Is the metric signature really electromagnetic in origin?

The programme of ‘pre-metric electromagnetism’, developed by Hehl and collaborators, seeks to derive certain aspects of the spacetime geometry of the world (in particular, metrical signature) from elementary, empirically-informed axioms regarding electromagnetic ﬁelds. The programme should, therefore, be of profound interest to both empiricist and relationalist philosophers. Up to this point, however, pre-metric electromagnetism has received very little attention within the philosophy of physics; this paper seeks to rectify the situation, by engaging in a detailed foundational study of the programme. In particular, in this article, we (a) present a streamlined version of the programme, identifying key input assumptions, (b) consider the connections between this programme and other notable projects in the foundations of spacetime theories, e.g. (i) the dynamical approach to spacetime and (ii) constructive axiomatics, and (c) consider in detail the extent to which this programme can be generalised beyond electromagnetism. In achieving these tasks, we hope to open up to philosophers an extraordinarily fecund—but lamentably little-known!—project within the foundations of physics.


INTRODUCTION
No debate in the foundations of spacetime theories has more prominence than that between substantivalists and relationalists.According to the former, space and time are fundamental and exist independently from material bodies; according to the latter, space and time are (in some way or another) ontologically subordinate to material bodies (and the relations which obtain between those bodies). 1 Debates between substantivalists and relationalists often have a metaphysical flavour somewhat removed from the details of the dynamics of specific physical theories.To take two examples: (i) according to Maudlin's (1993) "Newtonian relationalism," the relationalist can shirk a commitment to Newtonian absolute space by countenancing trans-temporal relations between material bodies; this equips the relationalist with the tools to account for Newton's thought experiment of the globes as presented in the Scholium (say); 2 (ii) Sklar (1977) proposes that one needn't commit oneself to pieces of spatiotemporal structure in order to underwrite (say) absolute notions of acceleration, for one can instead take such accelerations to be primitive, monadic properties of material bodies.This general "pure metaphysics" emphasis notwithstanding, some recent versions of relationalism have been more sensitive to the details of particular physical theories.
One prominent example is the "dynamical approach" to spacetime theories of Brown   and Pooley (2001; 2006), according to which spatiotemporal geometrical structure just is a codification of the symmetries of the dynamical laws for material bodies: the laws of physics feature more prominently on this approach than on other, more traditional versions of relationalism. 3 That being said, all is not necessarily well with the dynamical approach, for it has faced charges of obscurity or circularity (or both) (see, e.g., Dewar   2020; Fletcher 2021; Norton 2008). 4e approach to subordinating spatiotemporal (geo)metrical structure to properties of the equations governing material bodies which one finds in the physics literature, and which has on occasion been mentioned in the same breath as the dynamical approach (see in particular Fletcher 2021; Read et al. 2018), is the program of pre-metric electromagnetism of Hehl and collaborators (see in particular Hehl et al. 2016; Hehl and Obukhov 2003;   2005, 2006; Itin and Hehl 2004).The question to be addressed by these authors is this: is there any way of reformulating or reconceptualising Maxwell's equations so that they in fact do not presuppose ab initio metrical structure, but rather come themselves to fix 1 Of course, there is a variety of ways of making more precise the difference between substantivalism and relationalism-see e.g.(Baker 2021; Dasgupta 2011; Earman 1977; North  2021; Pooley 2013)-but for now, this characterisation is sufficient.

2
Recall that the challenge here is this: how is the relationalist to account for different tensions in the string yoking the two globes, when ex hypothesi those are the only material bodies in the universe?The Newtonian relationalist can ground facts about the string tension in these trans-temporal distance relations.

3
The term "relationalism" is ambiguous between attempts to reduce to material bodies (and the relations between those bodies) (i) manifold points, versus (ii) the distance relations obtaining between those points.In this article, we understand "relationalism" to take broad scope; on this understanding, attempts to reduce (ii) to material bodies (and the relations between those bodies) also count as relationalist projects.In this sense, the dynamical approach to spacetime, as well as the program of pre-metric electromagnetism (which constitutes the focus of this article), qualify as relationalist agendas.

4
Another example of relationalism in this spirit is the program of "shape dynamics" of Barbour and his collaborators.For a book-length introduction, see (Mercati 2018).
functionalism."In §5, we assess the prospects for generalising this program beyond electromagnetism.There is also an appendix, in which we consolidate some relevant results from differential geometry.
To answer the title question: while the program of pre-metric electromagnetism offers prima facie a technically sound means of reducing metrical signature to material bodies, its advantages over the standard approach are on reflection unclear.For example, this approach is not any more ontologically parsimonious than the "standard" formulation of electromagnetism which it seeks to replace.It is also not clear that this approach has achieved more "background independence" than the standard formulation of electromagnetism.In addition, there are significant obstacles for generalizing the approach beyond electromagnetism, although there remains interesting and important technical work to be undertaken on this front.While overall we remain sceptical regarding the conceptual merits of pre-metric electromagnetism, nevertheless we would like to emphasize that we regard the program as being of significant intrinsic interest and importance in the foundations of physics in its attempt to develop empirically and conceptually perspicuous physical theories and to reduce their undue dependence upon the spacetime metric.

PRE-METRIC ELECTROMAGNETISM
In this section, we first remind the reader of the relevant aspects of standard, textbook electromagnetism ( §2.1), before turning our attention to pre-metric electromagnetism ( §2.2).

5
See Callender, 2017,  §6.2.1, for discussion.6  See footnote 3 for the sense in which the dynamical approach counts as a version of relationalism (cf.also Pooley 2013).Chen and Read  Philosophy of Physics DOI: 10.31389/pop.45

TEXTBOOK ELECTROMAGNETISM
We begin with standard electromagnetism.When written in terms of differential forms, Maxwell's equations for electromagnetism read as follows: = , * d F J (1) = 0; dF (2)   here and throughout, d denotes the exterior derivative acting on differential forms; F is the Faraday tensor, which encodes facts about the electric and magnetic fields and which is a spacetime 2-form (by the Poincaré lemma, which states that every closed form is locally exact, it follows from (2) that locally one can write F = dA for some 1-form A, which is the electromagnetic potential); J represents a current, and is a spacetime 3-form; d denotes the exterior derivative operator, and * denotes the Hodge star operator. 7For our purposes, the crucial point to note is that the Hodge star operator presupposes a metric field on the manifold-intuitively, given an n-dimensional (pseudo)-Riemannian manifold (M, g ab ), the Hodge star operator 'completes' a p-form to the volume form of (M, g ab ), and thereby defines an (n-p)-form on M. 8 The point is that presupposition of a metric field is baked into the standard formulation of electromagnetism. 9 In standard presentations of electromagnetism, in addition to Maxwell's equations, one has the Lorentz force law, which, in the language of differential forms, can be written Here,  denotes the interior product operation on differential forms.( 3) is the equation of motion for a point charge with charge q, describing the electromagnetic force f on said charge.On the right hand side, uF is the interior product of the tangent vector u to the worldline of the charge with the 2-form F. 10

THE PRE-METRIC PROGRAM
The question to be addressed, then, is this: is there any way of reformulating or reconceptualising Maxwell's equations such that they in fact do not presuppose primitive metrical structure?Although-as already mentioned-the question has a venerable pedigree, it has over the past twenty or so years been taken up in a particularly intriguing manner by Hehl and his collaborators.It is upon this modern version of the pre-metric approach which we focus in this article; the presentation of Itin and Hehl (2004) will prove 7 Note that in the covariant tensor formulation , the current J v is a 4-vector.To convert the 3-form J in (1) to J v , we use the Hodge star to convert the 3-form to a 1-form, and then use the metric to raise the index.The differential form formulation of Maxwell's equations has the advantage of minimizing the role of the metric and is therefore a good starting point for the pre-metric program.For further details on the inter-conversion between these two different formulations of Maxwell's equations, see Baez and Muniain (1994).

8
One might reasonably ask here: which volume form, since any top-ranked form can in principle serve to define a volume element?Typically in these discussions, one means the volume form , which is the top-ranked form "adapted" to the Lorentzian metric g ab (and in standard electromagnetism, this is the Minkowski metric, which is a non-dynamical entity).

