Spacetime Matters Conference

Supersymmetry is one of the most widespread features of theories extending the standard model of particle physics. At the same time, philosophical reflection on supersymmetry still lags behind the sophisticated theories developed by high energy physicists. An important theory in the landscape of supersymmetric field theories is supergravity, where supersymmetry is treated as a gauge symmetry, and which constitute the low-energy effective field theories arising from string theory. Philosophically, these theories are especially interesting since they challenge the usual understanding of the split between matter and spacetime. This is because in supergravity we have a gauge symmetry (the gauged supersymmetry transformation) which relates fermions, usually encoding matter, and bosons, including the graviton, which encodes general relativity’s gravitational DoFs, and hence relativistic spacetime.

In this talk, we will explore the extent to which supergravity creates problems for the distinction between spacetime and matter. We will start by framing this distinction, following Cinti and Sanchioni (2024), as about two competing ontological readings of the supermultiplets of supersymmetric field theories: (1) one grounded in Wigner’s classification of particles, which takes supermultiplets (and superfields) as fundamental, thereby erasing the distinction between matter and spacetime; and (2) one that upholds the spin-statistics theorem, thereby preserving the separateness of bosons and fermions, at the price of introducing a redundancy in the ontological framework by treating components of superfields as ontologically primary.

We will first point out that in supergravity, since supersymmetry is gauged, the supermultiplet is most naturally treated as fundamental. We will then challenge this view, and so defend the split between spacetime and matter, by appealing to rheonomy constraints, which allow to express the supermultiplet only in terms of bosonic DoFs (Castellani et al., 1991; Fré and Grassi, 2008; Donagi and Witten, 2013). In this way, we can save the distinction between spacetime and matter in supergravity, but at the cost of removing fermionic degrees of freedom from the theory. Finally, we will look at supergravity as a low-energy effective theory of string theory. This changes the situation again: while at tree-level the rheonomy constraints still apply, at five-loops they do not, 2 as proved by Donagi and Witten (2013). Hence, the distinction between spacetime and matter falls apart again.

Indeed, looking at string theory’s connection to supergravity allows us to show how questions on the spacetime-matter distinction intersect questions regarding emergence and fundamentality, of spacetime in particular. The superfield-based ontology favored by looking at supergravity through the lens of strung theory fits well with treating spacetime and matter as emergent from a deeper, unified structure, whereas the component-based ontology of (2), which retains the spacetime-matter distinction, favors an emergentist account of supersymmetry itself, with spacetime and matter remaining ontologically distinct at fundamental levels; at the price, however, of complicating our understanding of how supergravity itself can emerge from a more fundamental theory.

Overall, we show that supergravity provides a compelling case to evaluate the distinction between spacetime and matter in high-energy physics, highlighting that the tenability of this distinction hinges on subtle mathematical and interpretational issues.

It has been argued that black holes are thermal systems. This claim seems to be highly sensitive to what we mean by ‘black holes.’ The notion commonly refers to the semi-classical black hole that is circumscribed by an event horizon, in a stationary spacetime. Many criticisms [1] attack the validity of the claim because of the overly restricted assumptions about the stationarity of the black hole spacetime and the defects of the event horizon. It is correct that it is difficult to define an even approximate Killing vector field, and that the use of an event horizon causes some inconsistencies between black hole thermodynamics (BHT) and ordinary thermodynamics. Regarding the latter point, it will be shown that two fatal issues arise when an event horizon is used as a black hole’s boundary: (1) in a physical scenario when a shell of mass collapses and forms a black hole, the metric inside the shell is still flat so that the area of the event horizon increases with no flux of energy and momentum passing across it. Hence, the first law of BHT cannot generally give us the quantitative relation among the changes of mass, area, and angular momentum; (2) In the usual form of the first law of BHT, there is an inconsistency in defining each term that appears in the formula: mass M and angular momentum J are defined in the asymptotically distant region while the surface gravity k and the angular velocity Ω are defined on the event horizon. So, it raises a natural question: can we develop BHT based on other (quasi-local) horizons, instead of the event horizon?

