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I present an approach to gravity in which the spacetime metric is constructed from a non-Abelian gauge potential with values in the Lie algebra of the group ^{3} cosmological metric.

The observational evidence in favor of Einstein's general theory of relativity has clarified that the spacetime manifold is not flat, and hence that it can be approximated by the flat Minkowski spacetime only over limited regions. Quantum Field Theory, and in particular the perturbative approach through the Feynman's integral, has shown the importance of expanding near a “classical” background configuration. Although we do not have at our disposal a quantum theory of gravity, it would be natural to take a background configuration which approximates as much as possible the homogeneous curved background that is expected to take place over cosmological scales accordingly to the cosmological principle. Therefore, it is somewhat surprising that most classical approaches to quantum gravity start from a perturbation of Minkowski's metric in the form _{μv}_{μv}_{μv}

Expanding over the flat metric is like Taylor expanding a function by taking the first linear approximation near a point. It is clear that the approximation cannot be good far from the point and that no firm global conclusion can be drawn from similar approaches. A good global expansion should be performed in a different way, taking into account the domain of definition of the function. So, a function defined over an interval would be better approximated with a Fourier series than with a Taylor expansion. Despite of these simple analogies, much research has been devoted to quantum gravity by means of expansions of the form

Actually, some years ago [

To start with let us observe that general relativity seems to privilege in its very formalism the flat background. Indeed, the Riemann curvature ℛ measures the extent by which the spacetime is far from flat, namely far from the background

Gauge theories were axiomatized in the fifties by Ehresmann [^{−1}(^{−1} (_{a}

A connection over

The tangent space at ^{−1}(_{U}^{μ}^{−1}(

We are used to define a manifold through charts ^{4}, ^{4}. Let us instead take them with value in a four-dimensional canonical manifold with enough structure to admit some natural metric. We shall use a matrix Lie group

We take as background metric the expression
^{−1}d_{g}_{g}G

Of course, we demand that _{B}_{4} or to the group

Thus let us consider the group ^{i}^{λ}r^{λ}r, λ_{g}_{u′gu′}_{†}(^{†},^{†}) = _{g}^{†} = ^{†} = ^{i}^{λ}r, r^{†}d^{†}d^{iλ}^{†}d^{3} section.

More specifically, let _{0} = _{i}, i_{μ}_{μ}

These calculations, first presented in [

In this section we shall suppose that _{g}

We mentioned that we wish to use charts

A second section ^{−1}(

We observe that
^{μ}^{a}^{b}

One can further ask whether the Einstein equations can be rephrased as dynamical equations for the potential

We recall that a tetrad field (vierbein)
_{a}

In order to obtain a dynamics for _{a}_{ab}

In the Abelian case _{4} (not in the

The

This work has been partially supported by GNFM of INDAM.

The author declares no conflicts of interest.