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We revisit the old question of what distinguishes the formulation ofspacetime geometry in terms of a Lorentzian metric physically from more generalgeometric structures. Our approach to this question is operational and leads usto also revisit the notion of local inertial frames, arising in operationalformulations of the equivalence and Mach's principle, both of which can beinterpreted in generalized geometries. We extend the notion of inertiallaboratory frames by taking serious that all matter inside the lab isfundamentally quantum and considering how it may couple to the quantumgravitational degrees of freedom generating the ambient effective spacetimestructure. This revolves around the question of which structures an agentinside the inertial laboratory has available to operationally define theorientation of their reference frame, an aspect on which both the equivalenceand Mach's principle are silent. We then contemplate the situation of acompletely inertial laboratory, which, in terms of the quantum matterexperiments inside it, is not only isolated from any matter outside it, butalso from a direct coupling to effective quantum gravitational degrees offreedom. We formulate this as a local Mach principle (LMP): a local inertiallaboratory has to be self-sufficient, so that an agent can only resort torelations among the quantum matter systems inside it to self-generate anyreference structures relative to which to orient their frame. Thetransformations between different frame orientations thereby originate in thelocal quantum matter structures. Combining this with dispersion relations leadsto various non-trivial compatibility conditions on the spacetime structuresencoded by them. This permits us to formulate additional operationalassumptions under which the LMP singles out Lorentzian metric spacetimes withingeneralized geometries defined by dispersion relations.

A local quantum Mach principle and the metricity of spacetime