9
For background on the notions of differential forms introduced in this paragraph, see the appendix.
10 There are well-known foundational issues regarding the consistency of electromagnetism when taken as this complete package-see Frisch (2005).We'll pass over these issues almost entirely in this article, since they're not relevant for the points which we seek to make.
Axiom 1. Spacetime is a four-dimensional differentiable manifold M that admits a foliation into codimension-1 hypersurfaces Σ.
Let the hypersurfaces be labelled by a smooth parameter σ Î (-∞, ∞), which is called "topological time." 12 Given σ, we additionally choose a vector field on the manifold v such that vdσ = 1. 13The above is an axiom about manifold topology: it is a presupposition of the pre-metric approach, which must be taken for granted if one is to proceed. 14ving laid down this axiom, so as to achieve independence from any background metric we next use differential forms in order to formalize all mathematical objects representing the electromagnetic field.The foliation of the manifold postulated in Axiom 1 allows for convenient decompositions of differential forms.These will play a central role in the formalism of pre-metric electromagnetism.More specifically, for an n-form α, we have where β is an (n-1)-form and ɣ is an n-form, which is a unique decomposition if we require β to be 'tangential' and ɣ to be "orthogonal" to the direction of "time."15combination of p-wedge products of dx, dy, dz, and dσ.For example, a 2-form can be written as adx Ù dy + bdy Ù dz + cdz Ù dσ + rdσ Ù dx for some a, b, c, r.This can be rearranged into the decomposition form specified by (4).A decomposition satisfying the form of (4) alone is not unique-since, for example, it does not prohibit ɣ to contain dσ.The further constraint that ɣ is "orthogonal" to the direction of time means that ɣ(v) = 0, which ensures the uniqueness of the decomposition.We say β Ù dσ is "tangential" to time since it contains dσ, and by extension we also say that β is "tangential" to time since it is wedged with dσ.Thus, we may call β and ɣ respectively the (spatio)temporal and spatial component of α.The next axiom deployed is physical in nature (regarding, as it does, the charge current density), and reads as follows: 16 Axiom 2. The charge current density is a conserved twisted 3-form J, so that when J is integrated over a closed 3D submanifold C Ì M (so that ∂C = ∅), one has 17 = 0.
From this, it follows that dJ = 0, via Stokes' theorem.Intuitively, Axiom 2 states that the rate of the change of the electric charge ρ (i.e., ρ ⋅ ) in a region is equal to the electric flux dj into the region (d is the spacelike exterior derivative and dots denote derivatives with respect to σ)-this is clear once one recalls that J can be decomposed into components as (ρ, j): ρ and j result from the decomposition of J with respect to "time," where ρ is a 3-form considered as the charge density in "space" and j is a 2-form considered as the current density of ρ.We can derive from dJ = 0 (along with other plausible conditions, which have to do with the numerical coefficients in the time-space decomposition of J-which, however, can be normalized away) that dj + ρ = 0, which amounts to the usual continuity equation for the conservation of electric charge.
By de Rham's theorem (which has to do with the theory of de Rham cohomology groupssee Hehl and Obukhov 2003, 96-101), one can write 18 = , J dH (6)   where H is a twisted 2-form called the "electromagnetic excitation"; this can be decomposed using (4) into two parts  and , respectively (these are called the "electric" and "magnetic" 16 Twisted differential forms are defined in the appendix.17 As is standard in the context of Stokes' theorem, by "boundary," we do not mean the notion of "topological boundary" but "boundary of a manifold," where "manifold" is understood as the generalized notion of "manifold with boundary" (namely that a Hausdorff space in which every point has a neighborhood homeomorphic to an open subset of 1 2 1 := {( , , ..., ) , ..., 0} for some n).The boundary of an n-dimensional manifold M is the complement of the interior of M in M, where the interior of M is the set of all points in M that have neighborhoods homeomorphic to open subsets of n  .
It is important to note that the "closed" 3D submanifold C here refers to a compact boundaryless manifold, such as a 3-sphere, rather than one of the folia (which is not compact).The paradigmatic case in this context is a 3D "cylinder" with its height along the time dimension, its top and bottom surface being each a 3-ball, and its cross-sections 2-spheres.The total flux into or out of this region being zero is an elegant formulation of charge conservation, as explained in the text.Note that if the region in question is one of the folia, then the integral of the flux may not be zero (but rather equals the total charge of the region).
18 To be clear, de Rham cohomology addresses (at least in part) the question: when can a spacetime n-form be considered the curvature of some (n-1)-form potential?The Poincaré lemma-which we also consider below-regards all closed n-forms, and finds them to be locally exact (i.e., of the form dη, for some (n-1)-form η).In the pre-metric approach, when we are dealing with a closed 2-form F (given (2)), both de Rham theorems and the Poincaré lemma are potentially relevant.If the conditions of de Rham theorems are satisfied, F is globally of the form dA; otherwise, given the Poincaré lemma and the fact that F is closed, F can still be regarded as being locally of the form dA. For the twisted 3-form J, dynamics do not impose that dJ = 0 (unlike the case of F); therefore, only the de Rham theorems are relevant.In this case, however, the conditions of those theorems are satisfied, and so the above result obtains.Chen and Read Philosophy of Physics DOI: 10.31389/pop.45 excitations, respectively). 19Of course, H is not unique: the addition to H of any exact form would still satisfy (6).The physical significance of  and  is this: both can be measured by ideal electric conductors and superconductors, respectively (Hehl and Obukhov 2003,   116)-as Hehl and Obukhov state, "the electric excitation  can be measured by means of Maxwellian double plates as charge per unit area, the magnetic excitation  by means of a small test coil, which compensates the -field to be measured, as current per unit length.
[…] In other words, the extensive quantities  and -and thus the 4-dimensional excitation H-have an operationally [sic] significance of their own, since they are related to charge at rest or in motion, respectively" (Hehl and Obukhov 2001, 7).In this sense, Axiom 2 can be regarded as being imbued with empirical content-the significance of this will be explored further below, in §4.3.
We turn now to the third axiom of the pre-metric approach, which has to do with the behaviour of point charges: where e α is a frame and F is an untwisted 2-form.
This is an alternative formulation of the Lorentz force law (3) that appeals to the frame field. 20Note that no metric is required in order to write down this law; thus, treating it as an axiom is consistent with the pre-metric methodology.In the pre-metric approach, Axiom 3 is treated as imbuing the electromagnetic field strength tensor F with empirical content (see Hehl and Obukhov 2003, 123); accordingly, its conceptual status is in fact quite subtlewe'll return to this in §4.3. 21Once again, one can decompose F into (spatio)temporal and spatial parts-these are the electric and magnetic fields, E and B, respectively. 22e fourth axiom of pre-metric electromagnetism is again physical in nature, and involves the conservation of magnetic flux:23 19 That is, = - + H dσ    (Itin and Hehl 2004, 66).This is analogous to the decomposition of F into E and B, namely F = E Ù dt + B (see footnote 22); note here that B is to be understood as a 2-form which encodes the components of the magnetic vector field.Note, however, that the role between the magnetic and the electric components of H is somewhat exchanged when compared to those of F:  is temporal and  spatial while E is temporal and B spatial-this is perhaps not surprising since H is standardly the Hodge dual of F (note, though, that this condition has yet to be imposed in the pre-metric approach).We can derive that  and  satisfy inhomogeneous Maxwell equations , which are respectively the Gauss law and the Ampère Law, although these are standardly formulated in terms of the electric and magnetic fields.See Hehl and Obukhov (2005)  Note that this differs from Axiom 5 as presented by Itin and Hehl (2004)-this, indeed, is the only point at which we deviate substantially from the presentation of that article.Axiom 5 as presented by Itin and Hehl (2004) can be understood as a corollary to Axiom 5 presented above; Hehl and Itin therefore take Axiom 5 as presented above to be a further postulate.
Since it is the above condition which will ultimately be required in order to derive the Lorentzian metric signature, we have elected to present this condition as an axiom directly.
Axiom 5 is used, first of all, to derive an expression for a quantity Σ α , called the "energymomentum current" of the electromagnetic field, which is a covector-valued 3-form (the object is defined using a frame field e α , hence the index).This quantity is introduced as a generalized potential of the Lorentz force-that is, the Lorentz force describes how the energymomentum current varies under spatiotemporal variation of the electromagnetic field: where X α involves the derivative of the frame field e α (for explicit expressions, see (Itin and   Hehl, 2004, p. 69)). 27Then, substituting (7) from Axiom 3 into (9) and using the properties magnetic fields expressed in §4.3.
27 The frame field eα consists of four tangent vectors at each point.Note that dΣα is not defined explicitly by Itin and Hehl (2004), which is problematic because Σα is not an ordinary differential form.We can infer from the source that the dΣα is implicitly defined as d(eαΣα), namely that we first evaluate Σα at each eα and then apply d to each of them, resulting in a 4-tuple of 4-forms.Without this explicit definition, the role of the frame field eα is obscured.Similarly, "the derivative of the frame field" referred to by Itin and Hehl (2004) does not literally mean the exterior derivative of the field since the latter is not a form, but rather the operator d(eα), which first contracts a form with the frame field and then applies d to the result.
which gives rise to This also means that the trace := Σ (where α J is the coframe field associated to e α ) of the energy-momentum current Σ α vanishes, which ensures that the photon be massless (see Itin and Hehl 2004, 70).This trace would not vanish were it not for the conditions imposed via Axiom 5; therefore, this can be considered (at least some of) the extra empirical content delivered by Axiom 5, over and above that of Axiom 3 alone.
From these axioms, there follows the "electric-magnetic reciprocity relation" that preserves the energy-momentum current, which is where ζ is a twisted 0-form (which is a scalar function save that it changes signs under parity inversion-see appendix) (Itin and Hehl 2004, 71).

