  Recently, a proposal has been made to figure out the expected discrete nature
of spacetime at the smallest scales in terms of atoms of spacetime, capturing
their effects through a scalar $\rho$, related to their density, function of
the point $P$ and vector $v^a$ at $P$. This has been done in the Euclideanized
space one obtains through analytic continuation from Lorentzian sector at $P$.
$\rho$ has been defined in terms of a peculiar `effective' metric $q_{ab}$, of
quantum origin, introduced for spacelike/timelike separated events. This metric
stems from requiring that $q_{ab}$ coincides with $g_{ab}$ at large
(space/time) distances, but gives finite distance in the coincidence limit, and
implements directly this way one single, very basic aspect associated to any
quantum description of spacetime: length quantization. Since the latter appears
a quite common feature in the available quantum descriptions of gravity, this
quantum metric $q_{ab}$ can be suspected to have a rather general scope and to
be re-derivable (and cross-checkable) in various specific quantum models of
gravity, even markedly different one from the other.
  This work reports on an attempt to introduce a definition of $\rho$ not
through the Euclidean but directly in the Lorentz sector. This turns out to be
not a so trivial task, essentially because of the null case, meaning when $v^a$
is null, as in this case it seems we lack even a concept of $q_{ab}$. A notion
for the quantum metric $q_{ab}$ for null separated events is then proposed and
an expression for it is derived. From it, a formula for $\rho$ is deduced,
which turns out to coincide with what obtained through analytic continuation.
This virtually completes the task of having quantum expressions of any kind of
spacetime intervals, with, moreover, $\rho$ defined directly in terms of them
(not in the Euclideanized space).
