### The Implications of General Covariance for Spacetime Ontology

John S. Earman

University of Pittsburgh

Dept. of History and Philosophy of Science

**Abstract**

While any well-posed spacetime theory can be written in generally covariant notation, Einstein’s general theory of relativity (GTR) is the first theory of modern physics to satisfy the substantive requirement of general covariance which demands that a theory admit the diffeomorphism group as a gauge symmetry. One implication is that the “observables” (a.k.a genuine physical magnitudes) of GTR–or of any theory that satisfies substantive general covariance–must be diffeomorphic invariants. Philosophers of science have not appreciated the force of this implication for spacetime ontology and ideology. I argue that the ontology/ideology that emerges does not fit comfortably with either the traditional relations vs. substantival alternatives. Next I note that while the observables of GTR are “constants of the motion,” this result does not flow from substantive general covariance alone; for example, it does not hold in unimodular gravity, a variant of standard GTR in which the cosmological constant is treated as a variable parameter. I offer some conjectures about what, in addition to general covariance, is needed to derive this result.