VANISHING VIERBEIN IN GAUGE THEORIES OF GRAVITATION 1
A. Jadczyk 12
Institut für Theoretische Physik
der Universität Goettingen
Bunsenstrasse 9, D 3400 Göttingen
We discuss the problem of a degenerate vierbein in the framework of gauge theories of gravitation. We show that a region of space-time with vanishing vierbein but smooth principal connection can be, in principle, detected by scattering experiments.
This paper was send for publication in February 1984. It then took almost a year to get Referee Report. Here its is:
This paper contains a summary of some of the known aspects of gravity as a gauge theory and addresses, without substantive results, the phenomena associated with regions where the vierbein vanishes. What is new in the paper is connected with this latter question, but I find the discussion misleading and in any case not sufficiently well developed to justify publication in the paper's present form. Coordinate regions where the metric or vierbein vanishes must be treated with considerable care, as there are generally identifications of points to be taken into account. Pictures such as that of Fig. 2 are thus very misleading. The vanishing of the vierbein is not in general a coordinate-independent statement and what seems to be two points in one coordinate frame could well be seen to be just one in another. This is the case, e.g., with a uniformly accelerated frame in the x direction, where all points with and finite are to be identified.
December 17, 1984
My comments on the above report - as of today September 18, 1999:
In retrospect, I realize that the anonymous referee did not, in fact, reject the paper. He suggested that it did not justify publication in its present form. However, I was sufficiently discouraged by his comments to decide not to work on improving the form any further. Also in retrospect, it seems clear that the referee missed my point: when all the information that we have is included in the metric then, we can try to play the game of identifying the points to get rid of the "singularity." But within the framework I was discussing in this paper, there was also a principal connection using extra dimensions and giving extra-information. Gluing some space-time points together would create discontinuity of the connection. Vierbein, in the paper, was defined as one-form with values in an associated vector bundle and its vanishing was a coordinate independent statement. Yet, the referee was right on one point: the ideas of the paper could have been developed better. Although I have shied away from the subject in the intervening years, at the time I was suggesting that "faster-than-light teleportation" is possible but, indeed, I failed to provide the explicit description of a transdimensional remolecularizer working on this principle.
Hanson and Regge  (see also D'Auria and Regge  ) suggested that a gravitational Meissner effect might exist producing torsion vortices accompanied by vanishing of a vierbein. This in turn may indicate a phenomenon of "unglueing" of a principal (or ) bundle from the bundle of frames of the space-time manifold The idea that a proper arena for gauge theories of gravitation is an external principal bundle over rather than the frame bundle of has been put forward by many authors (see e.g. [3,4,5]). It is not our aim to justify this belief here. In fact what is relevant is not so much the choice of a bundle but the choice of dynamical variables and their dynamics. As long as a vierbein is thought of as representing a homomorphism of the two bundles, any distinction between and a subbundle of does not really matter. The distinction becomes important when the vierbein is interpreted as a linear map from the tangent spaces of into the fibers of a vector bundle associated with Such an interpretation is natural in gauge theories of the Poincaré group (see e.g. [4,5,6]) and of , where appears as a composite object.
The paper is organized as follows. in Sect. 3 we formulate a gauge theory of the Lorentz group and point out that the requirement of smoothness of the Lagrangian at a degenerate vierbein is a strong selection criterion. Only three terms survive the test, one of them having a "wrong parity", and two others are (cosmetically improved) the standard Einstein-Hilbert Lagrangian and the cosmological term. In Sect. 4 we briefly discuss the structure of the gauge theory of the Poincare group, and of the gauge theory considered by Stelle and West . In Sect. 5 we analyse relations between these abstract gauge theories and the conventional ones based on metric tensor and an affine connection. The relation can be visualized as follows. In an open region where the vierbein is nondegenerate the external bundle is glued with the help of to the frame bundle and the bundle connection can be interpreted as an affine connection on . When becomes degenerate the external bundle detaches from the frame bundle. The principal connection chooses to live in the external bundle and the affine connection dies. Two examples are given in Sect. 6 . The first is a kind of gravitational instanton considered by Hanson and Regge  and D'Auria and Regge . A zero of a vierbein is accompanied by non-zero torsion. The second example is an adaptation of a model discussed by Einstein and Rosen . The vierbein vanishes here on a 3-dimensional "bridge" connecting two mirror copies of the exterior Schwarzschild universe. It is interesting to notice that the principal connection continues smoothly over the bridge and need not be regularized as in the first example. The Einstein-Rosen bridge is therefore a torsion-free regular solution of vacuum field equations of gauge theory.
