3.1. Representing Newton-Cartan Connections

In the physics literature, it is a commonplace that a Newton-Cartan connection can be specified by means of an observer vector field N and a 2-form F, at least up to ‘Milne gauge transformations’. But while this fact is useful for computations, it is hardly illuminating, and moreover, it is difficult to see what its relationship might be (if any) to what philosophers refer to as the ‘gauge symmetry’ of Neo-Newtonian models.Footnote ⁴ We must thus seek out a more incisive analysis of this gauge symmetry if we are to achieve our end; I claim that such an analysis will stem from the recent work of Bekaert and Morand (Reference Bekaert and Morand2016) on ‘representing’ a Newton-Cartan connection.

Bekaert and Morand (Reference Bekaert and Morand2016) note that although the requirement of compatibility with a Leibnizian geometry does not determine a unique Newton-Cartan connection (cf. the Levi-Civita connection), there is still room for an analogy between Newton-Cartan connections and torsionful Lorentzian connections, for the latter, too, are not uniquely determined by the requirement of compatibility. On this basis, they argue that a familiar strategy for representing torsionful Lorentzian connections can be transferred to the case of Newton-Cartan connections.

Their ‘physically perspicuous’ representational strategy takes the following form: First, notice that the affine space $C$ of Newton-Cartan connections is modeled on a vector space $V$ (whose vectors are differences between connections), which is in turn canonically isomorphic to the vector space Ω² of 2-forms on M. We can thus view $C$ as an affine space that is modeled on Ω².

Next, endow $C$ with a choice of origin (call the resulting vector space $C_0$) by means of the following vector space isomorphism:

(7)$\begin{array}{}& \stackrel{N}{\Theta }:C_0\to {\Omega }^2,\nabla \mapsto {\stackrel{N}{F}}_{\mathrm{ab}}≔\mathrm{-2}{\stackrel{N}{h}}_{c[a}{\nabla }_b]N^c,\text{where}N\in ℱ\mathrm{.}\end{array}$

Evidently, the origin or ‘reference point’ is $\stackrel{N}{{\Theta }^{\mathrm{-1}}}\left(0\right)$; that is, it is the connection such that $\stackrel{N}{F}\in {\Omega }^2$ vanishes. The Newton-Cartan literature dignifies such reference connections by means of the following definition: A (Newton-Cartan) connection is special just in case for some $N\in ℱ$, $\stackrel{N}{F}$ vanishes with respect to that connection. We denote a special connection by $\stackrel{N}{\nabla }$, which also helps us keep track of relevant observer vector field N.

Having made this choice of origin, it follows that an arbitrary Newton-Cartan connection $\nabla \in C$ can be uniquely specified as a vector from the origin. We thus say that the pair ($\stackrel{N}{\nabla }$, $\stackrel{N}{F}$) represents ∇ by providing (i) the reference point data $\stackrel{N}{\nabla }$ with respect to which ∇ is being represented and (ii) the directional data $\stackrel{N}{F}$ that unambiguously pick ∇ out with respect to the reference point. In fact, we obtain a slightly more efficient formulation if we recall a classical result; that is, for any $N\in ℱ$, there exists a unique special connection $\stackrel{N}{\nabla }$. The representation can then be expressed as (N, $\stackrel{N}{F}$).Footnote ⁵

Why should this representational strategy be deemed physically perspicuous? To see this, note first that the directional data $\stackrel{N}{F}$ have a physical interpretation: their physical content consists of the (linear) acceleration $\stackrel{N}{\alpha }$ and the vorticity $\stackrel{N}{\omega }$ experienced by a field of observers N.Footnote ⁶ To remind ourselves of this fact, we will call $\stackrel{N}{F}$ a Newton-Coriolis 2-form and (N, $\stackrel{N}{F}$) a Newton-Coriolis representation of ∇.

Next, we consider the physical interpretation of the reference point $\stackrel{N}{\nabla }$ relative to the directional data. Since $\stackrel{N}{\nabla }$ is special, it can be represented by (N, 0). It thus follows from the preceding discussion that N is geodesic and twistless with respect to $\stackrel{N}{\nabla }$. In other words, $\stackrel{N}{\nabla }$ can be interpreted as encoding a ‘generalized inertial structure’ (call this ‘inertial*’) with respect to which N is the vector field corresponding to inertial* trajectories. Thus, Newton-Coriolis representations are physically perspicuous because their reference point data provide a background inertial* structure with respect to which further physics (i.e., the forces encoded in a nonvanishing Newton-Coriolis 2-form) can be described; furthermore, by specifying such forces, we can pick out an arbitrary Newton-Cartan connection. This insight also gives us the resources to understand how Newton-Coriolis representations implicitly encode Saunders’s ‘vector relationist dynamics’. As articulated by Wallace (Reference Wallace2016), vector relationism can be divided into two components: (i) test particles that define idealized background inertial structures (and relative accelerations between pairs of test particles; see eqq. [56]–[58] of Wallace [Reference Wallace2016]) and (ii) motions of particles relative to the test particles. But the vector field of the trajectories of a test particle is nothing other than some $N\in ℱ$, and so it is evident that the background inertial structure of i is encoded in the reference point data $\stackrel{N}{\nabla }$, with respect to which the directional data correspond to part ii of Wallace’s schema. As a special case, the relative acceleration between two inertial* structures $\stackrel{N}{\nabla }$ and $\stackrel{N^{\prime }}{\nabla }$ can be expressed as $\stackrel{N}{F}$, that is, the directional data of $\stackrel{N^{\prime }}{\nabla }$ when they are represented by (N, $\stackrel{N}{F}$).

