220 5 Einstein Versus Yang-Mills Field Equations

The Stress-Energy Tensor of the Yang-Mills Field Let us consider the YangMills action in vielbein formalism spelled out in (5.3.35). The variation with respect to a vielbein δEu vielbein produces the following result:

where as usual the symbol denotes a contracted index. The reader will notice that the stress-energy tensor of a Yang-Mills field is traceless in 4-space-time dimensions m = 4.

The Stress-Energy Tensor of a Scalar Field The stress-energy tensor obtained from the scalar Lagrangian Ascalar defined in (5.6.21) is computed in a similarly easy way. First we get:

Having reviewed these examples let us now turn to the derivation of Einstein equations from the particle theorist’s view point. To do so let us begin with the weak field approximation of the geometrical theory.

5.7 Weak Field Limit of Einstein Equations

Resuming the discussion of previous sections and focusing on the physically interesting dimension m = 4, the Einstein equation can be written as follows:

The weak field approximation consists of assuming that the metric differs by a very small perturbation from the flat Minkowski metric. In other words we set:

where hμν (x) is a symmetric tensor field with respect tot the Lorentz group SO(1, 3). Under an infinitesimal diffeomorphism:

where ξ μ(x) denotes a four-vector of arbitrary functions, the transformation of the fluctuation field is immediately derived by linearizing the tensor transformation of

To first order in the perturbation h we can write the corresponding vielbein as follows:

Obviously this is not a unique solution. Infinitely many other can be obtained by performing infinitesimal Lorentz rotations. Each of them, however, will influence only the antisymmetric part of the matrix δEμa ≡ Eμa − δμa . We have gauge-fixed Lorentz symmetry by imposing that δEμa is symmetric. Inserting (5.7.6) into the soldering condition Ta = 0 we immediately obtain the solution for the spin connection which is calculated at the first order in the fluctuation field h:

Inserting this result into the expression for the curvature two-form Rab and calculating the Einstein tensor Gab to first order in the perturbation h, we obtain:

Notice that at this level of approximation we no longer make any distinction between the Latin and the Greek indices. The tangent bundle is soldered to the principal Lorentz bundle and the unperturbed vielbein is a Kronecker delta δμa which precisely identifies Greek and Latin indices. Hence the linearized field equation of Einstein gravity reduces to:

As the reader can easily check, the right hand side of the above equation is invariant under the gauge transformation (5.7.5).

So let us temporarily forget the geometrical origin of the gauge invariant linear equation (5.7.9) and investigate its properties.

5.7.1 Gauge Fixing

According to a general strategy, in presence of a local symmetry, such as (5.7.5), we look for gauge fixing conditions to be imposed on the field, in order to break such an invariance and eliminate the unphysical degrees of freedom. The most commonly used gauge-fixing for the gravitational field is the Hilbert-de Donder7 gauge that we also use in the second volume, while discussing gravitational waves. It is defined as follows. We first introduce the following useful combination of the hμν tensor and its trace

and then we impose the differential constraint:

Using this constraint the field equation (5.7.9) simplifies dramatically and reduces to:

which will be our starting point in the study of gravitational waves in Chap. 7.

The Hilbert-de Donder gauge breaks gauge invariance, yet not completely, because there still exist local transformations that preserve the gauge condition (5.7.11). Indeed consider the gauge transformation of (5.7.11); we have

Any quadruplet of functions ξ μ(x) which satisfy the following differential equation (where ∂ · ξ denotes the divergence of ξ μ):

corresponds to residual gauge transformations that can be utilized to further reduce the number of degrees of freedom of hμν . We have to dispose of all the spurious

7Théophile Ernest de Donder (1872–1957) was a Belgian mathematician and physicist. His most famous work dating 1923 concerns a correlation between the Newtonian concept of chemical affinity and the Gibbsian concept of free energy. Prof. de Donder was among the very first scientists who studied Einstein General Relativity and was one of its sustainers since its very beginning. He was a personal close friend of Einstein. His main field of activity was the thermodynamics of irreversible processes and the his work can be considered the early basis of the full-fledged development of this subject performed by Ilya Prigogine, the famous Russian born Belgian chemical-physicist who received the 1977 Nobel Prize.

degrees of freedom in a systematic way in order to single out the true physical ones carried by the gravitational field.

The simplest way to do so is to introduce light-cone coordinates and consider the propagation equation of the metric fluctuation in a specific arbitrarily chosen direction. Null coordinates in an m-dimensional Minkowski space are defined as follows:

where we have conventionally selected the first as the axis along which the gravitational quantum propagates. Correspondingly the Minkowskian metric takes the form:

In the vacuum, namely in a space-time region where the stress-energy tensor essentially vanishes, the metric perturbation γμν subject to the de Donder gauge condition, obeys the d’Alembert free propagation equation:

Plane waves are the simplest solutions of (5.7.19) and correspond to perturbation propagating at the speed of light in the conventionally chosen direction (the first

Fig. 5.4 A graphical representation of a transverse wave. The amplitude oscillates in the transverse direction to the propagation direction

axis). We have incoming and outgoing waves: in both cases the perturbation γμν depends only on one of the light-cone coordinates u, v: outgoing waves depend only on u, while incoming waves depend only of v. The discussion is absolutely similar in both cases, mutatis mutandis. Let us choose outgoing waves and set:

Such a position automatically satisfies (5.7.19). We have to consider the implementation of the Hilbert-de Donder gauge: ∂μγμν = 0. Since both hμν and γμν depend only on the variable u we have:

The last of the above equations is fixed by our physically chosen boundary conditions. At infinity, namely at very remote future times and in very distant space locations where the wave has not yet arrived, the metric is just Minkowski. Hence there is no constant part of γμν .

An m-tuplet of functions which satisfies the harmonic condition (5.7.14) and correspondingly preserves the Hilbert de Donder gauge is given by arbitrary function ξ μ(u) of the light-cone coordinate u. Let us see which further degrees of freedom of the tensor γμν can be removed by exploiting such transformations. It suffices to consider the explicit transformation of γ component-wise. We find:

It is evident that by means of these transformations we can set γ−μ = 0. Indeed ξi suffices to remove γ−i , the divergence of ξ− suffices to remove γ−− and γ−+ = γ +− was already zero on the basis of the previous argument. Furthermore the divergence of ξ+ suffice to remove the trace of the transverse tensor γij . Hence in every spacetime dimensions m the physical degrees of freedom of a quantum of the metric field, hereafter named the graviton are those of a traceless transverse tensor (see Fig. 5.4):