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2 Vierbeins, et cetera 3 General theory |
It was developed by Élie Cartan, as part of (and a way of formulating) his method of moving frames. It operates with differential forms and so is computational in character, but has two other major aspects, both more geometric.
The first of these looks first to the theory of principal bundles (which one can call the general theory of frames). The ideal of a connection on a principal bundle for a Lie group G is relatively easy to formulate, because in the 'vertical direction' one can see that the required datum is given by translating all tangent vectors back to the identity element (into the Lie algebra), and the connection definition should simply add a 'horizontal' component, compatible with that. If G is a type of affine group with respect to another Lie group H - meaning that G is a semidirect product of H with a vector translation group T on which H acts, an H-bundle can be made into a G-bundle by the associated bundle construction. There is a T-bundle associated, too: a vector bundle, on which H acts by automorphisms that become inner automorphisms in G.
The first type of definition in this set-up is that a Cartan connection for H is a specific type of principal G-connection.
The second type of definition looks directly at the tangent bundle TM of the smooth manifold M assumed as the base. Here the datum is a certain type of identification of TM, as a bundle, as the 'vertical' tangent vectors in the T-bundle mentioned before (where M is natural identified as the zero section). This is called a soldering (sometimes welding): we now have TM within a richer setting, expressed by the H-valued transition data. A major point here, as with the previous discussion, is that it is not assumed that H acts faithfully on T. That immediately allows spinor bundles to take their place in the theory, with H a spin group rather than simply an orthogonal group.
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.
If you're looking for a basis-dependent index notation, see tetrad (index notation).
Suppose given differential manifold M of dimension n, and fixed natural numbers p and q with p+q = n. We suppose given a SO(p,q) principal bundle B over M (called the frame bundle), and a vector SO(p,q)-bundle V associated to B by means of with 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:TM->V where TM is the tangent bundle of M.
A (pseudo)Riemannian metric is defined over M as the push forward of η by e. To put it in other words, if we have two sections of TM, X and Y,
In the tetrad formulation of general relativity, the action, as a functional of the cotetrad e and a connection A over a four dimensional differential manifold M is given by
Cartan reformulated the differential geometry of (pseudo) Riemannian geometry; and not just those (metric) manifolds, but theories for an arbitrary manifold, including Lie group manifolds. This was in terms of "moving frames" (repère mobile) as an alternative reformulation of general relativity.
The main idea is to develop expressions for connectionss and curvature using orthogonal frames.
Cartan formalism is an alternative approach to covariant derivatives and curvature, using differential forms and frames. Although it is frame dependent, it is very well suited for computations. It can also be understood in terms of frame bundles, and it allows generalizations like the spinor bundle.
See also: Riemannian geometry, General relativityAspects of the theory
A general theory of frames
Identifying the tangent bundle
Vierbeins, et cetera
The basic ingredients
Constructions
A connection over V, A is defined as the unique connection satisfying these two conditions:
Now that we've specified A, we can use it to define a connection over TM by the pullback (or is it push forward?) by e;
Since what we now have here is a SO(p,q) gauge theory, the Riemann curvature F defined as is pointwise gauge covariant. This is simply the Riemann tensor in a different guise.The Palatini action
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.General theory