A positive answer is rarely suggested in philosophical literature. This paper aims to fill the lacuna by calling for attention on advances of black hole studies, namely the more suitable notion of a black hole’s horizon in BHT—the isolated/dynamical horizons. I will present and discuss the properties of the isolated/dynamical horizons [2] (which help overcome the difficulties in developing thermodynamics in stationary black holes), and the refinement of laws of BHT accordingly. I argue that results based on stationary black holes can be reduced to a special case in those of isolated/dynamical horizons, with the latter offering grounding dynamics for equilibrium black holes. This agreement gives genuine support in securing the current achievements of BHT based on the stationary black holes. Therefore, understanding the consistency of results on different notions of black holes present a robustness argument for the claim that black holes are thermal bodies. An example of deriving a covariant temperature from the Tolman effect for black holes is then given to show the close interconnection among the use of quasi-local horizons, the event horizon, and the stretched horizon in the membrane paradigm. In particular, I show that (1) the globally stationary states are a limiting case—hence an approximation—in light of the quasi-local framework, that (2) various thermodynamic laws defined on quasi-local horizons reveal their deeper physical significance, and that (3) the quasilocal framework allows a more general discussion of black hole thermodynamics which could potentially pave the way for the underlying statistical mechanical descriptions.

The debate over geometric conventionalism – whether determining the world’s geometry involves a conventional element – needs an update. On the Helmholtzian assumption of physical geometry, Reichenbach’s (1928) (in)famous “theorem theta” proposed that any metric tensor can do an empirically equivalent job to any other, if suitably compensated for by “universal forces”. Thus, combinations of geometry and universal forces, { G+F , G’+F’ , G’’+F’’ , … }, are empirically indistinguishable and only picked out by convention.

Especially in the relativistic context, few issues are as divisive as this now century-old question whether to take such alternative models physically serious. Part of the problem is semantic: “force” is an unfortunate word, called “misleading” (Sklar 1974), “funny” (Malament 1985), better seen as “universal effects” (Carnap 1956, Sexl 1970). some find theorem theta trivial, others call universal effects “problematic” (Acuña 2013), “unacceptable” (Weatherall & Manchak 2014), like “fairies that hide only when I look” (Norton 1994). This is surprising, for the physics is straightforward and the philosophy textbook level.

In this talk, I make a distinction between authors with philosophical aims (e.g., Reichenbach, Carnap, Sklar, Dürr & Ben-Menahem 2022) from those with scientific aims (e.g., Einstein 1921, Acuña, Norton, Weatherall & Manchak). The former see universal effects purely as the difference between two metric tensors – they need not even be sourced by matter-energy. The latter constrain universal effects to resemble known physical forces. Weatherall and Manchak (2014), for example, require them to be rank-2 tensors in a geodesic equation. Yet, such constraints often break the universality of theorem theta by fiat, missing the conventionalist’s conceptual point of making magnitudes measurable in the first place (cf. Padovani 2017).

Regarding the philosophical aims, I suggest the absolute disagreement stems not just from differing ideas about mathematical representation of physical concepts, but from two convictions on the side of conventionalists: (i) a hyperrealism about a (too) fine-grained conception of physical geometry I call Universal Physical Geometry and (ii) that these universal effects are like dynamical forces in the sense of causing a geometrical change.Departing from Reichenbach’s operationalist semantics, I explore the sense in which universal effects and physical geometry can be seen as theoretically equivalent (both conceptually and mathematically), and argue that a realist should commit to the logically weaker concept of “Differentiated Physical Geometry”, relative to non-universal forces only, as Helmholtz intended. Seen this way, dynamical talk of universal forces in relativity amounts to geometric talk about spacetime.

This leaves intact the distinction, firmly within the Democritean-Newtonian tradition, between spacetime and differential forces. Without deciding whether this distinction could be collapsed, I draw on Einstein’s “Geometry and Experience” (1921), often read as an endorsement of conventionalism about G+F pairs. However, Einstein’s scientific aim was unification of spacetime and dynamics. Conventionalism only applied to a final theory describing simply one unified field, from which matter, dynamics and spacetime emerge, conventionally divisible into different containers and what they contain.

I discuss key issues of conventionalism on the field theoretic approach to GR (see e.g. Petrov and Pitts (2020)). First, I consider whether the field theoretic approach indeed supports the claim that the standard metric g of GR is only a conventional choice of ‘spacetime geometry’ (D¨urr and Read, 2024). Indeed, on the field theoretic view, a chosen background metric g ′ (and not g) is the decisive technical geometry (say, for lowering and raising indices). However, if ‘spacetime geometry’ is understood more substantially as ‘physical geometry’ (the notion of interest to the authors initially), I maintain that, on the field theoretic approach, physical geometry is still represented by the (now effective) g metric. [My argument proceeds via establishing that whether a structure represents physical geometry, is not a question of its non-derivative nature in the representing formalism but solely of its functional role within that formalism and its interpretation.] This view on physical geometry in the field theory approach accords with various points that the background metric is only auxiliary (Petrov and Pitts, 2020): for instance, a trajectory is gauge-related to what counts as different trajectories relative to that background.]

With this in place, I turn to what I regard as the central issue of conventionality on the field-theoretic approach—the choice of the background metric. Various authors have raised the issue: Salimkhani has argued for a Minkowski background as a way of incorporating (thus re-formulated) GR into the Brown’s dynamical approach to SR (among other things, leaving open how to swallow (i) the import of an additional gauge group, (ii) the commitment to an unphysical nature of that background — as discussed in the first part —, and (iii) the need for a dS background to incorporate a cosmological constant); and Pitts (2022) has argued for a Minkowski metric background that is, however, only lightly interpreted as a numerical matrix (thereby partly dodging his own bullets).1 I maintain that (i) the field-theoretic approach is a complementary view to that of standard GR; and that (ii) there are no good non-pragmatic reasons for breaking the conventional freedom of choosing the background. [The argument en bref: the physical geometry is in any case represented by g (as argued before). The field theoretic approach provides different representational perspectives on the two degrees of freedom linked to the gravitational field (as precisely identifiable non-perturbatively in the ADM 1See W¨uthrich and Huggett (2025) for an analogous issue in perturbative string theory. 1 formalism). Breaking this pluralism from conventionalism about the background metric, limits GR (and also low-energy-quantum gravity; see Wallace (2022)) illegitimately; for instance, breaking this pluralism by commitment to a substantial sense of background spacetime, limits GR to a perspective that concerns isolating strong gravitational regimes within that flat background spacetime; one that is only interesting for a low-energy partial interpretation (Schneider, 2024) in the spirit of a constructivist bottom-up account (Adlam et al., 2024).]

There are two current and arguably controversial projects in the philosophy of physics that go hand in hand. First, inspired by contemporary physics, Niels Martens and collaborators challenge what they dub the "Democritean-Newtonian tradition of assuming a strict conceptual dichotomy between spacetime and matter". Second, tipped off by what has become jargon, Rasmus Jaksland and I urge philosophers of quantum gravity to specify their claims on the 'emergence of spacetime' in terms of concrete spatiotemporal aspects. I will reflect on these developments and explore to what extent traditional categorizations have to give.

General Relativity offers the possibility to model attributes of matter, like mass, momentum, angular momentum, spin, chirality etc. from pure space, endowed only with a single field that represents its Riemannian geometry. I review this picture of ‘Geometrodynamics’ and comment on various developments after Einstein. 

This presentation is based on this paper.

Suppose that you’re in a room with a lightbulb. A curtain is drawn between you and the lightbulb. The lightbulb cannot be seen, but some light still filters through the curtain into the room. You and your friend Feld have a disagreement. Feld thinks that there can be free (i.e., sourceless) fields, whereas you hold that all radiation is sourced. Feld claims that the room you’re in is evidence that they are correct, for one can model just the room with the curtain as a boundary condition and the representation of that scenario corresponds to a solution of the relevant laws of nature. That argumentative move feels too cheap, you say. All Feld did was replace the source with a special boundary condition, the state of the curtain. That state carries on it an echo or trace of the real source. The free solution exists as an artifact of modeling, or so you think.

