## Abstract

Eleanor Knox has argued that our concept of spacetime applies to whichever structure plays a certain functional role in the laws (the role of determining local inertial structure). I raise two objections to this inertial functionalism. First, it depends on a prior assumption about which coordinate systems defined in a theory are reference frames, and hence on assumptions about which geometric structures are spatiotemporal. This makes Knox’s account circular. Second, her account is vulnerable to several counterexamples, giving the wrong result when applied to topological quantum field theories and parity- and time-asymmetric theories. I advance an alternative account on which our spacetime concept is a cluster concept. On this view, the notion of metaphysical fundamentality may feature in the cluster, in which case spacetime functionalism may be uninformative in the absence of answers to fundamental metaphysical questions like the substantivalist/relationist debate.

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## Notes

Or more precisely, to borrow a term of art from Brown, as having

*chronogeometric significance,*i.e. as being measured directly by physical rods and clocks.Knox does not address the question of what we should say when there are multiple, inequivalent but equally simple structures which are all sufficient to define inertial frames; the question of what she should say in such a case is beyond the scope of this paper, so I will assume a unique such structure exists in the specific cases discussed.

In general relativistic field theories exhibiting so-called minimal coupling, such frames will not always exist even local to a point (see Read et al. 2018). Although this is a serious problem for the scope of Knox’s inertial functionalism, it is far enough removed from the considerations I wish to raise here that I will set it aside.

Knox cites Ehlers et al. (2012) here.

More may need to be said to recover non-inertial structure in other theories aside from general relativity—and indeed, more may need to be said about what licenses the assumption on Knox’s part that all of this is happening within the specific theoretical framework of general relativity rather than some more general one—but rather than pursue this point here, let us move on.

This is not obviously correct, though, since (for example) in a spatially closed universe the scale factor is required to determine the total volume of space, which certainly seems like a physically significant quantity.

The sense in which the dynamical symmetries preserve the laws may differ from theory to theory; typically this means something like leaving the Hamiltonian and/or the Lagrangian unchanged.

As Knox notes, this should only go for external symmetries—we do not expect internal symmetries such as gauge transformations to correspond to symmetries of spacetime (Knox 2018, p. 11). Unfortunately, this introduces an ambiguity in Earman’s principles, since as Knox notes, there is no accepted spacetime-independent definition of which symmetries count as “external.” But Knox treats this as a peripheral complication; therefore, so will I.

To reiterate, she has suggested in discussion that these theories are her view’s intended domain.

Knox (2013, p. 348) notes that Erik Curiel and James Weatherall have objected to her view in conversation along these lines.

To avoid the complication of Knox’s first criterion for inertial frames, I assume here a version of electrodynamics which treats it purely as a field theory, either without charged matter or with charged matter represented by a field rather than point particles.

The details of the example can be filled in in a couple of different ways. In particular, the particles could be assumed to be labeled, or possess identity over time, in which case the laws literally govern their discontinuous motion. Alternatively, the particles could be assumed to lack transtemporal identity, in which case the laws simply govern the facts about the distribution of particles at any given time. Either way there is no apparent way to extract inertial structure from the laws.

This holds if we understand ‘three-dimensional general relativity’ to mean the theory governed by the Einstein field equations in three dimensions. Interesting questions arise as to whether this is the correct way to identify general relativity in lower dimensions, given the vastly different qualitative behavior of the resulting theory (Fletcher et al. 2018). This is an interesting question in the semantics of scientific theories, but it is not necessary to answer it for present purposes, since the important question for present purposes is only whether the lower-dimensional Einstein field theory describes spacetime—which Fletcher et al. agree that it does.

The probability for a given value of the geodesic’s length is given by integrating the amplitude over all metrics in the superposition assigning that value to the length (Barrett 2003). These probabilities, and hence the observable’s expectation value, are topological invariants, even though the individual metrics in the superposition are not.

An analogy that may help: In ordinary quantum mechanics, the singlet state \(1/\sqrt{2} (|\uparrow \downarrow \rangle -|\downarrow \uparrow \rangle )\), and the observables defined on it, are permutation invariant even though the two terms in the superposition are individually not permutation invariant. Similarly, the state in three-dimensional quantum gravity, and the observables defined on it, are topologically invariant even though the state is a superposition of metrics which are not individually topologically invariant.

I say this is “arguable” because we are verging into delicate territory about what it is for the laws to take “the same form” in different frames/coordinates. For example, suppose one writes down relativistic laws with a metric signature \((+ - - -)\), then writes down the same laws except with metric signature \((- + + +)\). Has one changed the “form” of the laws? (Of course this is not a change in coordinates, but it is similar in that it is a conventional change in the geometric representation of the laws.)

Not the biological concept of a cat as a natural kind whose evolutionary descent or genetic structure is essential to it, but rather the folk concept of a cat on which (e.g.) the Cheshire cat is a cat.

In cases where there is a tie, it remains an option to treat the different equally-good realizers as distinct coexisting spacetimes, as the straightforward variant would have it. But plausibly the criteria are many and fine-grained enough to prevent ties except in unusual examples.

That is to say, the state of the structure does not differ between different states of the theory; or if it does differ, its state is fully determined by a “mass-energy” source like the mass in Newtonian gravity or the stress-energy in general relativity.

As another rough guide,

*T*is a more fundamental theory than \(T'\) if \(T'\) reduces to*T*in some sense.As Lewis points out, Putnam’s model theoretic argument (Putnam 1977) implies that there is no way to distinguish an intended model for a theory by adding posits to the theory itself (by adding “more theory”). Thus fundamental structures in the domain of the theory, which are more apt subjects of reference for our theoretical terms than non-fundamental structures, are necessary to avoid the unacceptable result that all possible theories (or perhaps all possible empirically adequate theories) are true. Adapting a classic example: On this view, it is the relative fundamentality of green as compared with grue that explains why our word ‘green’ refers to the former property and not the latter, despite the fact that its use is ambiguous between those two interpretations.

I take relationism to be the denial of substantivalism—that is, following North, the claim that spacetime is less fundamental than material objects.

As North puts it: “The relationalist says that material bodies, and certain of their properties and relations, are fundamental, and a world’s spatiotemporal structure holds in virtue of them. [...] So, for example, the fact that a world has a Euclidean spatial structure is grounded in, holds in virtue of, the fact that its particles are, and can be, arranged in various ways, with various distance relations between them.” (North 2018, p. 13) On this sort of picture, the distance relations are treated as fundamental properties even though spatial structure itself is not a fundamental substance.

Note that this means Knox’s approach and Albert’s may turn out to be compatible in a sense, since there is nothing in Albert’s work to rule out the possibility that Knox’s inertial functionalism is the correct way to identify the space of interactive distances.

Again, assuming for the sake of argument (out of charity to Knox) that a unique simplest structure determining local inertial frames exists.

Following North, I take this possibility to be equivalent to substantivalism about the spacetime of Newton–Cartan theory.

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## Acknowledgements

Thanks to Tushar Menon and James Read for illuminating comments on a previous draft, and to Gordon Belot, David Wallace and (especially) Eleanor Knox for helpful discussions and correspondence.

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Baker, D.J. Knox’s inertial spacetime functionalism (and a better alternative).
*Synthese* **199**
(Suppl 2), 277–298 (2021). https://doi.org/10.1007/s11229-020-02598-z

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DOI: https://doi.org/10.1007/s11229-020-02598-z