We now consider the sense in which the above representational strategy provides a perspicuous account of how the physicist’s ‘Milne gauge symmetry’ arises and an incisive notion of ‘gauge choice’ with respect to this symmetry. The root of this descriptive freedom can be found in the fact that isomorphism (7) manifestly depends on data over and above the Leibnizian geometry, that is, a choice of N. It is thus clear that there is a whole family ${\left\{\stackrel{N}{\nabla }\right\}}_{N\in ℱ}$ of reference connections that provide distinct—but equally good—representations of a Newton-Cartan connection.

What is the relationship between these representations? The answer is evident when we recall the Milne boost $N\mapsto N^{\prime }=N+V$ of section 2, which in turn induces the following Milne symmetry action on the space of Newton-Coriolis representations:

(8)$\begin{array}{}& \left(N,\stackrel{N}{F}\right)\mapsto \left(N^{\prime },\stackrel{N^{\prime }}{F}=\stackrel{N}{F}+d\stackrel{N,V}{\Phi }\right),\text{where}\stackrel{N,V}{{\Phi }_c}{=}_{h_{\mathrm{cb}}}^N{V}^b-\frac{1}{2}{}_{h_{\mathrm{ab}}}^N{V}^a{V}^b{t}_c.\end{array}$

This is of course nothing other than what we referred to earlier as the physicist’s ‘Milne gauge symmetry’. However, our intellectual path has given us a much deeper understanding of this concept: that it is underwritten by the above representational strategy and that a gauge choice (with respect to the Milne symmetry) is given by the representation (N, $\stackrel{N}{F}$). As we will soon see, the fact that a gauge choice refers to the reference data $\stackrel{N}{\nabla }$ will be crucial to our argument.

We are now in a position to discuss the invariance of the connection with respect to the Milne gauge symmetry and the sense in which this provides a ‘standard of sameness’ for Newton-Cartan connections. First, note that the connection is a well-defined object in its own right; it does not need to be described using the notion of Milne symmetry. However, given that we have adopted a descriptive scheme in which many different gauge choices ${\left\{(N,\stackrel{N}{F})\right\}}_{N\in ℱ}$ can be used to pick out the same connection, the question arises as to how the Milne symmetry can be used to describe the connection invariantly, that is, in a manner that is independent of any particular choice of gauge. As one might expect, this can be done by identifying a Newton-Cartan connection with the Milne orbit (i.e., gauge equivalence class) of one of its Newton-Coriolis representations (i.e., gauge choices). That such an identification makes sense is the content of the following proposition:

We thus have a Milne symmetry-based standard by which to judge that a set of models possesses a certain connection structure; that is, we require that the set of models has sufficient data to define the Milne orbit [(N, $\stackrel{N}{F}$)] corresponding to that connection.

3.2. Representing Newtonian Connections

The models of interest in this article all use Newtonian connections, that is, connections whose Newton-Coriolis 2-forms are closed.Footnote ⁷ Thus, given a Newtonian connection ∇, the space of its Newton-Coriolis representations is $ℱ\times {\Omega }_{\mathrm{closed}}^2$, and the invariant content of a Newtonian connection can be identified with the corresponding Milne orbit in this space. (Note that since, locally, a connection is Newtonian just in case it is special, the same point applies to special connections.)Footnote ⁸

Let us emphasize that this method of imposing the Newtonian condition does not change anything about the observer vector fields over which we quantify in order to describe different representations of the connection. Rather, it changes the structure of the directional data $\stackrel{N}{F}$ that we use to pick out the connection. We now explore the implications of this change.

From Poincaré’s lemma, we know that any closed form is locally an exact form. Thus, given a Newtonian connection, each of its Newton-Coriolis 2-forms $\stackrel{N}{F}$ has a local description as the ‘gauge field’ $\stackrel{N}{A}$, where $\stackrel{N}{A}$ is a 1-form such that $\stackrel{N}{F}=\stackrel{N}{\mathrm{dA}}$. It will also be clear from the gauge theory analogy that $\stackrel{N}{F}$ does not correspond to a single gauge field $\stackrel{N}{A}$ but rather to the equivalence class $\stackrel{N}{\left[A\right]}$ whose equivalence relation is given by the Maxwell gauge transformation $A\mapsto A+\mathrm{df}$, where f is an arbitrary 0-form. We will call (N, $\stackrel{N}{\left[A\right]}$) a principal connection representation, and we will call each of its Maxwell representatives (N, $\stackrel{N}{A}$) a gauge field representation.

