5.6 The Action of Gravity

Let us next consider the implication of the differential Bianchi identity (5.5.4). Taking into account the vanishing torsion condition and expanding along the vielbein basis we get:

If in the above equation we take the double contraction f ↔ a and b ↔ d we obtain the following immediate result:

So from the study of Bianchi identities, in presence of the soldering equation, we have come to the conclusion that there exists a unique linear combination of the Riemann tensor components which is symmetric and fulfills the unique property of being divergenceless D a Gab = 0. Such a property guarantees that it can function as the left hand side of an equation, whose right hand side is a conserved symmetric tensor. The object Gab is named the Einstein tensor for very good reasons. The intuitive idea, summarized by Eddington’s metaphor (see Fig. 5.1), that the space-time curvature is proportional to its energy-matter content, could find a precise mathematical formulation when Einstein discovered the tensor Gab . Indeed it was clear to him from the beginning that the conserved current sitting on the right hand side of the gravitational equation had to be the symmetric stress-energy tensor, namely the conserved current of the relativistic momentum P μ, which substitutes the Newtonian gravitational mass. The left hand side had similarly to be a symmetric conserved tensor, encoding the curvature of space-time, which should be quadratic in the derivatives of the metric gμν . The answer was clearly provided by Gab which allowed Einstein to write an equation of the form [2, 3]:

where G is Newton’s constant and Tab is the stress-energy tensor describing all matter components filling space-time. It is very much rewarding that (5.5.18) and the soldering condition can be simultaneously obtained as variational equations from a single action principle whose structure can be uniquely determined under few reasonable and compelling assumptions.

5.6 The Action of Gravity

The Yang-Mills action, as we described it in the previous sections is quadratic in the curvature field strength so that it leads to second order differential equations

for the gauge field Aμ. Yet in order to be written it requires either the use of the Hodge-duals or of the first order formulation based on the vielbein formalism we explained in Sect. 5.3.2. In any case knowledge of the gravitational field is intrinsically involved. In the case of gravity a quadratic action in the Riemann tensor is excluded since this would lead to differential equations of order four for the gravitational field which is the metric gμν or its substitute, the vielbein Eμa . This would violate the basic principles of classical mechanics where positions and velocities of the field should define a complete set of initial data. Hence the gravitational action has to be at most linear in the Riemann curvature tensor. Here is the first striking difference with Yang-Mills theory. Such a difference streams from the fact that the basic dynamical degrees of freedom of the gravitational field are not encoded in the connection itself (the Levi Civita connection) rather in the metric from which it derives. On the other hand the whole discussion of the present chapter taught us that the vielbein is in its own sake a connection, simply of a larger algebra, namely the Poincaré Lie algebra. Yet gravity cannot be simply regarded as the Yang-Mills theory of the Poincaré connection since it should include, as a built in principle the soldering (5.4.1) which allows to eliminate the spin connection ωab in terms of the vielbein.

So we come up with the following apparently very difficult task: we should construct an action functional

for the vielbein Ea and the spin connection ωab possessing the following obligatory properties:

(a)Lgrav(E, ω) should be an m-form (for m-dimensional gravity) constructed only with wedge products of Ea , ωab and their exterior differentials dEa , dωab , without the use of any Hodge duals, which would imply the necessary use of a metric tensor. Fulfilling these specified properties, the action Agrav will be automatically invariant against arbitrary diffeomorphisms of the base manifold M .

(b)Lgrav(E, ω) should be at most linear in the curvature two-form Rab .

(c)Lgrav(E, ω) should be invariant against arbitrary local Lorentz transformations defined as follows:

(d)The variation of the action with respect to the spin connection one-form ωab

should just implement the soldering condition, namely should have as unique solution the vanishing of the torsion two-form Ta = 0.

(e)The variation of the action with respect to the vielbein Ea should result in the unique algebraic condition that the Einstein tensor vanishes: Gab = 0.

Such a hard task has indeed a simple and unique solution. Before presenting it let us illustrate the reasons behind all the listed requirements.