Diffeomorphism Invariance and General Relativity

TL;DR

The paper clarifies the nature of diffeomorphism invariance in general relativity, distinguishing between two definitions and explaining why GR's invariance is unique compared to other classical spacetime theories.

Contribution

It introduces a formal criterion for immutable spacetime structure and demonstrates its relation to the presence of preferred coordinate systems in spacetime theories.

Findings

GR is not characterized by diff-invariance$_2$

GR's diffeomorphism invariance is manifest due to its inhomogeneous and mutable spacetime structure

A theory has a preferred coordinate system if and only if it has immutable spacetime structure

Abstract

Diffeomorphism invariance is often considered to be a hallmark of the theory of general relativity (GR). But closer analysis reveals that this cannot be what makes GR distinctive. The concept of diffeomorphism invariance can be defined in two ways: under the first definition (diff-invariance$_1$), both GR and all other classical spacetime theories turn out to be diffeomorphism invariant, while under the second (diff-invariance$_2$), neither do. Confusion about the matter can be traced to two sources. First, GR is sometimes erroneously thought to embody a "general principle of relativity," which asserts the relativity of all states of motion, and from which it would follow that GR must be diff-invariant$_2$. But GR embodies no such principle, and is easily seen to violate diff-invariance$_2$. Second, GR is unique among spacetime theories in requiring a general-covariant formulation, whereas other classical spacetime theories are typically formulated with respect to a preferred class of global coordinate systems in which their dynamical equations simplify. This makes GR's diffeomorphism invariance (in the sense of diff-invariance$_1$) manifest, while in other spacetime theories it lies latent -- at least in their familiar formulations. I trace this difference back to the fact that the spacetime structure is inhomogeneous within the models of GR, and mutable across its models. I offer a formal criterion for when a spacetime theory possesses immutable spacetime structure, and using this criterion I prove that a theory possesses a preferred class of coordinate systems applicable across its models if and only if it possesses immutable spacetime structure.