  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.
