Quantum Gravity Meets Structuralism: Interweaving Relations in the Foundations of Physics

Abstract

The physics of gravity is inextricably connected to the geometry of space and time. In Einstein’s theory of general relativity—the best theory of classical gravity that we have—the geometry (curvature) of spacetime, as encoded in the metric tensor g μν, is identified with the gravitational field. But the metric field is also responsible for the characteristic structures of space and time too (causal structure, notions of distance, and so on). Hence, the metric plays a dual role in general relativity: it serves to generate both the gravitational field structures and the chronometric, spatio-temporal structures (cf. Stachel 1993). In the context of general relativistic physics, of course, the metric—and, therefore, the geometry of space—is dynamical : the metric on spacetime is not fixed across the physically admissible models of the theory (as it is in, for example, Newtonian and specially relativistic theories). The geometry of spacetime is affected by matter in such a way that different distributions of matter yield different geometries—the coupling and the dynamics is described by Einstein’s field equation. In other words, general relativity does not depend on the fixed metrical structure of spacetime; rather, the metric itself, and hence the geometry, comes only once a matter distribution has been specified (and the dynamical equation has been solved). Classically, this feature, called background independence,1 is rather