## Abstract

I argue that a criterion of theoretical equivalence due to Glymour (Noûs 11(3):227–251, 1977) does not capture an important sense in which two theories may be equivalent. I then motivate and state an alternative criterion that does capture the sense of equivalence I have in mind. The principal claim of the paper is that relative to this second criterion, the answer to the question posed in the title is “yes”, at least on one natural understanding of Newtonian gravitation.

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## Notes

Throughout the paper I use the abstract index notation, explained in Malament (2012, §1.4).

Here \(\nabla ^a\varphi = h^{ab}\nabla _b\varphi .\)

The notation \(\nabla ' = (\nabla ,C^a{}_{bc})\) is explained in Malament (2012, Prop. 1.7.3). Briefly, the action of the derivative operator \(\nabla '\) on any tensor field can be expressed as the sum of the action of \(\nabla \) on that field and terms involving a “connecting field” \(C^a{}_{bc}\). Specifying \(\nabla \) and \(C^a{}_{bc}\) is thus sufficient to define \(\nabla' .\)

Note that throughout this section, one could substitute “gravitational field” for “gravitational potential” by replacing every instance of \(\nabla ^a\varphi \) with a smooth vector field \(\varphi ^a\) satisfying \(\nabla ^{[a}\varphi ^{b]} = \mathbf {0}\). The choice makes no difference to the results below, though some readers may think a theory committed to a gravitational field is more plausible than one committed to a gravitational potential.

Glymour does not state that empirical equivalence is a necessary condition for theoretical equivalence, though he does appear to take theoretical equivalence to be strictly stronger than empirical equivalence, and, as Sklar (1982) emphasizes, empirical equivalence is a substantive interpretive constraint that goes beyond any formal relations between two theories.

For more on this sort of translation, see Barrett and Halvorson (2015a).

It is essential that one can go from a model \(A_1\) of \(T_1\) to a model \(A_2\) of \(T_2,\) and then back to the

*same*model \(A_1\) of \(T_1.\) See Andréka et al. (2005).Actually, all Glymour claims is that clause (2) of this criterion is a necessary condition for theoretical equivalence. I am extrapolating when I say that the two clauses together are also sufficient.

Minkowski spacetime is a (fixed) relativistic spacetime \((M,\eta _{ab})\) where

*M*is \(\mathbb {R}^4,\) \(\eta _{ab}\) is a flat Lorentzian metric, and the spacetime is geodesically complete. For more on these two formulations of electromagnetism, see Weatherall (2015c).Here and in what follows, we do not include the charge-current density in specifications of models of electromagnetism, as this field can be uniquely reconstructed from the other fields, given Maxwell’s equations.

The status of the vector potential arguably changes in quantum mechanics. See Belot (1998).

I am particularly grateful to Thomas Barrett and Jeff Schatz for discussions about and suggestions on this paragraph and the next two. But they should not be held responsible for what I say!

The classic results here are Beth’s theorem and Svenonius’ theorem. See Hodges (1993, Theorem 6.64 & Corollary 10.5.2) and the surrounding discussion.

For instance, if I do not know what the “language of electromagnetism” is, I can hardly write down an explicit definition in that language!

A

*category*consists of (1) a collection of objects \(A,B,C\ldots ;\) (2) a collection of arrows \(f,g,h\ldots ;\) and (3) assignments to each arrow*f*of a pair of objects,*dom*(*f*) and*cod*(*f*), called the domain and codomain of the arrow, respectively. (We abbreviate this by \(f:dom(f)\rightarrow cod(f)\).) We require that for any arrows*f*,*g*such that \(cod(f) = dom(g) ,\) there exists an arrow \(g\circ f : dom(f)\rightarrow cod(g)\) called the*composition of f and g*; and for any object*A*, there exists an arrow \(1_A: A\rightarrow A\) called the*identity arrow*. Together, these must satisfy: (1) for any arrows*f*,*g*,*h*, if \((h\circ g)\circ f\) exists, then \((h\circ g)\circ f = h\circ (g\circ f)\); and (2) for any arrow \(f:A\rightarrow B\), \(f\circ 1_A = f = 1_B\circ f\). The category of models of a theory*T*has models of*T*as objects and elementary embeddings as arrows. For more on categories and the related notions described below, see Mac Lane (1998), Borceux (2008), or Leinster (2014), among many other excellent texts. For more on the present proposal for understanding theoretical equivalence using category theory, see Halvorson (2012), Halvorson (2015), Barrett and Halvorson (2015b), and Weatherall (2015a).A

