Which additional morphisms detect the scale? Natural rescalings of Lorentzian metrics and categories with lower-dimensional isometric embeddings

Let $\mathrm{Lor}_n$ be the category of connected $n$-dimensional Lorentzian manifolds with open isometric embeddings. A natural rescaling is an endofunctor that leaves manifolds and maps unchanged and replaces each metric by a new Lorentzian metric. Besides constant multiples, every smooth positive function of a local curvature invariant gives one; those built from the squared norm of the Weyl tensor ($n \ge 4$) or the Cotton tensor ($n = 3$) even generate an infinite-dimensional group of automorphisms. Such rescalings are conformal, so they are invisible to causal data. We ask which lower-dimensional isometric embeddings, modelling worldlines, Cauchy surfaces or submanifold probes, must be adjoined so that only constant multiples survive. As a baseline, unit-speed non-null geodesic segments of one sign force every natural rescaling to be constant. Our main results concern higher-dimensional sources. First, adjoining all isometric embeddings of one fixed source of dimension at least two, even with degenerate induced metric, forces every rescaling depending on the scalar curvature and the Weyl or Cotton norm to be constant, because jet perturbations that fix the induced metric vary these invariants independently over all real values. Second, for totally geodesic embeddings a dichotomy appears: a spacelike normal direction forces constancy, whereas totally geodesic spacelike hypersurfaces (time-symmetric data, where the magnetic Weyl part vanishes) preserve a non-trivial subgroup of automorphisms. Third, in dimension three, totally geodesic surfaces of either non-degenerate signature remove all non-constant Cotton rescalings.