EMERGENCE OF METRIC SIGNATURE
So far, we have recapitulated the axiomatic version of pre-metric electromagnetism, as presented by Hehl et al. (and with particular attention to the presentation by Itin and Hehl   2004).Supposedly, in this approach no metric occurs in the fundamental posits. 28Butperhaps surprisingly!-thetheory already contains the seeds of metrical information; in particular, the seeds of the signature. 29With the work of the previous section in hand, we are now ready to derive the metric signature from Axioms 1-5 of pre-metric electromagnetism. 30It is worth flagging that we intend this section to be fairly uncritical; a thoroughgoing foundational and philosophical assessment of this supposed derivation of the metric signature is deferred to §4.
Given the reciprocity relation ( 12) and ( 13) and the constitutive relation H = κ(F), it is not hard to derive that κ is proportional to the identity operator, and in particular 2 2 = -κ ζ  (see Itin and Hehl 2004,  §4.3).In fact, we can now replace ζ by an ordinary (rather than twisted) scalar function λ, since the differences between ordinary scalars and twisted scalars disappear when the two are squared.We can assimilate the scalar factor by defining This reminds us straightforwardly of the Hodge star operator * 28 We will assess in §4 whether, strictly, this is true.
29 We mean this in the sense of Le Bihan and Linnemann (2019), according to which the Lorentizan signature is the "seed" of spacetime.Le Bihan and Linnemann argue that (again, perhaps surprisingly) this "seed" can be found in many approaches to quantum gravity (despite claims in the philosophy literature that spacetime disappears from the fundamental commitments of such theories-on which see, for example, Huggett and Wüthrich 2013).
30 It is possible that all of the background metric-not just the signature-can be derived.
But we are silent in this paper about whether the other parts of a metric can be derived from electrodynamics or other matter theories.
(see appendix), which has the property that 2 ind * = (-1)  when acting on 2-forms in 4-dimensional spacetime, where "ind" denotes the number of minus signs in the metric signature.We hereby now reverse the order of derivation: instead of deriving the Hodge star from the metric signature, we derive the metric signature sig(g ab ) from , which plays the formal role of the Hodge star up to a scalar factor.In particular, we have: It follows that the sig(g ab ) is Lorentzian, given Axioms 1-5 of pre-metric electromagnetism as presented in the previous section.
Of course, the difference between signatures (+, -, -, -) and (-, +, +, +) is typically (and quite reasonably) regarded as being a conventional (rather than factual) one; by contrast, the difference between signatures (+, +, -, -) and (+, +, +, +) is far from trivial or a matter of convention alone. 31For example, in the former case, one (perhaps surprisingly) has wellposed Cauchy problems (see Craig and Weinstein 2009; Weinstein 2008, for the mathematical details) and so in a straightforward sense can model temporal evolution; this is not so in the latter case. 32This is fortuitous, since the different choices of signature associated with 2 = -  thereby turn out to be physically immaterial; not so for the case 2 =  . 33 While the Lorentzian metric signature is thereby determined uniquely from assumptions regarding the constitutive relation encoded in Axiom 5, we may ask the question as to whether other metric signatures are derivable from pre-metric electromagnetism given different assumptions regarding the constitutive relation.The answer is affirmative: case considered previously, however, in this case the signature (and, hence, the spacetime structure) would not be fixed uniquely by this particular form of the constitutive relationin this regard, a Lorentzian signature appears to be special. 34 For presentational clarity, we're setting aside other possibilities in the second of the above categories-for example, (-, -, +, +), (-, +, +, -), or (-, -, -, -).For some brief philosophical discussion of these issues, see Read et al. (2018, fn.35) and (separately) Callender (2017,  §6.3).
32 That being said, the connection between metric signature and the possibility of temporal evolution in fact turns out to be quite subtle-see Read and Cheng (2022) and, relatedly, Linnemann and Read (2021c).
33 There is, however, a potential problem for the program of pre-metric electromagnetism in this vicinity which, to our knowledge, has thus far gone unacknowledged: even supposing that 2 = - , why should it be the case that the "time" direction identified in the resulting Lorentzian signature be the same as that of the vector field v, orthogonal to dσ?That the two coincide appears to be "miraculous"-cf.Read et al. (2018, fn.35)

PHILOSOPHICAL APPRAISAL
Technically, the approach to pre-metric electromagnetism of Hehl et al. appears to have achieved the following.First, it is (or at least, purports to be) a reformulation of classical electromagnetism without presupposing a metric or any structure that is conceptually equivalent to, or derivative upon, metrical structure. 35Second, the Lorentzian signature is derived from the theory.These points, however, are not completely obvious or indubitable.
The derivation of the signature from the theory hinges upon the constitutive relation between F and H.One may object from this single step of derivation that the constitutive relation selected by Hehl et al. is conceptually equivalent to choosing a Lorentzian signature along with the standard formula H = * F. In this sense, the thought would go, we already introduce the metric signature in disguise via Axiom 5, and therefore this is not metricfree electromagnetism after all.But this criticism seems unfair, since there is no a priori conceptual connection between F, H, and metrical signature.The former are well-defined independently of any geometric concept, and their relation being local, linear and symmetric are reasonable requirements independent of any considerations to do with a metric.
Although this response seems to us to be reasonable, it's worth expanding upon some conceptual puzzles which arise out of the derivation presented in §3.One significant concern is this: as written, ( 14) implies that that there is some antecedently given piece of ontology-the metric field g ab -with some associated property (i.e., ideology, in the sense of Quine (1951); cf.Pooley ( 2013))-its signature sig(g ab )-not to be regarded as being fundamental, but rather to be regarded as having been derived via the program of premetric electromagnetism. 36This would seem to imply that the pre-metric program regards there as existing ab inito a metric field after all; it is just that certain properties of said metric field are to be derived on the basis of dynamical considerations (cf.our discussion of the related program of Schuller et al. in  §4.4).This strikes us as an acceptable understanding of what is going on in the pre-metric derivation of §3-although it is an understanding the discussion of which is often not emphasized in the writings of Hehl et al. on this topic. 37,38 It is also worth noting that there seem to be at least two other alternative ways in which one could interpret the derivations presented in the previous two subsections.First: one could make some attempt to separate out explicitly the signature degrees of freedom of g ab from all others (this proposal is discussed in greater detail in §4.1); if successful, one could thereby treat those degrees of freedom as derivative ontology.Second, one could continue 35 One might question whether it's really possible to understand the approach as having proceeded without presupposing a metric; we'll return shortly to this worry.
36 In the foregoing, we've plumped for an understanding of the metric field g ab on which it is a piece of ontology-a field "living" on the manifold M.But there is an alternative understanding of g ab according to which it too is a piece of ideology, for it encodes properties of the spacetime points comprising M, in particular distances and angles between said points.For the sake of simplifying the narrative, in the above we have elected to identify g ab as a piece of ontology.
37 This might be the preferred position of Hehl et al., since they want to leave room for a separate theory of gravity that interacts with other matter fields including the electromagnetic field (see Hehl and Obukhov 2003, ch.F).
38 Continuing on the theme of footnote 36: if the metric field g ab -qua object encoding properties of points of M-were indeed to be regarded as a piece of ideology, then one would, in analogy with our discussion in the main text, wonder what it could mean to retain a commitment to g ab in general relativity, yet reject a commitment to the manifold of points M-this being the interpretative stance urged by Earman and Norton (1987)  to think of sig(g ab ) as a piece of derived ideology, but reject the view that this need be a property of anything in the model: rather, in this case, one would regard that derived property as having modal import: if there were a metric in the model, then it would have this signature.We confess that we do not find this latter option particularly compelling, for if there were indeed a metric in the model (not to mention the possibility of multiple metrics, which would complicate the story yet further), what would compel it to have that particular derived signature, rather than some other?39But in any case, our purpose here isn't to defend one of these options, but rather to chart the space of interpretative questions which arise for Hehl et al., having presented the derivation of the metric signature within the context of pre-metric electromagnetism which we have recapitulated in § §2-3. 40en setting aside these issues (at least for the time being), there remains a great deal more to explore regarding the philosophical significance of pre-metric electromagnetism.
In this section, we take up these matters with respect to: (i) the question as to whether it is possible to liberate electromagnetism from a commitment to "background" structures ( §4.1), (ii) the significance of pre-metric electromagnetism for the version of spacetime relationism associated with the "dynamical approach" of Brown and Pooley (2001; 2006)   (some of the issues here were outlined already in §1) ( §4.2), (iii) the relation between premetric electromagnetism and the program of "constructive axiomatics" promulgated by Reichenbach in 1924 ( §4.3), and (iv) the connections between this work and the à la mode topic of spacetime functionalism ( §4.4).

BACKGROUND INDEPENDENCE
It is sometimes claimed that general relativity occupies a privileged point in the space of spacetime theories by virtue of its "substantive general covariance" or "background independence" (for reviews of such issues, see, e.g., Norton 1993; Pooley 2017; Read 2023); we use the two terms interchangeably in what follows, and will for simplicity tend to use the latter term). 41Clearly, this presupposes several things: 1.A precise definition of these terms.

