To begin, the models of relativity theory are relativistic spacetimes, which are pairs (M,gab) consisting of a 4-manifold M and a smooth, Lorentz-signature metric gab. The metric represents geometrical facts about spacetime, such as the spatiotemporal distance along a curve, the volume of regions of spacetime, and the angles between vectors at a point. It also characterizes the motion of matter: the metric gab determines a unique torsion-free derivative operator ∇, which provides the standard of constancy in the equations of motion for matter. Meanwhile, geodesics of this derivative operator whose tangent vectors ξa satisfy gabξaξb > 0 are the possible trajectories for free massive test particles, in the absence of external forces. The distribution of matter in space and time determines the geometry of spacetime via Einstein’s equation, Rab − 1/2Rgab = 8πTab, where Tab is the energy-momentum tensor associated with any matter present, Rab is the Ricci tensor, and R = Raa. Thus, as in Yang-Mills theory, matter propagates through a curved space, the curvature of which depends on the distribution of matter in spacetime.
The most widely discussed topic in the philosophy of general relativity over the last thirty years has been the hole argument, which goes as follows. Fix some spacetime (M,gab), and consider some open set O ⊆ M with compact closure. For convenience, assume Tab = 0 everywhere. Now pick some diffeomorphism ψ : M → M such that ψ|M−O acts as the identity, but ψ|O is not the identity. This is sufficient to guarantee that ψ is a non-trivial automorphism of M. In general, ψ will not be an isometry, but one can always define a new spacetime (M, ψ∗(gab)) that is guaranteed to be isometric to (M,gab), with the isometry realized by ψ. This yields two relativistic spacetimes, both representing possible physical configurations, that agree on the value of the metric at every point outside of O, but in general disagree at points within O. This means that the metric outside of O, including at all points in the past of O, cannot determine the metric at a point p ∈ O. General relativity, as standardly presented, faces a pernicious form of indeterminism. To avoid this indeterminism, one must become a relationist and accept that “Leibniz equivalent”, i.e., isometric, spacetimes represent the same physical situations. The person who denies this latter view – and thus faces the indeterminism – is dubbed a manifold substantivalist.
One way of understanding the dialectical context of the hole argument is as a dispute concerning the correct notion of equivalence between relativistic spacetimes. The manifold substantivalist claims that isometric spacetimes are not equivalent, whereas the relationist claims that they are. In the present context, these views correspond to different choices of arrows for the categories of models of general relativity. The relationist would say that general relativity should be associated with the category GR1, whose objects are relativistic spacetimes and whose arrows are isometries. The manifold substantivalist, meanwhile, would claim that the right category is GR2, whose objects are again relativistic spacetimes, but which has only identity arrows. Clearly there is a functor F : GR2 → GR1 that acts as the identity on both objects and arrows and forgets only structure. Thus the manifold substantivalist posits more structure than the relationist.
Manifold substantivalism might seem puzzling—after all, we have said that a relativistic spacetime is a Lorentzian manifold (M,gab), and the theory of pseudo-Riemannian manifolds provides a perfectly good standard of equivalence for Lorentzian manifolds qua mathematical objects: namely, isometry. Indeed, while one may stipulate that the objects of GR2 are relativistic spacetimes, the arrows of the category do not reflect that choice. One way of charitably interpreting the manifold substantivalist is to say that in order to provide an adequate representation of all the physical facts, one actually needs more than a Lorentzian manifold. This extra structure might be something like a fixed collection of labels for the points of the manifold, encoding which point in physical spacetime is represented by a given point in the manifold. Isomorphisms would then need to preserve these labels, so spacetimes would have no non-trivial automorphisms. On this view, one might use Lorentzian manifolds, without the extra labels, for various purposes, but when one does so, one does not represent all of the facts one might (sometimes) care about.
In the context of the hole argument, isometries are sometimes described as the “gauge transformations” of relativity theory; they are then taken as evidence that general relativity has excess structure. One can expect to have excess structure in a formalism only if there are models of the theory that have the same representational capacities, but which are not isomorphic as mathematical objects. If we take models of GR to be Lorentzian manifolds, then that criterion is not met: isometries are precisely the isomorphisms of these mathematical objects, and so general relativity does not have excess structure.
This point may be made in another way. Motivated in part by the idea that the standard formalism has excess structure, a proposal to move to the alternative formalism of so-called Einstein algebras for general relativity is sought, arguing that Einstein algebras have less structure than relativistic spacetimes. In what follows, a smooth n−algebra A is an algebra isomorphic (as algebras) to the algebra C∞(M) of smooth real-valued functions on some smooth n−manifold, M. A derivation on A is an R-linear map ξ : A → A satisfying the Leibniz rule, ξ(ab) = aξ(b) + bξ(a). The space of derivations on A forms an A-module, Γ(A), elements of which are analogous to smooth vector fields on M. Likewise, one may define a dual module, Γ∗(A), of linear functionals on Γ(A). A metric, then, is a module isomorphism g : Γ(A) → Γ∗(A) that is symmetric in the sense that for any ξ,η ∈ Γ(A), g(ξ)(η) = g(η)(ξ). With some further work, one can capture a notion of signature of such metrics, exactly analogously to metrics on a manifold. An Einstein algebra, then, is a pair (A, g), where A is a smooth 4−algebra and g is a Lorentz signature metric.
Einstein algebras arguably provide a “relationist” formalism for general relativity, since one specifies a model by characterizing (algebraic) relations between possible states of matter, represented by scalar fields. It turns out that one may then reconstruct a unique relativistic spacetime, up to isometry, from these relations by representing an Einstein algebra as the algebra of functions on a smooth manifold. The question, though, is whether this formalism really eliminates structure. Let GR1 be as above, and define EA to be the category whose objects are Einstein algebras and whose arrows are algebra homomorphisms that preserve the metric g (in a way made precise by Rosenstock). Define a contravariant functor F : GR1 → EA that takes relativistic spacetimes (M,gab) to Einstein algebras (C∞(M),g), where g is determined by the action of gab on smooth vector fields on M, and takes isometries ψ : (M, gab) → (M′, g′ab) to algebra isomorphisms ψˆ : C∞(M′) → C∞(M), defined by ψˆ(a) = a ◦ ψ. Rosenstock et al. (2015) prove the following.
Proposition: F : GR1 → EA forgets nothing.