Einstein gravity as a gauge theory for the conformal group

General Relativity in dimension $n = p + q$ can be formulated as a gauge theory for the conformal group $\SO\left(p+1,q+1\right)$, along with an additional field reducing the structure group down to the Poincaré group $\ISO\left(p,q\right)$. In this paper, we propose a new variational principle for Einstein geometry which realizes this fact. Importantly, as opposed to previous treatments in the literature, our action functional gives first order field equations and does not require supplementary constraints on gauge fields, such as torsion-freeness. Our approach is based on the ``first order formulation'' of conformal tractor geometry. Accordingly, it provides a straightforward variational derivation of the tractor version of the Einstein equation. To achieve this, we review the standard theory of tractor geometry with a gauge theory perspective, defining the tractor bundle a priori in terms of an abstract principal bundle and providing an identification with the standard conformal tractor bundle via a dynamical soldering form. This can also be seen as a generalization of the so called Cartan-Palatini formulation of General Relativity in which the ``internal'' orthogonal group $\SO\left(p,q\right)$ is extended to an appropriate parabolic subgroup $P\subset\SO\left(p+1,q+1\right)$ of the conformal group.

Gauge Field Theory and Gravitation

The notion of 2-spinor soldering form allows a neat formulation, called the ‘tetrad-affine setting’, of a theory of matter and gauge fields interacting with the gravitational field. The latter is represented by a couple constituted by the soldering form and a 2-spinor connection. This approach is suited to describe matter fields with arbitrary spin and generic further internal structure. In particular one gets an approach to interacting Einstein-Cartan-Maxwell-Dirac fields, in which the only assumption is a complex bundle with 2-dimensional fibers: the needed bundles are obtained from it by natural geometric contructions, and any object which is not determined from these ‘minimal geometric data’ is assumed to be a dynamical field.

Spinor Bundles and Spacetime Geometry

Spinor bundles and other related bundles are constructed by exploiting the algebraic notions introduced in the previous chapter. The linear connections of these bundles and their mutual relations are studied. The notion of 2-spinor soldering form, or ‘tetrad’, yields the fundamental link between spinor algebra and spacetime geometry. The Fermi transport of spinors is studied in view of the definition of free states of particles with spin. The notion of Lorentzian distance is examined in relation to 2-spinor geometry, obtaining simplifications in regard to certain issues which are discussed, in the literature, in relation to the ‘algebraic approach’ to spacetime geometry.

On the symplectic formalism for general relativity

A covariant formalism for the hamiltonian formulation of general relativity in arbitrary dimensions is presented. Specifically, a presymplectic form on the solution space for the vacuum equations in n dimensions is given. The basic variables are taken to be a soldering form and a torsion-free connection on an SO ( p , q )-bundle over the space-time manifold M . It is shown how the present formalism is related to the standard ADM-formalism.