A local quantum Mach principle and the metricity of spacetime
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.