*functor*\(F:\mathbf {C}\rightarrow \mathbf {D}\) is a map between categories that takes objects to objects and arrows to arrows, and which preserves identity arrows and composition. Given functors \(F:\mathbf {C}\rightarrow \mathbf {D}\) and \(G:\mathbf {D}\rightarrow \mathbf {E}\), the composition \(G\circ F\), defined in the obvious way, is always a functor. A functor \(F:\mathbf {C}\rightarrow \mathbf {D}\) is an*isomorphism*of categories if there is a functor \(F^{-1}:\mathbf {D}\rightarrow \mathbf {C}\) such that \(F\circ F^{-1} = 1_{\mathbf {D}}\) and \(F^{-1}\circ F = 1_{\mathbf {C}}\), where \(1_{\mathbf {C}}:\mathbf {C}\rightarrow \mathbf {C}\) and \(1_{\mathbf {D}}:\mathbf {D}\rightarrow \mathbf {D}\) are functors that act as the identity on objects and arrows. The result cited in the text is proved by Barrett and Halvorson (2015b). (The result they state concerns*equivalence*of categories, to be discussed below, but in fact they show the stronger thing as well).Again, this is discussed in full detail in Barrett and Halvorson (2015b). It is not known how much weaker categorical isomorphism is than definitional equivalence, or Morita equivalence, which is a weakening of definitional equivalence that allows one to define new sorts. Note that the model theoretic criterion Glymour begins with is actually equivalent to definitional equivalence (de Bouvere 1965b), at least in simple cases, though that is of little comfort if, as I have argued, it cannot actually be applied in realistic cases. See Glymour (2013) and Halvorson (2013) for a recent discussion of these issues.

What is meant by “preserve the equivalence classes” is described in more detail in Lemma 5.3, below.

Here \(\chi _*\) is the pushforward along \(\chi \), defined for differential forms because \(\chi \) is a diffeomorphism.

An

*equivalence of categories*is a pair of functors \(F:\mathbf {C}\rightarrow \mathbf {D}\) and \(G:\mathbf {D}\rightarrow \mathbf {C}\) that are*almost inverses*in the sense that given any object*A*of \(\mathbf {C},\) there is an isomophism \(\eta _A:A\rightarrow G\circ F(A),\) where these isomorphisms collectively satisfy the requirement that for any arrow \(f:A\rightarrow B\) of \(\mathbf {C}\), \(\eta _B\circ f = G\circ F(f)\circ \eta _A\); and likewise,*mutatis mutandis*, for any object of \(\mathbf {D}.\)More generally, Barrett and Halvorson (2015b) show that if two theories are

*Morita*equivalent, which is similar to definitional equivalence, but with the flexibility to define new sorts, then their categories of models are equivalent, but*not*necessarily isomorphic. So there is reason to think that even in the first order case, we should be interested in categorical equivalence, rather than isomorphism.Given a diffeomorphism \(\chi :M\rightarrow M'\) and derivative operators \(\nabla \) and \(\nabla '\) on

*M*and \(M'\) respectively, we say that \(\chi \) preserves \(\nabla \) if for any tensor field \(\lambda ^{a_1\cdots a_r}_{b_1\cdots b_s}\) on*M*, \(\chi _*(\nabla _n\lambda ^{a_1\cdots a_r}_{b_1\cdots b_s}) = \nabla '_n\chi _*(\lambda ^{a_1\cdots a_r}_{b_1\cdots b_s})\).Alternatively, one could replace “gravitational potential” with “gravitational field” to yield a distinct, and perhaps more plausible, option. (Recall footnote 8.) But the difference does not matter for the present discussion.

