Geometrical Constructivism and Modal Relationalism: Further Aspects of the Dynamical/Geometrical Debate

ABSTRACT

I draw together some recent literature on the debate between dynamical versus geometrical approaches to spacetime theories, in order to argue that (i) there exist defensible versions of the geometrical approach; (ii) these versions of the geometrical approach can provide constructive explanations (in the sense of Einstein) of dynamical effects; (iii) light can be shed upon different relationalist views about spacetime which have been articulated in the context of this debate by appeal to the distinction between modal versus non-modal relationalism.

Acknowledgments

I am very grateful to Catherine Ashworth, Harvey Brown, Kyle Butcher, Neil Dewar, Patrick Dürr, Tushar Menon, Vladimir Mikulik, Brian Pitts, Oliver Pooley, Nic Teh, Jim Weatherall, and the anonymous referees, for invaluable discussions on these matters.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 It is now widely agreed that, whatever is the relevant sense of explanation at play, it must be non-causal. See Reutlinger and Saatsi (2018) for a recent collection on ‘explanation beyond causation’.

2 Two points here. First: throughout this paper abstract index notation is deployed. Second: the notion of a fixed field is still rather unclear in the literature; I have included an appendix on this notion, to which I refer the reader for further details. Reference to the notion of a fixed field is part of my reconstruction of the dynamical approach, but I think that other classes of objects—in particular, those which violate what Brown calls the ‘action-reaction principle’ (cf. Brown and Lehmkuhl 2015)—would serve equally well. (I thank an anonymous referee for prompting me to write the appendix on fixed fields.)

3 There are many open questions here. For example: which symmetries are relevant—‘internal’ or ‘external’? (For discussion of this issue, see Dewar 2020.) Second: how is this ontological reduction supposed to work? (To my mind, one of the most promising approaches is the generalised Human strategy defended in Stevens (2020).) Third: how is the dynamical view supposed to distinguish between spacetimes without symmetries? And fourth: can the ontological reduction aspect of the dynamical view be extended from geometry to topology? (For discussion on this matter, see Menon 2019; Norton 2008.)

4 This latter point is what Butterfield calls in Butterfield (2007) ‘Brown's moral.’

5 Or at least, legitimate explanations of the symmetry properties of the dynamical equations governing matter. Even for a geometrician who makes the above further stipulations, there is no guarantee that there will exist (say) stable rods and clocks capable of surveying the piece of spatiotemporal structure under consideration (see Read 2020). This is the distinction between what is referred to in Read and Menon (2019) as ‘theoretical spacetime’ (that structure whose symmetries coincide with the symmetries of the dynamical laws) and ‘operational spacetime’ (that structure surveyed by rods and clocks). One case in which theoretical spacetime seems to come apart from operational spacetime is massive scalar gravity—a case study explored extensively by Pitts (2011a, 2011b, 2016, 2019). To be clear though: I do not regard such cases as being problematic for the viable versions of the geometrical approach discussed in this paper, which seek to appeal to certain pieces of geometrical structure to explain (whether constructively or not—see below) some aspects of the dynamics (in particular, the symmetries of the laws), and not (as made clear above) why that structure has chronogeometric significance (and, moreover, do so without assuming that said structure is the ‘One True Geometry’, thereby evading concerns raised in Pitts (2019)).

6 Oliver Pooley has argued to me that advocates of the dynamical approach should extend (1) to dynamical fields also—so that they should seek to ontologically reduce the metric field of general relativity also to the dynamics of matter. Though I agree in principle with this verdict, it remains to be shown in detail how this strategy can be realised in practice: Pooley has suggested a generalisation of the Humean strategy articulated in Stevens (2020).

7 Cf. Lange (2018).

8 Although Acuña writes, ‘It would be better to use the term “absolutism” for this basic thesis, so that “substantivalism” would refer to a particular form of the absolutist thesis.’ Acuña (2016, 6), I am not convinced that ‘absolutism’ is better nomenclature—it is too reminisient of the closely-related issue of the existence of ‘absolute objects’ in spacetime theories (cf. Pitts 2006). Perhaps simply ‘generalised substantivalism’ would be a better choice.

9 It is worth pointing out that Acuña is not the only author to observe this cosmic coincidence—it was labelled in Read, Brown, and Lehmkuhl (2018) the ‘second miracle of relativity’.

10 And I have sympathies with him—cf. footnote 6 above.

11 On this matter too I am in agreement with (Acuña 2016).

12 One attempt to give such an explanation was provided in Read (2019).

13 Frisch remarks that ‘it is not clear how well Janssen's classification of a spacetime account as constructive sits with Einstein's own characterisation of constructive theories as building “up a picture of the more complex phenomena out of the materials of the relatively simple formal scheme from which they start out.”’ Frisch (2011, 179) Frisch is correct here, if we read Einstein as identifying constructive and fundamental explanations. We have, however, already seen that these two notions can be teased apart.

