Is spacetime fundamental or emergent? If emergent, is it wholly or partially emergent? From what does it emerge? According to one influential reading, these questions lie at the heart of the classic debate between substantivalism and relationalism. Standard lore tells us that in the context of Newtonian gravitation, the score between substantivalism and relationalism remains close to tied. Whether spacetime is fundamental, existing independently of the material bodies it contains, or, rather, if it emerges from relations between these bodies is effectively a toss-up. With the advent of general relativity, however, things shift in favour of substantivalism. Gravity is ‘geometrized’, reduced to the effects of spacetime curvature, and there are dynamically interesting vacuum solutions entirely devoid of matter. This suggests that spacetime is indeed fundamental. But once again the situation changes when we shift to quantum gravity. Here it seems increasingly likely that spacetime emerges from something non-spatiotemporal and non-material—a coherent graviton background, quantum spin foam, causal sets, and so on.

In his sharp, incisive new book, The Non-fundamentality of Spacetime, Kian Salimkhani challenges this standard lore. Insofar as quantum gravity provides evidence in favour of spacetime emergentism, according to Salimkhani it is weak evidence at best. There are too many different quantum gravity programmes, with different motivations and ontological ramifications, all of which remain speculative and empirically untested. Nevertheless, Salimkhani contends that we can formulate a better argument for spacetime emergentism at the level of semi-classical general relativity. This argument has three stages. First, he deploys the dynamical interpretation of general relativity championed by Harvey Brown, Oliver Pooley, and James Read as a serious alternative to the usual geometric interpretation. Next, he shows that by recasting general relativity as a classical spin-2 field theory, we can resolve a major challenge to the dynamical interpretation, putting it on equal footing with the geometric interpretation. Third, he argues that if we view the classical theory as the low-energy limit of an effective spin-2 quantum field theory the scales tip decisively in favour of the dynamical interpretation. In the quantum theory, the metrical properties of spacetime emerge from a background of interacting gravitons, rendering these features non-fundamental. Once all the dust settles, if Salimkhani is right, the outlook for relationalism looks far rosier than standard lore suggests.

This argument unfolds slowly over the book’s six chapters, although most of it is concentrated in chapters 4 and 5. Chapter 1 gives an overview of the main argument, while chapter 2 provides a serviceable introduction to contemporary analytic views on fundamentality. Salimkhani favours views that treat fundamentality as ontological independence, and he argues that it is philosophically fruitful to reframe the substantivalism–relationalism debate along these lines. The main issue is whether spacetime is ontologically independent or dependent on matter, not whether spacetime exists or not. One word of caution: Salimkhani has an expansive view of matter. It need not be massive and fermionic; it includes photons, gluons, and other kinds of fields as well. Accordingly, the spin-2 view he advocates counts as a kind of relationalism, although not everyone will see it this way. For example, Martens ([2019]) classifies it as a different kind of spacetime emergentism, distinct from relationalism. Not much turns on this, but to appease everyone Salimkhani switches to using spacetime fundamentalism to label substantivalism and spacetime non-fundamentalism to label any view on which spacetime emerges from something non-spatiotemporal.

Chapter 3 introduces the dynamical interpretation of relativity (Brown [2005]) and articulates two significant challenges that it faces. According to the dynamical interpretation, the fact that rods and clocks reliably measure the metrical properties of spacetime, the so-called chronogeometricity of the metric field, requires a detailed physical argument rooted in the strong equivalence principle. These measurement devices are made from matter governed by dynamical laws that happen to share certain symmetry properties (local Poincaré-invariance) with the spacetime metric. (In contrast, the geometric interpretation maintains that the spacetime metric actually constrains and explains the dynamical behaviour of matter fields.) This guiding idea evolves into two distinct metaphysical pictures: one for special relativity and one for general relativity. In the case of special relativity, the Minkowski metric (η_μν) is entirely ontologically dependent on the matter fields, it merely ‘codifies’ their dynamical symmetries. In the case of general relativity, the metric g_μν is an ontologically independent field, on par with the matter fields. What makes g_μν a metric field—what explains its chronogeometricity—however, is the fact that all matter fields have the same local symmetry properties as g_μν.