Evidently, using a gauge field representation in place of a Newton-Coriolis representation allows us to describe the structure of ∇ in a way that increases the gauge symmetry acting on the directional data of a representation. On the one hand, the Milne symmetry (eq. [8]) acts on principal connection representations as follows:

(9)$\begin{array}{}& \left(N,\stackrel{N}{\left[A\right]}\right)\mapsto \left(N+V,\left[\stackrel{N}{A}+\stackrel{N,V}{\Phi }\right]\right),\end{array}$

where Φ is defined in (8). And on the other hand, each representation carries within it the action of the Maxwell gauge symmetry, as indicated by the Maxwell orbit [A]. Thus, an orbit of (9) is the equivalence class [(N, $\stackrel{N}{\left[A\right]}$)], where $\left(N,\stackrel{N}{A}\right)\sim \left(N^{\prime },\stackrel{N^{\prime }}{A}\right)$ just in case $N^{\prime }=N+V$ and $\stackrel{N^{\prime }}{A}=\stackrel{N}{A}+\stackrel{N,V}{\Phi }+\mathrm{df}$; let us call this a Milne-Maxwell orbit. By similar reasoning to that used in proposition 1, one can show

We will need one last important fact about gauge field representations, that is, that they afford us a way of constructing Milne-invariant quantities in terms of the gauge fields $\stackrel{N}{A}$. As is well known in the literature, there are three such quantities: a scalar, a vector, and a metric Milne-invariant, respectively:Footnote ⁹

(10)$\begin{array}{}& ɸ≔2\stackrel{N}{A_b}\left(N^b\right)-h^{\mathrm{ab}}\stackrel{N}{A_a}\stackrel{N}{A_b}\end{array}$

(11)$\begin{array}{}& Z^a≔N^a-h^{\mathrm{ab}}\stackrel{N}{A_b}\end{array}$

(12)$\begin{array}{}& g_{\mathrm{ab}}≔h_{\mathrm{ab}}^N+2t_{(a}{\stackrel{N}{A}}_b).\end{array}$

These quantities are Milne-invariant in the sense that they remain unchanged under the action of the Milne symmetry on N, $\stackrel{N}{A}$, and $\stackrel{N}{h}$, that is, the N-dependent quantities in terms of which they are exclusively defined (over and above the Leibnizian structure). Notice that Z is a vector field of observers and that g is nondegenerate.Footnote ¹⁰ Notice further that the invariants can be defined in terms of each other and the Leibnizian structure; for example, $g_{\mathrm{ab}}{Z}^b=t_aɸ$. While these Milne-invariants are interesting for a variety of reasons, their primary interest for us is that they allow us to construct a third representation: one that will finally let us make contact with the Trautman Recovery Theorem. In the next section, we use the insights that we have developed about ‘gauge’ to pursue this goal.

4.1. Motivating Proto-recovery

The Trautman Recovery Theorem assumes that its (curved) Newton-Cartan models involve a Newtonian connection ∇ that satisfies the curvature condition R2. It is furthermore easy to see that (R2-Equiv) R2 holds of ∇ if and only if R2′ holds of ∇ and there exists a twistless and rigid vector field of observers $N\in F$ with respect to ∇: $\stackrel{N}{\omega }=\stackrel{N}{\theta }=0$.Footnote ¹¹

In light of this equivalence, a well-known result called the Künzle-Ehlers Recovery Theorem proceeds to generalize Trautman Recovery by dropping the twistlessness condition on such a vector field; that is, it only requires the existence of a rigid vector field of observers.Footnote ¹² And since a rigid vector field of observers exists just in case R2′ holds, this generalization is tantamount to replacing the stronger curvature condition R2 with the weaker spatial flatness condition R2′.Footnote ¹³

I remind the reader of these classical results because I wish to highlight the fact that there exists a complementary, albeit previously unexplored, direction of generalization to the one pursued by Künzle-Ehlers Recovery, that is, where one drops rigidity and retains twistlessness instead of dropping twistlessness and retaining rigidity. Might it be possible to obtain a different generalized recovery result (from Künzle-Ehlers) if we required only the existence of a twistless vector field of observers? It turns out that it is indeed possible to prove such a result, which we call ‘Proto-recovery’ in order to highlight its more general and highly schematic nature. Furthermore, reconceiving Trautman Recovery as a special case of Proto-recovery will be precisely what is needed to illuminate the relationship between Milne symmetry and Trautman symmetry.

