Scale Invariance in Gravity On The Light-Front

TL;DR

This paper reformulates general relativity using a double null foliation, revealing a hidden local conformal invariance and introducing a dual theory with enhanced gauge symmetries, including Weyl invariance.

Contribution

It presents a novel reformulation of GR that makes dynamical degrees of freedom and hidden symmetries explicit, employing symmetry trading and linking to scalar-tensor theories.

Findings

Revealed a hidden local conformal invariance in the null foliation formulation.

Constructed a dual theory with Weyl and diffeomorphism invariance restricted to null surfaces.

Established equivalence to a class of scalar-tensor theories with conformal coupling.

Abstract

In general relativity, the double null foliation is one for which $d$-dimensional spacetime is foliated by two families of intersecting null hyper surfaces (i.e. surfaces whose normal vectors are null) of $(d-1)$ dimensions. Their intersection is at space like surfaces of dimension $(d-2)$. This means that the leaves of this foliation are the space like surfaces of two dimensions lower than that of spacetime which are located by looking at the bundles of light rays that are going into, and emanating from them. Using this foliation, we present a reformulation of the theory which makes explicit the true dynamical degrees of freedom. This is accompanied by making manifest a hidden local conformal invariance. Revealing this local symmetry comes at the cost of preferring a parameterisation of the null hyper surfaces. More precisely, a preferred ruling of the null surfaces by their generators needs to be chosen so that the theory whose gauge symmetries are enhanced by Weyl invariance restricted to preserve the foliation, is equivalent to general relativity. I therefore find a dual theory that is locally equivalent to general relativity but possessing enhanced local gauge symmetries: Weyl local scale invariance and diffeomorphism invariance, both of which are restricted to preserve the two families of null surfaces. This theory is constructed employing the so called symmetry trading algorithm, which shall be described in detail in this essay. Alternatively, the theory can also be seen as a `phase' of a particular Scalar-Tensor theory, because it is equivalent to a particular class of configurations of a scalar field conformally coupled to general relativity.