A first-order purely frame-formulation of general relativity

Abstract

In the gauge natural bundle framework a new space is introduced and a first-order purely frame-formulation of General Relativity is obtained. In some of our recent works [1, 2, 3] a new geometrical framework for YangMills field theories and General Relativity in the tetrad-affine formulation has been developed.The construction of the new geometrical setting has been obtained quotienting the first-jet bundles of the configuration spaces of the above theories in a suitable way, resulting into the introduction of a new family of fiber bundles.In this letter we show that these new spaces allow a (covariant) first-order purely frame-formulation of General Relativity.The whole geometrical construction will be developed within the gauge natural bundle framework [4], which provides the suitable mathematical setting for globally describing gravity in the tetrad formalism.To start with, let M be a space-time manifold, allowing a metric tensor g with signature η = (1, 3): the manifold M will be called a η-manifold and the metric tensor canonical representation will be η µν := diag (−1, 1, 1, 1). Moreover, let L(M ) be the frame-bundle over M and P → M a principal fiber bundle over M with structural group SO (1, 3).The configuration space of the theory (the tetrad space) is a GL(4, ℜ) bundle π : E → M , associated to P × M L(M ) through the left-action λ : (SO(1, 3) × GL(4, ℜ)) × GL(4, ℜ) → GL(4, ℜ), λ(Λ, J; X) = Λ · X · J −1 (1) 1

General Relativity as a Constrained Gauge Theory

“…The recent developments in the study of J -bundles [3][4][5] allowed to build a firstorder frame formulation of General Relativity [1] on the one hand and a regular Hamiltonian formulation of Gauge theories [2] on the other. The interaction of the two aspects will allow to deduce General Relativity as a constrained variational problem for a Hamiltonian function of a SO(1, 3)-gauge theory.…”

On the Nonsymmetric Purely Affine Gravity

“…and the field equations (76) reproduce the field equations (35). Therefore, the additive term in the Lagrangian density (71) containing the second Ricci tensor Q µν does not affect the spacetime structure of the purely affine gravity.…”

“…This statement can be generalized to theories of gravitation with Lagrangians that depend on the full Ricci tensor and the second Ricci tensor [20,21], and to a general connection with torsion [21]. There also exist formulations of gravity in which the dynamical variables are: metric and torsion (Einstein-Cartan theory) [22,23,24,25,26,27], metric and nonsymmetric connection [28,29], tetrad and spin connection (EinsteinCartan-Kibble-Sciama theory) [30,31,32,33,34,35,36], spin connection [37,38,39], and spinors [40,41].…”

f(R) GRAVITY WITH TORSION: A GEOMETRIC APPROACH WITHIN THE $\mathcal{J}$-BUNDLES FRAMEWORK

“…J -bundles have been recently used to provide new geometric formulations of gauge theories and GR [1][2][3][29][30][31][32].…”