It was pointed out in  that "the vanishing vierbein at some point is not a disastrous feature of theory". It is one of the aims of the present note to point out that with an appropriate dynamics vanishing vierbein in a whole region need not be a disaster either. It is shown in Sect. 7 that such a "dead" region can have observable effects seen from outside, and that it introdu ces statistical elements (that is "freedom of choice") already on a classical level.
Let be a principal bundle, let be a representation of on a vector space and let denote the derived representation of the Lie algebra of . For every let denote the fundamental vector field on generated by An -valued -form, , on is called tensorial of type if and for all (see [10, p. 75] ). Let be the vector bundle associated with via the representation ( [10, p. 55] ). One can identify -valued tensorial -forms of type on with -valued -forms on (see [10, p. 76], [12, Ch. VII. 1], [11, Ch XX. 5]). This identification is extensively used in the literature and we shall use it too without further references. In particular the curvature 2-form of a principal connection on can be thought of as a 2-form on with values in , where denotes the adjoint representation of on
be a principal bundle over a -dimensional base manifold The fibers of are to be thought of as being a priori completely detached from fibers
of the frame bundle of Let denote the natural representation of on . The dynamical variables of a generalized Einstein-Cartan
theory are: a principal connection on , and a -valued -form on In a "pure gauge theory" 3 a Lagrangian -form should be constructed out of and alone. We demand that the action should be a smooth (and thus
nonsingular) function of field configuration variables
In particular should be continuous at Among geometrical objects at our disposal we find only six
candidates which have this property:
Variations of and are exact -forms and do not contribute to classical field equations. Owing to
the first Bianchi identity differs from by an exact form only. has internal parity different from that of and , and should not be combined with them unless is reduced to , which would imply dynamically preferred orientation of - a viable possibility.4 Introducing arbitrary coupling constants
, Euler-Lagrange equations for
where and are -forms representing the sources: energy-momentum and spin. It is important to notice that field equations (8) and (9) make sense for all smooth configurations , in particular for those with degenerate -s. However, the predictive power of these equations falls down with the rank of .
Replacing with one gets a gauge theory of the Poincaré group. The two important representations of on are and The dynamical variables of an gauge theory are (cf. [4,5]): a principal connection on , and a -valued -form on being the affine bundle over associated to The admissible Lagrangians are those of the gauge theory but now is not a primitive field - it is defined by . Given a field configuration one can always adapt a gauge (that is to use the translational freedom and choose a cross section of the principal bundle) in such a way that - thus, effectively, eliminating the field from the theory. In this gauge the soldering form , defined above as , coincides with the translational part of the connection This closes our discussion of the Poincaré group gauge theory: after gauge fixing it effectively reduces to Lorentz group gauge theory.
Another viable possibility is a gauge theory of , with five dimensional fibres, as discussed in . The dynamical variables of this theory are: an
- connection and a -valued -form where denotes the natural representation of on The Lagrangian 4-form is 
together with a constraint
Given a configuration one can always adapt gauge in such a way that The (generalized) vierbein can be defined by imposing the condition
in the adapted gauge.
be a field configuration of an gauge theory as discussed in Sect. 3. A point is said to be a critical point of if
otherwise is called a regular point. The set of all regular points of a smooth is an open subset of When
then is called degenerate. A linear space-time connection in (sometimes also called an affine connection on ; notice that we admit torsion here) is said to be compatible
with a given field configuration
i. e. if is parallel with respect to
when considered as a cross section of the fiber product
We have then
and so the - valued -form may be considered as a generalized torsion.
