On the other hand, varying the gravitational part of the action in the connection field, we easily see that it is just of the form discussed before but with a Kpqf tensor of the following form:

Hence, also in this case, there is an addition to the Levi Civita part of the spin connection which is contributed by the matter field, in this case the dilaton.

We will see in the second volume that this kind of couplings can be easily reabsorbed by means of a so called Weyl transformation. Indeed it suffices to introduce a new vielbein:

and, regarded as a functional of the new vielbein E2a , the gravitational part of the action becomes the standard one written in (5.6.3). Obviously the Weyl transformation (5.6.25) implies modifications of the scalar part of the action.

5.6.2 The Einstein Equation

Let us now consider the variation of the gravitational action (5.6.3) with respect to the vielbein Ea . For convenience we consider at once also the contributions from the matter action. Hence we obtain the following variational equation:

of the Lorentz group SO(1, m − 1). Correspondingly we can parameterize them in terms of two rank two tensors Tf g and Xf g , according to the following formulae:

where Gf g is the Einstein tensor defined in (5.5.17). Hence we obtain the equation:

Hence if we fix the numerical parameter β introduced above to the value β = 4π and we interpret Tf g as the stress energy tensor of the matter system, we get κ = 4πc2G reproducing exactly the Einstein equation anticipated in (5.5.18).

In this way we have shown that the gravitational action (5.6.3) does indeed the job it was required to do:

•Realizes the soldering of the Lorentz bundle with the tangent bundle by imposing the vanishing of the torsion Ta = 0 as a dynamical equation.

•Yields the Einstein equation (5.5.18).

In the case there are source terms for the torsion we shew that these lead to a modification of the Levi Civita spin connection by the addition of a Δωab term depending only on the matter fields which, once replaced back into the second order action, produces additional interactions between matter fields, that are just a gravitational effect.

5.6.3Conservation of the Stress-Energy Tensor and Symmetries of the Gravitational Action

A natural question which arises at this point concerns the conservation of the stressenergy tensor as defined by (5.6.27), namely as the variation of the matter action with respect to the vielbein Ea . The answer to such a question relates with the symmetries of the gravitational and of the matter actions. The main starting point of Einstein search for the correct right hand side of the gravitational field equation was the search of a symmetric tensor, linear in the components of the Riemann tensor whose divergence, in force of the Bianchi identities, should vanish in order to couple to the stress-energy tensor that, in Einstein’s approach, was by its own definition a conserved, divergenceless tensor. So let us consider the symmetries of the action functional (5.6.3).

As many times emphasized, the multiplet made by the spin connection plus the vielbein {ωab, Ea } can be regarded as a principal connection on a Principal Poincaré bundle that has space-time Mm as its base manifold. Yet the action we have constructed is not invariant under local gauge transformation of the full Poincaré Lie algebra whose infinitesimal form is:

where τ a is the gauge parameter associated with the translation generator P a while the antisymmetric εab are the gauge parameters associated with the Lorentz generators J ab . Enumerating the requirements that the gravitational action should satisfy, we asked for local Lorentz invariance which corresponds to formulae (5.6.31), (5.6.32) with εa = 0 and whose finite form was displayed in (5.6.3), but we did not enforce gauge translation invariance. Indeed by means of an integration by parts, an immediate calculation shows that under a gauge-translation the action functional (5.6.3) varies as follows: