  We show how to reformulate Variable Speed of Light Theories (VSLT) in a
covariant fashion as Variable Light-Cone Theories (VLCT) by introducing two
vierbein bundles each associated with a distinct metric. The basic
gravitational action relates to one bundle while matter propagates relative to
the other in a conventional way. The variability of the speed of light is
represented by the variability of the matter light-cone relative to the
gravitational light-cone. The two bundles are related locally by an element M,
of SL(4,R). The dynamics of the field M is that of a SL(4,R)-sigma model gauged
with respect to local (orthochronous) Lorentz transformations on each of the
bundles. Only the ``massless'' version of the model with a single new coupling,
F, that has the same dimensions as Newton's constant $G_N$, is considered in
this paper. When F vanishes the theory reduces to standard General Relativity.
  We verify that the modified Bianchi identities of the model are consistent
with the standard conservation law for the matter energy-momentum tensor in its
own background metric.
  The implications of the model for some simple applications are examined, the
Newtonian limit, the flat FRW universe and the spherically symmetric static
solution.