As we have just seen, facts linking curvature conditions to the existence of vector fields satisfying certain physical properties are crucial to understanding the Trautman and Künzle-Ehlers recovery results. Thus, in order to obtain our desired generalization, we need to understand a similar relationship that involves only twistlessness. This comes in the form of R1-Equiv: ∇ is Newtonian just in case there exists a family of twistless (with respect to ∇) vector fields whose associated Newton-Coriolis 2-forms are closed. We now turn to an explanation of this statement and its relationship with the notion of ‘gauge choice’ developed in section 3.

4.2. Twistless Gauge-Fixing and the Maxwell Representation

Recall that in section 3.2 the Newtonian condition was implemented in a way that placed no restriction on the observer vector fields that label representations. By contrast, R1-Equiv suggests a radically different way of interpreting the Newtonian condition: it invites us to view the condition as guaranteeing the existence of a special (because twistless) set of observer vector fields and, thus, also a special set of reference connections corresponding to these vector fields. Let us illustrate this point in the case of gauge field representations: R1-Equiv tells us that if ∇ is Newtonian, we can represent ∇ by means of the set of twistless representations ${\left\{(Z,\stackrel{Z}{A})\right\}}_{Z\in {ℱ}_{\mathrm{twistless}}}$, where ${ℱ}_{\mathrm{twistless}}$ denotes the set of twistless vector fields of observers. In other words, the Newtonian condition guarantees the existence of a particular ‘gauge-fixing’ for ∇, that is, a restriction to the set of twistless representations.

I now provide an explicit prescription for constructing these twistless representations. We will need the following proposition, which characterizes twistless vector fields:

Proposition 3. A vector field $Z^{\prime }\in ℱ$ is twistless with respect to a Newtonian connection ∇ just in case it has the form

(13)$\begin{array}{}& Z^{\prime }=Z-h\left(\mathrm{df}\right),\end{array}$

where f is an arbitrary 0-form and Z is the Milne-invariant vector field defined in equation (11).

Indeed, the proposition also characterizes Milne-invariant vector fields: evidently, Z is itself a twistless vector field, and any Z′ generated by exact 1-form shifts will be Milne-invariant.Footnote ¹⁴

Next, recall from equation (11) that the Milne-invariant vector field Z is constructed by means of a Milne boost $N\mapsto Z$ from an arbitrary vector field of observers $N\in F$, albeit one that is parameterized by the 1-form $\stackrel{N}{A}$. Thus, this Milne boost induces a Milne symmetry transformation from an arbitrary (N, $\stackrel{N}{A}$) to a twistless (Z, $\stackrel{Z}{A}$):

(14)$\begin{array}{}& N^a\mapsto Z^a=N^a+h^{\mathrm{ab}}\left(-\stackrel{N}{A_b}\right),\stackrel{N}{A}\mapsto \stackrel{Z}{A}=\stackrel{N}{A}+\stackrel{N,A}{\Phi }.\end{array}$

Upon evaluating the result of this transformation, we find that $\stackrel{Z}{A}=(1/2)ɸt$; that is, it is defined exclusively in terms of the clock 1-form t and the Milne-invariant scalar ɸ (eq. [10]). Furthermore, since Z is Milne-invariant, this result is independent of the representative of the Milne symmetry orbit ${\left[(N,\stackrel{N}{A})\right]}_{\mathrm{Milne}}$ that one chooses to apply the boost to.

We have just used the Newtonian condition to construct a twistless gauge choice, labeled by $Z\in ℱ$. However, we also know from proposition 3 that an entire family of such choices exists and that they are related by exact 1-form shifts.Footnote ¹⁵ Thus, in the parlance of physicists, the Newtonian condition only guarantees the existence of a ‘partial’ gauge-fixing; that is, we are left with a residual descriptive freedom after implementing this restriction. Analyzing this residual gauge freedom now leads us to the construction of a third kind of representation for Newtonian connections.

Following Bekaert and Morand (Reference Bekaert and Morand2016), let us consider the pair (Z, ɸ), which is canonically and mutually definable from the twistless representation $\left(Z,\stackrel{Z}{A}=(1/2)ɸt\right)$. We will call (Z, ɸ) a Maxwell representation of a Newtonian connection. The Maxwell representation is manifestly Milne-invariant because both Z and ɸ are Milne-invariant objects. However, notice that it is not invariant under the standard Maxwell symmetry action on gauge fields (i.e., $\stackrel{N}{A}\mapsto \stackrel{N}{A}+\mathrm{df}$), which is precisely the residual symmetry that was not ‘used up’ in our prescription for constructing a twistless representation.Footnote ¹⁶ In fact, it is easy to check that the standard Maxwell action on the gauge fields in the definition of Z induces the exact 1-form shifts (13) that parameterize the twistless vector fields (see prop. 3). Similarly, the Maxwell action on gauge fields induces the following transformation of ɸ to another Milne-invariant scalar ɸ′:

(15)$\begin{array}{}& ɸ\mapsto {ɸ}^{\prime }=ɸ+2{\left(\mathrm{df}\right)}_a{Z}^a-h^{\mathrm{ab}}{\left(\mathrm{df}\right)}_a{\left(\mathrm{df}\right)}_b.\end{array}$

By combining these transformations, we arrive at the induced Maxwell action $\left(Z,ɸ\right)\mapsto \left(Z^{\prime },{ɸ}^{\prime }\right)$ on Maxwell representations. This action parameterizes all twistless representations of a Newtonian connection; in other words, it is precisely the descriptive freedom that remains after we have gauge-fixed to the twistless representations.

