartan [29].

We assume that the reader is acquainted with the simpler aspects of Riemannian geometry in the holonomic formalism (see for instance [30,31,32]) and in this Section we present some basic concepts of the anholonomic formalism, necessary for the motivation of the more general scheme introduced in Section 2.

The matrices $e^{\mu}_i$ and their inverses $e_{\mu}^i$ can be used to transform holonomic indices into anholonomic ones and vice-versa, namely to perform a change of basis. In particular, the formula

$$g_{\mu \nu} = e_{\mu}^i e_{\nu}^k g_{ik}$$ (1.2)

shows that the tetrads determine the metric tensor. The quantities $e_{\mu}^i$ are the components of a dual tetrad of covariant four-vectors $e^i$ that form a basis in the cotangent space $T^*_x \mathcal{M}$.

One can show that the set of all the tetrads is a differentiable manifold, which we indicate by $\mathcal{S}$. The tetrads with a common origin $x$ form a fiber and one can consider the differentiable projection mapping $\pi : \mathcal{S} \to \mathcal{M}$ associating to every tetrad $s \in \mathcal{S}$ its origin $x = \pi(s) \in \mathcal{M}$.

If $\Lambda^i{}_k$ is a $4 \times 4$ Lorentz matrix, it transforms a tetrad into another tetrad according to the formula

$$e_i \to e'_i = e_k \Lambda^k{}_i, \qquad e^i \to e'^i = (\Lambda^{-1})^i{}_k e^k,$$ (1.3)

which can also be written in the abbreviated form $s \to s' = s \Lambda$. We see that there is a right action of the Lorentz group $\mathcal{L}$ on the manifold $\mathcal{S}$. This means that $s (\Lambda \Lambda') = (s \Lambda) \Lambda'$ and the parentheses are not necessary. The action of $\mathcal{L}$ preserves the fibers (namely it commutes with $\pi$) and acts freely and transitively on every fiber. As a consequence, every fiber is diffeormorphic to $\mathcal{L}$. In this situation one says [25,26,27] that $\mathcal{S}$ is a principal fiber bundle with base $\mathcal{M}$ and structural group $\mathcal{L}$.

There is an important detail to be clarified. It is natural to assume that in every tangent space $T_x \mathcal{M}$ it is possible to choose in a continuous way a future cone, namely that $\mathcal{M}$ is time orientable [33]. This is a physically justified restriction to the topology of $\mathcal{M}$. Then it is natural to consider only tetrads with the timelike four-vector $e_0$ belonging to the future cone. As a consequence, $\mathcal{L}$ has to be the orthochronous Lorentz group.

The group $\mathcal{L}$ contains the space inversion and has two connected components. Of course, the fibers have the same property. If the connected manifold $\mathcal{M}$ is orientable, $\mathcal{S}$ has two connected components containing the left-handed and the right-handed tetrads. Otherwise, $\mathcal{S}$ is connected.

A principal fiber bundle is the natural arena for gauge field theories. In particular it has been shown [34,35,36,37,38] that General Relativity and other theories of gravitation can be formulated as gauge theories of the Lorentz group $\mathcal{L}$ or of the Poincaré group $\mathcal{P}$.