5.2 Locally Inertial Frames and the Vielbein Formalism

calculus of Ricci and Levi Civita there is no place for spinors, so that the gravitational interaction of, say, two electrons, cannot be described within the original metric formulation of General Relativity: an obvious absurdity. Cartan’s vielbein formalism allows instead for the gravitational coupling of spinor fields by means of the spin-connection, which replaces the Levi Civita connection. From this it follows that the vielbein formalism is actually more fundamental than the metric formalism and extends the latter which is not just equivalent to, rather it is contained in the former.

Finally comes the question of the Lagrangian formulation of Einstein field equations, namely their derivation from a variational principle. Also in this respect the vielbein formalism shows its superiority in comparison with the metric formalism since within the former framework both the action and its variation are much simpler.

Yet the particle interpretation of the theory is much clearer in the metric formulation and the spin two quantum of the gravitational field, the graviton, is best described as an infinitesimal fluctuation of the metric tensor gμν (x).

In view of these facts the present chapter is organized as follows. First, taking a purely geometric view-point we recast Riemannian and pseudo-Riemannian geometry in the vielbein formalism à la Cartan. In this framework the derivation of Bianchi identities which will play a fundamental role in establishing the form of Einstein equations is particularly simple.

Secondly using electro-magnetic theory and Yang-Mills theory as a term of comparison we discuss the linearized form of Einstein field equations and we argue why and how they should be non-linearly extended as a consequence of logical selfconsistency. The full non-linear form of the field equations is then uniquely determined with the help of the previously derived Bianchi identities.

Thirdly we discuss the action principle and we study its structural and symmetry properties, emphasizing similarities and fundamental differences with Yang-Mills Theories.

The main goal of the present chapter will be attained if the reader will develop a clear-cut consciousness that, without gravity the other gauge theories of fundamental interactions could not even be formulated. Yet, although a gauge theory by itself, gravity is very specific and all of its peculiarities can be traced back to the spin two of its fundamental quantum, opposed to the spin one of the non-gravitational interaction messengers.

5.2 Locally Inertial Frames and the Vielbein Formalism

As we anticipated in Sect. 3.2.5, the very idea of a Cartan repère mobile has its distant historical roots in the work of Frenet and Serret. In the context of (pseudo-) Riemannian geometry, it naturally emerges from the concept of Locally Inertial Frames, which is physically related to the Equivalence Principle and to Einstein’s Gedänken Experiment of the free falling lift. The whole matter is summarized in the following mathematical lemma.

Lemma 5.2.1 Let (M , g) be a (pseudo-)Riemannian manifold of dimension N and p M an arbitrary point of that manifold. Moreover let Up be an open neighborhood of p,whose points are labeled by coordinates xμ. In the same neighborhood we can always construct a new coordinate system xμ with the following properties:

(a)The coordinates xμ of the point p vanish: xμ(p) = 0.

(b)The value of the metric tensor at p in the x-coordinates is just equal to the signature of the metric:

(c) The coefficients of the Levi Civita connection vanish at p in the x system:

Γ λμν (p) = 0.

The coordinate system xμ with the above properties is named a Locally Inertial Frame.

Proof To prove the lemma we just need to consider the transformation properties of the metric tensor and of the Levi Civita connection. For the first we have:

while for the second, adapting to the case of the tangent bundle the general transformation rule (3.4.14) of a principal connection, we find:

Let us now implicitly define the new coordinates xλ as the solutions of the following quadratic equations:

where xμ(p) are the coordinates of the point p M in the old chart and Q and Ω are coefficients to be determined. By means of the position (5.2.3), condition (a) of the lemma is automatically satisfied: x(p) = 0. Consider next point (b) of the lemma and let us evaluate the metric coefficients gμν at p in the new frame. By using (5.2.3) in (5.2.1) we find:

where we purposely introduced matrix notation. Recalling the results of Sect. 3.6.1, we know that, whatever the non-degenerate symmetric matrix g(p) might be, we can

always find a non-degenerate matrix Q that reduces g(p) to be a diagonal matrix with eigenvalues that are all either plus one or minus one. Hence the coefficients Qμν in the transformation (5.2.3) are defined by the existing solution of such a

problem. Incidentally let us remark that we actually have an N (N −1) -dimensional