We conclude this section by noting that, just as with the previous representational strategies, Maxwell representations allow us to describe the invariant content of Newtonian connections. The relevant notion of invariance is given by the notion of a Maxwell orbit, that is, an equivalence class [(Z, ɸ)] generated by the Maxwell action, whose identification with a Newtonian connection is expressed in the following proposition:

4.3. Proto-recovery

We now combine the resources developed in previous sections to prove a generalized recovery result that we call ‘Proto-recovery’.

Proof. The main idea of the proof is contained in the Milne Triangle of figure 1, which depicts the relationships between three connections ∇, $\stackrel{Z}{\nabla }$, and $\stackrel{N}{\nabla }$: $\stackrel{Z}{\nabla }$ is used as a reference point to represent ∇, and $\stackrel{N}{\nabla }$ is used as a reference point to represent both ∇ and $\stackrel{Z}{\nabla }$. The edges of the Milne Triangle represent the directional data of the representations: we decompose these data into the acceleration α (represented by solid arrows) and the vorticity ω (represented by dotted arrows) relative to the background inertial* structure described by the respective reference connections. The vorticity and acceleration are omitted on the left and right edges of the Milne Triangle, respectively, to indicate that these quantities vanish. Furthermore, we will see that the vorticities of the bottom and right edges are the same and that the accelerations of the bottom and left edges are the same up to a sign.

Figure 1. Milne Triangle.

We first prove part 1 of Proto-recovery, that is, that the data of an A-model ($L$, ∇, N) can be identified with a Maxwell orbit of B-models [($L$, $\stackrel{Z}{\nabla }$, ɸ_B, N)]. The argument can be divided into three parts:

1a The Left Edge: Using ∇ to construct ($\stackrel{Z}{\nabla }$, ɸ_B) up to Maxwell symmetry.

This part of the argument follows directly from the fact (see sec. 4.2) that any Newtonian connection ∇ has a Maxwell representation (Z, ɸ). We then define the scalar field data ɸ_B of a B-model by identifying it with the Milne-invariant scalar ɸ: ${ɸ}_B\equiv ɸ$. We have thus constructed the left edge of the Milne Triangle, and it is immediate from the considerations of section 4.2 that this construction holds up to Maxwell symmetry.

1b The Right and Bottom Edges: Constructing the relationship between the A-model and B-model dynamics.

The A-dynamics is captured by the statement that some vector field of observers N is geodesic with respect to the A-connection ∇. This fact can be perspicuously expressed by means of a Newton-Coriolis representation, that is, by representing ∇ through the reference data $\stackrel{N}{\nabla }$ and the directional data $\stackrel{\nabla ,N}{\omega }$ and $\stackrel{\nabla ,N}{\alpha }$. Since N is by hypothesis geodesic with respect to ∇, it is clear that the vorticity $\stackrel{\nabla ,N}{\omega }$ is the only free directional parameter; we have thus constructed the right edge of the Milne Triangle.

We now observe two facts. First, given the Maxwell representation (Z, ɸ) in 1a, we know that its corresponding gauge field representation is (Z, $\stackrel{Z}{A}=(1/2)ɸt$) and that its corresponding Newton-Coriolis representation is (Z, $\stackrel{Z}{\mathrm{dA}}$). Thus, we can readily compute the acceleration $\stackrel{\nabla ,Z}{\alpha }=-(1/2)h\left(dɸ\right)$. Second, by studying the way in which acceleration transforms under the Milne boost $N\mapsto Z$, we immediately obtain the following pair of equations for the accelerations relative to ∇ and $\stackrel{Z}{\nabla }$:Footnote ¹⁷

(16)$\begin{array}{}& \stackrel{\nabla ,Z}{\alpha }=\stackrel{\nabla ,N}{\alpha }+K\end{array}$

(17)$\begin{array}{}& \stackrel{\stackrel{Z}{\nabla },Z}{\alpha }=\stackrel{\stackrel{Z}{\nabla },N}{\alpha }+K.\end{array}$

Since $\stackrel{\stackrel{Z}{\nabla },Z}{\alpha }=0$ by the definition of a ‘special connection’, it follows that $\stackrel{\nabla ,N}{\alpha }=0$ just in case $\stackrel{\stackrel{Z}{\nabla },N}{\alpha }=-\stackrel{\nabla ,Z}{\alpha }=(1/2)h\left(dɸ\right)$. Notice two features of this argument: (i) it is independent of the choice of N; (ii) it holds for any Maxwell representation (Z, ɸ) of ∇.

