Abstract
We extend the established Galilean/relativistic structural divider in algebraic quantum field theory, namely, the absence of Reeh-Schlieder and of Tomita-Takesaki modular flow on local algebras of any Galilean Haag-Kastler net satisfying a natural axiom set augmented by the Bargmann-charge hypotheses (G7^*)(a) and (G7^*)(d) to curved backgrounds via the Newton-Cartan (c→∞) limit. We show, for the free Klein-Gordon field on Minkowski and on static globally hyperbolic spacetimes admitting a Post-Newtonian expansion, that a position-independent rest-energy rescaling produces in the limit a Galilean Haag-Kastler net satisfying the axioms of Ref. [1] in flat-space form (Minkowski) or in a curved-space modification (Killing-flow invariance and uniqueness of the vacuum replacing full translation invariance) appropriate to the static case. The Bargmann central charge equals the Klein--Gordon mass~m; the gravitational potential V(x) enters the limiting Schr\"odinger Hamiltonian but not the algebraic structure obstructed by the Galilean Reeh-Schlieder no-go theorem. The strengthened obstruction theorem of Ref. [1] extends to the modified curved-space setting on Fock representations, and the limiting net carries no modular flow on local algebras. Schwarzschild is treated as a worked example: the Killing horizon shrinks to a point, the Hartle-Hawking thermal state has no c→∞ limit, and the Boulware vacuum limits to the gravitational hydrogenic ground state. The Reissner-Nordstr\"om metric is included as a sanity check confirming that leading Post-Newtonian misses the electromagnetic content of the background. We discuss how Newton's constant~G enters the present (background-metric) framework only at the level of the limiting Hamiltonian, and indicate where dynamical-metric extensions would require G to play a structural role.