Gravitation and spatial conformal invariance

TL;DR

This paper explores a modified theory of gravity derived from gauge formulations of General Relativity with a null internal vector, revealing Lorentz violation and conformal invariance under spatial transformations.

Contribution

It introduces a novel null-vector configuration in a gauge theory of gravity, linking de Sitter symmetry to spatial conformal invariance and analyzing resulting Lorentz-violating dynamics.

Findings

The theory exhibits invariance under all conformal symmetries except spatial translations.

Null vector configuration leads to Lorentz-violating modifications of gravity.

Establishes a connection between de Sitter group and spatial conformal group.

Abstract

It is well-known that General Relativity with positive cosmological constant can be formulated as a gauge theory with a broken SO(1,4) symmetry. This symmetry is broken by the presence of an internal space-like vector $V^A$, $A=0,...,4$, with SO(1,3) as a residual invariance group. Attempts to ascribe dynamics to the field $V^{A}$ have been made in the literature but so far with limited success. Regardless of this issue we can take the view that $V^A$ might actually vary across spacetime and in particular become null or time-like. In this paper we will study the case where $V^A$ is null. This is shown to correspond to a Lorentz violating modified theory of gravity. Using the isomorphism between the de Sitter group and the spatial conformal group, $SO(1,4)\simeq C(3)$, we show that the resulting gravitational field equations are invariant under all the symmetries, but spatial translations, of the conformal group C(3).