  The canonical description is based on the prior choice of a spacelike
foliation, hence making a reference to a spacetime metric. However, the metric
is expected to be a dynamical, fluctuating quantity in quantum gravity. After
presenting the developments in the History Projection Operator histories theory
in the last seven years--giving special emphasis on the novel temporal
structure of the formalism--we show how this problem can be solved in the
histories formulation of general relativity. We implement the 3+1 decomposition
using metric-dependent foliations which remain spacelike with respect to all
possible Lorentzian metrics. This allows us to find an explicit relation of
covariant and canonical quantities which preserves the spacetime character of
the canonical description. In this new construction we have a coexistence of
the spacetime diffeomorphisms group Diff(M) and the Dirac algebra of
constraints.