2

manifold of such solutions. Indeed, suppose that Q0 is such that:

and let Λ SO(p, N − p) be an arbitrary element of the pseudo-orthogonal group preserving the flat metric η, i.e. ΛT ηΛ = η. For any such Λ, the matrix Q = Q0Λ is another solution of the posed problem. Hence, as far as the linear terms are concerned, the transformation (5.2.3) is fixed up to SO(p, N − p) transformations. Consider next point (c) of the lemma. From (5.2.2) we get:

As anticipated the existence of locally inertial frames is the very content of the Equivalence Principle and the starting point of General Relativity. At any point in space-time, as long as we can make measures only in a very small neighborhood around us, we cannot decide whether gravitational accelerations felt by falling bodies are due to the presence of a genuine gravitational field or they are simply due to the non-inertial character of the chosen reference frame. Indeed local accelerations and other bending forces acting at point p on the world line of any test particle crossing through such a point are provided by the connection coefficients Γμνλ (p). As we have seen the latter can be set to zero by choosing an appropriate reference frame. Hence can we conclude that gravity is just an illusion? By no means! The subtle point in the above discussion is inherent to the word locally which precedes the qualifier inertial. In the proof of the lemma we have shown how to construct, for any point p M , a coordinate system in which Γμνλ vanishes at p, yet nothing was stated about the vanishing of Γμνλ at neighboring points. The opposite is true. In the coordinate system where the connection coefficients vanish at p, they are generically non-zero at all other points of the neighborhood Up . Obviously, choosing an infinitesimally close point p + dp we can repeat the above construction and find a new coordinate transformation xμ(x, p + dp) such that Γ λμν vanishes now at p + dp, yet the transformation xμ(x, p + dp) and the old one xμ(x, p) are not the same. This elementary observation is what leads to the concept of vielbein or, as Cartan named it in French, repère mobile.

5.2.1 The Vielbein

The word vielbein comes from the German viel = many and bein = legs. Originally this nomenclature was introduced in classical General Relativity as vierbein from the German vier = four and the already explained bein. It denoted the mobile reference frame of a four-dimensional Lorentzian space-time. Sometimes the same concept was referred to by means of the more classically sounding word tetrad. With the development of unified theories in diverse dimensions, there appeared in the literature all sort of German multiple enumerations of legs, i.e. the fünfbein, the zehnbein, the elfbein up to the sechsundfünfzigbein. It was finally tacitly agreed to use the word vielbein which covers all possible dimensions.

Consider a (pseudo-)Riemannian manifold (M , g) of dimensions m and in any local chart U M , let us construct the family of coordinate transformations which, point by point p U realize the three conditions mentioned in the lemma. The result is a multiplet of m bi-local functions ξ a (y, x) (a = 1, . . . , m) where xμ denote (in the old frame) the coordinates of the point p where the connection is required to vanish and the metric to become η, while yμ are the coordinates (also in the old frame) of a generic point belonging to the neighborhood of U . By construction we have ξ a (x, x) = 0. Let us now define the following m × m square matrix which smoothly depends on the point p, namely on the coordinates xμ:

where δba is the Krönecker delta, namely the identity matrix. By means of this construction we realize that Eaμ(x) are, at each point x, nothing else but the entries of the matrix Q introduced in the proof of Lemma 5.2.1 and that we have:

which is more inspiring since it reveals that the metric tensor can be regarded as a quadratic form in terms of another possibly more fundamental object. Indeed Cartan’s brilliant idea, of which we gave a first anticipation in Sect. 3.2.5, was precisely the following one: instead of utilizing the metric tensor as the primary field describing gravity, let us introduce a non-degenerate square matrix Eμa (x), which is no longer requested to be symmetrical, and let us regard (5.2.9) not as an equation, rather as a definition of the metric tensor gμν (x) in terms of the new primary object, named the vielbein. Let us single out the distinctive advantages of such a viewpoint.

First of all, multiplying by the coordinate differentials, we realize that: