Looking at spacetime atoms from within the Lorentz sector

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 $\rho$, function of the point $P$ and the vector $v^a$ at $P$, expressing their density. This has been done in the Euclideanized space one obtains through analytic continuation from Lorentzian sector at $P$. $\rho$ is defined in terms of a peculiar `effective' metric $q_{ab}$, also recently introduced, which stems from a careful request that $q_{ab}$ coincides with $g_{ab}$ at large (space/time) distances, but gives finite distance in the coincidence limit. This work reports on an attempt to introduce a definition of $\rho$ directly in the Lorentz sector. This turns out to be not a so trivial task, essentially because of the null case, i.e. when $v^a$ is null, as in this case we lack even a concept of $q_{ab}$. A notion for $q_{ab}$ in the null case is here proposed and an expression for it is derived. In terms of it, an expression for $\rho$ can be derived, which turns out to coincide with what obtained from analytic continuation. This, joined with the consideration of timelike/spacelike cases, potentially completes a description of $q_{ab}$ and $\rho$ within Lorentz spacetimes.