$δ$-flatness and its natural extension

Abstract: What, if anything, makes Minkowski spacetime a privileged local reference geometry in General Relativity? Recent work by \cite{WeatherallFletcher} argues: nothing. In response, I proposed a criterion of ``$δ$-flatness'', which bounds the magnitude of tidal acceleration within a tubular neighborhood by $δ$, and showed that Minkowski uniquely saturates the bound. Here I ask how $δ$-flatness generalises, and distinguish two kinds of extension. Reference-metric generalisations compare physical tidal acceleration with a reference tidal-force term constructed from another metric. Lorentzian signature obstructs every such comparison in which the reference fails to share the lightcones of the physical metric. Criterion modifications instead build the criterion from the physical geometry alone. The space of such modifications is wide, and I focus on those closest to $δ$-flatness, where one bounds the deviation of tidal acceleration from a non-zero target. The minimal candidate, which I call $δ$-MSS, has a singular limit that picks out the maximally symmetric spacetimes (Minkowski, de Sitter, Anti-de Sitter) via Schur's theorem. A restricted variant, which compares $g$ with a conformal rescaling of itself, either collapses to $δ$-MSS or fails to pick out a unique geometry. The chief result is that $δ$-flatness distinguishes Minkowski with no free parameter. $δ$-MSS, a weaker variant, picks out the maximally symmetric family at the cost of an externally supplied scalar.