In the debates between substantivalists and relationists in general relativity, the existence of distinct nontrivial vacuum solutions to Einstein’s field equations is often taken to present a quick and serious challenge to relationism. For example, Minkowski and Schwarzschild spacetimes have different metric fields but the same matter content. Their existence suggests that the metric field is underdetermined by the matter content, which would conflict with versions of relationism and Machianism that claim that curvature or the metric is always “sourced” by matter. I will argue that one can respond to this famous threat to relationism in a way similar to how one may respond to the above challenge to the claim that all radiation is sourced. The boundary condition contains suspicious “echoes” of forgotten material sources. This response will not prove that relationism is true any more than the earlier move proved that all radiation is sourced. But it will deflate one worry about relationism.

To see the general idea, note that the Schwarzschild solution is Ricci-flat but not Riemann-flat. Minkowski is both. What makes Schwarzschild not Riemann-flat is that it has non-vanishing Weyl curvature. The Weyl tensor can be thought to encapsulate the “free” gravitational degrees of freedom whereas Ricci the “source” degrees of freedom. Yet as above in our imagined dialogue, one might look with suspicion upon Schwarzschild. All we’ve done is draw a curtain around the matter to make it a vacuum, yet we do this for the sake of modeling. The boundary conditions contain the trace of ignored material sources. Arguably the underdetermination arises from this modeling.

I’ll then move to what is perhaps the most hostile environment for my view, the Ozsvath-Schucking metric, the “ultimate” anti-Machian solution. This solution describes a pp-wave spacetime containing only gravitational radiation. Here I’ll show that the dimensional parameters of the model can be viewed as echoes of forgotten matter. But one can also pose a dilemma based on the Ehlers-Kundt conjecture (1962) and a theorem by Penrose (1965) to the effect that the spacetime is either incomplete or a mere idealization. Even the most hostile vacuum solution can be Machianized.

In this talk I develop an account of the connection between chronogeometry and matter dynamics in spacetime theories that explores a third way between the two dominant views: the dynamical and the geometrical approaches. Some of the most recent works published in this area can be also interpreted in this key (see, for instance, Acuña (2016), Weatherall (2021), Myrvold (2019), Read (2021), Sus (2019)).

My proposal shares some important features with the relativized a priori framework, mostly defended in recent times by Michael Friedman, adapting ideas of Reichenbach, and rooted in those parts of Kant's work that contains the resources to connect the geometry of space and some dynamical concepts. Friedman (2012) reconstructs the Kantian connection between two notions of space (geometrical and perspectival) using the mathematical notion of group of transformations. Kant's transcendental insight, under this perspective, consists on relating the geometry of space to certain properties of the observer's intuition, which leads him to the, inadequate from our vantage point, characterisation of space and time in Newtonian terms. Other authors like Helmholtz and Lie, from different posits, apply this strategy to tackle the question of the determination of spatial geometry. The starting point for my proposal is that this strategy can be generalised to be also effective in relativistic settings. My idea to do so passes by making explicit what in the Kantian approach is hidden under the notion of intuition. This is related to, and must be contrasted with, Weyl’s justification of his pure infinitesimal geometry.

The proposal can be presented in two steps. The first one consists on developing a notion of ideal observer that plays the role that intuition plays in a Kantian scheme. This will be generally expressed by the identification of physical principles that characterise the measuring devices in the theory. The second step consists on, from a minimal characterisation of what is involved in measurement, deriving consequences for the formulation of dynamical laws. This is achieved by making contact with the discussion about the empirical significance of the notion of symmetry when applied to subsystems that can act as measuring devices. With this, I define a notion of “dynamical congruence” that selects a subset of the dynamical symmetries that can be interpreted as conforming the group of spacetime symmetries. What this provides, then, is a connection between certain dynamical symmetries and structures that are going to play a spatiotemporal role in the theory or, differently expressed, it presents a dynamical definition of spacetime symmetry.

