Classical Theory of Fields

Galilean spacetime consists in a quadruple (M, t_a, h^ab, ∇), where M is the manifold R4; t_a is a one form on M; h^ab is a smooth, symmetric tensor field of signature (0, 1, 1, 1), and ∇ is a flat covariant derivative operator. We require that t_a and h^ab be compatible in the sense that t_ah^ab = 0 at every point, and that ∇ be compatible with both tensor fields, in the sense that ∇_at_b = 0 and ∇_ah^bc = 0.

The points of M represent events in space and time. The field t_a is a “temporal metric”, assigning a “temporal length” |t_aξ^a| to vectors ξ^a at a point p ∈ M. Since R4 is simply connected, ∇_at_b = 0 implies that there exists a smooth function t : M → R such that t_a = ∇_at. We may thus define a foliation of M into constant – t hypersurfaces representing collections of simultaneous events – i.e., space at a time. We assume that each of these surfaces is diffeomorphic to R3 and that h^ab restricted these surfaces is (the inverse of) a flat, Euclidean, and complete metric. In this sense, h^ab may be thought of as a spatial metric, assigning lengths to spacelike vectors, all of which are tangent to some spatial hypersurface. We represent particles propagating through space over time by smooth curves whose tangent vector ξ^a, called the 4-velocity of the particle, satisfies ξ^at_a = 1 along the curve. The derivative operator ∇ then provides a standard of acceleration for particles, which is given by ξⁿ∇_nξ^a. Thus, in Galilean spacetime we have notions of objective duration between events; objective spatial distance between simultaneous events; and objective acceleration of particles moving through space over time.

However, Galilean spacetime does not support an objective notion of the (spatial) velocity of a particle. To get this, we move to Newtonian spacetime, which is a quintuple (M, t_a, h^ab, ∇, η_a). The first four elements are precisely as in Galilean spacetime, with the same assumptions. The final element, η_a, is a smooth vector field satisfying η^at_a = 1 and ∇_aη^b = 0. This field represents a state of absolute rest at every point—i.e., it represents “absolute space”. This field allows one to define absolute velocity: given a particle passing through a point p with 4-velocity ξ^a, the (absolute, spatial) velocity of the particle at p is ξ^a − η^a.

There is a natural sense in which Newtonian spacetime has strictly more structure than Galilean spacetime: after all, it consists of Galilean spacetime plus an additional element. This judgment may be made precise by observing that the automorphisms of Newtonian spacetime – that is, its spacetime symmetries – form a proper subgroup of the automorphisms of Galilean spacetime. The intuition here is that if a structure has more symmetries, then there must be less structure that is preserved by the maps. In the case of Newtonian spacetime, these automorphisms are diffeomorphisms θ : M → M that preserve t_a, h^ab, ∇, and η^a. These will consist in rigid spatial rotations, spatial translations, and temporal translations (and combinations of these). Automorphisms of Galilean spacetime, meanwhile, will be diffeomorphisms that preserve only the metrics and derivative operator. These include all of the automorphisms of Newtonian spacetime, plus Galilean boosts.

It is this notion of “more structure” that is captured by the forgetful functor approach. We define two categories, Gal and New, which have Galilean and Newtonian spacetime as their (essentially unique) objects, respectively, and have automorphisms of these spacetimes as their arrows. Then there is a functor F : New → Gal that takes arrows of New to arrows of Gal generated by the same automorphism of M. This functor is clearly essentially surjective and faithful, but it is not full, and so it forgets only structure. Thus the criterion of structural comparison may be seen as a generalization of the latter to cases where one is comparing collections of models of a theory, rather than individual spacetimes.

To see this last point more clearly, let us move to another well-trodden example. There are two approaches to classical gravitational theory: (ordinary) Newtonian gravitation (NG) and geometrized Newtonian gravitation (GNG), sometimes known as Newton-Cartan theory. Models of NG consist of Galilean spacetime as described above, plus a scalar field φ, representing a gravitational potential. This field is required to satisfy Poisson’s equation, ∇^a∇_aφ = 4πρ, where ρ is a smooth scalar field representing the mass density on spacetime. In the presence of a gravitational potential, massive test point particles will accelerate according to ξⁿ∇_nξ^a = −∇^aφ, where ξ^a is the 4-velocity of the particle. We write models as (M, t_a, h^ab, ∇, φ).

The models of GNG, meanwhile, may be written as quadruples (M,t_a,h^ab,∇ ^̃), where we assume for simplicity that M, t_a, and h^ab are all as described above, and where ∇ ^̃ is a covariant derivative operator compatible with t_a and h^ab. Now, however, we allow ∇ ^̃ to be curved, with Ricci curvature satisfying the geometrized Poisson equation, R_ab = 4πρt_at_b, again for some smooth scalar field ρ representing the mass density. In this theory, gravitation is not conceived as a force: even in the presence of matter, massive test point particles traverse geodesics of ∇ ^̃ — where now these geodesics depend on the distribution of matter, via the geometrized Poisson equation.

There is a sense in which NG and GNG are empirically equivalent: a pair of results due to Trautman guarantee that (1) given a model of NG, there always exists a model of GNG with the same mass distribution and the same allowed trajectories for massive test point particles, and (2), with some further assumptions, vice versa. But in an, Clark Glymour has argued that these are nonetheless inequivalent theories, because of an asymmetry in the relationship just described. Given a model of NG, there is a unique corresponding model of GNG. But given a model of GNG, there are typically many corresponding models of NG. Thus, it appears that NG makes distinctions that GNG does not make (despite the empirical equivalence), which in turn suggests that NG has more structure than GNG.

This intuition, too, may be captured using a forget functor. Define a category NG whose objects are models of NG (for various mass densities) and whose arrows are automorphisms of M that preserve t_a, h^ab, ∇, and φ; and a category GNG whose objects are models of GNG and whose arrows are automorphisms of M that preserve t_a, h^ab, and ∇ ^̃. Then there is a functor F : NG → GNG that takes each model of NG to the corresponding model, and takes each arrow to an arrow generated by the same diffeomorphism. This results in implying

F : NG → GNG forgets only structure.

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