Volume Article Contents. Citing Article s :. Oxford Academic. Google Scholar. Cite Citation. Permissions Icon Permissions. Google Preview. Search ADS. Kyoto, Issue Section:. Download all figures. View Metrics. Email alerts New issue alert. Keywords: Differential geometry, Nonmetricity, Gauge theories of gravity, Metric-affine gravity. The renowned article by T. The former  puts gravity in a Riemann-Cartan geometry RCG , where torsion, T a , is allowed and spinorial matter can couple to gravity.
The second case [2, 3], describes a metric-affine geometry MAG where not only torsion, but also nonmetricity, ab , is allowed. In this case, the relation between the spin and affine connections entails also the deviation tensor, M ab. As it will become evident in this work, the deviation tensor is directly related to the nonmetricity. However, the nonmetricity cannot be derived from the algebra of the covariant derivatives, and thus, from the gauge procedure standpoint, it seems unnatural to consider it as a field strength.
In that context, the nonmetricity appears explicitly in the action of a gravity theory. Therefore, to highlight the issue of the physical role of the nonmetricity, we discuss in this work if the nonmetricity may or may not be eliminated from the geometry. This equation is taken here as the fundamental equation of the MAG. Further, we consider only Euclidean metrics on the tangent space.
This restriction enforces a particular case of the MAG. We then show that the nonmetricity eliminates, in a natural way, the symmetric degrees of freedom of the spin connection in the full covariant derivative of the vielbein. To be more specific, the symmetric part of the deviation tensor and the symmetric part of the spin connection cancel, eliminating all the symmetric degrees of freedom of the referred equation.
Thus, we have on this equation just Lorentz algebra valued quantities. The antisymm! Thus, as a consequence one identify a kind of geometric reduction in the form A d , SO d , on the tangent manifold. Obviously, the restriction to Euclidean tangent metrics plays a fundamental hole on the cancelation of the symmetric degrees of freedom. A more general and formal proof is left for a future work .
We also discuss the consequences of this geometric transition from the curvature and torsion points of view, as well as a from a general gravity action, not explicitly depending on the nonmetricity. It is shown that the nonmetricity is not completely eliminated, but it appears as a matter field de-coupled from the geometry. The same result is obtained by considering as starting point an action with an explicit dependence on the nonmetricity. This work is organized as follows: In Sect. II, we provide a brief review of the properties, notation and conventions of the MAG, used in this work.
In Sect. After that, the physical consequences of the reduction are discussed. Finally, in Sect.
IV, we display our conclusions. This is the so called metric-affine formalism.
The geometry is associated with a A d , diffeomorphism invariance. We define the covariant derivative, , by its action on a tensor field v according to. The curvature and torsion can be identified from. One may also study the MAG through the isometries of the affine group in the tangent space, , the well-known Einstein-Cartan formalism.
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The gauge covariant derivative D acts on the tangent space according to. In 5 the vielbein does not appear as a connection due to fact that, strictly speaking, it is not a connection. The vielbein is associated to the connection of translations modulo a compensating part that ensures the vector behavior of the vielbein. In fact, as discussed in  and references therein, the equivalence between translational connections and vielbein occurs when one assumes a breaking on the local translational invariance to a global one.
As said before, the results on this work are restricted to this particular class of the MAG and, therefore, iss taken as a heuristic proof of the geometric reduction from the MAG to the RCG. In order to characterize the MAG by a single geometric equation, we adopt to work with the full covariant derivative, , acting on a - mixed object. Here, for the sake of convenience, we take the vielbein itself,. Now, by defining the deviation tensor M as. Let us develop some useful algebraic properties of the symmetry of the tangent manifold.
The group decomposition of the affine group is. The space S d is formally defined as the coset space,. The affine group decomposition might be used to decompose the algebra-valued spin connection. For that, we expand it on the generators of the GL d , group, T ab ,.
It is important to emphasize again that we are considering Euclidean tangent metric tensors.
This condition is fundamental to obtain the first relation in Where q , associated to the nonmetricity through 20 , is the symmetric part of the spin connection. The fact that we are restricted to Euclidean tangent metrics, implies that the vielbein transforms always through O d group transformations. Thus, to preserve 20 one has to restrict the GL d , transformations to its Lorentz sector also for the symmetric part of the spin connection. Thus, a kind of symmetry breaking is enforced by the Euclidean condition on the tangent metric at the same level of the case of the vielbein and translations.
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This property is of remarkable importance in what follows. We start with the most general MAG based on the local affine gauge group, including local translations. Thus, we demand two ad hoc requirements:. The vielbein transforms as a vector under the Affine group transformations. As discussed at the Introduction, see , this requirement implies that the local translations breaks in favor of global ones. This second requirement is consistent with the topological nature of the GL d , group, which is noncompact. The compact sector is the SO d subgroup.
Moreover, this requirement ensures the validity of 20 in any gauge. Those requirements suggests that the general MAG is automatically driven to a subclass of it in which the gauge connection is the Lorentz spin connection while the vielbein and the symmetric spin connection are tensors. In fact, that is what occurs and the proof is straightforward:. Proof : From the decomposition 15 into a trivial part and a compact one, we see that the connection character of the spin connection lives at the Lorentz sector. Also, the Lorentz group is the stability group of the affine group.
This property establishes that the Lorentz group is the essential ingredient to define a geometry on the tangent manifold. Physically, it means that the Lorentz group is the sector which establishes a gauge theory for gravity. Thus, the vielbein and the symmetric part of the GL d , spin connection are taken as tensors under SO d gauge transformations. This statement is very supportive for the two requirements above stated. The cancelation of the nonmetric degrees of freedom with the symmetric sector of the spin connection and the redefinition of the spin connection according to 23 provides then a kind of natural geometric reduction of the tangent manifold.
We can then interpret the cancelation between the symmetric spin connection and the symmetric deviation tensor as an evidence of the decoupling of the nonmetric degrees of freedom from the MAG. Also, the redefinition 23 seems to be compatible with background field methods , since is Lorentz algebra valued.
In expression 22 , the relevant quantity of the geometry is the spin connection, which is algebra-valued on the Lorentz group.
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The tensor field is irrelevant for the geometry, since it can be absorbed into the spin connection. Thus, to carry or not is just a matter of convenience. Absorbing it, we are just changing the tetrad, e , in other to fit it into geodesic curves. Therefore, Statements 2 and 3 follows naturally.