A Theory of Gravitation Covariant under $Sp(4, \mathbf{R})$}

TL;DR

This paper develops a covariant gravitational theory on a 10-dimensional space of tetrads, invariant under the symplectic group $Sp(4, R)$, incorporating Goldstone fields called augmentons that relate to variable gravitational coupling.

Contribution

It introduces a novel $Sp(4, R)$-covariant Lagrangian framework for gravity with spontaneously broken symmetries and Goldstone fields linked to scalar-tensor theories.

Findings

Goldstone fields behave as components of an $SO(2,3)$ 5-vector.

The scalar square of augmentons can be interpreted as a Brans-Dicke scalar.

The theory incorporates Dirac fields as sources for augmentonic fields.

Abstract

We present a Lagrangian theory of gravitation that develops some ideas proposed several years ago. It is formulated on the 10-dimensional space $\mathcal{S}$ of the local Lorentz frames (tetrads) and it is covariant under the symplectic group $Sp(4, \mathbf{R})$, locally isomorphyic to the anti-de Sitter group $SO(2, 3)$. The corresponding transformation formulas contain a constant $ \ell$, besides the light velocity $c$. We also assume the covariance under the "total dilatations" of all the coordinates of the tangent spaces of $\mathcal{S}$. These symmetries, that we may call "augmented Lorentz covariance", are spontaneously broken and the corresponding (generalized) Goldstone fields, that we call "augmentons", behave as the components of a 5-vector of $SO(2, 3)$. Its square can be interpreted as the Brans-Dicke scalar field, that describes a variable gravitational coupling. The source of the augmentonic fields is provided by the Dirac fields. Finally, we discuss the physical relevance of the theory and its possible further developments.