  The differing concepts of time in general relativity and quantum mechanics
are widely accused as the main culprits in our persistent failure in finding a
complete theory of quantum gravity. Here we address this issue by constructing
ADM-type theories \emph{in a particular time gauge} directly from first
principles. The principles are expressed as conditions on phase space
constraints: we search for two sets of spatially covariant constraints, which
generate symmetries (are first class) and gauge-fix each other leaving two
propagating degrees of freedom.
  One of the sets is the Weyl generator tr$(\pi)$, and the other is a
one-parameter family containing the ADM scalar constraint $\lambda R-
\beta(\pi^{ab}\pi_{ab}+(\mbox{tr}(\pi))^2/2))$. The two sets of constraints can
be seen as defining ADM-type theories with a maximal slicing gauge-fixing. This
work provides an independent, first principles derivation of ADM gravity.
  The principles above are motivated by a heuristic argument relying in the
relation between symmetry doubling and exact renormalization arguments for
quantum gravity, aside from compatibility with the spatial diffeomorphisms. As
a by-product, these results address one of the most popular criticisms of Shape
Dynamics: its construction starts off from the ADM Hamiltonian formulation.
  The present work severs this dependence: the set of constraints yield reduced
phase space theories that can be naturally represented by either Shape Dynamics
or ADM. More precisely, the resulting theories can be naturally "unfixed" to
encompass either spatial Weyl invariance (the symmetry of Shape Dynamics) or
refoliation symmetry (ADM).