Given a configuration
there exists a unique on , being the open submanifold of consisting of all regular points of the vierbein, compatible with
Outside of - on the set of critical space-time points - the affine connection
will not, in general, exist. To see this observe that at regular
point we have, as a consequence of Eq. (13):
where are the coefficients of in a coordinate system From Eq. (13) we get
so that the part of which is responsible for the parallel transport of the length scale depends on the vierbein only (and not on the connection ). When becomes degenerate then , and therefore also , diverge. If is identically zero then the argument does not apply, and inside such a region any affine connection is compatible with such a configuration.
For every configuration
one defines a covariant "metric" tensor
is the diagonal constant matrix. The induced
scalar product in is nondegenerate if and only if is regular. On we have then
, where is a unique affine connection compatible with
. The standard Einstein-Hilbert Lagrangian density of is
where is the scalar curvature of . One easily finds that
where denotes the - component of a -form .
The identity (18) holds on . Outside of this region the left hand side of Eq. (18) is not defined. The right hand side is "almost" defined - if not for the . It is thus clear why taking (7) for the Lagrangian four-form is a better choice.
It is nondegenerate everywhere except at the origin where it vanishes.
The unique torsion-free connection form (compatible with ) given in
is flat and singular at A regularization
results in a non-zero torsion
Thus we get a globally defined non-singular connection in an external bundle. It induces space-time affine connection everywhere except at the origin. The induced connection has non-vanishing torsion.
Example 2.: Einstein and Rosen  considered two examples of solutions of (modified) vacuum
field equations of general relativity with a degenerate metric. Let us
show how both examples can be easily reformulated in terms of an gauge theory. The first model discussed in  describes a uniformly accelerated frame in a flat Minkowski
space. The second, described below, brings similar features with a non-flat
connection. One takes here
, and the vierbein is given by
It degenerates at (the "bridge"). The connection form can be represented by
At first sight it seems that is singular on the bridge in an analogy to (21).
However, since the vierbein is degenerate, the -forms do not form a basis at and therefore not every -form can be represented in terms of -s. In fact, we have
We have seen that certain Lagrangians are smooth functions of configurations even for degenerate -s, The requirement of smoothness is a strong selection criterion. In particular it rules out quadratic terms essential in theories with propagating torsion . It has to be understood whether a singularity at stabilizes a theory. Stability properties of metric theories of gravitation have been discussed by many authors . Theories admitting vanishing vierbein need, however, special considerations. Hanson and Regge  speculated about a possibility of having a "torsion foam" - a region with vanishing vierbein and well defined principal connection. No dynamical mechanism which would make such configurations stable is known. Nevertheless it can be interesting to look for a possible method of detecting such a phenomenon. Here we discuss motion of a test particle which meets on its way a domain with vanishing vierbein.
Consider scattering of spinless particles on a double-cone shaped region
where the vierbein vanishes (see Fig. 1). Equations of motion for a non-spinning
test particle are (see )
Notice that four-momentum and four-velocity are a priori considered as independent variables. It is only through equation (26) that momentum becomes co-linear with velocity, but this co-linearity needs to obeyed at regular space-time only. Not inside of the vanishing vierbein domain.
In our case, in regions and , where is invertible, momentum must be proportional to the velocity, as it follows from Eq. (26). In consequence trajectories and are geodesics. In region , where degenerates, momentum and velocity may be totally decoupled. Let (resp. ) denote the position and momentum of the particle when it enters (resp. leaves) vanishing-vierbein region . For given denote by the collection of all continuous paths with the following property: when paralelly transported along from to coincides with . Observe that a past trajectory , and the future trajectory are uniquely determined by and respectively. Observe also that two trajectories and both belong if and only if is an eigenvector belonging to the eigenvalue zero of the flux of the curvature operator through the surface enclosed between and .5 Let be the number of elements (measure) of . Would everywhere inside region then , where and are uniquely determined by the initial data and the geometry. If and in region then where is uniquely determined by the initial data. If so, then any observed non-zero dispersion of implies degeneracy of in region and reflects curvature effects in this region. Can some statistical effects that are normally attributed to quantum fluctuations be accounted for through the above mechanism?
The picture shown in Fig. 1 can be misleading in two ways. First, the
time of emerging of the probe from region is random. Second, a path of the particle should be continuous
but need not be differentiable.