Superconformal Spin/Field Theories: When Vector Spaces have same Dimensions: Part 1, Note Quote.


A spin structure on a surface means a double covering of its space of non-zero tangent vectors which is non-trivial on each individual tangent space. On an oriented 1-dimensional manifold S it means a double covering of the space of positively-oriented tangent vectors. For purposes of gluing, this is the same thing as a spin structure on a ribbon neighbourhood of S in an orientable surface. Each spin structure has an automorphism which interchanges its sheets, and this will induce an involution T on any vector space which is naturally associated to a 1-manifold with spin structure, giving the vector space a mod 2 grading by its ±1-eigenspaces. A topological-spin theory is a functor from the cobordism category of manifolds with spin structures to the category of super vector spaces with its graded tensor structure. The functor is required to take disjoint unions to super tensor products, and additionally it is required that the automorphism of the spin structure of a 1-manifold induces the grading automorphism T = (−1)degree of the super vector space. This choice of the supersymmetry of the tensor product rather than the naive symmetry which ignores the grading is forced by the geometry of spin structures if the possibility of a semisimple category of boundary conditions is to be allowed. There are two non-isomorphic circles with spin structure: S1ns, with the Möbius or “Neveu-Schwarz” structure, and S1r, with the trivial or “Ramond” structure. A topological-spin theory gives us state spaces Cns and Cr, corresponding respectively to S1ns and S1r.

There are four cobordisms with spin structures which cover the standard annulus. The double covering can be identified with its incoming end times the interval [0,1], but then one has a binary choice when one identifies the outgoing end of the double covering over the annulus with the chosen structure on the outgoing boundary circle. In other words, alongside the cylinders A+ns,r = S1ns,r × [0,1] which induce the identity maps of Cns,r there are also cylinders Ans,r which connect S1ns,r to itself while interchanging the sheets. These cylinders Ans,r induce the grading automorphism on the state spaces. But because Ans ≅ A+ns by an isomorphism which is the identity on the boundary circles – the Dehn twist which “rotates one end of the cylinder by 2π” – the grading on Cns must be purely even. The space Cr can have both even and odd components. The situation is a little more complicated for “U-shaped” cobordisms, i.e., cylinders with two incoming or two outgoing boundary circles. If the boundaries are S1ns there is only one possibility, but if the boundaries are S1r there are two, corresponding to A±r. The complication is that there seems no special reason to prefer either of the spin structures as “positive”. We shall simply choose one – let us call it P – with incoming boundary S1r ⊔ S1r, and use P to define a pairing Cr ⊗ Cr → C. We then choose a preferred cobordism Q in the other direction so that when we sew its right-hand outgoing S1r to the left-hand incoming one of P the resulting S-bend is the “trivial” cylinder A+r. We shall need to know, however, that the closed torus formed by the composition P ◦ Q has an even spin structure. The Frobenius structure θ on C restricts to 0 on Cr.

There is a unique spin structure on the pair-of-pants cobordism in the figure below, which restricts to S1ns on each boundary circle, and it makes Cns into a commutative Frobenius algebra in the usual way.


If one incoming circle is S1ns and the other is S1r then the outgoing circle is S1r, and there are two possible spin structures, but the one obtained by removing a disc from the cylinder A+r is preferred: it makes Cr into a graded module over Cns. The chosen U-shaped cobordism P, with two incoming circles S1r, can be punctured to give us a pair of pants with an outgoing S1ns, and it induces a graded bilinear map Cr × Cr → Cns which, composing with the trace on Cns, gives a non-degenerate inner product on Cr. At this point the choice of symmetry of the tensor product becomes important. Let us consider the diffeomorphism of the pair of pants which shows us in the usual case that the Frobenius algebra is commutative. When we lift it to the spin structure, this diffeomorphism induces the identity on one incoming circle but reverses the sheets over the other incoming circle, and this proves that the cobordism must have the same output when we change the input from S(φ1 ⊗ φ2) to T(φ1) ⊗ φ2, where T is the grading involution and S : Cr ⊗ Cr → Cr ⊗ Cr is the symmetry of the tensor category. If we take S to be the symmetry of the tensor category of vector spaces which ignores the grading, this shows that the product on the graded vector space Cr is graded-symmetric with the usual sign; but if S is the graded symmetry then we see that the product on Cr is symmetric in the naive sense.