Note that, since

*all*of the derivative operators considered in NG and GNG agree once one raises their index, one can characterize the gauge transformation with regard to any of them without ambiguity.For more on this point, see the debate between John Norton (1992, 1995) and David Malament (1995). Arguably, Newton himself recognized the empirical equivalence of models related by these transformations—for instance, see the discussions of Corollary VI to the laws of motion in DiSalle (2008); see also Saunders (2013), Knox (2014), Weatherall (2015b).

Note, however, that one could construct an alternative presentation of NG\(_2\) analogous to EM\(_2'\), in such a way that this

*would*be equivalent to GNG by criterion 1\(^\prime \). Moreover, if one restricts attention to the collections of models of NG and GNG in which (1) the matter distribution is supported on a spatially compact region and (2) the gravitational field (for models of NG) vanishes at spatial infinity, then \(\mathbf {NG}_1\), \(\mathbf {NG}_2\), and \(\mathbf {GNG}\) are all equivalent by*both*criteria.Once again, one could substitute “gravitational field” for “gravitational potential,”

*mutatis mutandis*. Recall footnotes 8 and 29.I am grateful to Oliver Pooley for pressing this point.

Knox (2011) makes a closely related point.

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## Acknowledgments

This material is based upon work supported by the National Science Foundation under Grant No. 1331126. Thank you to Steve Awodey, Jeff Barrett, Thomas Barrett, Ben Feintzeig, Sam Fletcher, Clark Glymour, Hans Halvorson, Eleanor Knox, David Malament, John Manchak, Colin McLarty, John Norton, Cailin O’Connor, Oliver Pooley, Sarita Rosenstock, Jeff Schatz, Kyle Stanford, and Noel Swanson for helpful conversations on the topics discussed here, and to audiences at the Southern California Philosophy of Physics Group, the University of Konstanz, and Carnegie Mellon University for comments and discussion. I am particularly grateful to Erik Curiel, Clark Glymour, Hans Halvorson, David Malament, and two anonymous referees for comments on a previous draft, and to Thomas Barrett for detailed discussion and assistance concerning the relationship between categorical equivalence and definitional equivalence.

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## Proofs of Propositions

### Proofs of Propositions

###
*Proof of Prop. 5.1*

Suppose there were vector potentials \(A_a\) and \(\tilde{A}_a\) such that \([A_a]\ne [\tilde{A}_a],\) but for every \(X_a\in [A_a],\) \(\nabla _{[a}X_{b]} = \nabla _{[a}\tilde{A}_{b]} = F_{ab}.\) Then \(\nabla _{[a}(X_{b]}-\tilde{A}_{b]}) = \mathbf {0}\) for every \(X_a\in [A_a],\) and thus \(X_a-\tilde{A}_a\) is closed for every \(X_a\in [A_a].\) Thus \([A_a]\subseteq [\tilde{A}_a].\) A similar argument establishes that \([\tilde{A}_a]\subseteq [A_a].\) \(\square \)

###
*Proof of Lemma 5.3*

Suppose we have an isometry \(\chi \) s.t. \(\chi _*(F_{ab}) = F'_{ab}.\) Then for every \(X_a\in [A_a],\) we have \(\chi _*(\nabla _{[a}X_{b]}) = \chi _*(F_{ab}) = F'_{ab}.\) But exterior derivatives commute with pushforwards along diffeomorphisms, and so \(\chi _*(\nabla _{[a}X_{b]}) = \nabla _{[a}\chi _*(X_{b]}) = F'_{ab}.\) Thus by Prop. 5.1, \([\chi _*(A_a)] = [A'_a].\) Conversely, if \(\chi _*(A_a)\in [A_a],\) then \(F'_{ab} = \nabla _{[a}\chi _*(A_{b]}) = \chi _*(\nabla _{[a}A_{b]} = {\chi _*(F_{ab})}).\) \(\square \)

###
*Proof of Prop. 5.2*

The isomorphism is given by \(F:EM'_2\rightarrow EM_1\) acting on models as \((M,\eta _{ab},[A_a])\mapsto (M,\eta _{ab},\nabla _{[a}A_{b]})\) and acting on arrows as the identity. That this yields an isomorphism is an immediate consequence of Prop. 5.1, Lemma 5.3, and basic facts about the composition of pushforward maps. \(\square \)