2.
A clear assessment that general relativity satisfies said definitions.

3.
A clear assessment that other theories do not satisfy said definitions.Now, when it comes to (1), a cornucopia of definitions present themselves-see Read (2023) for a survey and assessment-however, we will focus here on a definition in terms of "absolute objects" (going back to Anderson (1967), and subject to significant critical scrutiny by Pitts (2006)), since (a) it is one of the most popular and plausible of such definitions, and (b) as we will see, it interacts in particularly interesting ways with the above-presented results from pre-metric electromagnetism.
The "no absolute objects" definition of background independence says this: a theory is background independent just in case it has no absolute objects in its formulation, where an Chen and Read Philosophy of Physics DOI: 10.31389/pop.45 absolute object is an object fixed (up to isomorphism) in all dynamically possible models of that theory. 42To illustrate the thought behind this proposal, consider the following two versions of special relativity, which Pooley (2017) dubs SR1 and SR2.SR1 has kinematical possibilities given by triples (M, η ab , φ) for some fixed Minkowski metric field η ab (i.e., η ab fixed identically in all kinematically possible models of the theory) and scalar field φ, and dynamical possibilities (a subset, of course, of the kinematical possibilities) given by the Klein-Gordon equation = 0, where Ñ is the Levi-Civita derivative operator of η ab .By contrast, SR2 has kinematical possibilities given by triples (M, g ab , φ) for some generic Lorentzian metric field g ab in this case not fixed identically across kinematical possibilities, and dynamical possibilities given by = 0, where now Ñ is the Levi-Civita derivative operator of g ab .(In both SR1 and SR2, we take it to be the case that The point of introducing these two different versions of special relativity is this: intuitively, SR2 is no more 'dynamical' in any interesting sense than SR1, for in essence all that one has done is convert the metric field's being fixed at the level of kinematics to its being fixed at the level of dynamics, by way of introduction of a suitable additional dynamical equation (imposing flatness of the associated derivative operator).The "no absolute objects" definition captures this intuition correctly, for it adjudicates that neither SR1 nor SR2 is background independent: both theories fail the definition.
Turn now to general relativity.At first blush, one's verdict might be that this theory passes the "no absolute objects" definition, for in this case the metric field is dynamical, and varies from dynamical possibility to dynamical possibility.However, Pitts (2006) demonstrates correctly that this verdict is false, for consider the fact that a Lorentzian metric field g ab can generically be decomposed as where g is the determinant of the metric, ˆab g is the conformal metric density, and n is the dimension of the metric manifold under consideration (for us, n = 4).Note, in particular, that locally the scalar density g is indeed fixed (up to isomorphism) in all models of general relativity-so, in fact, general relativity does have an absolute object!(For further detailed discussion of this point, see Read (2023,  §3.3).) This is very closely related to the fact that the signature of the metric in general relativity is fixed (up to isomorphism) in all dynamical (indeed, kinematical) possibilities.To see this, note that the metric determinant encodes this signature: if g < 0, one can infer that the metric has Lorentzian signature (recall again our discussion of these issues in §3; note also that g < 0 underdetermines the particular Lorentzian signature convention).Of course, g encodes more than just the metric signature: the magnitude of g encodes in addition facts about global volume which, alongside the conformal structure encoded in ˆab g (which 42 For a more formal presentation of Anderson's account, also commenting on the differences between Anderson's original presentation (Anderson 1967, 83) and Friedman's later reformulation (Friedman 1983, 58-60), see Pitts (2006, 348-353).Cf. also Hiskes (1984).Chen and Read Philosophy of Physics DOI: 10.31389/pop.45 essentially fixes the lightcone structure at each point), together via (18) suffice to fix the metric g ab .But the point is that one can see straightforwardly how the metric signature is the same (up to isomorphism) in all dynamical possibilities of general relativity, even if the metric itself is not so fixed.
If one could identify a means of modifying general relativity so as to incorporate dynamical signature change, then this concern regarding the exhibition of absolute objects even in general relativity-insofar as it is indeed a concern!-could be overcome.There is, indeed, a precedent for considering signature change in (modified versions of) general relativity (see, e.g., Bojowald and Brahma 2018; Brahma 2020; Hartle and Hawking 1983; Huggett and   Wüthrich 2018).Arguably though, such approaches are lacking, in the sense that they really seem to be changing the subject, by moving to theories distinct from general relativity.What would seem to be preferable would be an account as to how the metric signature need not be regarded as being fundamental in a relativistic theory: in this way, concerns regarding any commitment to the fundamentality of this particular absolute object could be avoided. 43is is exactly what the machinery of pre-metric electromagnetism seems to provide: at least within the framework of something like Einstein-Maxwell theory, the thought would be that the metric signature can be derived from the electromagnetic sector.But is it really so?At this point, it might again help to think through the issues in terms of through which the Lorentzian metric field g μν can be taken to be defined in terms of the frame fields; η ab = diag (-1, 1, 1, 1) is here to be regarded as being a mere matrix.
Thus, kinematical possibilities of electromagnetism set in a relativistic spacetime can be construed such that it is the frame fields which are primitive; thereby, the models can be written (M, η ab , a a e , F ab , J abc ). 45The merit of this particular decomposition over that discussed before is that η ab encodes all and only signature degrees of freedom-so, on the pre-metric approach, it is exactly this degree of freedom which is expunged. 46In this case, dynamics are 43 Of course, there's still the open possibility that the theory turns out to have other absolute objects.It's also important to be clear that, in viewing said absolute object as being non-fundamental, one has not thereby excised said object (one is, in other words, not being an eliminativist about that object, but rather simply reducing it to something else); nevertheless, the thought would be that one has made some progress by not having to regard this object as being fundamental.(For further discussion of issues regarding the fundamentality of absolute objects, see Read (2023, ch.3).)Chen and Read Philosophy of Physics DOI: 10.31389/pop.45 picked out by (a) the constitutive relations discussed in §3 which suffice to fix a Lorentzian signature and thereby the metric signature (now understood as, in part, a derived object), and (b) the Einstein-Maxwell equations (where now, of course, some constituents of such equations are to be regarded as being derived, rather than fundamental). 47w does this compare with the case of the move from SR1 to SR2? There, one retains the same ontology (of a Lorentzian metric field) and ideology (flatness of the derivative operator compatible with said metric field) in both theories-it is simply that the ideology is imposed at the level of kinematics in SR1, while at the level of dynamics in SR2. 48By contrast, in the move from "standard" electromagnetism to pre-metric electromagnetism, one in a certain sense trades ontology for ideology: the metric signature in the former theory is encoded in extra relations imposed upon the remaining geometric objects which comprise the kinematical possibilities of the latter theory. 49These differences notwithstanding, an analogy with the case of SR1 and SR2 is fairly plain: in both cases, one has moved kinematical impositions to dynamical ones.By analogy, then, if the original version of the theory qualifies as not being background independent in virtue of its possession of an absolute object, one might worry that this reformulated version, appealing to pre-metric electromagnetism, should also fail to be background independent.But there are ways to push back against this: one might argue that because there is an ontological reduction in the case of electromagnetism in a way that there is not in the case of SR1 and SR2, there is a better case to be made that there is genuine liberation from background structures here.On the other hand, if one notes that the metric signature and g are still implicitly definable (cf.Barrett 2017), then, as already indicated, one can levy the charge of a lack of background independence (understood in terms of the absence of absolute objects) in both cases.
We'll close this subsection with two more criticisms in the vicinity.First: in Read et al. (2018,   fn.34), it is argued that in theories such as Einstein-Maxwell (standardly formulated), it's a miracle (by which Read et al. (2018) mean: something surprising and unexplained) why the signature associated with the Lorentizan metric field which codifies the symmetries of the dynamical equations governing matter coincides with the signature of the primordial metric field g ab .One might think that the move to pre-metric electromagnetism can help to resolve this issue, for one could argue that the signature of g ab is nothing other than that derived from the equations governing material fields.In fact, however, it's at least not obvious to us that there would be a reduction in "miracles" on making this move, for it still appears to be outstanding why the signature obtained from the pre-metric approach should interact with the other metrical degrees of freedom (encoded-redundantly-in the frame fields) just so as to yield a well-behaved Lorentizan metric field g ab .
Second: in standard electromagnetism, one has the constitutive relation H = * F, where * is the Hodge star operator associated with g ab .Given this, one needn't commit to H or F as being primitive: both can be derived from J. By contrast, in pre-metric electromagnetism, 47 As usual in foundational discussions of electromagnetism, there are questions to be asked regarding whether to take F or its potential A (mutatis mutandis J or its potential H) as ontological primitives.It's principally for the sake of simplicity that we elect to formulate our theories above in terms of F and J.
48 For more on the kinematics/dynamics distinction in modern theories of physics, see Curiel (2016) and Linnemann and Read (2021c).
49 We say "in a certain sense," because one could argue that the metric signature is not an entity, but a property of the metric field to begin with.Fair enough: in this case, the correct thing to say is that one moves from primitively stipulated ideological commitments to ones which are derived on the basis of further dynamical considerations, much (again) as in the case of the move from SR1 to SR2. (Cf.our discussion in the introduction to this section.)Chen and Read Philosophy of Physics DOI: 10.31389/pop.45 one must retain a commitment to H (or J, as presented above) and F as primitive ontological posits, in order to derive the Hodge operator for a Lorentzian metric field from the constitutive relation between these objects.Thus, ontological parsimony and Occamist norms (cf.Dasgupta 2016) do not favour pre-metric electromagnetism in the way that one might have expected.Since Occamist motivations are some of the strongest militating in favour of relationalism in general (cf.(Babic, 2023; Pooley, 2013)), this point is, therefore, arguably a blow to some of the foundational motivations underlying the premetric program.