There is an analogue for spin theories of the theorem which tells us that a two-dimensional topological field theory “is” a commutative Frobenius algebra. It asserts that a spin-topological theory “is” a Frobenius algebra C = (Cns ⊕ CrC) with the following property. Let {φk} be a basis for Cns, with dual basis {φk} such that θCkφm) = δmk, and let βk and βk be similar dual bases for Cr. Then the Euler elements χns := ∑ φkφk and χr = ∑ βkβk are independent of the choices of bases, and the condition we need on the algebra C is that χns = χr. In particular, this condition implies that the vector spaces Cns and Cr have the same dimension. In fact, the Euler elements can be obtained from cutting a hole out of the torus. There are actually four spin structures on the torus. The output state is necessarily in Cns. The Euler elements for the three even spin structures are equal to χe = χns = χr. The Euler element χo corresponding to the odd spin structure, on the other hand, is given by χo = ∑(−1)degβkβkβk.

A spin theory is very similar to a Z/2-equivariant theory, which is the structure obtained when the surfaces are equipped with principal Z/2-bundles (i.e., double coverings) rather than spin structures.

It seems reasonable to call a spin theory semisimple if the algebra Cns is semisimple, i.e., is the algebra of functions on a finite set X. Then Cr is the space of sections of a vector bundle E on X, and it follows from the condition χns = χr that the fibre at each point must have dimension 1. Thus the whole structure is determined by the Frobenius algebra Cns together with a binary choice at each point x ∈ X of the grading of the fibre Ex of the line bundle E at x.

We can now see that if we had not used the graded symmetry in defining the tensor category we should have forced the grading of Cr to be purely even. For on the odd part the inner product would have had to be skew, and that is impossible on a 1-dimensional space. And if both Cns and Cr are purely even then the theory is in fact completely independent of the spin structures on the surfaces.

A concrete example of a two-dimensional topological-spin theory is given by C = C ⊕ Cη where η2 = 1 and η is odd. The Euler elements are χe = 1 and χo = −1. It follows that the partition function of a closed surface with spin structure is ±1 according as the spin structure is even or odd.

The most common theories defined on surfaces with spin structure are not topological: they are 2-dimensional conformal field theories with N = 1 supersymmetry. It should be noticed that if the theory is not topological then one does not expect the grading on Cns to be purely even: states can change sign on rotation by 2π. If a surface Σ has a conformal structure then a double covering of the non-zero tangent vectors is the complement of the zero-section in a two-dimensional real vector bundle L on Σ which is called the spin bundle. The covering map then extends to a symmetric pairing of vector bundles L ⊗ L → TΣ which, if we regard L and TΣ as complex line bundles in the natural way, induces an isomorphism L ⊗C L ≅ TΣ. An N = 1 superconformal field theory is a conformal-spin theory which assigns a vector space HS,L to the 1-manifold S with the spin bundle L, and is equipped with an additional map

Γ(S,L) ⊗ HS,L → HS,L

(σ,ψ) ↦ Gσψ,

where Γ(S,L) is the space of smooth sections of L, such that Gσ is real-linear in the section σ, and satisfies G2σ = Dσ2, where Dσ2 is the Virasoro action of the vector field σ2 related to σ ⊗ σ by the isomorphism L ⊗C L ≅ TΣ. Furthermore, when we have a cobordism (Σ,L) from (S0,L0) to (S1,L1) and a holomorphic section σ of L which restricts to σi on Si we have the intertwining property

Gσ1 ◦ UΣ,L = UΣ,L ◦ Gσ0


Is There a Philosophy of Bundles and Fields? Drunken Risibility.

The bundle formulation of field theory is not at all motivated by just seeking a full mathematical generality; on the contrary it is just an empirical consequence of physical situations that concretely happen in Nature. One among the simplest of these situations may be that of a particle constrained to move on a sphere, denoted by S2; the physical state of such a dynamical system is described by providing both the position of the particle and its momentum, which is a tangent vector to the sphere. In other words, the state of this system is described by a point of the so-called tangent bundle TS2 of the sphere, which is non-trivial, i.e. it has a global topology which differs from the (trivial) product topology of S2 x R2. When one seeks for solutions of the relevant equations of motion some local coordinates have to be chosen on the sphere, e.g. stereographic coordinates covering the whole sphere but a point (let us say the north pole). On such a coordinate neighbourhood (which is contractible to a point being a diffeomorphic copy of R2) there exists a trivialization of the corresponding portion of the tangent bundle of the sphere, so that the relevant equations of motion can be locally written in R2 x R2. At the global level, however, together with the equations, one should give some boundary conditions which will ensure regularity in the north pole. As is well known, different inequivalent choices are possible; these boundary conditions may be considered as what is left in the local theory out of the non-triviality of the configuration bundle TS2.

Moreover, much before modem gauge theories or even more complicated new field theories, the theory of General Relativity is the ultimate proof of the need of a bundle framework to describe physical situations. Among other things, in fact, General Relativity assumes that spacetime is not the “simple” Minkowski space introduced for Special Relativity, which has the topology of R4. In general it is a Lorentzian four-dimensional manifold possibly endowed with a complicated global topology. On such a manifold, the choice of a trivial bundle M x F as the configuration bundle for a field theory is mathematically unjustified as well as physically wrong in general. In fact, as long as spacetime is a contractible manifold, as Minkowski space is, all bundles on it are forced to be trivial; however, if spacetime is allowed to be topologically non-trivial, then trivial bundles on it are just a small subclass of all possible bundles among which the configuration bundle can be chosen. Again, given the base M and the fiber F, the non-unique choice of the topology of the configuration bundle corresponds to different global requirements.

A simple purely geometrical example can be considered to sustain this claim. Let us consider M = S1 and F = (-1, 1), an interval of the real line R; then ∃ (at least) countably many “inequivalent” bundles other than the trivial one Mö0 = S1 X F , i.e. the cylinder, as shown


Furthermore the word “inequivalent” can be endowed with different meanings. The bundles shown in the figure are all inequivalent as embedded bundles (i.e. there is no diffeomorphism of the ambient space transforming one into the other) but the even ones (as well as the odd ones) are all equivalent among each other as abstract (i.e. not embedded) bundles (since they have the same transition functions).

The bundles Mön (n being any positive integer) can be obtained from the trivial bundle Mö0 by cutting it along a fiber, twisting n-times and then glueing again together. The bundle Mö1 is called the Moebius band (or strip). All bundles Mön are canonically fibered on S1, but just Mö0 is trivial. Differences among such bundles are global properties, which for example imply that the even ones Mö2k allow never-vanishing sections (i.e. field configurations) while the odd ones Mö2k+1 do not.

The Only Maximally Extended, Future-directed, Null and Timelike Geodesics in Gödel Spacetime are Confined to a Submanifold. Drunken Risibility.

Let γ1 be any maximally extended, future-directed, null geodesic confined to a submanifold N whose points all have some particular z ̃ value. Let q be any point in N whose r coordinate satisfies sinh2r = (√2 − 1)/2. Pick any point on γ1. By virtue of the homogeneity of Gödel spacetime, we can find a (temporal orientation preserving) global isometry that maps that point to q and maps N to itself. Let γ2 be the image of γ1 under that isometry. We know that at q the vector (t ̃a + kφa) is null if k = 2(1 + √2). So, by virtue of the isotropy of Gödel spacetime, we can find a global isometry that keeps q fixed, maps N to itself, and rotates γ2 onto a new null geodesic γ3 whose tangent vector at q is, at least, proportional to (t ̃a + 2(1 + √2)φa), with positive proportionality factor. If, finally, we reparametrize γ3 so that its tangent vector at q is equal to (t ̃a + 2(1 + √2)φa), then the resultant curve must be a special null geodesic helix through q since (up to a uniform parameter shift) there can be only one (maximally extended) geodesic through q that has that tangent vector there.

The corresponding argument for timelike geodesics is almost the same. Let γ1 this time be any maximally extended, future-directed, timelike geodesic confined to a submanifold N whose points all have some particular z ̃ value. Let v be the speed of that curve relative to t ̃a. (The value as determined at any point must be constant along the curve since it is a geodesic.). Further, let q be any point in N whose r coordinate satisfies √2(sinh2r)/(cosh2r) = v. (We can certainly find such a point since √2 (sinh2r)/(cosh2r) runs through all values between 0 and 1 as r ranges between 0 and rc/2) Now we can proceed in three stages, as before. We map γ1 to a curve that runs through q. Then we rotate that curve so that its tangent vector (at q) is aligned with (t ̃a + kφa) for the appropriate value of k, namely k = 2 √2/(1 − 2 sinh2r). Finally, we reparametrize the rotated curve so that it has that vector itself as its tangent vector at q. That final curve must be one of our special helical geodesics by the uniqueness theorem for geodesics.

The special timelike and null geodesics we started with – the special helices centered on the axis A – exhibit various features. Some are exhibited by all timelike and null geodesics (confined to a z ̃ = constant submanifold); some are not. It is important to keep track of the difference. What is at issue is whether the features can or cannot be captured in terms of gab, t ̃a, and z ̃a (or whether they make essential reference to the coordinates t ̃, r, φ themselves). So, for example, if a curve is parametrized by s, one might take its vertical “pitch” (relative to t ̃) at any point to be given by the value of dt ̃/ds there. Understood this way, the vertical pitch of the special helices centered on A is constant, but that of other timelike and null geodesics is not. For this reason, it is not correct to think of the latter, simply, as “translated” versions of the former. On the other hand, the following is true of all timelike and null geodesics (confined to a z ̃ = constant submanifold). If we project them (via t ̃a) onto a two-dimensional submanifold characterized by constant values for t ̃ as well as z ̃, the result is a circle.

Here is another way to make the point. Consider any timelike or null geodesic γ (confined to a z ̃ = constant submanifold). It certainly need not be centered on the axis A and need not have constant vertical pitch relative to t ̃. But we can always find a (new) axis A′ and a new set of cylindrical coordinates t ̃′, r′, φ′ adapted to A′ such that γ qualifies as a special helical geodesic relative to those coordinates. In particular, it will have constant vertical pitch relative to t ̃′.

Let us now consider all the timelike and null geodesics that pass through some point p (and are confined to a z ̃ = constant submanifold). It may as well be on the original axis A. We can better visualize the possibilities if we direct our attention to the circles that arise after projection (via t ̃a). The figure below shows a two-dimensional submanifold through p on which t ̃ and z ̃ are both constant. The dotted circle has radius rc. Once again, that is the “critical radius” at which the rotational Killing field φa is null. Call this dotted circle the “critical circle.” The circles that pass through p and have radius r = rc/2 are projections of null geodesics. Each shares exactly one point with the critical circle. In contrast, the circles of smaller radius that pass through p are the projections of timelike geodesics. The figure captures one of the claims – namely, that no timelike or null geodesic that passes through a point can “escape” to a radial distance from it greater than rc.


Figure: Projections of timelike and null geodesics in Gödel spacetime. rc is the “critical radius” at which the rotational Killing field φa centered at p is null

Gödel spacetime exhibits a “boomerang effect.” Suppose an individual is at rest with respect to the cosmic source fluid in Gödel spacetime (and so his worldline coincides with some t ̃-line). If that individual shoots a gun at some point, in any direction orthogonal to z ̃a, then, no matter what the muzzle speed of the gun, the bullet will eventually come back and hit him (unless it hits something else first or disintegrates).

Revisiting Twistors

In twistor theory, α-planes are the building blocks of classical field theory in complexified compactified Minkowski space-time. The α-planes are totally null two-surfaces S in that, if p is any point on S, and if v and w are any two null tangent vectors at p ∈ S, the complexified Minkowski metric η satisfies the identity η(v,w) = vawa = 0. By definition, their null tangent vectors have the two-component spinor form λAπA, where λA is varying and πA is fixed. Therefore, the induced metric vanishes identically since η(v,w) = λAπA μAπA = 0 = η(v, v) = λAπA λAπA . One thus obtains a conformally invariant characterization of flat space-times. This definition can be generalized to complex or real Riemannian space-times with non-vanishing curvature, provided the Weyl curvature is anti-self-dual. One then finds that the curved metric g is such that g(v,w) = 0 on S, and the spinor field πA is covariantly constant on S. The corresponding holomorphic two-surfaces are called α-surfaces, and they form a three-complex-dimensional family. Twistor space is the space of all α-surfaces, and depends only on the conformal structure of complex space-time.

Projective twistor space PT is isomorphic to complex projective space CP3. The correspondence between flat space-time and twistor space shows that complex α-planes correspond to points in PT, and real null geodesics to points in PN, i.e. the space of null twistors. Moreover, a complex space-time point corresponds to a sphere in PT, and a real space-time point to a sphere in PN. Remarkably, the points x and y are null-separated iff the corresponding spheres in PT intersect. This is the twistor description of the light-cone structure of Minkowski space-time.

A conformally invariant isomorphism exists between the complex vector space of holomorphic solutions of  ◻φ = 0 on the forward tube of flat space-time, and the complex vector space of arbitrary complex-analytic functions of three variables, not subject to any differential equation. Moreover, when curvature is non-vanishing, there is a one-to-one correspondence between complex space-times with anti-self-dual Weyl curvature and scalar curvature R = 24Λ, and sufficiently small deformations of flat projective twistor space PT which preserve a one-form τ homogeneous of degree 2 and a three-form ρ homogeneous of degree 4, with τ ∧ dτ = 2Λρ. Thus, to solve the anti-self-dual Einstein equations, one has to study a geometric problem, i.e. finding the holomorphic curves in deformed projective twistor space.

Relationist and Substantivalist meet by the Isometric Cut in the Hole Argument


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.

Classical Theory of Fields


Galilean spacetime consists in a quadruple (M, ta, hab, ∇), where M is the manifold R4; ta is a one form on M; hab is a smooth, symmetric tensor field of signature (0, 1, 1, 1), and ∇ is a flat covariant derivative operator. We require that ta and hab be compatible in the sense that tahab = 0 at every point, and that ∇ be compatible with both tensor fields, in the sense that ∇atb = 0 and ∇ahbc = 0.

The points of M represent events in space and time. The field ta is a “temporal metric”, assigning a “temporal length” |taξa| to vectors ξa at a point p ∈ M. Since R4 is simply connected, ∇atb = 0 implies that there exists a smooth function t : M → R such that ta = ∇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 hab restricted these surfaces is (the inverse of) a flat, Euclidean, and complete metric. In this sense, hab 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 ξa, called the 4-velocity of the particle, satisfies ξata = 1 along the curve. The derivative operator ∇ then provides a standard of acceleration for particles, which is given by ξnnξa. 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, ta, hab, ∇, ηa). The first four elements are precisely as in Galilean spacetime, with the same assumptions. The final element, ηa, is a smooth vector field satisfying ηata = 1 and ∇aηb = 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 ξa, the (absolute, spatial) velocity of the particle at p is ξa − ηa.

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 ta, hab, ∇, and ηa. 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, ∇aaφ = 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 ξnnξa = −∇aφ, where ξa is the 4-velocity of the particle. We write models as (M, ta, hab, ∇, φ).

The models of GNG, meanwhile, may be written as quadruples (M,ta,hab,∇ ̃), where we assume for simplicity that M, ta, and hab are all as described above, and where ∇ ̃ is a covariant derivative operator compatible with ta and hab. Now, however, we allow ∇ ̃ to be curved, with Ricci curvature satisfying the geometrized Poisson equation, Rab = 4πρtatb, 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 ta, hab, ∇, and φ; and a category GNG whose objects are models of GNG and whose arrows are automorphisms of M that preserve ta, hab, 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.