Subadditive Average Distances and Quantum Promptness

Abstract A central property of a classical geometry is that the geodesic distance between two events additive. When considering quantum fluctuations in metric or statistical superposition different spacetimes, additivity generically lost at level expectation values. In presence metrics, distances can be made diffeomorphism invariant by frame family free-falling observers pressureless fluid, provided we work sufficiently low energies. We propose to use average squared 〈d^2(x,y)〉 as proxy for understanding effective (or statistical) and emergent causal relations among such observers. At each point, 〈d^2(x,y)〉defines an tensor. However, due non-additivity, not (squared) associated with it. show departures from conveniently captured bi-local quantity C(x, y). Violations build up mutual separation x y correspond C< 0 (subadditive) C > (superadditive). Euclidean are always subadditive: they satisfy triangle inequality but generally fail saturate Lorentzian signature there no definite result about sign C, most physical examples give < exist counterexamples. The causality induced subadditive unorthodox pathological. Superadditivity violates transitivity relations. On these bases, argue expected outcome dynamical evolution, if relatively generic initial conditions considered.