DYNAMICAL RELATIVITY
One of the central tenets of the "dynamical approach" to spacetime theories of Brown   and Pooley (2001; 2006)  When it comes to the Minkowski metric field of special relativity, Brown and Pooley's ambitions align, and both authors seek to reduce the metric field to symmetries of the dynamical equations via the dynamical approach.However, when it comes to the metric field g ab of general relativity, the two authors diverge: Pooley is still motivated to expunge a commitment to (what's represented by) this structure on Occamist grounds; by contrast, since g ab is dynamical, Brown is explicitly uninterested in such a reduction (see Brown 2005, ch.9).This being said, Brown recognizes that the signature of the metric in general relativity is indeed still fixed, non-dynamical, and action-reaction principle violating-so, his reductionist ambitions continue to track this more modest target.
On the face of it, pre-metric electromagnetism may seem to avail Brown of exactly the resources he requires in order to expunge any commitment to the fundamentality of the metric signature in general relativity.However, for the reasons already articulated in the previous subsection, one mustn't be too hasty here, for (a) this program clearly requires very particular material content, so cannot be fully general in the sense of being applicable to all possible worlds regardless of their material content (see §5 for prospects of generalisation), and (b) as already discussed, arguably the suspect absolute objects are "waiting in the wings" even when this program can be applied-so it is at least not obvious that Brown should be satisfied with it.Note also the reduction of the metric signature to dynamical 50 Actually, this is equivocal between eliminativism about the Minkowski metricaccording to which said object simply does not exist in one's roster of fundamental ontological commitments-and reductionism about the Minkowski metric-according to which said object exists, but is reducible to other entities in one's ontology.In our view, it is better to view this version of the dynamical approach as a form of reductionism rather than eliminativism; this squares with our comments below on functionalism.Chen and Read Philosophy of Physics DOI: 10.31389/pop.45 considerations will not satisfy authors (such as Pooley, on the above presentation) who strive for such a reduction of the full metrical structure of spacetime (for this, some other strategy-perhaps a generalisation of Huggett's "regularity relationalism" (Huggett 2006)   (see Pooley 2013; Stevens 2020)-may be more effective).Indeed, this is especially so given that the Occamist stripes of pre-metric electromagnetism, as we have just seen, are not as evident as one might think initially.
Finally, even granting that the metric signature can be derived from electromagnetic considerations, this result could, in principle, be co-opted for the purposes of those with a more "geometrical" outlook (see, e.g., (Brown and Read 2021) for more on the dynamical/ geometrical debate in the philosophy of spacetime).In particular, having derived said signature, one could argue subsequently that it acts as a constraint on the metrical signature associated with all other material fields-in this sense, the signature as it is manifest in all physics would indeed be specifically electromagnetic in origin.On the other hand, those with a dynamical outlook may prefer a more egalitarian stance, according to which each set of material fields yields its own distinct metrical signature-these may or may not agree, but this is ultimately a contingent matter, or at the very least is dependent upon the interactions between those fields.(Of course, this assumes that the pre-metric approach can be generalized to other material fields-more on this in §5.)This point is consistent with that of Geroch ( 2010), who stresses that each set of material fields may come equipped with its own light cone structure.

CONNECTION TO CONSTRUCTIVE AXIOMATICS
The dynamical approach discussed in the previous subsection occupies one corner of the literature on what is known as the "epistemology of spacetime" (of course, being an ontological reduction account, the approach is also metaphysical in nature-but the point is that, having secured said reduction, one would thereby also have come to settle an epistemological issue-see Dewar et al. (2022)).In another corner of this literature, one finds the program of what's known as "constructive axiomatics."The idea of constructive axiomatics, which is an agenda going back to Reichenbach in 1924 (Reichenbach 1969), is that one should build up the entire edifice of one's theories (including in many theories inter alia a commitment to a spacetime metric) from axioms which (supposedly) have indubitable empirical content.Thereby, the thought goes, one comes to set one's theories on solid empirical footing.The most famous constructive axiomatisation of general relativity is the 1972 approach of Ehlers, Pirani, and Schild (henceforth EPS) (Ehlers et al. 2012), 52 who purport to build up the Lorentzian manifold of general relativity from the trajectories of light rays (yielding, the claim goes, conformal structure) and freely falling particles (yielding, the claim goes, projective structure).The EPS axiomatisation has recently been subjected to detailed foundational scrutiny by Adlam et al. (2022) and Linnemann and   Read (2021b); in this subsection, we will assume some familiarity with the approach-the reader is referred to the cited articles for more background.
The fact that Hehl et al. formulate axioms for pre-metric electromagnetism also affords their approach the flavour of a project in constructive axiomatics; Hehl and Obukhov   (2006), indeed, explicitly identify the approach as such.But the very different nature of the axioms of the pre-metric approach as compared with those of EPS means that the former warrants some further consideration qua constructivist project.The first question to settle is this: which of the axioms of pre-metric electromagnetism are in fact empirical in nature?Tackling each in turn, the empirical content of the pre-metric axioms can-very briefly!be summarized as follows: Axiom 1: This is an axiom about the foliability of the manifold into hypersurfaces of constant (topological) time.Insofar as locally we think we can distinguish (even phenomenologically) a time direction, it seems to have empirical status; however, the global assumption seems to go beyond anything which we think we can access empirically. 53 Axiom 2: The empirical significance of  and  (which, recall, are the magnetic and electric components of H, which in turn is the potential for the current J) is this: both can be measured by ideal electric conductors and superconductors in the lab (Hehl and Obukhov, 2003, p. 116).J is measured indirectly through these objects-so this is the operational significance of the axiom.This, indeed, allows the axiom to make contact with the empirical, albeit in a much more theory-laden and indirect manner than e.g. the axioms of EPS. 54 Axiom 3: The motions of charged test bodies can be considered observable (albeit perhaps indirectly); such is the empirical justification for this axiom. 55In turn, this is used to afford F with empirical meaning. 56 Axiom 4: The empirical content of this axiom-which regards the conservation of F-derives from measuring in the lab the electric and magnetic parts of F, which are the usual electric (E) and magnetic (B) fields, respectively.
Axiom 5: As discussed in §2, the empirical content of this axiom is encoded in the energy-momentum trace being zero, which corresponds to the photon being massless. 57 The weakening of Axiom 1 to "local" foliations presented in footnote 14 thereby appears to be better justified on empirical/phenomenological grounds than the "global" version of this axiom.Note that (empirical/phenomenological, local) evidence for there being some foliation does not imply uniqueness--so there is no tension here with, for example, footnote 27.
55 Of course, there's more work to be done if this is to be compelling-one might appeal to, for example, (a) the fact that we can "track" the trajectories of outgoing bodies in particle scattering experiments (such as those at the LHC), or (b) the fact that an operational prescription for "tracking" test bodies using radar coordinates is given by Ehlers et al. (2012)  (see Linnemann and Read 2021b, for discussion).
56 It might not, however, be sufficient to afford an operational definition of F-see footnote 21.
57 By "massless," we mean that the field in question is dispersion-free-that is, propagates on null geodesics.There may ultimately be some complications to this understanding, given that it has recently been argued in Asenjo and Hojman (2017) and Linnemann and Read (2021a) that electromagnetic fields in curved spacetimes can exhibit dispersion; nevertheless, we'll set aside these complications here, on the grounds that (a) such dispersion relations remain to be fully understood, and (b) these dispersion relations appear only to be non-vanishing in the context of particular spacetimes (e.g., Gödel, Kerr) in any case.
It is worth noting here, however, that the characterisation of masslessness in terms of dispersion-freeness is not necessarily completely unproblematic, given that the latter notion makes sense only once one has to hand a particular metric field (with its associated signature)-but this is what one is supposed to be constructing in the pre-metric program!The right thing to say here, it seems to us, is this: the motions of photons are empirically detectable in any case (at least granting the points raised in footnote 55) but can only be characterized as dispersion-free once one has to hand this additional structure.But ultimately, it is the motions which are empirically-detectable, and which are underwritten by Axiom 5-not their status as dispersion-free/those of massless bodies.Chen and Read Philosophy of Physics DOI: 10.31389/pop.45Adlam et al. 2022, for further discussion).Given that pre-metric electromagnetism has a much richer empirical base (potentially encompassing a range of empirical phenomena to do with electromagnetism), a proponent of the dynamical view need not levy the same complaint in this case.
Hehl and Obukhov's own main complaint against EPS is that the latter appeal to the notion of a point particle, which they claim is no longer tenable in light of e.g.quantum mechanics (Hehl and Obukhov 2006).Here, they may have failed to appreciate that point particles and light rays are meant to be empirical starting points rather than fundamental, theoretical descriptions of matter that predict these empirical outcomes, which is not the concern of EPS. 61But still the complaint against EPS hints at a perhaps legitimate concern about the paradoxical nature of their starting points.How we formulate the observational statements is not devoid of theoretical postulates: we neither observe a point particle nor that it follows a smooth trajectory, as assumed by EPS.It is unclear why these theoretical postulates are less problematic, or more indubitable, than (say) the waves and the dynamical equations of electromagnetism, especially considering that the latter has the virtue of guiding and correcting the empirical observations and may thereby have an elevated epistemic status. 62