The first challenge to the dynamical interpretation concerns the asymmetry between these pictures. As Read et al. ([2018]) note, dynamical general relativity must accept two unexplained brute facts: the dynamical laws governing matter fields are locally Poincaré-invariant (MR1), and all matter fields have the same local symmetry properties as the metric field (MR2). Dynamical special relativity, in contrast, explains away MR2 by reducing η_μν to properties of the matter fields. Dynamical general relativity explains less, and is therefore worse off compared to its special relativistic counterpart. The second challenge is the ‘problem of pregeometry’ (Norton [2008]). Whether the dynamical interpretation succeeds in giving a relationalist explanation for the metrical properties of spacetime, it does not explain the topological properties of the underlying spacetime manifold and is therefore only a half-way kind of relationalism. Based on joint work with Niels Linnemann (Linnemann and Salimkhani [unpublished]), Salimkhani argues against existing strategies to handle the problem of pregeometry, and proposes two new ones. Although much of chapter 3 surveys existing literature, this final section makes a significant new contribution to the debate over the dynamical interpretation of relativity.

Chapters 4 and 5 are the heart of the book, and it is here that we meet the classical and quantum spin-2 field theories head on. Viewing gravity as a spin-2 field theory has been a tremendously influential idea in the particle physics and quantum gravity communities, but it will be new to many philosophers of physics. The idea traces back to early work by Markus Fierz and Wolfgang Pauli in the 1930s, gaining momentum in the period between 1950 and 1970, with important input from Suraj Gupta, Robert Kraichnan, Richard Feynman, and Stanley Deser, among others. We begin with a classical, Poincaré-covariant, massless spin-2 field, h_μν, with two polarization modes living in Minkowski spacetime with flat metric η_μν. The free dynamics are given by a linear wave equation determined by the Fierz–Pauli Lagrangian. If we wish to couple h_μν to matter fields, we find ourselves highly constrained. The wave equation for h_μν is divergence-free (unlike, say, the wave equation for a free scalar field) and, as a result, energy–momentum conservation requires incorporating a self-coupling term for h_μν. Assuming the spin-2 field couples to itself and all other matter fields universally via the total energy–momentum tensor, various arguments then entail that the resulting non-linear dynamics must take the form of the Einstein field equations with an effective curved metric defined by g_μν = η_μν + h_μν.

The resulting spin-2 field theory gives us a theory of gravity that, at least for certain universes, is empirically equivalent to classical general relativity. (Not every general relativistic universe fits into the spin-2 framework; there are difficulties involved with incorporating a cosmological constant term, and non-globally hyperbolic solutions are ruled out.) Moreover, Salimkhani shows that this reformulation allows us to dissolve the first challenge to the dynamical interpretation noted above. Instead of treating g_μν as a fundamental field, we can reduce it to a combination of the flat Minkowski metric, η_μν, and the spin-2 graviton field, h_μν. The latter is just one of several matter fields in the theory, while the former merely codifies the common local symmetry properties of all matter fields. Just like dynamical special relativity, dynamical spin-2 general relativity therefore explains away MR2 and is only faced with MR1 as a brute fact.

This is a major advancement, putting the dynamical interpretation on arguably equal footing with the geometric one. Salimkhani suggests additional reasons to favour the classical spin-2 theory, but concedes that his opponents may not be convinced. The identity relation g_μν = η_μν + h_μν is symmetric after all. While the dynamical interpretation views g_μν as emergent, ontologically dependent on η_μν and h_μν, the geometric interpretation can plausibly maintain that g_μν is fundamental, with both η_μν and h_μν emerging from a splitting of the metric field into a flat background and curved dynamical part. In chapter 5, however, Salimkhani argues that this defensive move is no longer available once we treat the classical spin-2 theory as the low-energy limit of a spin-2 quantum field theory. In this context, g_μν is revealed to be ontologically dependent on the properties of interacting quantum matter, but not vice versa. We cannot recover the full dynamics of the quantum graviton field merely by splitting g_μν as the geometric interpretation suggests. What is more, the quantum spin-2 theory has a richer set of explanatory resources to draw on compared to the classical spin-2 theory. For example, Steven Weinberg’s ‘soft graviton argument’ can be used to explain why the graviton field must couple universally (something that is typically assumed in the classical spin-2 theory), and Salimkhani contends this can provide a new way to understand the weak and strong equivalence principles. Taken as a whole, the argument in this chapter is a case study in deploying inter-theoretic relations and unification as a constraint on metaphysical inference. When general relativity is viewed in isolation, spacetime fundamentalism and non-fundamentalism both appear to be tenable views. When it is viewed as continuous with modern particle physics, however, Salimkhani concludes that only the latter is tenable. Chapter 6 offers a short coda further developing and generalizing these methodological ideas for naturalized metaphysics.

