Reconsiderations on the formulation of general relativity based on Riemannian structures

Abstract

We prove that some basic aspects of gravity commonly attributed to the modern connection-based approaches, can be reached naturally within the usual Riemannian geometry-based approach, by assuming the independence between the metric and the connection of the background manifold. These aspects are: 1) the BFlike field theory structure of the Einstein-Hilbert action, of the cosmological term, and of the corresponding equations of motion; 2) the formulation of Maxwellian field theories using only the Riemannian connection and its corresponding curvature tensor, and the subsequent unification of gravity and gauge interactions in a four dimensional field theory; 3) the construction of four and three dimensional geometrical invariants in terms of the Riemann tensor and its traces, particularly the formulation of an anomalous Chern-Simons topological model where the action of diffeomorphisms is identified with the action of a gauge symmetry, close to Witten's formulation of threedimensional gravity as a Chern-Simon gauge theory. 4) Tordions as propagating and non-propagating fields are also formulated in this approach. This new formulation collapses to the usual one when the metric connection is invoked, and certain geometrical structures very known in the traditional literature can be identified as remanent structures in this collapse.

Second-order Lagrangians admitting a first-order Hamiltonian formalism

“…The E-L equations for an affine second-order Lagrangian L, given as in the formula (4), are of third order and they are of second order if and only if the equations (5) hold (cf. [23,Proposition 2.2]).…”

“…Remark 1.1. The equations (5) simply means that for every index h the form η h = L hi α dy α i is d 10 -closed, namely d 10 η h = 0. Hence, there exist functions L i ∈ C ∞ (J 1 E) such that locally, the equations (ii) above being a consequence of the symmetry L hi α = L ih α .…”

“…The equations (5) admit a variational meaning. The Euler-Lagrange (or E-L for short) operator of an arbitrary second-order Lagrangian can be written in terms of the coefficients of the P-C form (see the formulas (1), ( 2)) as follows:…”

“…In this section we consider a new approach to BF Lagrangians (cf. [2], [5], [8], [14], [15]) generalizing the E-H functional. Let π : F (N ) → N be the principal Gl(n, R)-bundle of linear frames on N .…”

Post-Riemannian approach for the symplectic and elliptic geometries of gravity

“…These post-Riemannian approaches have been profusely explored in the past, and reference [1] is an excellent source backward (references therein) and forward (the recent literature quoting it). However, these explorations are in principle inexhaustible and, for example, along this direction the more recent results established by the present authors provide mainly a BF field theory structure for general relativity, including the BF gauge symmetries of the theory, which provides general relativity with the structure of a gauge field theory [2]; also the unification of gravity and gauge interactions in a four-dimensional field theory is reached using only the Riemannian connection as the fundamental field; further results concern with geometrical invariants in three and four dimensions using also the Riemannian connection, and particularly the formulation of an anomalous Chern-Simons topological model where the action of diffeomorphisms is identified with the action of a gauge symmetry group [3]. In this paper, we extend these results along the following lines: (a) the symplectic geometry of the covariant phase space of the new formulation, and the study of the fields at the asymptotic region; the results obtained in this part of the paper can be considered as an updating (from a post-Riemannian point of view) of the classical results established in [4]; additionally the torsion plays a new role, different from those considered in the classical treatment [1] and in the stringy point of view [5].…”