CONNECTION TO FUNCTIONALISM
Spacetime functionalism is sometimes presented in the contemporary philosophy of physics literature as the thesis that "spacetime is as spacetime does" (Lam and Wüthrich   2018).Absent further details, this is of course vague: Butterfield and Gomes (2020) put flesh on the bones by understanding "functionalism as a species of reduction" in the sense of Lewis (1970), so that if the primitive posits of one's theory fix uniquely some piece of (putatively) spatiotemporal structure, then the latter is reduced to the former, and (in this sense) one can be a functionalist about said latter structure.Butterfield and Gomes give several examples of this approach to spacetime functionalism, including inter alia the functional reduction of the temporal metric in the "Machian relationalist" theories of Barbour and collaborators (see Mercati 2018, for an introduction).The case which they discuss which is most relevant for our purposes is the program of Schuller et al. (see Düll   et al. 2018; Schuller 2011).As in previous subsection vis-à-vis the EPS axiomatisation, here we assume some basic familiarity with this work of Schuller et al.; Butterfield and Gomes   (2020) provides an accessible introduction.
In Schuller's program, one begins with a Lagrangian theory of matter coupled to some tensorial geometrical structure, the exact nature of which (e.g., index structure) is left unspecified.By imposing certain constraints on such theories (e.g., "predictivity" and "quantizability": note that these constraints are yet more theoretical-that is, less directly connected to empirical phenomena-than the axioms deployed in Hehl's premetric approach (cf.again Adlam et al. 2022)), Schuller et al. can, for example, rule out birefingence (according to which there are multiple possible lightcone structures at each spacetime point) 63 and ultimately arrive at the conclusion that the geometrical structure 61 Although note that authors such as Lämmerzahl have sought to modify the EPS approach by using more realistic-but theoretical!-quantummechanical models of matter as inputs (see Lämmerzahl 1998 and Adlam et al. 2022, for discussion).
62 For more on different degrees of theoretical inputs in constructive axiomatic approaches, see Adlam et al. (2022).
63 Notably, some of Hehl's axioms can also be understood as ruling out birefrigence-see Perlick (2011).Chen and Read Philosophy of Physics DOI: 10.31389/pop.45 postulated at the outset must in fact be that of a Lorentzian metric field (with its associated unique lightcone structure at each spacetime point), satisfying the Einstein equation. 64 The (Lewisian) functionalist strategy of Schuller et al. does indeed seem to afford another means of not regarding the metric field of (say) general relativity as being fundamental.
However, as already indicated in parentheses above, the assumptions which it makes are more theoretical than those of Hehl's approach; moreover, the approach begins by assuming that there is some geometrical structure before proceeding to pin down its properties, whereas the pre-metric electromagnetism approach of Hehl et al. does not begin with an ontological assumption of this form (at least explicitly-although recall the concerns raised at the beginning of this section that this may in fact be the case after all).This being said, on the other hand the payoff from the approach of Schuller et al. is arguably larger, insofar as the entirety of the metrical structure-rather than just the metric signature-is derived alongside the salient dynamics. 65Moreover, the approach of Schuller et al. is (at least at present) more general than that of Hehl et al., in the sense that it is not wedded to electromagnetism in particular.(Though, again, see below for the prospects for generalising Hehl's approach.)And finally: insofar as the approach of Schuller et al. proceeds in terms of action principles rather than in terms of equations of motion, it might ultimately have the merit of being more amenable to, say, path integral quantisation.

PROSPECTS FOR GENERALISATION
There are compelling motivations for seeking to extend the pre-metric approach to electromagnetism to other matter theories, such as non-abelian Yang-Mills theories. 66e important such motivation is to avoid a dependence of the metric field upon the electromagnetic field in particular-the issue here is that if other matter theories necessarily were to involve the metric field, then they would also be counterfacturally dependent on the electromagnetic field; however, there is little reason to think that the electromagnetic field should play such a privileged role amongst all matter fields.Additionally, the desire to isolate and minimize the role of the metric field in the formulation of electromagnetism applies to other matter theories as well.By reasoning analogous to that of Hehl et al. in   the case of electromagnetism, this leads to a conceptually clearer formulation of those theories.
In this section, we investigate the feasibility of generalising the pre-metric approach to non-abelian Yang-Mills theories.We demonstrate that while this is possible in principle, the resulting formulation is less straightforward or attractive due precisely to the nonabelian nature of a generic Yang-Mills theory.We close this section with some more general reflections on the prospects for a philosophy of non-abelian Yang-Mills theories.
64 Schuller considers the conditions which he imposes to be "classically hardly negotiable necessary requirements for the field equations and their geometric optical limit" (Schuller,  2022, 685).But arguably there is a parallel between the approach of Schuller et al. on the one hand, and that of Hehl et al. on the other, in the sense that just as one might argue that the latter's postulates regarding the relationships obtaining between H and J bake in what one is after (and so are no more conceptually perspicuous than simply postulating that the metric signature be Lorentzian), one might also argue that the former's primitive postulates again bake in what one is after.
65 Hehl et al. in fact regard the metric field as being on a par ontologically with all material fields.As outlined in their book, Hehl and Obukhov (2003) aim to purge the metric from electromagnetism so that the electromagnetic current density can be a pure matter contribution to the Einstein field equations, interacting with the metric field.
66 Yang-Mills theories are gauge theories based upon a special unitary gauge group SU(N).
For N = 1, we have electromagnetism.Chen and Read Philosophy of Physics DOI: 10.31389/pop.45

PRELIMINARIES FOR YANG-MILLS THEORIES
A generic Yang-Mills theory is characterized by two equations of motion which look very similar to the Maxwell equations of electromagnetism ( 1) and (2) (see, e.g., Skinner 2022): Despite this apparent similarity, there are two crucial differences between ( 20) and ( 21) and the Maxwell equations of electromagnetism.The first is that the field strength F, current J, and "connection" A are here not ordinary differential forms, but rather Lie algebravalued differential forms (for technical background here, see the appendix). 67The second is that instead of using ordinary exterior derivative operators on physical quantities, in the above equations one uses a "covariant exterior derivative" d A associated with a connection A, which is a g-valued 1-form (where g denotes the Lie algebra in question).For any g -valued form P, we have where d is the exterior derivative applied to such forms (see appendix) and [A, P] is the Lie bracket defined by that of g (again, see appendix).Furthermore, F is a g-valued 2-form, which is also called a "curvature form," but physically represents the field strength of a given Yang-Mills field.Technically, this means that the curvature can be obtained by applying the covariant exterior derivative with respect to the connection