The scale and complexity of Salimkhani’s book-length argument is both a strength and a weakness. There are several points where I suspect defenders of spacetime fundamentalism will want to get off the boat. For instance, in chapter 3 there is ongoing debate about what exactly it means for fields to share local Poincaré invariance and this threatens to undermine the version of the strong equivalence principle that the dynamical interpretation relies on (Weatherall [2021]). In chapter 4, one might worry about hidden assumptions and a general lack of rigour in the derivation of the Einstein field equations from the classical spin-2 theory (Linnemann et al. [2023]). In chapter 5, great weight is placed on the ‘Heisenberg collapse argument’ to supply evidence in favour of interpreting the Minkowski metric, η_μν, dynamically rather than geometrically. As Salimkhani’s own discussion makes clear, the interpretation of this argument is fraught. Meanwhile, the problem of pregeometry looms in the background throughout the book. Perhaps the new resolution Salimkhani sketches in chapter 3 will work, perhaps not.

Nevertheless, Salimkhani makes a strong case for the spin-2 dynamical view, and his argument provides a unifying theme linking several topics that are often debated separately: substantivalism versus relationalism, geometric versus dynamical interpretations of relativity, classical and semi-classical gravity, classical and quantum field theory. While the armour has more joints to exploit, it also has more overlapping protective plates. For example, treating gravity as a spin-2 field theory in flat Minkowski spacetime might allow us to better understand the thorny concept of local Poincaré invariance. Similarly, we might plausibly draw on novel features of the quantum spin-2 theory to patch holes in the classical derivation of the Einstein field equations. What is more, the book’s twists and turns supply much of its philosophical richness. Not only will it be of interest to philosophers working on spacetime emergence, dynamical relativity, and spacetime functionalism, but the spin-2 theory supplies a new, sophisticated case study for those investigating questions about theoretical equivalence. Philosophers of quantum field theory will be intrigued by the idea that general relativity reduces to quantum field theory, a direct challenge to the adage that familiar Minkowski QFT is merely a stepping stone to QFT in curved spacetime. Philosophers of science will appreciate the frequent, nuanced methodological detours, and metaphysicians will revel in seeing cutting-edge ideas about fundamentality deployed in debates about cutting-edge physics. All in all, there is much to recommend and much to learn in this excellent read.

Noel Swanson
University of Delaware
nswanson@udel.edu

References

Brown, H. [2005]: Physical Relativity: Space-Time Structure from a Dynamical Perspective, Oxford: Oxford University Press.

Linnemann, N. and Salimkhani, K. [Unpublished]: ‘The Constructivist’s Program and the Problem of Pregeometry’, available at <philsci-archive.pitt.edu/20035/>.

Linnemann, N., Smeenk, C. and Baker, M. R. [2023]: ‘GR as a Classical Spin-2 Theory?’, Philosophy of Science, 90, pp. 1363–73.

Martens, N. C. M. [2019]: ‘The Metaphysics of Emergent Spacetime Theories’, Philosophy Compass, 14, available at <doi.org/10.1111/phc3.12596>.

Norton, J. [2008]: ‘Why Constructive Relativity Fails’, British Journal for the Philosophy of Science, 59, pp. 821–34.

Read, J., Brown, H. and Lehmkuhl, D. [2018]: ‘Two Miracles of General Relativity’, Studies in History and Philosophy of Modern Physics, 64, pp. 14–25.

Weatherall, J. O. [2021]: ‘Two Dogmas of Dynamicism’, Synthese, 199, pp. S253–75.