5.5 Einstein Field Equations

In approaching the issue of Einstein field equations we follow two alternative paths. First we discuss Bianchi identities which played a fundamental role in electrodynamics and have a similar fundamental relevance in gravity, as well. Then we construct a geometrical action motivating its uniqueness and from that action we derive the field equations. Secondly, following an inverse procedure, we provide the physical arguments that lead to an essentially unique form of the linearized field equations and we argue why and how they should be non-linearly extended. The iterative procedure, known under the name of Noether coupling, which reconstructs the full non-linear structure of the geometrical theory will be outlined.

Our starting point is provided by the definition of the Poincaré curvatures split into the Torsion and the Lorentz curvature two-form, respectively. Summarizing we write:

from which we can deduce very important consequences in case we assume the soldering condition Ta = 0. Let us begin by expanding the curvature two form Rab along the vielbein basis, which introduces the notion of Riemann tensor:

From the point of view of representation theory of the Lorentz group SO(1, m − 1), m being the number space-time dimension, the tensor Rabcd is not irreducible, rather it is the tensor product of two irreducible rank two antisymmetric represen-

tations, denoted , in the language of Young tableaux. A priori the total number of independent components is 14 m2(m − 1)2 and the decomposition into irreducible representations is as follows:

In the first line of (5.5.6) we just listed the available symmetries of the four-index tensors appearing in the decomposition. These are not yet irreducible, since for the pseudo-orthogonal group we can still construct invariants by taking traces with the invariant metric ηab . In the second line we enumerated the complete list of irreducible representations, the hat over a young tableau meaning that the corresponding tensor is irreducible, since it has been made traceless. In this way the total number of irreducible representations contained in Rabcd turns out to be six. However, inserting the soldering condition Ta = 0 into the Bianchi identity (5.5.3) we obtain:

This simply implies that any irreducible tensor contained in Rabcd that has at least three antisymmetric indices should vanish. In other words, the algebraic Bianchi identity (5.5.7) translates into the statement:

Taking this into account, the Riemann tensor of a Riemannian torsionless manifold decomposes into the following three irreducible representations of the Lorentz group SO(1, m − 1):

where Wcdab is named the Weyl tensor, Ric8 ab is the symmetric traceless Ricci tensor and R is the scalar curvature. Explicitly we can write:

where the normalization factors are chosen in such a way that by taking a contraction of the original Riemann tensor, for instance contracting a ↔ c, we obtain:

Instead of the traceless Ricci tensor it is customary to use in General Relativity the full Ricci tensor defined as follows: