  We revisit the old question of what distinguishes the formulation of
spacetime geometry in terms of a Lorentzian metric physically from more general
geometric structures. Our approach to this question is operational and leads us
to also revisit the notion of local inertial frames, arising in operational
formulations of the equivalence and Mach's principle, both of which can be
interpreted in generalized geometries. We extend the notion of inertial
laboratory frames by taking serious that all matter inside the lab is
fundamentally quantum and considering how it may couple to the quantum
gravitational degrees of freedom generating the ambient effective spacetime
structure. This revolves around the question of which structures an agent
inside the inertial laboratory has available to operationally define the
orientation of their reference frame, an aspect on which both the equivalence
and Mach's principle are silent. We then contemplate the situation of a
completely inertial laboratory, which, in terms of the quantum matter
experiments inside it, is not only isolated from any matter outside it, but
also from a direct coupling to effective quantum gravitational degrees of
freedom. We formulate this as a local Mach principle (LMP): a local inertial
laboratory has to be self-sufficient, so that an agent can only resort to
relations among the quantum matter systems inside it to self-generate any
reference structures relative to which to orient their frame. The
transformations between different frame orientations thereby originate in the
local quantum matter structures. Combining this with dispersion relations leads
to various non-trivial compatibility conditions on the spacetime structures
encoded by them. This permits us to formulate additional operational
assumptions under which the LMP singles out Lorentzian metric spacetimes within
generalized geometries defined by dispersion relations.