At this point, we have shown that given an A-model, we can recover a B-model. We now extend this statement to the orbit of the B-model under the action of Maxwell symmetry.

1c. Showing that an A-model can be identified with a Maxwell-orbit of B-models.

We first stipulate that the Maxwell action on a B-model ($L$, $\stackrel{Z}{\nabla }$, ɸ, N) is to be defined as the Maxwell action on the Maxwell representation (Z, ɸ) part of the B-model data. From point ii of 1b, it is clear that this Maxwell action is compatible with the definition of a B-model, in the sense that every representative of the orbit [(Z, ɸ)] is also a B-model when conjoined with N; that is, it satisfies the dynamical relationship described in 1b. Furthermore, it is straightforward to show that a Maxwell transformation $\left(Z,ɸ\right)\mapsto \left(Z^{\prime },{ɸ}^{\prime }\right)$ of B-models satisfies part of the definition (eq. [5]) of a Trautman symmetry (where we set $\psi ≔{ɸ}^{\prime }-ɸ$); however, we do not have enough data to show that it satisfies the constraint (6).

It now follows from proposition 4 that an A-model can be identified with its corresponding Maxwell orbit of B-models. In this sense, the dynamics of the models decouples from the standard by which we judge an A-model to be ‘equivalent’ to a set of B-models, that is, the canonical identification between a Newtonian connection ∇ and its corresponding Maxwell orbit.

2. The Right and Bottom Edges: Showing that all models share the same vorticity.

First, we show that an A-model and one of its recovered B-models share the same vorticity; that is, we compare the dotted arrows on the right and bottom edges of the Milne Triangle. Just as in the case of acceleration in 1b, we can study how vorticity transforms under the Milne boost $N\mapsto Z$, thus obtaining the following pair of equations for the vorticities with respect to ∇ and $\stackrel{Z}{\nabla }$:

(18)$\begin{array}{}& \stackrel{\nabla ,Z}{\omega }=\stackrel{\nabla ,N}{\omega }+d\chi \end{array}$

(19)$\begin{array}{}& \stackrel{\stackrel{Z}{\nabla },Z}{\omega }=\stackrel{\stackrel{Z}{\nabla },N}{\omega }+d\chi ,\end{array}$

where χ is a 1-form parametrizing the boost. But since $\stackrel{\stackrel{Z}{\nabla },Z}{\omega }=0$ by the definition of ‘special connection’ and $\stackrel{\nabla ,Z}{\omega }=0$ by the fact that a Maxwell representation is a twistless gauge choice, it follows that the A-model vorticity (the dotted right edge of the triangle) is equal to the B-model vorticity (the dotted bottom edge of the triangle):

(20)$\begin{array}{}& \stackrel{\nabla ,N}{\omega }=\stackrel{\stackrel{Z}{\nabla },N}{\omega }=-d\chi .\end{array}$

Notice that the argument that the A-model and the B-model share the same vorticity is independent of our choice of N; however, the value of their vorticity does depend on N.

Since our choice of B-model in the above argument was arbitrary, it follows that all B-models share the same vorticity with respect to N. As a consistency check, we now show that the relationship (20) is invariant under the Maxwell transformation $\left(Z,ɸ\right)\mapsto \left(Z^{\prime },{ɸ}^{\prime }\right)$. This is so because the Maxwell boost $Z\mapsto Z^{\prime }$ induces the following relationship between vorticities:

(21)$\begin{array}{}& \stackrel{Z^{\prime }}{\omega }=\stackrel{N}{\omega }+d\left(\chi +\mathrm{df}\right),\end{array}$

where f is a 0-form and χ is the same 1-form as in equations (18) and (19). Since $d^2=0$, the B-model vorticities are invariant under Maxwell transformations. QED

We now discuss the ways in which gauge symmetry and gauge-fixings are implicated in Proto-recovery. Two kinds of partial gauge-fixings are relevant: first, the twistless gauge-fixing that stems from the Newtonian condition is used to define the Maxwell representation data of the B-models. Thus, the Maxwell gauge symmetry acts on the left edge of the Milne triangle, thereby parametrizing the set of B-models. Second, we can view the geodesic condition on the dynamical vector field $N\in F$ as a gauge-fixing condition on Newton-Coriolis representations of the A-model connection ∇. Milne symmetry transformations that preserve this ‘geodesic gauge’ thus act on the right edge of the triangle (while preserving the dynamical relationship between an A-model and a B-model). Since a fixed N is itself part of the dynamical data of the models, this symmetry in fact parameterizes a family of Proto-recovery results: it shows that the structure of the result is invariant under such transformations of the dynamical data (however, note that such transformations of N will in general change the value of the vorticity). We thus see that the recovery result involves a subtle interplay between the Maxwell and Milne gauge symmetries acting on the left and right edges of the triangle, respectively.

