Gravity and Gauge Theory∗

Gauge theories are theories that are invariant under a characteristic group of “gauge” transformations. General relativity is invariant under transformations of the diffeomorphism group. This has prompted many philosophers and physicists to treat general relativity as a gauge theory, and diffeomorphisms as gauge transformations. I argue that this approach is misguided. The theories of three of the four fundamental interactions of nature are “gauge” theories. A central feature of such theories is their invariance under a group of local transformations, i.e., transformations which may vary from spacetime point to spacetime point. The characteristic group of these “gauge” transformations is called the “gauge group.” The theory of the fourth interaction, gravity, is general relativity. General relativity has its own invariance group, the diffeomorphism group. Insofar as one understands “gauge theory” to mean a theory in which “the physics” (more on the ambiguity of this term later) is invariant under a certain group of transformations, one might be tempted to construe general relativity as a gauge theory.1 Just such a construal Þgures in recent work of Belot (1996) and Belot & Earman (1999a,b), who follow many (but not all) physicists in treating the diffeomorphism group as a gauge group, and who draw implications for the “hole argument.”2 In this paper, I show that general relativity is not a gauge theory at all, in the speciÞc sense that “gauge theory” has in elementary particle physics. This issue is of crucial importance to attempts to quantize general relativity, because in quantum theory, the generators of gauge transformations are emphatically not treated as observables, while the generators of spatiotemporal (e.g., Lorentz) transformations are in fact the canonical observables. Thus the discussion in this paper sheds light on the origin of some of the deep and longstanding difficulties in quantum gravity, including the “problem of time”, a familiar form of which arises from treating the parametrized time-evolution of canonical general relativity as a gauge transformation.3 ∗Thanks to Arthur Fine, Chris Isham, and Bob Wald for helpful discussions.