  Considering the scale dependent effective spacetimes implied by the
functional renormalization group in d-dimensional Quantum Einstein Gravity, we
discuss the representation of entire evolution histories by means of a single,
(d + 1)-dimensional manifold furnished with a fixed (pseudo-) Riemannian
structure. This "scale-space-time" carries a natural foliation whose leaves are
the ordinary spacetimes seen at a given resolution. We propose a universal form
of the higher dimensional metric and discuss its properties. We show that,
under precise conditions, this metric is always Ricci flat and admits a
homothetic Killing vector field; if the evolving spacetimes are maximally
symmetric, their (d + 1)-dimensional representative has a vanishing Riemann
tensor even. The non-degeneracy of the higher dimensional metric which
"geometrizes" a given RG trajectory is linked to a monotonicity requirement for
the running of the cosmological constant, which we test in the case of
Asymptotic Safety.