It is worth noting that Proto-recovery is a highly schematic result: it needs to be ‘filled in’ with more determinate content (e.g., source equations) before it can describe real physical scenarios. However, what we are concerned to emphasize here is the striking number of features of Trautman Recovery that can be traced to the Newtonian condition when it is interpreted as the existence of a twistless gauge-fixing: the relationship between the equations of motion (1) and (3), part (5) of the Trautman symmetry, and the nonuniqueness of recovered models all flow from this. In addition, we have the result of part 2, that is, that all models share the same vorticity. As we discuss in section 5, this result—in conjunction with proposition 4—provides a generalization of recent results (Weatherall Reference Weatherall2016; Dewar Reference Dewar2018) concerning the equivalence of Maxwell gravitation and Newton-Cartan gravitation.

The above discussion will continue to apply, mutatis mutandis, in our version of Trautman Recovery. To conclude this section, let us highlight a feature of Proto-recovery that will not carry over to Trautman Recovery: Recall that since every Newtonian connection ∇ is also a special connection (i.e., $\nabla \equiv \stackrel{N^{\prime }}{\nabla }$), ∇ can be represented by the Maxwell representation (N′, $ɸ=0$), where N′ is not only twistless but also geodesic. Thus, supposing that ∇ is the connection of an A-model, it is itself contained (under the description of a special connection) in the element ($L$, $\stackrel{N^{\prime }}{\nabla }$, 0, N) of the corresponding Maxwell orbit of B-models. As we are about to see, the additional gauge-fixing of the next section will pry the represented connection apart from its orbit of recovered models.

4.4. Recovering Trautman Recovery

We now argue that Trautman Recovery can be obtained as a special case of the Proto-recovery result from the previous section.

5.1. The Relationship with Maxwell Gravitation

We now comment on the relationship between our work and that of Dewar (Reference Dewar2018). Let us take a Maxwell space-time to be the structure ($L$, [∇_flat]), where [∇_flat] is a ‘vorticity equivalence class’ of flat connections; that is, two flat connections are equivalent just in case for any $N\in ℱ$, they share the same vorticity with respect to N.Footnote ¹⁹ If we adopt this terminology, then it is evident that Proto-recovery implicitly defines a generalized notion of Maxwell space-time: Let a Proto-Maxwell space-time be the structure ($L$, [∇]), where [∇] is a vorticity equivalence class of Newtonian connections. (Thus, a Maxwell space-time is an example of a Proto-Maxwell space-time, but the latter concept is more general because its connection representatives need not be flat.) But by the reasoning in part 2 of Proto-recovery, Newtonian connections fall into the same vorticity class just in case they are the reference data (i.e., special connections {$\stackrel{Z}{\nabla }$}, where Z is a twistless time-like vector field) of the Maxwell representations in a Maxwell orbit. Thus, a Proto-Maxwell space-time corresponds to a Maxwell orbit of these reference connections, and the ‘vorticity class’ structure of the space-time is preserved by the corresponding Maxwell transformations.

We have just seen that a Proto-Maxwell space-time can be obtained from an orbit of Maxwell representations by omitting the directional data (i.e., the Milne-invariant scalars) of these representations. We now consider how including these directional data allows one to ‘set a gravitational model’ on Proto-Maxwell space-time.

One of the contributions of Dewar (Reference Dewar2018) is that it defines a clear notion of a Maxwell model of gravitation: this is taken to be the structure ($L$, [∇_flat], ρ, N), where any representative $\nabla \in \left[{\nabla }_{\mathrm{flat}}\right]$ satisfies the source equation ${\nabla }_a{\alpha }^a=\mathrm{-4}\pi \rho$ and the equation of motion $\nabla {[}^a{\alpha }^c]=0$. While the B-models of Proto-recovery are not equipped with sources, they do give us the resources to provide a deeper explanation for these equations of motion. To see this, consider the generalization of a Maxwell model to what we call a Proto-Maxwell model, that is, a structure ($L$, [∇], N), whose equations of motion are defined as follows: for any representative $\nabla \equiv \stackrel{Z}{\nabla }\in [\nabla ]$, we can use its directional data ɸ to compute that ∇^[aα^c] equals a sum of terms linear in ∇^aZ^b, which we will take to be our Proto-Maxwell equations of motion. If we apply the further gauge-fixing of section 4.4, that is, by taking the twistless Z to be in addition rigid (so ${\nabla }^a{Z}^b=0$), we then recover precisely the equations of motion for Maxwell gravitation.

Finally, note that proposition 4 furnishes us with a ‘proto’ analog of propositions 4 and 5 of Dewar (Reference Dewar2018): by means of the correspondence with its Maxwell orbit of B-models, a Proto-Maxwell model is equivalent to an A-model. As before, the more specific results concerning Maxwell gravitation can then be obtained by the further gauge-fixing of section 4.4.

5.2. The Analogy between Electromagnetism and Newton-Cartan Gravitation

The following surface-level analogy between the Maxwell symmetry of electromagnetic gauge fields and the Trautman gauge symmetry of Neo-Newtonian models is often noted: each theory has an ‘invariant’ formulation such that, locally, there exists a gauge symmetry orbit of empirically equivalent models corresponding to the invariant formulation.