The concept of dynamically interacting fields is applicable to all fundamental interactions, but in the case of gravitational interaction, it might appear incomplete and reliant upon additional geometrical notions. This foundational significance of geometry is a recurring theme in modern physics [e.g. Weyl, 1918, Yang, 1980]. Geometrical concepts facilitate coordination between theory and chrono-geometric measurements, aid interpretative clarity, and serve as crucial heuristic devices both historically and in contemporary theory construction. The significance of these geometric frameworks notably intensifies in advanced gravitational theories, particularly when dealing with gravitational coupling to spinors (which fundamentally describe most form of matter), as well as theoretical extensions, including torsion and supergravity. In such cases, geometrical framework often serves as a central guideline and form of justification.

The research presented here addresses the heuristic dispensability of geometrical notions, demonstrating how empirically guided invariance heuristics suffices to introduce gravitational coupling, including in cases involving spinorial coupling and tetrads. In this derivation, neither the relationship between the tetrad and metric fields nor the ’tetrad postulate’—the statement that the full covariant derivative of the tetrad vanishes—is presupposed on geometric grounds. Instead, both emerge naturally from an empirically motivated ”methodological equivalence principle” (that has already been proven useful in simpler cases, see Hetzroni and Read [2024]), that is supported by the requirement for the universality of the gravitational interaction. Furthermore, it is demonstrated how these heuristics extend elegantly to the case of supergravity, introduced through the localization of supersymmetry. In all these cases a geometrical interpretation can be added retrospectively. However, it is shown that while the standard route, in which gravitational theories are constructed based on a geometrical conceptualization, leads to wide class of possible theories, the empirically guided invariance heuristics leads to a much more constrained class of theoretical possibilities. This distinction highlights a sense in which certain theoretical possibilities (e.g. torsion free general relativity) are better motivated by the evidence than alternatives that are currently empirically indistinguishable. The talk concludes by highlighting how a refined understanding of heuristic considerations is relevant for the justification of metaphysical claims regarding the nature and interpretation of spacetime and matter.

A popular view in the foundations of general relativity and quantum gravity is the ‘observables’ view, mostly influenced by the work of Carlo Rovelli1 . According to this view, general relativity ought to be interpreted as a gauge theory, and its physical content should be found in the gauge-invariant quantities of the theory, also known as the ‘observables’. By studying the phase-space structure of diffeomorphism invariant models similar to general relativity, Rovelli and collaborators argue that the predictions of this sort of model are correlations, i.e., the values that some physical quantities take when other quantities take some other values. In this way, the ‘observables’ view dismisses many spatiotemporal relations and claims that the metric is just a gravitational field which does not play a privileged role. That is, while in other understandings of general relativity one defines evolution in relation with spacetime, according to the observable view, any variable or field is as good as any other for defining correlations, and this includes the metric field. In this sense, the observables view is a view that makes the distinction between matter and spacetime fields thinner.

In this talk I want to oppose this view by showing some limitations of the analysis of the diffeomorphism invariance in which the position is based. First, I will show how for some systems, the technical definition of observable fails, in the sense that there are some correlations that do not satisfy the definition employed by Rovelli and collaborators. The reason for this failure is technical: this definition of observables works fine for continuous and differentiable functions in phase space, but I show how correlations do not satisfy these conditions. Then, I provide a generalization of the definition so that all invariant quantities under diffeomorphisms are included.

However, I argue that some of the quantities that are identified as observables are not correlations, but they correspond with temporal relations. That is, I argue that there are gauge invariant quantities that tell us that event A happens before event B, or that the proper time in between them is T. In this sense, I conclude that the technical argument for the observables view fails, as one can very reasonably argue that spatiotemporal relations are part of the gauge invariant content of diffeomorphism invariant models.

Finally, I conclude that in diffeomorphism invariant models spatiotemporal relations play an important role and cannot be dismissed as gauge artifacts. If the matter-spacetime distinction is to be eliminated or reduced, it is not because of the gauge structure of general relativity.

Although the standard description of gravity ties the mass-energy distribution of matter to the curvature of spacetime, this connection is not uniquely determined. By introducing additional geometric concepts, a relativist can construct various alternatives to GR, each with its own virtues and limitations.