THE PRE-METRIC APPROACH TO YANG-MILLS THEORIES
In generalising the pre-metric approach to generic Yang-Mills theories, we notice first that, as with electromagnetism, all the primary physical quantities (in this case g -valued forms) and all the operators other than the Hodge star are independent of the metric.Thus, to apply the methodology of Hehl et al., we mainly need to reconstrue the Hodge star in (21) as a relation between two physical fields, and from there to derive the metric signature.Furthermore, in order to implement fully the pre-metric methodology, we need to reformulate the differential equations in integral forms to highlight their empirical meaning. 69However, as we explain below, these tasks cannot be carried out straightforwardly in the case of a generic Yang-Mills theory because the techniques to which Hehl et al. appeal rely on the abelian property of electromagnetism.That said, we can make use of some less well-known techniques for non-abelian theories in order to make progress here.
We proceed now by imitating the axioms in §2 in formulating the axioms of a generic nonabelian Yang-Mills theory.For reasons which will become clear, however, this presentation does not follow the original order.
67 It is easiest to understand the objects under consideration here and the relations obtaining between them via the formalism of principle fibre bundles-on which see, for example, Nakahara (2003).The terminology of "connection" is appropriate to that setting.For our purposes in this article, the presentation in this paragraph will suffice.
Here, σ is a parameter on the closed curve ∂Σ introduced by the path-ordering P 1 ; τ is a second parameter on the surface Σ introduced by the surface-ordering P 2 , which roughly amounts to scanning Σ with τ-indexed loops, which in turn consist of σ-indexed points.
In this way, the order of integration is specified fully.On the LHS of ( 23), since A µ is a 1-form, we feed it the tangent vectors μ dx dσ on the path ∂Σ in integrating it over the path.This results in a Lie group element, since Lie algebra elements specify "infinitesimal paths" in the principal bundle space and adding them up produces a path which amounts to a Lie group element.The RHS of ( 23) is similar except for the addition of W and its inverse W -1 .The role of these objects is to parallel transport F μν (x) to the base point in the outmost loop. 72Intuitively, we parallel transport all F μν (x) with x Î Σ to a point using A, and then integrate them up in the order specified by P 2 .The result again is a Lie group element.It is important to note that (23) and more generally ( 22) are invariant under reparametrization (Ferreira and Luchini 2012b, 344). 73Otherwise, the equations would be unphysical, since the parameters introduced by the P n -ordering are arbitrary.
Note that under the non-abelian Stokes' theorem, (23) and more generally ( 22) are equivalent to their associated differential equations in the form of d A H = F for appropriate H,F (the derivation works in both directions).In this sense, they are integral forms of these equations, which we will utilize below.With (22) in hand, we are now ready to derive the integral form of the conservation laws.If the hypersurface in the RHS of ( 22) is a closed submanifold, and namely has no boundary, 70 There is a large literature here (see, e.g., Alvarez et al. 1998, 2009; Aref 'eva 1980; Bralić  1980; Constantinidis et al. 2017; Ferreira and Luchini 2012a,b; Fishbane et al. 1981).
71 Note that we have omitted the integration constant W R in this equation for simplicity, which is the value of W at the base point σ = τ = 0.The negative sign results from the differential equation relating W and Aµ (see Ferreira and Luchini 2012b, eq.2.1).See also footnote 72.The negative sign in this equation is a matter of convention adopted by Ferreira  and Luchini (2012b), which is related to how we define the relation between transported quantities and the connection that transports them.The negative sign convention implies that for a positive connection, the quantities decrease along the direction of transport.
72 Let L be a path joining x and the base point.Then W is defined as  74 Intuitively, since Ξ can be any closed 3D region, we can shrink it to an infinitesimal region (a 4-hypercube spanned by 4 tangent vectors), so the integral in YM Axiom 2 would become the desired covariant derivative.But we should be very cautious about such intuitions given that the high-dimensional case is notorious non-amenable to intuitions, not to mention the additional obstruction of non-abelianness.Even the simpler notion of exterior derivative is difficult to picture in the higher-dimensional case.Chen and Read Philosophy of Physics DOI: 10.31389/pop.45

OUTLOOK
What we have achieved here-modulo the substantial caveats mentioned above-is a metric-free reformulation of a generic Yang-Mills theory, from which we derive the Lorentzian signature.In this sense, the Lorentzian signature is not just grounded in electromagnetism, but in more general matter theories.But we should be completely explicit about the above-mentioned substantial caveats: 1.A satisfactory reformulation of the conservation laws hinges on technical developments on the non-abelian generalized Stokes' theorem in higher dimensions.

2.
The potential field H for the conserved current J may need to be posited as an axiom.
This is undesirable because it is less intuitive as a physical axiom than conservation laws, and moreover because the resulting theory is thereby committed to more ontological primitives.as those presented for the case of electromagnetism by Frisch (2005)?It goes without saying that answering any and all of these questions will have to remain a task for another day; nevertheless, the forgoing discussion should already make evident the philosophical and physical (as well as, of course, mathematical) significance of doing so.

A MATHEMATICS OF DIFFERENTIAL FORMS
In this appendix, we review some basics of differential forms, twisted differential forms, and Lie algebra-valued differential forms.

A.1 DIFFERENTIAL FORMS
Definition 1.A differential r-form is a totally antisymmetric tensor of rank (0, r).
As a tensor of type (0, r), a differential form takes r vectors and yields a number.As a totally antisymmetric tensor, it changes sign under the exchange of any two of its indices (for example, a 3-form T ijk is such that T ijk = -T jik = T jki = -T kji = …).Geometrically, an r-form in an n-dimensional space indicates the density and alignment of (n-r)-dimensional surfaces.
For example, a 2-form in 3D space can be represented by aligned lines.(For some beautiful diagrams illustrating this visualisation of differential forms, see Burke 1985.) Definition 2. The interior product of an r-form ω with a vector a, written as aω, is the (r - 1)-form such that (aω) (a 2 , a 3 , …a r ) = ω(a, a 2 , …, a r ).
In particular, the interior product (sometimes called the "contraction") of a 1-form with a vector amounts to evaluating the 1-form on the vector.For any scalar function f (which is a 0-form), we have that afω = faω = f(aω).
Definition 3.An exterior product Ù (also called the "wedge product") is a mapping from any p-form ω and any q-form ɣ to a (p+q)-form such that it satisfies associativity, and for any vector a, we have a(ω Ù ɣ) = (aω) Ù ɣ + (-1) p ω Ù (aɣ).
Such a mapping can be shown to be unique.It is useful to note that we have ω Ù ɣ = (-1) pq ɣ Ù ω, due to the antisymmetry of forms.Geometrically, as a simple example, the exterior product of 1-forms in 2D space can be visualized by arrays of points with an orientation (like screws).This requires assigning an orientation to the integration region given by dx 1 dx 2 …dx k , which one might find unnatural.This is an important reason for favoring twisted differential forms (see Burke 1985, 197-198).Chen and Read  Philosophy of Physics DOI: 10.31389/pop.45 Regarding the integration of forms on manifolds, there is the following famous result (where Γ is a region and ∂Γ is its boundary): Theorem 1 (Stokes' theorem).In physics, we are often concerned with the differential forms which can be understood to be the potentials (i.e., exterior derivatives) of other forms.For such matters, the following definitions and results are relevant.
Definition 6. (Exact form).A differential form ω is exact if there is a form H such that ω = dH.
The study of when a differential form is exact has to do with the theory of de Rham cohomology.
Definition 7. (Closed form).A differential form ω is closed if it is such that dω = 0.
For closed differential forms, even if they cannot be understood globally to be the potential of some other form, the following result underwrites this result locally: Theorem 2 (Poincaré lemma) In a space contractible to a single point, every closed form is exact.
Finally, on a manifold with metric, we can define the following important operations on differential forms: Definition 8. On manifold with metric g ab , the sharp operator  maps any 1-form α into a tangent vector α such that, for arbitrary vector , ( ) = The Hodge star operator can be shown to be unique.For the above definition to be welldefined, we have to define the Hodge star of a 0-form-so pick * 1 to be an n-form of unit density and positive orientation (Burke 1985, 160).The Hodge star contains information about the metric and the orientation of the space.Geometrically, we may visualize the Hodge star as mapping p-dimensional surfaces to (n-p)-dimensional orthogonal surfaces.As stated in the main text, the Hodge star operator "completes" a p-form to the metric-compatible volume form of (M, g) (where g is the metric on M), and thereby defines an (n-p)-form on M.

A.2 TWISTED DIFFERENTIAL FORMS
With the above basics of differential forms established, we turn now to twisted differential forms.To prepare, an orientation of a vector space is given by an equivalence class of all volume forms that have the same sign.For example, the orientation of the twisted form (dx, dx Ù dy Ù dz) is given by dy Ù dz.
Suppose we represent dx visually by the surface spanned by the y and z axis.Then we can consider the orientation of dx as pointing out from the surface.We can call this an "outer Chen and Read Philosophy of Physics DOI: 10.31389/pop.45 orientation."Now, the twisted form (dx, dx Ù dy Ù dz) has the orientation given by dy Ù dz, which can be visualized as pointing counterclockwise (or clockwise depending on the convention) inside the surface.We can call this an "inner orientation."Thus, twisting amounts to changing an outer orientation to an inner orientation.Note that it is often more natural to represent physical quantities using twisted forms than ordinary ones.For example, a twisted 3-form electric current in 4D spacetime can be visualized as directed lines, namely lines with inner orientations.(For more illustrations of twisting, see Burke   1985, 183ff.)As Burke writes of twisted differential forms, Do not let the fact that twisted forms are being represented in terms of ordinary forms lead you to think that twisted forms are less fundamental.The situation here is symmetric: we could define an ordinary vector as a pair consisting of a twisted vector and an orientation under the same equivalence relation.(Burke 1985, 189-90)   The point here is that one can define "ordinary" differential forms in terms of twisted forms or vice versa, but ultimately one can regard either as being the more primitive notion; in this sense, twisted differential forms need not be construed as being mathematically derivative upon "ordinary" differential forms, despite potential notational implications to the contrary.Under the operations defined in the previous subsection, twisted forms behave in the expected ways: As illustration: the twisted Hodge star maps an ordinary 1-form in 3D space to a twisted 2-form, and a twisted form to an ordinary form.Ironically, because the Hodge star depends upon an orientation, the twisted Hodge star in fact does not depend on the orientation of the space, despite its more complicated notation.
Despite its appearance, a twisted differential form can be integrated over in a space without orientation, for example, the Möbius strip.For twisted differential forms, we define Here, Ω is an orientation for the whole space, and (Γ, Ω) is a subspace that has an outer orientation (Burke 1985, 197).Given this, we then have the following result pertaining to the integration of twisted forms (see Burke 1985, 199-200)):