The analogy is of course correct so far as it goes. However, the analysis of the previous sections allows us to probe more deeply the analogies and the disanalogies between the two theories. First, notice that Newton-Cartan theory has a Milne gauge symmetry group that has no analog in electromagnetism. Second, if the Newton-Cartan connection is Newtonian, then 1-form gauge fields and the Maxwell symmetry group play a role in both theories. Indeed, they have the same formal source, that is, the application of Poincaré’s lemma to the equation $\mathrm{dF}=0$ (this, then, is the deep structural feature that gives rise to the surface-level analogy). Nonetheless, this equation has a very different interpretation in each theory: in electromagnetism it is a source-free equation of motion, whereas in Newton-Cartan theory, it is the Newtonian curvature condition, as expressed within the framework of Newton-Coriolis representations. Furthermore, the Newtonian condition is characterized by the existence of a twistless partial gauge-fixing for the Newton-Cartan connection (see R1-Equiv).

This brings us to a deeper, and perhaps unexpected, analogy between electromagnetism and Newton-Cartan theory. In the former, the field strength 2-form corresponds to a Maxwell orbit of gauge-dependent representations of the field strength (i.e., 1-form gauge fields). Along the same lines, one might have thought that a Newtonian connection could only correspond to a larger Milne-Maxwell orbit of representations. However, as explained in section 4.2, we can use the twistless partial gauge-fixing to construct a set of Milne-invariant representations such that a Newtonian connection corresponds to a Maxwell orbit of these representations (we thus called these the ‘Maxwell representations’). At this level of structure, then, the invariant content of both theories is in 1:1 correspondence with Maxwell orbits.

A final disanalogy presents itself when we consider the level of structure required by the Trautman Recovery Theorem, that is, where we impose R2 in addition to the Newtonian condition. As we saw, the Maxwell symmetry is insufficient to define the Trautman gauge symmetry acting on the recovered Neo-Newtonian models: in addition, one needs a rigidity-preserving gauge-fixing condition corresponding to R2. Thus, the surface-level analogy between electromagnetism and Neo-Newtonian models belies the fact that the gauge symmetry of the latter is somewhat more constrained than the Maxwell symmetry of electromagnetism. In this regard, it is interesting to compare the Trautman gauge to a partial gauge-fixing for electromagnetism such as the ‘Lorenz gauge’: formally, the two are very similar (compare, e.g., the rigidity-preserving condition ${\nabla }^a{\nabla }^{b}f=0$ to the harmonic condition for the Lorenz gauge), but the latter arises solely as a means of parameterizing the descriptive freedom, whereas in the former case, the existence of such a gauge needs to be guaranteed by the curvature conditions of the corresponding Newton-Cartan model.

There is much work that remains to be done in exploring these analogies. For instance, since Newton-Cartan theory has a ‘gauge field representation’, can one make sense of Wilson lines for these gauge fields? And to what extent would the philosophical discussion of holonomies and the Aharonov-Bohm effect carry over to such physical quantities?

5.3. Reflected Recovery

Given that a Newton-Cartan model does not geometrize away the vorticity of its recovered Neo-Newtonian models, it is worth asking whether there is an analog of Trautman Recovery in which the curved model does geometrize away the vorticity of its recovered models. In fact, the Milne Triangle of figure 1 suggests a visual strategy for obtaining such a scenario: instead of taking the (twistless) left edge of the triangle to define the data of a recovered model and using the right and bottom edges to encode the dynamical relationships, we can reflect this assignment about the center vertical axis of the triangle so that the (geodesic) right edge defines a recovered model, and the left and bottom edges encode the dynamical relationships.

Thus, in a Reflected Proto-recovery result, an A-model is defined as ($L$, ∇, Z), where ∇ is Newtonian and the dynamical vector field Z is twistless, and a B-model is defined as ($L$, $\stackrel{N}{\nabla }$, $\stackrel{\nabla ,N}{\omega }$, Z), where N is geodesic with respect to ∇. It is then immediate from (16), (17), (18), and (19) that from an A-model, we can recover a family of B-models (parametrized by Milne boosts preserving the geodesic gauge-fixing for N) each of which has the vorticity $\stackrel{\stackrel{N}{\nabla },Z}{\omega }=-\stackrel{\nabla ,N}{\omega }$, and that an A-model and its B-models share the same acceleration.

A Reflected Trautman Recovery can then be obtained by imposing rigidity on N (in conjunction with the geometrized source equation for ∇) and using the Künzle-Ehlers Recovery theorem (which assumes only the existence of a rigid vector field) to obtain the full set of equations of motion for these models. Clearly, more remains to be said about the additional curvature constraints that correspond to Reflected Recovery, as well as the (perhaps hydrodynamical) scenarios in which such a result might find application, but we leave such a task to a future work.