###
*Proof of Prop. 5.4*

\(\mathbf {EM}_2\) includes identity arrows, which are pairs of the form \((1_M,0);\) (2) it contains all compositions of arrows, since given any two arrows \((\chi ,G_a)\) and \((\chi ',G'_a)\) with appropriate domain and codomain, \((\chi ',G_a')\circ (\chi ,G_a) = (\chi '\circ \chi ,\chi ^*(G'_a)+G_a)\) is also an arrow; and (3) composition of arrows is associative, since given three pairs \((\chi ,G_a)\), \((\chi ',G_a')\), and \((\chi '',G_a'')\) with appropriate domain and codomain, \((\chi '',G_a'')\circ ((\chi ',G_a')\circ (\chi ,G_a)) = (\chi '',G_a'')\circ (\chi '\circ \chi ,\chi ^*(G'_a)+G_a) = (\chi ''\circ (\chi '\circ \chi ),\chi ^*\circ \chi '^*(G_a'') + \chi ^*(G_a)'+G_a) = ((\chi ''\circ \chi ')\circ \chi ,\chi ^*(\chi '^*(G_a'') + G_a')+G_a) = (\chi ''\circ \chi ',\chi '^*(G_a'') + G_a')\circ (\chi ,G_a) = ((\chi '',G_a'')\circ (\chi ',G_a'))\circ (\chi ,G_a)\). \(\square \)

###
*Proof of Prop. 5.5*

It suffices to show that there is a functor from \(\overline{\mathbf {EM}}_2\) to \(\mathbf {EM}_1\) that is full, faithful, and essentially surjective, and which preserves \(F_{ab}\). Consider the functor \(E:\overline{\mathbf {EM}}_2\rightarrow \mathbf {EM}_1\) defined as follows: *E* acts on objects as \((M,\eta _{ab},A_a)\mapsto (M,\eta _{ab},\nabla _{[a}A_{b]})\) and on arrows as \((\chi ,G_a)\mapsto \chi \). This functor clearly preserves \(F_{ab}\). It is also essentially surjective, since given any \(F_{ab}\), there always exists some \(A_a\) such that \(\nabla _{[a}A_{b]} = F_{ab}\). Finally, to show that it is full and faithful, we need to show that for any two objects \((M,\eta _{ab},A_a)\) and \((M,\eta _{ab},A'_a)\), the induced map on arrows between these models is bijective. First, suppose there exist two distinct arrows \((\chi ,G_a),(\chi ',G'_a):(M,\eta _{ab},A_a)\rightarrow (M,\eta _{ab},A'_a)\). If \(\chi \ne \chi '\) we are finished, so suppose for contradiction that \(\chi = \chi '\). Since by hypothesis these are distinct arrows, it must be that \(G_a\ne G'_a\). But then \(A_a+G_a\ne A_a + G'_a\), and so \(\chi _*(A_a+G_a)\ne \chi _*(A_a+G'_a)\). So we have a contradiction, and \(\chi \ne \chi '\). Thus the induced map on arrows is injective. Now consider an arrow \(\chi :E((M,\eta _{ab},A_a))\rightarrow E((M,\eta _{ab},A'_a))\). This is an isometry such that \(\chi _*(\nabla _{[a}A_{b]}) = \nabla _{[a}A'_{b]}\). It follows that \(\chi _*(\nabla _{[a}A_{b]}-\nabla _{[a}\chi ^*(A'_{b]})) = \mathbf {0}\), and thus that \(\nabla _{[a}A_{b]}-\nabla _{[a}\chi ^*(A'_{b]})\) is closed. So there is an arrow \((\chi ,\chi ^*(A'_a)-A_a) : (M,\eta _{ab},A_a) \rightarrow (M,\eta _{ab},A'_a)\) such that \(E((\chi ,\chi ^*(A'_a)-A_a)) = \chi \), and the induced map on arrows is surjective. \(\square \)