A.3 LIE ALGEBRA-VALUED DIFFERENTIAL FORMS
Finally, the formalism of non-abelian Yang-Mills theories mandates recourse to Lie algebravalued differential forms.To introduce these, first recall the definition of a Lie algebra: then it is underdetermined whether we have (+, +, -, -) or (+, +, +, +), both of which are non-Lorentzian signatures.Relatedly, in the recent work ofFecko (2022), Hodge star operators associated with Galilean spacetime (which is obtained by widening the lightcones of a Lorentzian spacetime subject to certain integrability conditions) and Carrollian spacetime (which is obtained by narrowing the lightcones of a Lorentizan spacetime) are constructed; in both cases, they are nilpotent operators, as one would expect, given 2 ind * = (-1) .This opens up further possibilities which could arise from pre-metric electromagnetism: if κ were nilpotent, one could in principle situate the theory in Galilean or Carrollian spacetime.As in the 2 =   kinematical and dynamical possibilities.For clarity of presentation separating out the two sub-constituents g and ˆab g of g ab , one could write the kinematical possibilities of standard EinsteinF J ; dynamical possibilities are given in terms of the Maxwell and Einstein equations written in terms of the relevant objects.This won't exactly do for our purposes, however, since we've already seen that g encodes more than merely signature degrees of freedom.44What we require, therefore, is a cleaner decomposition of a Lorentzian metric field, which allows us to isolate exclusively the signature degree(s) of freedom.One way to proceed would be to use frame fields a
, there is an important caveat: to derive the covariant derivative equation d A J = 0 from the integral form, we need to apply the non-abelian Stokes' theorem to a four-dimensional path-ordered integral.While we do not see an obvious reason why the four-dimensional case does not hold,74  we are not aware of any explicit proof for this in the literature (note that d in (22) is less than the dimension of spacetime and therefore at most d = 3).This technical lacuna may be attributed to the fact that the integral version of the standard Hodge star formulation of Yang-Mills theories does not require a higherdimensional non-abelian Stokes' theorem (the integral version of equation = * A d F J involves only up to a three-dimensional volume-ordered integral) as opposed to our premetric formulation in the spirit of Hehl et al.

Definition 4 .
An exterior derivative d is a mapping from an r-form to an (r+1)-form such that for any 0-form f, any p-form ω, and any q-form ɣ, (i) df is the differential of f, andd(df) = 0; (ii) d(ω Ù ɣ) = dω Ù ɣ + (-1) p ω Ù dɣ.Such a mapping can be shown to be unique.The integration of differential forms, which plays a central role in Hehl et al.'s formalism, is defined as follows.Let Γ be an embedding of k  into M, and let Σ be the region of k  such that Γ(Σ) is the integration region.Γ* is the pullback of Γ.Then we have: In an n-dimensional space, a Hodge star operator * is a linear operator from a p-form to an (n-p)-form such that, for β any r-form and α a 1-form, * (β Ù α) = α Ù * β.

Definition 10 .
A twisted form is an equivalence class of pairs consisting of an ordinary form ω and an orientation of whole space Ω under the equivalence relation ( , Ω) (-, -Ω) ω ω º .The twisted form represented by (ω, Ω) is said to have an orientation Ω ω which satisfies Ω = Ω ω ω  .
star operation itself presupposes an orientation, it too can be twisted, which is what allows for the possibility of a twisted Hodge star operator: Definition 13. (Hodge star (twisting)).( , Ω)( , Ω) = ; ( , Ω) = ( Chen and Read  Philosophy of Physics  DOI: 10.31389/pop.45 for more details.Chen and Read is that spacetime structure is to be understood as nothing more than a codification of the symmetries of the dynamical laws governing material fields: it is for this reason that Brown and Pooley declared famously that the Minkowski metric of special relativity is a "glorious non-entity." 50Of course, this is a relationalist thesis- however, when one gets into the weeds of their positions, different projects appear to be driving the ambitions of Pooley versus Brown.For the former, it is an Occamist ambition to expunge redundant structure from one's roster of metaphysical commitments(Pooley   2013). 51Brown, by contrast, what is problematic are, in particular, objects which violate the "action-reaction principle": which act on other bodies, but which are not acted back upon in turn (see the discussions byBrown (2005)andLehmkuhl and Brown (2016)for wider-ranging explorations of this principle).
the current potential H and the field strength F, from which we derive the metric signature.Assume for the time being that the existence of F and H are legitimized by other axioms.The good news is that this axiom also remains essentially the same for a generic Yang-Mills theory, with F and H now being Lie algebra-valued (twisted)As with Hehl et al., we would like to derive from this axiom that the energy-momentum current Σ α (which, in the absence of a force f, can be defined directly by H and F as Recall that d A F = dF + [A, F].In the generic case, the commutator [A, F] does not vanish because the Lie algebra in which A and F take their values is non-abelian.Thus, we cannot apply Stokes' theorem in order to generate a conservation law in the style ofHehl et al.Chen and ReadPhilosophy of Physics DOI: 10.31389/pop.45Theseproblems can be addressed to some extent by appealing to a non-abelian version of Stokes' theorem.70LetΞbe a simply-connected hypersurface of dimension d < n, where n is the dimension of the manifold.Let F and A be Lie algebra-valued d-forms and (d-1)-  are appropriate terms built out of F and A (which will be illustrated by a concrete example below).One non-abelian version of Stokes' theorem then implies(Ferreira and Luchini 2012b, eq.1.1): Chen and Read  Philosophy of Physics  DOI: 10.31389/pop.45then on the LHS the integral over the boundary is trivial-it yields the identity element in the Lie group.In this case, we have  is the identity element in the Lie group.It is easy to see that the curvature of a curvature vanishes by applying the non-abelian Stokes' theorem to the LHS of (24) twice in different directions (one involving d -1 and one involving d + 1).Note that this is different from claiming the covariant derivative d A is nilpotent, which is false, since two different connections are involved.As a special case, however, we do have 2 = 0 There cannot be a net flux into the region Ξ, thus the flux is conserved.Thus we have obtained a version of Axiom 4 suitable to non-abelian Yang-Mills theories.Note that, ∮.73 On this point, see alsoAlvarez et al. (1998, 2009)andFerreira and Luchini (2012a).unlikeAxiom 4, the above axiom is expressed in terms of  ; since, however, this object is built from F, one can still argue that YM Axiom 4 has empirical significance-albeit perhaps not quite as direct a manner as Axiom 4. In short: if F has empirical significance, then  will inherit this.Similarly, we can give an analogous version of Axiom 2, which concerns the conservation of electric current J.In Yang-Mill theories, J is a Lie-algebra-valued twisted 3-form.Let  be an appropriate term constructed from J, What can we conclude from these points?Barring further technical developments, this reformulation amounts to less than that to which Hehl et al.'s program aspired.Still, we may (a) accept this reformulation and conclude that the Lorentzian signature is grounded in a general matter theory and is therefore subject neutral.Or, we may (b) reject this reformulation and conclude that metrical signature is in nature electromagnetic and electromagnetic alone, precisely because the electromagnetic field commutes with the connection while a generic Yang-Mills theory does not.
Ad (a): Following this route seems to lead to further philosophical puzzles, for example: why is it, then, that the signatures derived from all different material Yang-Mills fields happen to coincide?(Recall§4.2, in which this question was already raised.)Ad(b):thisroute can lead to further pessimism about the pre-metric program in general-for why should we prescribe a special ontological status to the electromagnetic field (namely, providing the fabric of spacetime for all other material fields), just because of its seemingly conceptually insignificant feature of having this commutation relation?This may suggest that Hehl's program of reducing the metric signature to electromagnetism in particular is dialectically unstable.We may either achieve a fully satisfactory generalization or conclude that the metric signature is not reducible to matter fields in the wayHehl et al. envision after all.thenotoriouslyvexed issue of conservation of gravitational energy in general relativity (on which see, e.g., de Haro 2022, for a recent masterly review).Ad (iii): inter alia, this raises questions for theoretical equivalence (on which see, e.g., Weatherall 2019a, b, for recent surveys)-if a field strength tensor cannot always be written as the exterior derivative of some potential, does this thereby stand in the way of the two formulations' being equivalent?And ad (iv): one might ask, do such laws raise similar issues of inconsistencyChen and Read