In my talk, I examine two conflicting approaches - Geometric Trinity [1] and Metric Affine Gauge Gravity [2] - that enrich spacetime geometry with the additional structure of torsion and non-metricity. The Geometric Trinity approach claims an underdetermination of spacetime geometry [3] based on the existence of a dynamically equivalent trio of theories, each coupling matter exclusively to either curvature, torsion or non-metricity. However, I argue that from the Metric Affine Gauge Gravity (MAG) perspective the dynamical equivalence is deflated to mere empirical equivalence, and the different geometric attributes of spacetime become, at least in-principle, distinguishable. I highlight how the source of this incompatibility can be traced back to constraints on a quantity called 'Hyper-Momentum', computed by varying the action with the affine connection. While the Geometric Trinity formalism assumes a vanishing hyper-momentum, its non-vanishing in MAG reveals additional geometric degrees of freedom and matter coupling.

In the absence of decisive experimental results favouring either framework, I discuss an account of supra-empirical justifications in the form of a trade-off between parsimony and coherence: the former favouring the Geometric Trinity approach, while the latter leaning towards the MAG. Towards the end, I describe on-going work on using Ehlers’ frame theory constructions to understand the degenerate Newtonian limit of these formulations. A broader aim of this talk is to explicitly clarify the relationship (or the lack thereof) between the virtues of geometric conventionalism and the conceptual tension between the Geometric Trinity and Metric-Affine Gravity concerning dynamical equivalence.

Contemporary physics has steadily eroded the once-clear boundary between spacetime and matter. General relativity invites us to interpret the metric both as spacetime itself and as a dynamical field akin to matter, while quantum gravity promises to dissolve the dichotomy entirely—even though a coherent alternative framework has yet to be articulated. Against this backdrop, the recently proposed framework of Pure Shape Dynamics (PSD)— a reformulation of physics grounded in Leibnizian/Machian relationalism— recasts the ontological picture in a radical way, eliminating spacetime as a fundamental category already at the classical level (Vassallo et al., 2022).

Where conventional physical theories treat spacetime (or some variant thereof) as essential to the very intelligibility of physical description, PSD offers a fundamentally different outlook in which all physical content is encoded in the evolution of relational configurations within shape space (Koslowski et al., 2022). Within this framework, what exists fundamentally are selfsubsisting structures, i.e., material entities individuated purely through their intrinsic relational properties. Spatial and temporal facts, on this view, are not primitive ontological ingredients but are instead strongly reducible to patterns in the dynamics of these structures (Vassallo and Naranjo, 2025).

In this talk, I ask what this relationalist framework can teach us about materialism, understood here as the metaphysical stance that everything real is grounded in the material, physical world. The picture that emerges—what I call Leibnizian/Machian materialism—challenges both substantivalist and structuralist accounts of spacetime by reversing the explanatory hierarchy that has prevailed since Newton. I will argue that the PSD plus self-subsisting structures framework materializes geometry, rather than geometrizing matter: Spatial and temporal structures are best understood as higher-order 1 abstractions over changing material relations. This conceptual shift holds across theoretical contexts and offers a compelling metaphysical orientation for background-independent physics—one that may facilitate a more coherent path toward quantum gravity (Farokhi et al., 2024).

The upshot is clear: By introducing a perspective in which space and time are conceived as epistemic by-products of underlying material relations, PSD and the metaphysics of self-subsisting structures articulate a novel and coherent form of materialism—one in which the relational architecture of the world does all the explanatory work, including that which was once thought to require irreducible spatial and temporal concepts (Vassallo, 2025).

According to 'regularity relationalism', which is a generalised Humean strategy developed by Huggett (2006), both spacetime structure and dynamical laws are to be understood as being subordinate to a more fundamental non-spatiotemporal Humean mosaic. Although the approach works (modulo some caveats and subtleties) in the case of theories with fixed spacetime structure (see Pooley (2013) and Stevens (2020)), it has yet to be applied to the case of general relativity, and there are prima facie serious roadblocks to doing so. In this talk, I will show how this can in fact be achieved, thereby offering a novel relationalist perspective on general relativity. (Joint work with Henrique Gomes, Tushar Menon, and Oliver Pooley.)