###
*Proof of Prop. 6.1*

This argument follows the proof of Prop. 5.5 closely. Consider the functor \(E:\mathbf {NG}_2 \rightarrow \mathbf {GNG}\) defined as follows: *E* takes objects to their geometrizations, as in Prop. 2.1, and it acts on arrows as \((\chi ,\psi )\mapsto \chi \). This functor preserves empirical content because the geometrization lemma does; meanwhile, Prop. 2.2 ensures that the functor is essentially surjective. We now show it is full and faithful. Consider any two objects \(A = (M,t_a,h^{ab},\nabla ,\varphi )\) and \(A' = (M',t_a',h'^{ab},\nabla ',\varphi ')\). Suppose there exist distinct arrows \((\chi ,\psi ),(\chi ',\psi '):A\rightarrow A'\), and suppose (for contradiction) that \(\chi = \chi '\). Then \(\psi \ne \psi '\), since the arrows were assumed to be distinct. But then \(\varphi +\psi \ne \varphi +\psi '\), and so \((\varphi +\psi )\circ \chi \ne (\varphi +\psi ')\circ \chi \). Thus \(\chi \ne \chi '\) and *E* is faithful. Now consider any arrow \(\chi :E(A)\rightarrow E(A')\); we need to show that there is an arrow from *A* to \(A'\) that *E* maps to \(\chi \). I claim that the pair \((\chi ,\varphi '\circ \chi -\varphi ):A\rightarrow A'\) is such an arrow. Clearly if this arrow exists in \(\mathbf {NG}_2\), *E* maps it to \(\chi \), so it only remains to show that this arrow exists. First, observe that since \(\chi \) is an arrow from *E*(*A*) to \(E(A')\), \(\chi :M\rightarrow M'\) is a diffeomorphism such that \(\chi _*(t_a) = t'_a\) and \(\chi _*(h^{ab}) = h'^{ab}\). Moreover, \(\chi _*(\varphi + (\varphi '\circ \chi -\varphi )) = \chi _*(\varphi '\circ \chi ) = \varphi '\circ (\chi \circ \chi ^{-1}) = \varphi '\), so \(\chi \) maps the gauge transformed potential associated with *A* to the potential associated with \(A'\). Now consider the action of \(\chi \) on the derivative operator \(\nabla \). We need to show that for any tensor field \(\lambda ^{a_1\cdots a_r}_{b_1\cdots b_s}\), \(\chi _*(\tilde{\nabla }_n\lambda ^{a_1\cdots a_r}_{b_1\cdots b_s}) = \nabla '_n\chi _*(\lambda ^{a_1\cdots a_r}_{b_1\cdots b_s})\), where \(\tilde{\nabla } = (\nabla , t_bt_c\nabla ^a (\varphi '\circ \chi - \varphi ))\) is the gauge transformed derivative operator associated with *A*. We will do this for an arbitrary vector field; the argument for general tensor fields proceeds identically. Consider some vector field \(\xi ^a\). Then \(\chi _*(\tilde{\nabla }_n\xi ^a) = \chi _*(\nabla _n\xi ^a - t_n t_m\xi ^m\nabla ^a(\varphi '\circ \chi - \varphi )) = \chi _*(\overset{g}{\nabla }_n\xi ^a - t_nt_m\xi ^m\nabla ^a\varphi - t_n t_m\xi ^m\nabla ^a(\varphi '\circ \chi - \varphi )) = \chi _*(\overset{g}{\nabla }_n\xi ^a) - \chi _*(t_n t_m\xi ^m\nabla ^a(\varphi '\circ \chi ))\), where \(\overset{g}{\nabla } = (\nabla ,t_bt_c\nabla ^a\varphi )\) is the derivative operator associated with *E*(*A*). Now, we know that \(\chi :E(A)\rightarrow E(A')\) is an arrow of \(\mathbf {GNG}\), so \(\chi _*(\overset{g}{\nabla }_n\xi ^a) = \overset{g}{\nabla }_n \chi _*(\xi ^a)\). Moreover, note that the definitions of the relevant \(C^a{}_{bc}\) fields guarantee that \(\nabla ^a\lambda ^{a_1\cdots a_r}_{b_1\cdots b_s} = \overset{g}{\nabla }{}^a(\lambda ^{a_1\cdots a_r}_{b_1\cdots b_s})\) and similarly for \(\nabla '\) and \(\overset{g}{\nabla }{}'\). Thus we have \(\chi _*(\overset{g}{\nabla }_n\xi ^a)-\chi _*(t_n t_m\xi ^m\nabla ^a(\varphi '\circ \chi )) = \overset{g}{\nabla }{}_n' \chi _*(\xi ^a) - t'_nt'_m\chi _*(\xi ^m) \nabla '^a(\varphi '\circ (\chi \circ \chi ^{-1})) = \overset{g}{\nabla }{}_n'\chi _*(\xi ^a) - t'_nt'_m\chi _*(\xi ^m) \nabla '^a\varphi ' = \nabla '_n\chi _*(\xi ^a)\), where \(\overset{g}{\nabla }{}' = (\nabla ',-t_bt_c\nabla ^a\varphi ')\) is the derivative operator associated with \(E(A')\). So \(\chi \) does preserve the gauge transformed derivative operator. The final step is to confirm that \(\nabla ^a\nabla ^b(\varphi '\circ \chi -\varphi ) = \mathbf {0}\). To do this, again consider an arbitrary vector field \(\xi ^a\) on *M*. We have just shown that \(\nabla '_a\chi _*(\xi ^b)-\chi _*(\nabla _a\xi ^b) = -\chi _*(t_at_m\xi ^m\nabla ^b(\varphi '\circ \chi -\varphi ))\). Now consider acting on both sides of this equation with \(\nabla '^a\). Beginning with the left hand side (and recalling that \(\nabla \) and \(\nabla '\) are both flat), we find: \(\nabla '^n\nabla '_a\chi _*(\xi ^b)-\nabla '^n\chi _*(\nabla _a\xi ^b) = \nabla '_a\overset{g}{\nabla }{}'^n\chi _{*}(\xi ^b) - \chi _*(\nabla _a\overset{g}{\nabla }{}^n\xi ^b) = \overset{g}{\nabla }{}'_a\overset{g}{\nabla }{}'^n\chi _*(\xi ^a)- t'_at'_m(\nabla '^b\varphi ')\overset{g}{\nabla }{}'^n \chi _*(\xi ^m)-\chi _*(\overset{g}{\nabla }{}_a\overset{g}{\nabla }{}^n\xi ^b)+\chi _*(t_at_m(\nabla ^b\varphi )\overset{g}{\nabla }{}^n\xi ^m) = \chi _*(t_at_m(\nabla ^b(\varphi - \varphi '\circ \chi )) \overset{g}{\nabla }{}^n\xi ^m)\). The right hand side, meanwhile, yields \(-\nabla '^n(\chi _*(t_at_m\xi ^m\nabla ^b (\varphi '\circ \chi -\varphi )) = \chi _*(t_at_m(\nabla ^n\xi ^m) \nabla ^b(\varphi - \varphi '\circ \chi ))+\chi _*(t_at_m\xi ^m\nabla ^n \nabla ^b(\varphi - \varphi '\circ \chi ))\). Comparing these, we conclude that \(\chi _*(t_at_m\xi ^m\nabla ^n\nabla ^b (\varphi -\varphi '\circ \chi )) = \mathbf {0}\), and thus \(t_at_m\xi ^m\nabla ^n\nabla ^b(\varphi -\varphi '\circ \chi ) = \mathbf {0}\). But since \(t_a\) is non-zero and this must hold for *any* vector field \(\xi ^a\), it follows that \(\nabla ^a\nabla ^b(\varphi '\circ \chi -\varphi ) = \mathbf {0}\). Thus *E* is full. \(\square \)

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Weatherall, J.O. Are Newtonian Gravitation and Geometrized Newtonian Gravitation Theoretically Equivalent?.
*Erkenn* **81**, 1073–1091 (2016). https://doi.org/10.1007/s10670-015-9783-5

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DOI: https://doi.org/10.1007/s10670-015-9783-5