14 In the context of universally coupled massive scalar gravity (see Pitts 2011a, 2011b, 2016, 2019), or bimetric theories more generally, one might be able to point to one metric field (say ${\eta }_{ab}$) to explain some aspects of the laws—but perhaps not all of them, in light of the coupling of the other metric field. This I take to be compatible with the qualified geometrical approach.

15 That symmetry properties of dynamical laws can provide a suitable basis for constructive explanations is questioned in passing by Brown and Pooley, who write:

[O]ne might be tempted to deny that explanations which appeal to an explanans as non-concrete as the symmetries of the laws are genuinely constructive explanations. In other words, it turns out that there are even fewer contexts than one might have at first supposed in which length contraction stands in need of a constructive-theory explanation. Brown and Pooley (2006, 82–83)

I do not agree with this verdict: the truncated Lorentzian pedagogy appeals to certain aspects of the fundmental dynamics in order to explain special relativistic effects. I do not see what more could be required of a (partial/truncated—not fundamental) constructive explanation. Perhaps one reason for hesitance here is that one might think that, if symmetries of the laws can offer certain constructive explanations of dynamical effects, then, since spacetime structure codifies those dynamical symmetries, spacetime structure, even for advocates of the dynamical view, or for non-absolutist geometricians, would have to be understood as offering constructive explanations for certain dynamical effects. One natural response here is simply to deny that constructive explanation is conserved under codification. Then, on the dynamical view, although the symmetries of the laws can feature in constructive explanations, spacetime structure (which codifies those symmetries) cannot; similarly for non-absolutist geometrical views. In my view, such a response is reasonable: only aspects of the formalism of a theory with direct correlates in physical reality can offer constructive explanations. For dynamicists and non-absolutist geometricians, spacetime structure does not itself feature in constructive explanations—it only does so qua codification of certain other aspects of the physics.

16 This latter aspect was referred to in Read, Brown, and Lehmkuhl (2018) as the ‘first miracle of relativity’—cf. footnote 9.

17 Others may be extracted from Reutlinger and Saatsi (2018).

18 On this point, Pooley states that ‘the principles of principle theories are not explanatory’ (Pooley 2017b, 3), and subsequently suggests that ‘it would clearly be a mistake to think that the principles from which you derive something in the 1905 derivation are explanatory of what you go on to derive, in particular length contraction’ (Pooley 2017b, 5). I agree: the principles of principle theories (e.g. the light postulate) need not be explanatory (here: explanatory of special relativistic effects), but the principle theory itself, which is derived from those principles (and thus is logically weaker—in the context of special relativity, the principle theory is just that of universal Lorentz invariance of dynamical laws) can be explanatory of the relevant set of dynamical facts.

19 In light of the foregoing discussion, according to which even principle theories may sometimes be understood as offering (partial/truncated—not fundamental) constructive explanations of physical phenomena, it might be preferable to prefix ‘constructive theory’ in discussions of $P_2$ with (e.g.) ‘deeper’. I will pass over this subtlety in the remainder of this section.

20 Cartwright should not necessarily be understood as objecting to $P_2$; rather, she should only be understood as objecting to the stronger claim that there is always a fundamental theory associated with any given theory; this denial is consistent with $P_2$. What is true, however, is that the denial of $P_2$ entails the denial of this stronger claim. Thus, while Acuña and Cartwright might disagree on $P_2$, they nevertheless agree on the stronger claim. My thanks to Nic Teh for discussion on this point.

21 Although Brown himself would not place much weight upon the importance of the nomenclature ‘spacetime’—cf. Brown and Read (2020, §3.1).

22 At least assuming that the actual world is special relativistic!

23 As Weatherall stresses, “according to the theory of smooth manifolds, diffeomorphism is the standard of isomorphism for manifolds; just as other mathematical objects are only defined up to isomorphism, manifolds are only defined up to diffeomorphism” (Weatherall 2018, 335). Thus, supposing that we have two models $\mathcal{M}$ and $\sim \mathcal{M}$ related by some diffeomorphism ψ, but not by the identity map 1_M (so that they are, indeed, numerically distinct mathematical objects), we are not usually concerned with interpreting those models in a way which ‘cuts finer’ than diffeomorphism. Here is not the place to discuss Weatherall on the hole argument and representation in physics—I refer the reader to Pooley and Read (2020) for further discussion.

24 In this appendix, I’ll use O_i for generic geometric objects on M; F_i for fixed fields on M; and D_i for dynamical (i.e., non-fixed) fields on M.

25 Pitts, in turn, drew the notion of a confined object from (Thorne, Lee, and Lightman 1973).

26 One is only here entering rather than leaving the weeds—see (Pooley 2017a; Read 2016) for discussion.

27 See Kretschmann (1917) for Kretschmann's original paper, and e.g. (Teitel 2019, §2.1) for some recent discussion.

28 For further discussion, see (Hiskes 1984; Pitts 2006).

29 In Pooley (2017a), these theories are dubbed SR1 and SR2, respectively.

30 See Pooley (2002) for a detailed discussion of this view.

31 For some recent discussion in the philosophy of physics literature, see (Weatherall 2018); cf. footnote 23.